Book cover of Density Functional Theory

Mathematics and Molecular Modeling

Series Editors

Eric Cancès

CERMICS, Ecole des Ponts and Inria Paris, Marne-la-Vallée, France

Gero Friesecke

Department of Mathematics, Technische Universität München, Garching bei München, Germany

Lin Lin

Department of Mathematics, University of California, Berkeley and Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Jianfeng Lu

Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, NC, USA

Mathematics and Molecular Modeling offers an interdisciplinary forum for monographs and multi-author surveys related to the design, analysis, and simulation of molecular models at all scales. Topics covered range from electronic structure over molecular dynamics to deterministic and stochastic coarse-grained models as relevant for, e.g., conformation dynamics of biomolecules or evolution of defects in solids. Computational and mathematical aspects of electronic and atomic structure of materials, molecular optics, molecular scale imaging, control of molecular systems, crystal growth, supramolecular architectures and data-based modeling are also covered. Texts highlight the state of the art in a particular area, in a way accessible to graduate students and researchers in applied and computational mathematics as well as to interested researchers from other fields.

Editors

Eric Cancès and Gero Friesecke

Density Functional Theory

Modeling, Mathematical Analysis, Computational Methods, and Applications

The Springer logo.

Editors

Eric Cancès

CERMICS, Ecole des Ponts and Inria Paris, Marne-la-Vallée, France

Gero Friesecke

Department of Mathematics, Technische Universität München, Garching bei München, Germany

Mathematics and Molecular Modeling

ISBN 978-3-031-22339-6e-ISBN 978-3-031-22340-2

https://doi.org/10.1007/978-3-031-22340-2

Mathematics Subject Classiﬁcation (2010): 81-1035Q4065K1065H1781V5581Q0581V7065N25

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Preface

Density functional theory (DFT) provides the most widely used models for simulating molecules and materials based on the fundamental laws of quantum mechanics. It earned its main inventor Walter Kohn a Nobel prize, and nowadays plays a central role in a huge spectrum of applications in chemistry, physics, and materials science. DFT has become an interdisciplinary field as testified by the list of authors of this book, coming from chemistry, physics, materials science, mathematics, and scientific computing.

Quantum mechanics describes a system of N interacting particles in the physical three-dimensional space by a partial differential equation in 3N spatial variables. Hence, standard numerical methods incur an exponential increase of computational effort with N, a phenomenon known as the curse of dimensionality; in practice these methods already fail beyond N = 2. DFT overcomes this problem by (1) reformulating the N-body problem involving functions of 3N variables in terms of the electronic density, a function of 3 variables, (2) approximating it by a pioneering hybrid approach which keeps important ab initio contributions and re-models the remainder in a data-driven way.

In fact many different types of data were used by different researchers to parameterize the remainder, ranging from numerical results for reference systems over experimental properties of atoms and small molecules to exact constraints in asymptotic regimes.

Kohn–Sham DFT models a molecular system with N electrons as follows. For simplicity we describe the common case of a spin-unpolarized system. The quantum state of the electrons is described by a set of N∕2 doubly occupied spatial orbitals {φi}i=1,…,N∕2, which are functions on the physical space $$\mathbb {R}^3$$ that are orthonormal with respect to the L² inner product. The total electronic energy is

(1)

The first term models the kinetic energy of the electrons, vne is the electrostatic potential generated by the nuclei, and EHxc is the so-called Hartree-exchange-correlation functional which models the electron-electron interactions and interaction-induced corrections to the kinetic energy by some explicit functional of the density. The density ρ is expressed in terms of the orbitals as

(2)

The Kohn–Sham ground state is obtained by minimizing the energy functional E over {φi} subject to the orthonormality constraints $$\int _{\mathbb {R}^3} \varphi _i^*\varphi _j=\delta _{ij}$$ . The Euler–Lagrange equations (stationarity conditions) are the Kohn–Sham equations [1]

(3)

where the εi are Lagrange multipliers and vHxc(r) is the Hartree-exchange-correlation potential defined as the functional derivative of EHxc[ρ] with respect to ρ(r). The orbitals satisfying Eq. (3) are called the Kohn–Sham orbitals. Mathematically, the Kohn–Sham equations are a system of nonlinear elliptic eigenvalue problems, since the potential vHxc depends on the φi’s through the density ρ.

There are a great many challenges associated with this system, touching upon different areas of physical and chemical modeling, mathematical analysis, and scientific computing, such as:

Design of accurate yet computationally practical Hartree-exchange-correlation functionals, going beyond the local density approximation (LDA) introduced in Kohn and Sham’s seminal 1965 paper [1] and the—currently widely used—semilocal and hybrid functionals and remaining applicable in strongly correlated regimes (stretched chemical bonds, transition metal oxides, Mott insulators).

Design of effective numerical methods to solve the Kohn–Sham equations for large molecular systems, possibly coupled to an environment (linear scaling methods, coarse-graining, quantum-mechanics/molecular-mechanics (QM/MM), and other multilayer approaches).

Theoretical understanding of the errors incurred both by the modeling and the numerical treatment (comparison to exact many-body quantum mechanics in scaling limits, choice of discretization or background basis, iterative algorithms with stopping criteria, adaptive floating-point arithmetic, massive parallelization for hybrid CPU/GPU, and future exascale architectures).

Unearthing the intrinsic mathematical properties of different Kohn–Sham models in important basic situations such as bond dissociation, dispersion forces, or defects in crystals.

These challenges give rise to a variety of exciting and difficult problems for mathematicians and computational scientists, and bring one immediately to the edge of current knowledge in variational methods, nonlinear partial differential equations, large-scale numerical optimization and linear algebra, or model reduction. This book introduces and reviews the main models of DFT, covering their derivation and mathematical properties, numerical treatment, and selected applications. It intends to be an accessible yet state-of-art text on DFT for graduate students and researchers in applied and computational mathematics, physics, chemistry, and materials science. We hope that it helps to attract mathematicians and computer scientists willing to contribute to a very lively research topic and an essential tool to meet many scientific challenges of the twenty-first century, such as computer-aided drug design or computational materials engineering for green energy production and storage. We also hope that this volume will be useful to computational chemists and physicists seeking to better understand the mathematical foundations of DFT, and the state-of-the-art numerical methods used to solve the Kohn–Sham equations.

This book is organized as follows. This preface is followed by a prologue on the early days of DFT by Mel Levy, one of the pioneers of the field, who—among other things—introduced the celebrated concept of constrained search. The 11 chapters of the book can be classified into two groups:

DFT models and their derivation, mathematical justification and analysis: Chaps. 1 (J. Toulouse), 2 (A. Savin), 3 (M. Lewin, E.H. Lieb, R. Seiringer), 4 (G. Friesecke, A. Gerolin, P. Gori-Giorgi), 5 (S. Kvaal), 6 (D. Gontier, J. Lu, C. Ortner);

Numerical methods and applications: Chaps. 7 (E. Cancès, A. Levitt, Y. Maday, C. Yang), 8 (L. Lin, J. Lu, L. Ying), 9 (H. Chen, R. Schneider), 10 (X. Dai, A. Zhou), 11 (L. Genovese, T. Deutsch), 12 (K. Bhattacharya, V. Gavini, M. Ortiz, M. Ponga, P. Suryanarayana).

The chapters are self-contained and can be read independently, but we advise readers with mathematical or scientific computing backgrounds unfamiliar with DFT to get in shape with Sects. 1.​1 and 1.​3.​1 in Chap. 1 (basics of DFT) and the first three sections of Chap. 7 (mathematical formulation and discretization methods), and consult the first three sections of Chap. 3 for a mathematical justification of the LDA.

Let us finally mention that this book focuses on Kohn–Sham DFT methods for electronic ground-state calculation within the Born–Oppenheimer (clamped nuclei) approximation. It does not cover time-dependent DFT (computation of excited states, linear response theory…), orbital-free DFT (simulation of warm dense matter...), nor classical DFT (classical statistical mechanics of the liquid state). Likewise, post-DFT methods (e.g., Green’s function methods such as GW or Bethe–Salpeter), coupling of DFT with finer but computationally more expensive electronic structure methods for strongly correlated electrons (e.g., DFT+DMFT), coupling of DFT with molecular dynamics and implicit solvent models (multilayer models such as QM/MM), or machine learning methods trained on data generated by DFT calculations lie beyond the scope of this book. So are the other two most popular families of methods to compute ground-state electronic structures of molecules and materials, namely post-Hartree–Fock wave function methods (coupled-cluster, multiconfiguration methods, tensor network methods,…) and Monte Carlo methods. Each of these topics would be worth a book of their own in this series.

Reference

1.

W. Kohn and L.J. Sham. Self-consistent equations including exchange and correlation. Phys. Rev.140, A1133 (1965).

Eric Cancès

Gero Friesecke

Paris, FranceMunich, Germany

December 2021

Prologue: Early Days of Modern DFT (1964–1979)

It is a pleasure to write about the early days of modern DFT from a personal perspective.

At that time, the local density approximation dominated in approximating exchange-correlation for calculations. In basic theory, there was intense interest in the definition of F[ρ], the universal functional of the density for the kinetic energy plus the electron-electron repulsion energy, where, of course,

Along these lines, on a chalkboard at a special session of a theoretical chemistry conference in Boulder, Colorado, in the summer of 1975, Walter Kohn wrote out the proof, by contradiction, of the 1964 Hohenberg–Kohn theorem. As I recall, the audience applauded enthusiastically. Then there was a lively discussion about the F[ρ] in this HK paper of 1964, because this functional, born by this proof by contradiction, was identified specifically for only each trial density that is a non-degenerate ground state of some extremal potential (non-degenerate v-representable). There were questions. What happens when the trial density is not some ground state? What happens when the trial density is a ground state, but is a degenerate one? These questions were in my thoughts when I began the serious study of DFT in 1978.

Then, in looking through a 1978 issue of IJQC, I noticed a long paper by Jeremy Percus. In it he derived a number of interesting Legendre transform-type bounds, but, almost as an aside, he also happened to identify a noninteracting kinetic energy functional that was expressed as the minimum kinetic energy of all ensembles of idempotent density matrices that yield a noninteracting v-representable density. Influenced by this paper, it suddenly occurred to me a couple of weeks later that the original HK functional could be identified as a constrained search. Moreover, it could also be generalized by using this constrained search. Namely,¹

(1)

because

(2)

and because the same expectation value, involving vne, must be given by all wave functions that yield ρ(r).

I remember finding expressions (1) and (2) a bit frightening as I wrote them down and stared at them, because they were so simply expressed and because I knew that they generalized DFT to include densities that are degenerate and to densities that are not necessarily v-representable. I also realized that they led to a deeper understanding of DFT. For instance, I soon realized that the philosophy behind Eqs. (1) and (2) could be applied to generalize Gilbert’s functional of the electron-electron repulsion energy, as a functional of the one-matrix, to include a larger class of trial one matrices. All this appears in my 1979 paper.

Yes, the 15-year period from 1964 to 1979 was important indeed. Kohn–Sham theory and the thermal DFT of Mermin were formulated in 1965. Then von Barth and Hedin and Rajagopal and Calloway formulated spin-density functional theory in 1972 and 1973. In the mid-1970s, the adiabatic connection formula was derived by Langreth and Perdew and Gunnarsson and Lundqvist. This formula is, of course, important for the development of approximate exchange-correlation functionals. The concept of electronegativity was elucidated through DFT by Parr, Donnelly, Levy, and Palke in the 1978 paper that launched conceptual DFT. Then the time-independent ensemble theory for the density functional treatment of excited states was formulated by Theophilou in 1979.

Forty years ago, those of us immersed in DFT knew that we were gambling on a new field, especially since there were influential vocal skeptics. But we felt that there just might be good times ahead…

Mel Levy

List of Symbols

Parameters of the Electronic Problem

N

Number of electrons or electron pairs

Nocc

Number of electron occupied one-body electronic states

M, Nnuc  , Nn 

Number of nuclei

Zm  , Rm 

Nuclear charges, positions

β

Inverse temperature

Electron Coordinates

r

Single-electron position coordinate vector

rα 

Cartesian components of single-electron position coordinate vector

rij = |ri − rj |

Euclidean distance between electrons i and j

xi= (ri ,  si )

Space-spin coordinates of ith electron

Wavefunctions, Density Matrices, Densities, Orbitals

Ψ, Ψ(x₁ ,  … , xN )

Many-particle wavefunction

Γ

Many-particle density matrix

Γk 

k-particle reduced density matrix

γ

Single-particle density matrix/1-particle reduced density matrix

ρ, ρ(r )

Single-particle density

ρ₂  , ρ₂ (r₁ , r₂ )

Pair density

Φ

Kohn–Sham wavefunction

φi  , φa 

Occupied/unoccupied Kohn–Sham orbitals or spin-orbitals

i,  j,  a,  b

Summation indices: i, j for occupied orbitals, a, b for unoccupied orbitals

ΦI 

With I = (i₁, ⋯ , iN), or $$|\varphi _{i_1} \cdots \varphi _{i_N}\rangle $$ or $$\varphi _{i_1}\wedge \cdots \wedge \varphi _{i_N}$$ : Slater determinant built from orbitals $$\varphi _{i_1},...,\varphi _{i_N}$$

Functionals

F, F[ρ] 

General functional

EH 

Hartree energy functional

EHxc 

Hartree-exchange-correlation energy functional

Exc 

Exchange-correlation energy functional

Ex 

Exchange energy functional

Ec 

Correlation energy functional

T

Kinetic energy functional

TS 

Non-interacting kinetic energy functional

FLL 

Levy–Lieb constrained-search functional

FHK 

Hohenberg–Kohn functional

FDM 

Constrained-search density matrix functional

Energy Levels, Eigenvalues, Energy Densities, Fermi–Dirac Function

E₀ 

Ground state energy

εi  , εa 

Kohn–Sham eigenvalues

μ

Chemical potential, Fermi level

exc 

Exchange-correlation energy per unit volume

𝜖xc 

Exchange-correlation energy per particle

fβ 

Fermi–Dirac function

Potentials and Kernels

v

General (single-particle) potential

vext 

External potential

vs  , veff  , vKS 

Kohn–Sham effective potential

vH 

Hartree potential vH = ρo|⋅|⁻¹

vx  , vc  , vxc  , vHxc 

Exchange, correlation, exchange-correlation, Hxc potentials

fx  , fc  , fxc  , fHxc 

Exchange, correlation, exchange-correlation, Hxc kernels

Operators

Aor $$\widehat A$$

Generic linear operators

A∗  or A† 

Adjoint of A

H

Many-body Hamiltonian operators

h

General single-particle Hamiltonian

hρ  or hγ 

Kohn–Sham Hamiltonian

vC 

Coulomb operator

Δ = ∇² 

Laplace operator

χ

Reducible polarizability operator

χ₀ 

Irreducible polarizability operator

Fourier transform

Kohn–Sham map

Matrices

A⊤ 

Transpose of A

A∗ 

Conjugate transpose of A

Discretization Parameters, Basis Functions, Discretized Models

Nb 

Number of basis functions

Ng 

Number of grid points

ξμ 

Localized orbital (Gaussian, Slater, numerical,...)

Ξ

Set of localized orbitals

C, X

Coefficient matrices (of discretized KS orbitals in localized basis sets)

P

Discretized one-body density matrix

S

Overlap matrix

H

Hamiltonian matrices

V 

Generic matrix

VH 

Hartree matrix

Vx 

Exchange matrix

Vc 

Correlation matrix

Vxc  , VHxc 

xc, Hxc matrices

Contents

1 Review of Approximations for the Exchange-Correlation Energy in Density-Functional Theory1

Julien Toulouse

1.​1 Basics of Density-Functional Theory1

1.​2 Exact Expressions and Constraints for the Kohn–Sham Exchange and Correlation Functionals11

1.​3 Semilocal Approximations for the Exchange-CorrelationEnerg​y24

1.​4 Single-Determinant Hybrid Approximations45

1.​5 Multideterminant​ Hybrid Approximations53

1.​6 Semiempirical Dispersion Corrections and Nonlocal van der Waals Density Functionals66

1.​7 Orbital-Dependent Exchange-Correlation Density Functionals69

References78

2 On Connecting Density Functional Approximations to Theory91

Andreas Savin

2.​1 Introduction91

2.​2 Schrödinger Equation and Notations92

2.​3 The Density Functional Viewpoint95

2.​4 Practical Solutions for Density Functional Approximations96

2.​5 Outlook111

References111

3 Universal Functionals in Density Functional Theory115

Mathieu Lewin, Elliott H. Lieb and Robert Seiringer

3.​1 Introduction115

3.​2 Universal Functionals in Density Functional Theory117

3.​3 The Uniform Electron Gas and the Local DensityApproxima​tion128

3.​4 Kinetic Energy and Lieb–Thirring Inequalities132

3.​5 The Classical Interaction Energy and Lieb–Oxford Inequalities144

3.​6 Upper and Lower Bounds on the Levy–Lieb Functionals161

3.​7 The Hohenberg–Kohn Theorem163

3.​8 Proof of Theorem 3.​5168

References175

4 The Strong-Interaction Limit of Density Functional Theory183

Gero Friesecke, Augusto Gerolin and Paola Gori-Giorgi

4.​1 Introduction183

4.​2 The Many-Electron Schrödinger Equation and Universal Density Functional184

4.​3 The Strictly Correlated Electrons (SCE) Functional188

4.​4 Numerical Methods and Approximations231

4.​5 Kohn–Sham Combined with the Strong-Interaction Limit247

4.​6 Appendix:​ Kantorovich Duality258

References260

5 Moreau–Yosida Regularization in DFT267

Simen Kvaal

5.​1 Introduction267

5.​2 Exact DFT269

5.​3 Moreau–Yosida Regularization287

5.​4 Moreau–Yosida Regularized Exact DFT294

5.​5 Conclusion305

References305

6 Thermodynamic Limits of Electronic Systems307

David Gontier, Jianfeng Lu and Christoph Ortner

6.​1 Introduction307

6.​2 The Thomas–Fermi–von Weizsäcker Model309

6.​3 The Reduced Hartree–Fock Model318

6.​4 Scaling Limit for Kohn–Sham DFT323

References329

7 Numerical Methods for Kohn–Sham Models:​ Discretization, Algorithms, and Error Analysis333

Eric Cancès, Antoine Levitt, Yvon Maday and Chao Yang

7.​1 Introduction333

7.​2 Mathematical Structures of Discretized Kohn–Sham Problems336

7.​3 Discretization Methods352

7.​4 Pseudopotentials​357

7.​5 Algorithms363

7.​6 Error Analysis382

References393

8 Recent Progress in Evaluating the Kohn–Sham Map401

Lin Lin, Jianfeng Lu and Lexing Ying

8.​1 Introduction401

8.​2 Filtering Methods404

8.​3 Pole Expansion and Selected Inversion Method406

8.​4 Interpolative Separable Density Fitting Method414

8.​5 Hybrid Functionals417

8.​6 Conclusion and Future Directions422

References422

9 Augmented Plane Wave Methods for Full-Potential Calculations427

Huajie Chen and Reinhold Schneider

9.​1 Introduction427

9.​2 The Augmented Plane Wave Methods429

9.​3 Discretizations Based on Domain Decompositions434

9.​4 Convergence Analysis of the APW-Based Methods436

9.​5 Conclusions and Perspectives442

References443

10 Finite Element Methods for Density Functional Theory447

Xiaoying Dai and Aihui Zhou

10.​1 Introduction447

10.​2 Preliminaries452

10.​3 Finite Element Discretization460

10.​4 The A Priori and A Posteriori Analysis462

10.​5 Adaptive Finite Element Approximation466

10.​6 High Performance Computing471

10.​7 Concluding Remarks474

References474

11 Flexibilities of Wavelets as a Computational Basis Set for Large-Scale Electronic Structure Calculations479

Luigi Genovese and Thierry Deutsch

11.​1 Introduction479

11.​2 Operators of Generalized Kohn–Sham Formalism484

11.​3 Wavelets as a Computational Basis Set488

11.​4 Collocation Problems for Discretized Functions498

11.​5 Multipole-Preserving Collocations for Daubechies Wavelets:​ The Magic Filter505

11.​6 Representing the Charge Density on Real Space:​ An ISF Poisson Solver508

11.​7 The BigDFT Code Approach for Ground-State511

11.​8 Perspective518

References520

12 Accurate Approximations of Density Functional Theory for Large Systems with Applications to Defects in Crystalline Solids525

Kaushik Bhattacharya, Vikram Gavini, Michael Ortiz, Mauricio Ponga and Phanish Suryanarayana

12.​1 Introduction526

12.​2 Variational Formulation of Density Functional Theory528

12.​3 Filtering, Spectrum Splitting and Pseudopotentials​539

12.​4 Spatial Coarse-Graining:​ Finite-Element Discretization544

12.​5 Spectral Coarse-Graining:​ The Spectral Quadrature Method553

12.​6 Spatial and Spectral Coarse-Graining565

Appendix:​ Crystalline solids and the Cauchy–Born Rule573

References574

Glossary579

Contributors

Kaushik Bhattacharya

California Institute of Technology, Pasadena, CA, USA

Eric Cancès

CERMICS, Ecole des Ponts and Inria Paris, Marne-la-Vallée, France

Huajie Chen

School of Mathematical Sciences, Beijing Normal University, Beijing, China

Xiaoying Dai

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

Thierry Deutsch

Univ. Grenoble Alpes, CEA, IRIG-MEM, Grenoble, France

Gero Friesecke

Department of Mathematics, Technische Universität München, Garching bei München, Germany

Vikram Gavini

University of Michigan, Ann Arbor, MI, USA

Luigi Genovese

Univ. Grenoble Alpes, CEA, IRIG-MEM, Grenoble, France

Augusto Gerolin

Department of Chemistry and Biomolecular Sciences, University of Ottawa, Ottawa, ON, Canada

Augusto Gerolin

Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada

David Gontier

CEREMADE, Université Paris-Dauphine, PSL University, Paris, France

Paola Gori-Giorgi

Department of Theoretical Chemistry, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

Paola Gori-Giorgi

AI4Science Microsoft Research, Amsterdam, The Netherlands

Simen Kvaal

Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Oslo, Norway

Antoine Levitt

CERMICS, Ecole des Ponts and Inria Paris, Marne-la-Vallée, France

Mel Levy

Department of Chemistry and Quantum Theory Group, Tulane University, New Orleans, LA, USA

Mathieu Lewin

CNRS & CEREMADE, Université Paris-Dauphine, PSL University, Paris, France

Elliott H. Lieb

Departments of Mathematics and Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA

Lin Lin

Department of Mathematics, University of California, Berkeley and Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Jianfeng Lu

Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, NC, USA

Yvon Maday

Sorbonne Université, Université Paris-Cité, CNRS, Laboratoire Jacques-Louis Lions (LJLL), Paris, France

Yvon Maday

Institut Universitaire de France, Paris, France

Michael Ortiz

California Institute of Technology, Pasadena, CA, USA

Christoph Ortner

Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Mauricio Ponga

University of British Columbia, Vancouver, BC, Canada

Andreas Savin

Laboratoire de Chimie Théorique, CNRS and Sorbonne University, Paris, France

Reinhold Schneider

Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

Robert Seiringer

IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria

Phanish Suryanarayana

Georgia Institute of Technology, Atlanta, GA, USA

Julien Toulouse

Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, Paris, France

Julien Toulouse

Institut Universitaire de France, Paris, France

Chao Yang

Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Lexing Ying

Department of Mathematics and Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA

Aihui Zhou

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

Footnotes

1

See Sect. 1.​1.​1 for a precise definition of the notation.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_1

1. Review of Approximations for the Exchange-Correlation Energy in Density-Functional Theory

Julien Toulouse1, 2  

(1)

Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, Paris, France

(2)

Institut Universitaire de France, Paris, France

Julien Toulouse

Email: toulouse@lct.jussieu.fr

Abstract

In this chapter, we provide a review of the ground-state Kohn–Sham density-functional theory of electronic systems and some of its extensions, we present exact expressions and constraints for the exchange and correlation density functionals, and we discuss the main families of approximations for the exchange-correlation energy: semilocal approximations, single-determinant hybrid approximations, multideterminant hybrid approximations, dispersion-corrected approximations, as well as orbital-dependent exchange-correlation density functionals. The chapter aims at providing both a consistent bird’s-eye view of the field and a detailed description of some of the most used approximations. It is intended to be readable by chemists/physicists and applied mathematicians.

1.1 Basics of Density-Functional Theory

1.1.1 The Many-Body Problem

We consider an N-electron system (atom or molecule) in the Born–Oppenheimer and non-relativistic approximations. The electronic Hamiltonian in the position representation is, in atomic units,

(1.1)

where $$\nabla ^2_{{\mathbf {r}}_i} = \Delta _{{\mathbf {r}}_i}$$ is the Laplacian with respect to the electron coordinate ri and $$v_{{\mathrm {ne}}}({\mathbf {r}}_i)=- \sum _{\alpha =1}^{N_{\text{n}}} Z_{\alpha }/|{\mathbf {r}}_i - {\mathbf {R}}_{\alpha }|$$ is the nuclei-electron interaction depending on the positions {Rα} and charges {Zα} of the Nn nuclei. The stationary electronic states are determined by the time-independent Schrödinger equation,

(1.2)

where Ψ(x₁, x₂, …, xN) is a wave function written with space-spin coordinates $${\mathbf {x}}_i = ({\mathbf {r}}_i,\sigma _i) \in \mathbb {R}^3 \times \{ \uparrow ,\downarrow \}$$ (with $$\{ \uparrow ,\downarrow \} \cong \mathbb {Z}_2$$ being the set of spin coordinates) which is antisymmetric with respect to the exchange of two coordinates, and E is the associated energy.

Using Dirac notation, the Schrödinger equation (1.2) can be rewritten in a representation-independent formalism,

(1.3)

where the Hamiltonian is formally written as

with the kinetic-energy operator , the electron-electron interaction operator , and the nuclei-electron interaction operator .

The quantity of primary interest is the ground-state energy E₀. The variational theorem establishes that E₀ can be expressed as an infimum,

upper E 0 equals inf Underscript upper Psi element of script upper W Superscript upper N Baseline Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper H With caret vertical bar upper Psi right angle bracket comma

(1.4)

where the search is over the set of N-electron antisymmetric normalized wave functions Ψ having a finite kinetic energy,

(1.5)

where $$\bigwedge ^N$$ is the N-fold antisymmetrized tensor product, L² and H¹ are the standard Lebesgue and Sobolev spaces (i.e., respectively, the space of functions that are square integrable and the space of functions that are square integrable together with their first-order derivatives), and 〈⋅|⋅〉 designates the L² inner product. Density-functional theory (DFT) is based on a reformulation of the variational theorem in terms of the one-electron density defined as¹

(1.6)

which is normalized to the electron number, $$\int _{\mathbb {R}^3} \rho _\Psi ({\mathbf {r}}) \mathrm{d} {\mathbf {r}} = N$$ .

1.1.2 The Universal Density Functional

Building on the work of Hohenberg and Kohn [118], Levy [153] and Lieb [160] proposed to define the following universal density functional F[ρ] using a constrained-search approach,

upper F left bracket rho right bracket equals min Underscript upper Psi element of script upper W Subscript rho Superscript upper N Baseline Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi right angle bracket equals left angle bracket upper Psi left bracket rho right bracket vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi left bracket rho right bracket right angle bracket comma

(1.7)

where the minimization is done over the set of N-electron wave functions Ψ yielding the fixed density ρ [via Eq. (1.6)],

In Eq. (1.7), for a given density ρ, Ψ[ρ] denotes a minimizing wave function, which is known to exist [160] but is possibly not unique. This so-called Levy–Lieb functional F[ρ] is defined on the set of N-representable densities [160]:

(1.8)

We note that an alternative universal density functional can be defined by a Legendre–Fenchel transformation, or equivalently by a constrained-search over N-electron ensemble density matrices [160]. This so-called Lieb functional has the advantage of being convex but in this chapter we will simply use the Levy–Lieb functional of Eq. (1.7).

The exact ground-state energy can then be expressed as

(1.9)

and if a minimizer exists then it is a ground-state density ρ₀(r) for the potential vne(r). Hence, the ground-state energy can in principle be obtained by minimizing over the density ρ, i.e. a simple function of 3 real variables, which is a tremendous simplification compared to the minimization over a complicated many-body wave function Ψ. However, the explicit expression of F[ρ] in terms of the density is not known, and the direct approximations for F[ρ] that have been tried so far turn out not to be accurate enough.

If there is a unique wave function Ψ[ρ] (up to a phase factor) in Eq. (1.7), we can define kinetic and potential contributions to F[ρ],

where upper T left bracket rho right bracket equals left angle bracket upper Psi left bracket rho right bracket vertical bar ModifyingAbove upper T With caret vertical bar upper Psi left bracket rho right bracket right angle bracket ) and upper W Subscript e e Baseline left bracket rho right bracket equals left angle bracket upper Psi left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi left bracket rho right bracket right angle bracket ). The kinetic-energy functional T[ρ] is the contribution which is particularly difficult to approximate as an explicit functional of the density.

1.1.3 The Kohn–Sham Scheme

1.1.3.1 Decomposition of the Universal Functional

Following the idea of Kohn and Sham (KS) [135], the difficulty of approximating F[ρ] directly can be circumvented by decomposing F[ρ] as

(1.10)

where Ts[ρ] is the non-interacting kinetic-energy functional which can be defined with a constrained search,²

upper T Subscript normal s Baseline left bracket rho right bracket equals min Underscript upper Phi element of script upper S Subscript rho Superscript upper N Baseline Endscripts left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret vertical bar upper Phi right angle bracket equals left angle bracket upper Phi left bracket rho right bracket vertical bar ModifyingAbove upper T With caret vertical bar upper Phi left bracket rho right bracket right angle bracket comma

(1.11)

where the minimization is over the set of N-electron single-determinant wave functions Φ yielding the fixed density ρ:

Here, $$\mathcal {S}^N$$ is the set of N-electron single-determinant wave functions built from orthonormal spin orbitals

where ϕ₁ ∧ ϕ₂ ∧… ∧ ϕN designates the normalized N-fold antisymmetrized tensor product of N spin orbitals. The functional Ts[ρ] is defined over the entire set of N-representable densities $$\mathcal {D}^N$$ since any N-representable density can be obtained from a single-determinant wave function [80, 105, 160]. In Eq. (1.11), for a given density ρ, Φ[ρ] denotes a minimizing single-determinant wave function (again known to exist [160] but possibly not unique), also referred to as a KS wave function. The remaining functional EHxc[ρ] that Eq. (1.10) defines is called the Hartree-exchange-correlation functional. The idea of the KS scheme is then to use the exact expression of Ts[ρ] by reformulating the minimization over densities in Eq. (1.9) as a minimization over single-determinant wave functions Φ,

upper E 0 equals inf Underscript upper Phi element of script upper S Superscript upper N Baseline Endscripts left brace left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript n e Baseline vertical bar upper Phi right angle bracket plus upper E Subscript upper H x c Baseline left bracket rho Subscript upper Phi Baseline right bracket right brace comma

(1.12)

and if a minimum exists then any minimizing single-determinant wave function in Eq. (1.12) gives a ground-state density ρ₀(r). Thus, the exact ground-state energy can in principle be obtained by minimizing over single-determinant wave functions only. Even though a wave function has been reintroduced compared to Eq. (1.9), it is only a single-determinant wave function Φ and therefore it still represents a tremendous simplification over the usual variational theorem involving a correlated (multideterminant) wave function Ψ. The advantage of Eq. (1.12) over Eq. (1.9) is that a major part of the kinetic energy can be treated exactly with the single-determinant wave function Φ, and only EHxc[ρ] needs to be approximated as an explicit functional of the density.

In practice, EHxc[ρ] is decomposed as

(1.13)

where EH[ρ] is the Hartree energy functional,

(1.14)

representing the classical electrostatic repulsion energy for the charge distribution ρ(r) and which is calculated exactly, and Exc[ρ] is the exchange-correlation energy functional that remains to be approximated. If there is a unique KS wave function Φ[ρ] (up to a phase factor), we can further decompose Exc[ρ] as

(1.15)

where Ex[ρ] is the exchange energy functional,

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(1.16)

and Ec[ρ] is the correlation energy functional,

upper E Subscript c Baseline left bracket rho right bracket equals left angle bracket upper Psi left bracket rho right bracket vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi left bracket rho right bracket right angle bracket minus left angle bracket upper Phi left bracket rho right bracket vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi left bracket rho right bracket right angle bracket equals upper T Subscript c Baseline left bracket rho right bracket plus upper U Subscript c Baseline left bracket rho right bracket comma

which contains a kinetic contribution upper T Subscript c Baseline left bracket rho right bracket equals left angle bracket upper Psi left bracket rho right bracket vertical bar ModifyingAbove upper T With caret vertical bar upper Psi left bracket rho right bracket right angle bracket minus left angle bracket upper Phi left bracket rho right bracket vertical bar ModifyingAbove upper T With caret vertical bar upper Phi left bracket rho right bracket right angle bracket ) and a potential contribution upper U Subscript c Baseline left bracket rho right bracket equals left angle bracket upper Psi left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi left bracket rho right bracket right angle bracket minus left angle bracket upper Phi left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi left bracket rho right bracket right angle bracket ). Using the fact that Φ[ρ] is a single-determinant wave function, it can be shown that the exchange functional can be expressed as

(1.17)

where γσ, for σ ∈{↑, ↓}, is the spin-dependent one-particle KS density matrix,

(1.18)

which shows that $$E_{{\mathrm {x}}}[\rho ] \leqslant 0$$ . Moreover, from the variational definition of F[ρ], we see that $$E_{\text{c}}[\rho ] \leqslant 0$$ .

1.1.3.2 The Kohn–Sham Equations

The single-determinant wave function Φ in Eq. (1.12) is constructed from a set of N orthonormal occupied spin-orbitals {ϕi}i=1,…,N. To enforce Sz spin symmetry, each spin-orbital is factorized as $$\phi _i({\mathbf {x}}) = \varphi _i({\mathbf {r}}) \chi _{\sigma _i}(\sigma )$$ , where $$\varphi _i \in H^1(\mathbb {R}^3,\mathbb {C})$$ is a spatial orbital and $$\chi _{\sigma _i}$$ is a spin function from {↑, ↓} to {0, 1} such that $$\forall \sigma _i,\sigma \in \{ \uparrow ,\downarrow \}, \, \chi _{\sigma _i}(\sigma )=\delta _{\sigma _i,\sigma }$$ (σi is the spin of the spin-orbital i). Alternatively, when this is convenient, we will sometimes reindex the spatial orbitals, {φi}→{φiσ}, including explicitly the spin σ in the index. Writing the total electronic energy in Eq. (1.12) in terms of spin-orbitals and integrating over the spin variables, we obtain:

(1.19)

where the density is expressed in terms of the orbitals as

(1.20)

The minimization over Φ can then be recast into a minimization of E[{φi}] with respect to the spatial orbitals {φi} with the constraint of keeping the orbitals orthonormalized. The stationary condition with respect to variations of φi(r) leads to the KS equations [135],

(1.21)

where εi is the Lagrange multiplier associated to the normalization condition of φi and vHxc(r) is the Hartree-exchange-correlation potential defined as the functional derivative of EHxc[ρ] with respect to ρ(r),

(1.22)

which is itself a functional of the density. The orbitals satisfying Eq. (1.21) are called the KS orbitals. They are the eigenfunctions of the KS one-electron Hamiltonian,

(1.23)

where

(1.24)

is the KS potential, and εi are then the KS orbital energies. Note that Eq. (1.21) constitutes a set of coupled self-consistent equations since the potential depends on all the occupied orbitals {φi}i=1,…,N through the density [Eq. (1.20)]. The operator hs(r) defines the KS system which is a system of N non-interacting electrons in an effective external potential vs(r) ensuring that the density ρ(r) in Eq. (1.20) is the same as the exact ground-state density ρ₀(r) of the physical system of N interacting electrons. The exact ground-state energy E₀ is then obtained by injecting the KS orbitals in Eq. (1.19). The other (unoccupied) eigenfunctions in Eq. (1.21) define virtual KS orbitals $$\{\varphi _a\}_{a \geqslant N+1}$$ .

Note that to define the potential vHxc(r) in Eq. (1.22) a form of differentiability of the functional EHxc[ρ], also referred to as v-representability of the density, has been assumed. Justifying this is in fact subtle and has been debated [56, 57, 109, 142–144, 164] (see also Chap. 5 by Kvaal in this volume). Here, we will simply assume that a form of differentiability of EHxc[ρ] holds on at least a restricted set of densities that allows one to define the potential vHxc(r) up to an additive constant. For a further restricted set of densities that should include ground-state densities of electronic Hamiltonians of molecular systems [Eq. (1.1)], it is expected that the KS potential vs(r) tends to a constant as |r|→∞ and we choose this constant to be zero. Note also that the assumption of the existence of the KS potential vs(r) in Eq. (1.23), which does not depend on spin coordinates, implies that each spin-orbital must indeed have a definite Sz spin value.

Following the decomposition of EHxc[ρ] in Eq. (1.13), the potential vHxc(r) is written as

(1.25)

where is the Hartree potential and vxc(r) = δExc[ρ]∕δρ(r) is the exchange-correlation potential. Likewise, following the decomposition of Exc[ρ] in Eq. (1.15), and assuming that both Ex[ρ] and Ec[ρ] are differentiable with respect to ρ, the potential vxc(r) can be further decomposed as

(1.26)

where vx(r) = δEx[ρ]∕δρ(r) is the exchange potential and vc(r) = δEc[ρ]∕δρ(r) is the correlation potential. Thus, the KS equations are similar to the Hartree–Fock (HF) equations, with the difference that they involve a local exchange potential vx(r) instead of the nonlocal HF exchange potential, and an additional correlation potential. At least for ground-state densities of finite molecular systems, the exchange potential has the long-range asymptotic behavior (see, e.g., Ref. [93]),

StartLayout 1st Row gamma Subscript sigma Baseline left parenthesis bold r comma bold r Superscript prime Baseline right parenthesis 2nd Row equals upper N integral Underscript left parenthesis double struck upper R cubed times StartSet up arrow comma down arrow EndSet right parenthesis Superscript upper N minus 1 Baseline Endscripts upper Phi left bracket rho right bracket left parenthesis bold r prime comma sigma comma bold x 2 comma ellipsis comma bold x Subscript upper N Baseline right parenthesis Superscript asterisk Baseline upper Phi left bracket rho right bracket left parenthesis bold r comma sigma comma bold x 2 comma ellipsis comma bold x Subscript upper N Baseline right parenthesis normal d bold x 2 ellipsis normal d bold x Subscript upper N Baseline comma EndLayout

(1.27)

whereas the correlation potential decays faster [4].

1.1.3.3 Extension to Spin Density-Functional Theory

To deal with an external magnetic field, DFT has been extended from the total density to spin-resolved densities [13, 203]. Without external magnetic fields, this spin density-functional theory is in principle not necessary, even for open-shell systems (see, e.g., Ref. [263]). However, the dependence on the spin densities allows one to construct approximate exchange-correlation functionals that are more accurate, and is therefore almost always used in practice for open-shell systems.

The spin density ρσ,Ψ with σ ∈{↑, ↓} associated to a wave function Ψ is defined as

and integrates to the number of σ-spin electrons Nσ, i.e. $$\int _{\mathbb {R}^3} \rho _{\sigma ,\Psi }({\mathbf {r}}) \mathrm{d} {\mathbf {r}} = N_{\sigma }$$ . For $$\rho _\uparrow \in \mathcal {D}^{N_\uparrow }$$ and $$\rho _\downarrow \in \mathcal {D}^{N_\downarrow }$$ , the universal density functional is now defined as [200],

StartLayout 1st Row upper F left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket equals min Underscript upper Psi element of script upper W Subscript rho Sub Subscript up arrow Subscript comma rho Sub Subscript down arrow Subscript Superscript upper N Baseline Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi right angle bracket equals left angle bracket upper Psi left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket right angle bracket comma EndLayout

(1.28)

where the search is over the set of normalized antisymmetric wave functions Ψ with N = N↑ + N↓ electrons and yielding the fixed spin densities ρ↑ and ρ↓:

In Eq. (1.28), Ψ[ρ↑, ρ↓] designates a minimizing wave function.

A spin-dependent KS scheme is obtained by decomposing F[ρ↑, ρ↓] as

(1.29)

where Ts[ρ↑, ρ↓] is defined as

upper T Subscript normal s Baseline left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket equals min Underscript upper Phi element of script upper S Subscript rho Sub Subscript up arrow Subscript comma rho Sub Subscript down arrow Subscript Superscript upper N Baseline Endscripts left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret vertical bar upper Phi right angle bracket equals left angle bracket upper Phi left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket vertical bar ModifyingAbove upper T With caret vertical bar upper Phi left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket right angle bracket comma

(1.30)

with a constrained search over the set of single-determinant wave functions Φ yielding the fixed spin densities ρ↑ and ρ↓:

Here, Φ[ρ↑, ρ↓] denotes a minimizing KS single-determinant wave function, EH[ρ] is the Hartree energy which is a functional of the total density ρ = ρ↑ + ρ↓ only [Eq. (1.14)], and Exc[ρ↑, ρ↓] is the spin-dependent exchange-correlation energy functional. The ground-state energy is then obtained as

upper E 0 equals inf Underscript upper Phi element of script upper S Superscript upper N Baseline Endscripts left brace left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript n e Baseline vertical bar upper Phi right angle bracket plus upper E Subscript normal upper H Baseline left bracket rho Subscript upper Phi Baseline right bracket plus upper E Subscript xc Baseline left bracket rho Subscript up arrow comma upper Phi Baseline comma rho Subscript down arrow comma upper Phi Baseline right bracket right brace period

(1.31)

Writing the spatial orbitals of the spin-unrestricted determinant as {φiσ}i=1,…,N (with the index explicitly including the spin σ now for clarity), we arrive at the spin-dependent KS equations,

(1.32)

with the spin-dependent exchange-correlation potential,

(1.33)

and the spin density,

(1.34)

As before, if there is a unique KS wave function Φ[ρ↑, ρ↓] (up to a phase factor), we can decompose Exc[ρ↑, ρ↓] into exchange and correlation contributions,

(1.35)

with upper E Subscript normal x Baseline left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket equals left angle bracket upper Phi left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket right angle bracket minus upper E Subscript normal upper H Baseline left bracket rho right bracket ). It turns out that the spin-dependent exchange functional Ex[ρ↑, ρ↓] can be exactly expressed in terms of the spin-independent exchange functional Ex[ρ] [183],

(1.36)

which is known as the spin-scaling relation and stems directly from the fact the ↑- and ↓-spin electrons are uncoupled in the exchange energy [see Eq. (1.17)]. Therefore, any approximation for Ex[ρ] can be easily extended to an approximation for Ex[ρ↑, ρ↓]. Unfortunately, there is no such relation for the spin-dependent correlation functional Ec[ρ↑, ρ↓].

Obviously, in the spin-unpolarized case, i.e. ρ↑ = ρ↓ = ρ∕2, this spin-dependent formalism reduces to the spin-independent one.

1.1.4 The Generalized Kohn–Sham Scheme

An important extension of the KS scheme is the so-called generalized Kohn–Sham (GKS) scheme [222], which recognizes that the universal density functional F[ρ] of Eq. (1.7) can be decomposed in other ways than the KS decomposition of Eq. (1.10). In particular, we can decompose F[ρ] as

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(1.37)

where S[ Φ] is any functional of a single-determinant wave function $$\Phi \in \mathcal {S}^{N}$$ leading to a minimum in Eq. (1.37), and $$\bar {S}[\rho ]$$ is the corresponding complementary density functional that makes Eq. (1.37) exact. Defining the S-dependent GKS exchange-correlation functional as

(1.38)

we can express the exact ground-state energy as

StartLayout 1st Row upper E 0 equals inf Underscript upper Phi element of script upper S Superscript upper N Baseline Endscripts left brace left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript n e Baseline vertical bar upper Phi right angle bracket plus upper E Subscript normal upper H Baseline left bracket rho Subscript upper Phi Baseline right bracket plus upper E Subscript xc Superscript upper S Baseline left bracket upper Phi right bracket right brace comma EndLayout

(1.39)

and if a minimum exists then any minimizing single-determinant wave function in Eq. (1.39) gives a ground-state density ρ₀(r). Similarly to the KS equations [Eq. (1.21)], Eq. (1.39) leads to the one-electron GKS equations,

(1.40)

where $$v_{\bar {S}}({\mathbf {r}}) = \delta \bar {S}[\rho ]/\delta \rho ({\mathbf {r}})$$ is a local potential and $$\delta S[\Phi ]/\delta \varphi _{i\sigma }^*({\mathbf {r}})$$ generates a one-electron (possibly nonlocal) operator.

In the special case S[ Φ] = 0, we recover the KS exchange-correlation density functional:

(1.41)

Due to the freedom in the choice of S[ Φ], there is an infinity of GKS exchange-correlation functionals $$E_{{\mathrm {xc}}}^{S}[\Phi ]$$ giving the exact ground-state energy via Eq. (1.39). This freedom and the fact that Φ carries more information than ρΦ gives the possibility to design more accurate approximations for the exchange-correlation energy.

Of course, by starting from the density functional F[ρ↑, ρ↓] in Eq. (1.28), this GKS scheme can be extended to the spin-dependent case, leading to GKS exchange-correlation functionals of the form $$E_{{\mathrm {xc}}}^{S}[\Phi ] = S[\Phi ] + \bar {S}[\rho _{\uparrow ,\Phi },\rho _{\downarrow ,\Phi }]$$ .

1.2 Exact Expressions and Constraints for the Kohn–Sham Exchange and Correlation Functionals

1.2.1 The Exchange and Correlation Holes

Let us consider the pair density associated with the wave function Ψ[ρ] defined in Eq. (1.7),

(1.42)

which is a functional of the density, and normalized to the number of electron pairs, $$\int _{\mathbb {R}^3\times \mathbb {R}^3} \rho _2({\mathbf {r}}_1,{\mathbf {r}}_2) \mathrm{d} {\mathbf {r}}_1 \mathrm{d} {\mathbf {r}}_2 = N(N-1)$$ . The pair density is proportional to the probability density of finding two electrons at positions (r₁, r₂) with all the other electrons being anywhere. The pair density is useful to express the expectation value of the electron-electron interaction operator,

left angle bracket upper Psi left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi left bracket rho right bracket right angle bracket equals one half integral Underscript double struck upper R cubed times double struck upper R cubed Endscripts StartFraction rho 2 left parenthesis bold r 1 comma bold r 2 right parenthesis Over StartAbsoluteValue bold r 1 minus bold r 2 EndAbsoluteValue EndFraction normal d bold r 1 normal d bold r 2 period

(1.43)

Mirroring the decomposition of the Hartree-exchange-correlation energy performed in the KS scheme [Eq. (1.13)], the pair density can be decomposed as

(1.44)

The product of the densities ρ(r₁)ρ(r₂) corresponds to the case of independent electrons [up to a change of normalization, i.e. $$\int _{\mathbb {R}^3\times \mathbb {R}^3} \rho ({\mathbf {r}}_1) \rho ({\mathbf {r}}_2) \mathrm{d} {\mathbf {r}}_1 \mathrm{d} {\mathbf {r}}_2 = N^2$$ instead of N(N − 1)] and the exchange-correlation pair density ρ2,xc(r₁, r₂) represents the modification of the pair density due to exchange and correlation effects between the electrons. It can be further written as

(1.45)

where hxc(r₁, r₂) is the exchange-correlation hole. Introducing the conditional density $$\rho _{ }^{\text{cond}}({\mathbf {r}}_1,{\mathbf {r}}_2) =\rho _2({\mathbf {r}}_1,{\mathbf {r}}_2)/\rho ({\mathbf {r}}_1)$$ of the remaining N − 1 electrons at r₂ given that one electron has been found at r₁, the exchange-correlation hole can be interpreted as the modification of $$\rho _{ }^{\text{cond}}({\mathbf {r}}_1,{\mathbf {r}}_2)$$ due to exchange and correlation effects:

(1.46)

The positivity of ρ₂(r₁, r₂) implies that

Moreover, from Eq. (1.46), we have the following sum rule:

(1.47)

We can separate the exchange and correlation contributions in the exchange-correlation hole. For this, consider the pair density ρ2,KS(r₁, r₂) associated with the KS single-determinant wave function Φ[ρ] defined in Eq. (1.11). It can be decomposed as

(1.48)

where ρ2,x(r₁, r₂) is the exchange pair density, which is further written as

(1.49)

where hx(r₁, r₂) is the exchange hole. Just like the exchange-correlation hole, the exchange hole satisfies the conditions

and

(1.50)

Moreover, since the exchange hole can be written as [compare with Eq. (1.17)]

(1.51)

where $$\gamma _{\sigma }({\mathbf {r}}_1,{\mathbf {r}}_2) = \sum _{i=1}^{N_\sigma } \varphi _{i \sigma }^*({\mathbf {r}}_2) \varphi _{i \sigma }({\mathbf {r}}_1)$$ is the spin-dependent one-particle KS density matrix, it thus appears that the exchange hole is always non-positive,

(1.52)

From Eqs. (1.16), (1.43), (1.48), and (1.49), it can be seen that the exchange energy functional can be written in terms of the exchange hole,

(1.53)

leading to the interpretation of Ex as the electrostatic interaction energy of an electron and its exchange hole. It is useful to write the exchange energy functional as

(1.54)

where εx[ρ](r₁) is the exchange energy density per particle,

(1.55)

which is itself a functional of the density. It is also convenient to define the exchange energy density ex[ρ](r) = ρ(r)εx[ρ](r). For finite systems, we have the exact asymptotic behavior [18, 168]

rho 2 left parenthesis bold r 1 comma bold r 2 right parenthesis equals rho left parenthesis bold r 1 right parenthesis rho left parenthesis bold r 2 right parenthesis plus rho Subscript 2 comma x c Baseline left parenthesis bold r 1 comma bold r 2 right parenthesis period

(1.56)

The correlation hole is defined as the difference

and, from Eqs. (1.47) and (1.50), satisfies the sum rule

(1.57)

which implies that the correlation hole has negative and positive contributions.³ In contrast with the exchange hole which is a smooth function of the interelectronic coordinate r₁₂ = r₂ −r₁, the correlation hole satisfies the electron-electron cusp condition (i.e., it has a derivative discontinuity in r₁₂) [132, 252],

(1.58)

where $$h_{\text{c}}^{\prime }({\mathbf {r}}_1,{\mathbf {r}}_1) = (\partial \widetilde {h}_{\text{c}}({\mathbf {r}}_1,r_{12})/ \partial r_{12})_{r_{12}=0}$$ is the first-order derivative of the spherically averaged correlation hole and S(0, r₁₂) designates the sphere centered at 0 and of radius r₁₂ = |r₁₂|. The potential contribution to the correlation energy can be written in terms of the correlation hole:

(1.59)

In order to express the total correlation energy Ec[ρ] = Tc[ρ] + Uc[ρ] in a form similar to Eq. (1.59), we need to introduce the adiabatic-connection formalism.

1.2.2 The Adiabatic Connection

The idea of the adiabatic connection [102, 146, 147] (see, also, Ref. [106]) is to have a continuous path between the non-interacting KS system and the physical system while keeping the ground-state density constant. This allows one to obtain a convenient expression for the correlation functional Ec[ρ] as an integral over this path. An infinity of such paths are possible, but the one most often considered consists in switching on the electron-electron interaction linearly with a coupling constant λ. The Hamiltonian along this adiabatic connection is

rho 2 left parenthesis bold r 1 comma bold r 2 right parenthesis equals rho left parenthesis bold r 1 right parenthesis rho left parenthesis bold r 2 right parenthesis plus rho Subscript 2 comma x c Baseline left parenthesis bold r 1 comma bold r 2 right parenthesis period

(1.60)

where is the external local potential operator imposing that the ground-state density is the same as the ground-state density of the physical system for all $$\lambda \in \mathbb {R}$$ . Of course, Eq. (1.60) relies on a v-representability assumption, i.e. the external potential is assumed to exist for all λ. The Hamiltonian (1.60) reduces to the KS non-interacting Hamiltonian for λ = 0 and to the physical Hamiltonian for λ = 1.

Just as for the physical system, it is possible to define a universal functional associated with the system of Eq. (1.60) for each value of the parameter λ,

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(1.61)

where Ψλ[ρ] denotes a minimizing wave function. This functional can be decomposed as

(1.62)

where $$E_{{\mathrm {Hxc}}}^{\lambda }[\rho ]$$ is the Hartree-exchange-correlation functional associated with the interaction . One can write this functional as $$E_{{\mathrm {Hxc}}}^{\lambda }[\rho ]= E_{{\mathrm {H}}}^{\lambda }[\rho ] + E_{{\mathrm {x}}}^{\lambda }[\rho ] + E_{{\text{c}}}^{\lambda }[\rho ]$$ , where the Hartree and exchange contributions are simply linear in λ,

and

upper E Subscript normal x Superscript lamda Baseline left bracket rho right bracket equals left angle bracket upper Phi left bracket rho right bracket vertical bar lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi left bracket rho right bracket right angle bracket minus upper E Subscript normal upper H Superscript lamda Baseline left bracket rho right bracket equals lamda upper E Subscript normal x Baseline left bracket rho right bracket period

The correlation contribution is nonlinear in λ,

upper E Subscript c Superscript lamda Baseline left bracket rho right bracket equals left angle bracket upper Psi Superscript lamda Baseline left bracket rho right bracket vertical bar ModifyingAbove upper T With caret plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline left bracket rho right bracket right angle bracket minus left angle bracket upper Phi left bracket rho right bracket vertical bar ModifyingAbove upper T With caret plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi left bracket rho right bracket right angle bracket period

(1.63)

We will assume that Fλ[ρ] is of class C¹ as a function of λ for λ ∈ [0, 1] and that Fλ=0[ρ] = Ts[ρ], the latter condition being guaranteed for nondegenerate KS systems [see footnote on the definition of Ts[ρ] just before Eq. (1.11)]. Taking the derivative of Eq. (1.63) with respect to λ and using the Hellmann–Feynman theorem for the wave function Ψλ[ρ],⁴ we obtain

StartLayout 1st Row 1st Column StartFraction partial differential upper E Subscript c Superscript lamda Baseline left bracket rho right bracket Over partial differential lamda EndFraction 2nd Column equals 3rd Column left angle bracket upper Psi Superscript lamda Baseline left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline left bracket rho right bracket right angle bracket minus left angle bracket upper Phi left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi left bracket rho right bracket right angle bracket period EndLayout

(1.64)

Integrating over λ from 0 to 1, and using $$E_{{\text{c}}}^{\lambda =1}[\rho ]=E_{{\text{c}}}[\rho ]$$ and $$E_{{\text{c}}}^{\lambda =0}[\rho ]=0$$ , we arrive at the adiabatic-connection formula for the correlation energy functional of the physical system

upper E Subscript c Baseline left bracket rho right bracket equals integral Subscript 0 Superscript 1 Baseline normal d lamda left angle bracket upper Psi Superscript lamda Baseline left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline left bracket rho right bracket right angle bracket minus left angle bracket upper Phi left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi left bracket rho right bracket right angle bracket period

(1.65)

By introducing the correlation hole $$h_{{\text{c}}}^{\lambda }({\mathbf {r}}_1,{\mathbf {r}}_2)$$ associated to the wave function Ψλ[ρ], the adiabatic-connection formula for the correlation energy can also be written as

(1.66)

or, noting that $$h_{{\text{c}}}^{\lambda }({\mathbf {r}}_1,{\mathbf {r}}_2)$$ is the only quantity that depends on λ in Eq. (1.66), in a more compact way,

(1.67)

where $$\bar {h}_{{\text{c}}}({\mathbf {r}}_1,{\mathbf {r}}_2) = \int _{0}^{1} \mathrm{d} \lambda \; h_{{\text{c}}}^{\lambda }({\mathbf {r}}_1,{\mathbf {r}}_2)$$ is the coupling-constant-integrated correlation hole. This leads to the interpretation of Ec as the electrostatic interaction energy of an electron with its coupling-constant-integrated correlation hole. As for the exchange energy, the correlation energy functional can be written as

(1.68)

where εc[ρ](r₁) is the correlation energy density per particle

(1.69)

which is a functional of the density. We can also define the correlation energy density ec[ρ](r) = ρ(r)εc[ρ](r).

Finally, note that the sum-rule and cusp conditions of Eqs. (1.57) and (1.58) apply to the λ-dependent correlation hole in the form

(1.70)

and

(1.71)

1.2.3 One-Orbital and One-Electron Spatial Regions

For systems composed of only one spin-↑ (or, symmetrically, one spin-↓) electron (e.g., the hydrogen atom) with ground-state density ρ1e(r) = |φ1↑(r)|² where φ1↑(r) is the unique occupied KS orbital, the exchange hole in Eq. (1.51) simplifies to hx(r₁, r₂) = −ρ(r₂), and consequently the exchange energy cancels out the Hartree energy:

(1.72)

Furthermore, the correlation energy vanishes:

(1.73)

This must of course also be true for the spin-dependent version of the functionals introduced in Sect. 1.1.3.3, i.e.

(1.74)

and

(1.75)

For systems composed of two opposite-spin electrons (e.g., the helium atom or the dihydrogen molecule) in a unique doubly occupied KS orbital φ₁(r) = φ1↑(r) = φ1↓(r) with ground-state density $$\rho _{\text{2e}}^{\uparrow \downarrow }({\mathbf {r}}) = 2 |\varphi _1({\mathbf {r}})|{ }^2$$ , the exchange hole simplifies to hx(r₁, r₂) = −ρ(r₂)∕2, and consequently the exchange energy is equal to half the opposite of the Hartree energy:

(1.76)

These are constraints for the exchange and correlation density functionals in the special cases N = 1 and N = 2.

These special cases can be extended to more general systems. For systems with $$N\geqslant 1$$ electrons containing a spatial region $$\Omega _{\text{1o}}^\uparrow $$ in which, among the occupied KS orbitals, only one spin-↑ (or, symmetrically, one spin-↓) orbital is not zero (or, more generally, takes non-negligible values), we have again in this region

and therefore the contribution to the exchange energy density per particle coming from this region must locally cancel out the contribution to the Hartree energy density per particle coming from the same region,

(1.77)

where

Similarly, for systems with $$N \geqslant 2$$ electrons containing a spatial region $$\Omega _{\text{1o}}^{\uparrow \downarrow }$$ in which, among the occupied KS orbitals, only one doubly occupied orbital is not zero, we have in this region

and therefore the contribution to the exchange energy density per particle coming from this region must locally be equal to half the opposite of the contribution to the Hartree energy density per particle coming from the same region,

(1.78)

Thus, we see, particularly clearly for these $$\Omega _{\text{1o}}^\uparrow $$ or $$\Omega _{\text{1o}}^{\uparrow \downarrow }$$ regions, that the Hartree functional introduces a spurious self-interaction contribution which must be eliminated by the exchange functional. Even though the concepts of $$\Omega _{\text{1o}}^\uparrow $$ and $$\Omega _{\text{1o}}^{\uparrow \downarrow }$$ regions are formal, in practice they can be approximately realized in chemical systems. For example, the unpaired electron in a radical approximately corresponds to a $$\Omega _{\text{1o}}^\uparrow $$ , and an electron pair in a single covalent bond, in a lone pair, or in a core orbital approximately corresponds to a $$\Omega _{\text{1o}}^{\uparrow \downarrow }$$ region.

We can also consider one-electron regions Ω1e that we define as⁵

(1.79)

where $$\rho _2^\lambda ({\mathbf {r}}_1,{\mathbf {r}}_2)$$ is the pair density associated to the wave function Ψλ[ρ] along the adiabatic connection. This implies

where $$\bar {h}_{{\mathrm {xc}}}({\mathbf {r}}_1,{\mathbf {r}}_2) = h_{{\mathrm {x}}}({\mathbf {r}}_1,{\mathbf {r}}_2) + \bar {h}_{{\text{c}}}({\mathbf {r}}_1,{\mathbf {r}}_2)$$ and, consequently, the contribution to the exchange-correlation energy density per particle coming from this region must locally cancel out the contribution to the Hartree energy density per particle coming from the same region,

(1.80)

where . For regions that are simultaneously one-electron and one-orbital regions, this simply implies that the contribution to the correlation energy must vanish,

(1.81)

where $$\varepsilon _{\text{c}}^{\Omega } ({\mathbf {r}}_1) = (1/2) \int _{\Omega } \bar {h}_{{\text{c}}}({\mathbf {r}}_1,{\mathbf {r}}_2)/|{\mathbf {r}}_1 -{\mathbf {r}}_2| \mathrm{d} {\mathbf {r}}_2$$ , and we say that the correlation functional must not introduce a self-interaction error. However, the definition of Ω1e regions also includes the case of an electron entangled in several orbitals, such as the region around one hydrogen atom in the dissociated dihydrogen molecule. In this latter case, the Hartree functional introduces an additional spurious contribution (beyond the spurious self-interaction) which must be compensated by a static correlation (or strong correlation) contribution in the exchange-correlation functional.

1.2.4 Coordinate Scaling

1.2.4.1 Uniform Coordinate Scaling

We consider a norm-preserving uniform scaling of the spatial coordinates in the N-electron wave function along the adiabatic connection Ψλ[ρ] [introduced in Eq. (1.61)] while leaving untouched the spin coordinates [154, 155, 157],

where γ ∈ (0, +∞) is a scaling factor. The scaled wave function $$\Psi _{{\gamma }}^{\lambda }[\rho ]$$ yields the scaled density

with $$\int _{\mathbb {R}^3} \! \rho _{{\gamma }}({\mathbf {r}}) \mathrm{d} {\mathbf {r}} = \int _{\mathbb {R}^3} \! \rho ({\mathbf {r}}) \mathrm{d} {\mathbf {r}} = N$$ , and minimizes left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus lamda gamma ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi right angle bracket ) since

left angle bracket upper Psi Subscript gamma Superscript lamda Baseline left bracket rho right bracket vertical bar ModifyingAbove upper T With caret plus lamda gamma ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Subscript gamma Superscript lamda Baseline left bracket rho right bracket right angle bracket equals gamma squared left angle bracket upper Psi Superscript lamda Baseline left bracket rho right bracket vertical bar ModifyingAbove upper T With caret plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline left bracket rho right bracket right angle bracket period

We thus conclude that the scaled wave function at the density ρ and coupling constant λ corresponds to the wave function at the scaled density ργ and coupling constant λγ,

or, equivalently,

and that the universal density functional satisfies the scaling relation

or, equivalently,

(1.82)

At λ = 0, we find the scaling relation of the KS wave function Φ[ρ] introduced in Sect. 1.1.3.1:

This directly leads to the scaling relation for the non-interacting kinetic density functional [see Eq. (1.11)],

for the Hartree density functional [see Eq. (1.14)],

and for the exchange density functional [see Eq. (1.16)],

(1.83)

However, the correlation density functional Ec[ρ] has the more complicated scaling (as F[ρ]),

and, in particular for λ = 1,

(1.84)

These scaling relations allow one to find the behavior of the density functionals in the high- and low-density limits. In the high-density limit (γ →∞), it can be shown from Eq. (1.84) that, for nondegenerate KS systems, the correlation functional Ec[ρ] goes to a constant,

(1.85)

where $$E_{\text{c}}^{{\mathrm {GL2}}}[\rho ]$$ is the second-order Görling–Levy (GL2) correlation energy [90, 91] (see Sect. 1.7.2). This is also called the weak-correlation limit since in this limit the correlation energy is negligible with respect to the exchange energy which is itself negligible with respect to the non-interacting kinetic energy: |Ec[ργ]| = O(γ⁰) ≪|Ex[ργ]| = O(γ) ≪ Ts[ργ] = O(γ²). Equation (1.85) is an important constraint since atomic and molecular correlation energies are often close to the high-density limit. For example, for the ground-state density of the helium atom, we have Ec[ρ] = −0.0421 hartree and limγ→∞Ec[ργ] = −0.0467 hartree [119].

In the low-density limit (γ → 0), it can be shown from Eq. (1.82) that the Hartree-exchange-correlation energy EHxc[ρ] goes to zero linearly in γ,

upper E Subscript c Baseline left bracket rho Subscript gamma Baseline right bracket equals gamma squared upper E Subscript c Superscript 1 divided by gamma Baseline left bracket rho right bracket period

(1.86)

where upper W Subscript e e Superscript SCE Baseline left bracket rho right bracket equals inf Underscript upper Psi element of script upper W Subscript rho Superscript upper N Baseline Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi right angle bracket ) is the strictly-correlated-electron (SCE) functional [86, 220, 221, 223]. This is also called the strong-interaction limit since in this limit the Hartree-exchange-correlation energy dominates over the non-interacting kinetic energy: EHxc[ργ] = O(γ) ≫ Ts[ργ] = O(γ²). In this limit, the electrons strictly localize relatively to each other. In particular, for the uniform-electron gas, this corresponds to the Wigner crystallization. Thus, in this limit, each electron is within a one-electron region Ω1e [as defined in Eq. (1.79)]. For more information on the SCE functional, see Chap. 4 by Friesecke et al. in this volume.

1.2.4.2 Non-uniform Coordinate Scaling

We can also consider non-uniform one-dimensional or two-dimensional coordinate scalings of the density [156, 184],

(1.87)

and

(1.88)

which also preserve the number of the electrons. These non-uniform density scalings provide constraints for the exchange and correlation functionals. In particular, in the non-uniform one-dimensional high-density limit, the exchange functional remains finite and the correlation functional vanishes [89, 154]:

(1.89)

and

(1.90)

Also, in the non-uniform two-dimensional low-density limit, we have [89, 154]:

(1.91)

and

(1.92)

The conditions of Eqs. (1.89)–(1.92) are particularly useful because they also correspond to the limit of rapidly varying densities [158].

1.2.5 Atoms in the Limit of Large Nuclear Charge

A practical realization of the uniform high-density limit is provided by atomic ions in the limit of large nuclear charge, Z →∞, at fixed electron number N (see Refs. [65, 66, 123, 233]). In this limit, the exact ground-state atomic density ρN,Z(r) becomes the density of the isoelectronic hydrogenic (i.e., without electron-electron interaction) atom $$\rho _{N,Z}^{\text{H}}({\mathbf {r}})$$ , which obeys a simple scaling with Z:

StartLayout 1st Row limit Underscript gamma right arrow 0 Endscripts StartFraction 1 Over gamma EndFraction upper E Subscript c Baseline left bracket rho Subscript gamma Superscript left parenthesis 2 right parenthesis Baseline right bracket equals 0 period EndLayout

One can thus apply Eqs. (1.83) and (1.85) with γ = Z, which reveals that in an isoelectronic series the exchange functional scales linearly with Z,

StartLayout 1st Row limit Underscript gamma right arrow 0 Endscripts StartFraction 1 Over gamma EndFraction upper E Subscript c Baseline left bracket rho Subscript gamma Superscript left parenthesis 2 right parenthesis Baseline right bracket equals 0 period EndLayout

(1.93)

and, for nondegenerate KS systems, the correlation functional saturates to a constant,

(1.94)

Equations (1.93) and (1.94) are constraints for the exchange and correlation functionals, particularly relevant for highly ionized atoms but also for the core-electron regions of heavy atoms in neutral systems.

Another very interesting limit is the one of large nuclear charge of neutral atoms, N = Z →∞ (see, e.g., Ref. [129]). In this semiclassical limit, the exact ground-state atomic density ρN,Z(r) tends to the Thomas–Fermi (TF) density of a neutral atom $$\rho _{Z}^{\text{TF}}({\mathbf {r}})$$ which has a known scaling with Z [162, 163]:

StartLayout 1st Row limit Underscript gamma right arrow 0 Endscripts StartFraction 1 Over gamma EndFraction upper E Subscript c Baseline left bracket rho Subscript gamma Superscript left parenthesis 2 right parenthesis Baseline right bracket equals 0 period EndLayout

(1.95)

In this limit, it was suggested that the exact exchange and correlation energies have the approximate large-Z asymptotic expansions [28, 30, 51]

StartLayout 1st Row limit Underscript gamma right arrow 0 Endscripts StartFraction 1 Over gamma EndFraction upper E Subscript c Baseline left bracket rho Subscript gamma Superscript left parenthesis 2 right parenthesis Baseline right bracket equals 0 period EndLayout

(1.96)

and

StartLayout 1st Row limit Underscript gamma right arrow 0 Endscripts StartFraction 1 Over gamma EndFraction upper E Subscript c Baseline left bracket rho Subscript gamma Superscript left parenthesis 2 right parenthesis Baseline right bracket equals 0 period EndLayout

(1.97)

with the coefficients Ax = 0.220827, Ac = 0.020727, Bx ≈ 0.224, Bc ≈ 0.0372. Recently, it was argued that there is in fact a missing term in $$Z \ln Z$$ in the expansion of the exchange energy in Eq. (1.96) [10, 42].

1.2.6 Lieb–Oxford Lower Bound

Lieb and Oxford derived a lower bound for the indirect Coulomb energy (i.e., the two-particle Coulomb potential energy beyond the Hartree energy) [161], which, when expressed in terms of the exchange or exchange-correlation functional, takes the form [187]

(1.98)

where the optimal (i.e., smallest) constant CLO (independent of the electron number N) was originally shown to be in the range $$1.23 \leqslant C_{\text{LO}} \leqslant 1.68$$ [161]. The range was later successively narrowed to $$1.4442 \leqslant C_{\text{LO}} \leqslant 1.5765$$ [36, 41, 159, 187]. This bound is approached only in the low-density limit where the correlation energy becomes comparable to the exchange energy. Numerical results suggest that for densities of most physical systems the Lieb–Oxford lower bound on the exchange-correlation energy is far from being reached [182].

For two-electron densities, there is a specific tighter bound,

(1.99)

with the best known constant C₂ = 1.234 [161]. For one-electron densities, an even tighter bound is known for the exchange functional [74, 161],

(1.100)

with the optimal constant C₁ = 1.092. For two-electron spin-unpolarized densities, we have $$E_{{\mathrm {x}}}[\rho _{\text{2e}}^{\uparrow \downarrow }] = 2 E_{{\mathrm {x}}}[\rho _{\text{1e}}]$$ with $$\rho _{\text{1e}} = \rho _{\text{2e}}^{\uparrow \downarrow }/2$$ , and Eq. (1.100) implies [194]

(1.101)

which is a much tighter bound than the bounds of Eqs. (1.98) and (1.99).

1.3 Semilocal Approximations for the Exchange-Correlation Energy

We review here the different classes of semilocal approximations for the exchange-correlation energy.

1.3.1 The Local-Density Approximation

In the local-density approximation (LDA), introduced by Kohn and Sham [135], the exchange-correlation functional is approximated as

where $$e_{\text{xc}}^{\text{UEG}}(\rho )$$ is the exchange-correlation energy density of the infinite uniform electron gas (UEG) with the density ρ. The UEG represents a family of systems of interacting electrons with an arbitrary spatially constant density ρ ∈ [0, +∞) that acts as a parameter. Thus, in the LDA, the exchange-correlation energy density of an inhomogeneous system at a spatial point of density ρ(r) is approximated as the exchange-correlation energy density of the UEG of the same density.

In the spin-dependent version of LDA, sometimes specifically referred to as the local-spin-density approximation (LSDA), the exchange-correlation functional is approximated as [13]

where $$e_{\text{xc}}^{\text{UEG}}(\rho _\uparrow ,\rho _\downarrow )$$ is the exchange-correlation energy density of the UEG with spin densities ρ↑ and ρ↓. For spin-unpolarized systems, we recover the spin-independent LDA as $$E_{\text{xc}}^{{\mathrm {LDA}}}[\rho ] = E_{\text{xc}}^{{\mathrm {LSDA}}}[\rho /2,\rho /2]$$ .

The function $$e_{\text{xc}}^{\text{UEG}}$$ is a sum of exchange and correlation contributions, $$e_{\text{xc}}^{\text{UEG}} = e_{{\mathrm {x}}}^{\text{UEG}} + e_{\text{c}}^{\text{UEG}}$$ , and it is convenient to introduce exchange and correlation energies per particle, $$\varepsilon _{{\mathrm {x}}}^{\text{UEG}}$$ and $$\varepsilon _{\text{c}}^{\text{UEG}}$$ , such that $$e_{{\mathrm {x}}}^{\text{UEG}} = \rho \; \varepsilon _{{\mathrm {x}}}^{\text{UEG}}$$ and $$e_{\text{c}}^{\text{UEG}} = \rho \; \varepsilon _{\text{c}}^{\text{UEG}}$$ . The expression of the exchange energy per particle of the spin-unpolarized UEG is

(1.102)

where Cx = −(3∕4)(3∕π)1∕3, and the spin-polarized version is simply obtained from the spin-scaling relation [Eq. (1.36)], leading to

where ζ = (ρ↑− ρ↓)∕ρ is the spin polarization and ϕ₄(ζ) is defined by the general spin-scaling function

(1.103)

The LDA exchange functional is associated with the names of Dirac [44] and Slater [228]. For a rigorous mathematical derivation of Eq. (1.102), see Ref. [64].

The correlation energy per particle $$\varepsilon _{\text{c}}^{\text{UEG}}(\rho _\uparrow ,\rho _\downarrow )$$ of the UEG cannot be calculated analytically. This quantity has been obtained numerically for a sample of densities and fitted to a parametrized function satisfying the known high- and low-density expansions. Expressed in terms of the Wigner–Seitz radius rs = (3∕(4πρ))1∕3, the first terms of the high-density expansion (rs → 0) have the form

(1.104)

with spin-unpolarized coefficients $$A(0)=(1-\ln 2)/\pi ^2$$ , B(0) = −0.046921, C(0) = 0.009229, and fully spin-polarized coefficients A(1) = A(0)∕2, B(1) = −0.025738, C(1) = 0.004792. The first terms of the low-density expansion (rs → +∞) have the form

(1.105)

where the coefficients a = −0.895930, b = 1.325, and c = −0.365 are assumed to be independent of ζ. The low-density limit of the UEG corresponds to the Wigner crystallization. For a recent review of results on the UEG, see Ref. [165].

The two most used parametrizations are the one of Vosko, Wilk, and Nusair (VWN) [265] and the more recent one of Perdew and Wang (PW92) [199] which we give here. In this parametrization, the UEG correlation energy per particle is estimated using the approximate spin-interpolation formula

(1.106)

where εc(rs, ζ) is the UEG correlation energy per particle as a function of rs and ζ, f(ζ) = [(1 + ζ)4∕3 + (1 − ζ)4∕3 − 2]∕(24∕3 − 2) is a spin-scaling function borrowed from the exchange energy, and αc(rs) = (∂²εc(rs, ζ)∕∂ζ²)ζ=0 is the spin stiffness. This spin-interpolation formula was first proposed in the VWN parametrization based on a study of the ζ dependence of the UEG correlation energy per particle at the random-phase approximation (RPA) level. A unique parametrization function

is then used for approximating εc(rs, 0), εc(rs, 1), and − αc(rs), where

The form of G was chosen to reproduce the form of the high- and low-density expansions. The parameters Ai, β1,i, and β2,i (with i ∈{0, 1, 2}) are fixed by the first two terms of the high-density expansion, while the parameters α1,i, β3,i, and β4,i are fitted to quantum Monte Carlo (QMC) data [32] for εc(rs, 0) and εc(rs, 1), and to an estimation of − αc(rs) extrapolated from RPA data. The parameters are given in Table I of Ref. [199].

We now discuss the merits and deficiencies of the LDA. By construction, the LDA is of course exact in the limit of uniform densities. More relevant to atomic and molecular systems is that the LDA exchange and correlation energies are asymptotically exact in the limit of large nuclear charge of neutral atoms N = Z →∞. Indeed, in this semiclassical Thomas–Fermi limit, the LDA gives the exact coefficients Ax and Ac of the leading terms in the asymptotic expansions of Eqs. (1.96) and (1.97) [190]. However, the coefficients of the next terms are very different: $$B_{{\mathrm {x}}}^{\text{LDA}}\approx 0$$ instead of Bx ≈ 0.224 and $$B_{\text{c}}^{\text{LDA}}\approx -0.00451$$ instead of Bc ≈ 0.0372 [28].

Due to the scaling of the UEG exchange energy per particle,

the LDA exchange functional correctly scales linearly under uniform coordinate scaling of the density [Eq. (1.83)]. Similarly, due the scaling of the UEG correlation energy per particle in the low-density limit [Eq. (1.105)],

StartLayout 1st Row limit Underscript gamma right arrow 0 Endscripts StartFraction 1 Over gamma EndFraction upper E Subscript c Baseline left bracket rho Subscript gamma Superscript left parenthesis 2 right parenthesis Baseline right bracket equals 0 period EndLayout

(1.108)

the LDA correlation functional correctly scales linearly under uniform coordinate scaling to the low-density limit [Eq. (1.86)]. However, from the behavior of $$\varepsilon _{\text{c}}^{\text{UEG}}$$ in the high-density limit [Eq. (1.104)],

StartLayout 1st Row limit Underscript gamma right arrow 0 Endscripts StartFraction 1 Over gamma EndFraction upper E Subscript c Baseline left bracket rho Subscript gamma Superscript left parenthesis 2 right parenthesis Baseline right bracket equals 0 period EndLayout

(1.109)

we see that the LDA correlation functional diverges logarithmically under uniform coordinate scaling to the high-density limit whereas the exact correlation functional goes to a constant for nondegenerate KS systems [Eq. (1.85)]. Consequently, in the limit of large nuclear charge, Z →∞, at fixed electron number N, the LDA exchange energy correctly scales linearly with Z [Eq. (1.93)], albeit with an incorrect coefficient, and the LDA correlation energy does not reproduce the exact saturation behavior [Eq. (1.94)] for a nondegenerate isoelectronic series but incorrectly diverges [193]. Also, the LDA exchange and correlation functionals do not satisfy the non-uniform scaling conditions of Eqs. (1.89)–(1.92), but instead both diverge in these limits.

The LDA can also be thought of as approximating the exchange and the (coupling-constant-integrated) correlation holes of an inhomogeneous system in Eqs. (1.55) and (1.69) by the corresponding exchange and correlation holes of the UEG. Namely, considering the spin-independent version for simplicity, the LDA exchange hole is

(1.110)

with

(1.111)

where r₁₂ = |r₂ −r₁| is the interelectronic distance, kF = (3π²ρ)1∕3 is the Fermi wave vector, and j₁ is the spherical Bessel function of the first kind. Similarly, the LDA correlation hole is

Since the UEG is a physical system, the LDA exchange hole correctly fulfills the negativity and sum-rule condition [Eqs. (1.50) and (1.52)] and the LDA correlation hole correctly fulfills the sum-rule and electron-electron cusp condition [Eqs. (1.70) and (1.71)]. This constitutes a significant merit of the LDA. However, because the LDA exchange hole $$h_{{\mathrm {x}}}^{{\mathrm {LDA}}}({\mathbf {r}}_1,{\mathbf {r}}_2)$$ only depends on ρ(r₁) and not on ρ(r₂), the LDA exchange functional does not entirely eliminate the self-interaction contribution of the Hartree functional, in particular in one and two-electron systems [Eqs. (1.72) or (1.74), and (1.76)], or in one-orbital spatial regions of many-electron systems [Eqs. (1.77) and (1.78)]. Similarly, the LDA correlation functional does not vanish in one-electron systems [Eqs. (1.73) or (1.75)], or more generally in one-orbital one-electron regions [Eq. (1.81)]. Thus, the LDA introduces a self-interaction error. Moreover, the LDA exchange-correlation functional does not entirely cancel out the Hartree energy in entangled one-electron spatial regions [Eq. (1.80], i.e. it introduces a static-correlation error.

Another deficiency of the LDA is that the (spin-independent) LDA exchange potential

decays exponentially at infinity for finite molecular systems (since the density ρ(r) decays exponentially), i.e. much too fast in comparison to the − 1∕|r| asymptotic behavior of the exact exchange potential [Eq. (1.27)]. Since asymptotic spatial regions are dominated by the highest occupied molecular orbital (HOMO) and are thus one-orbital regions (assuming the HOMO is not degenerate), this is another signature of the incorrectness of the LDA exchange functional in these one-orbital regions.

For a review of mathematical results on the LDA, see Chap. 3 by Lewin et al. in this volume.

1.3.2 The Gradient-Expansion Approximation

The next logical step beyond the LDA is the gradient-expansion approximation (GEA) [135], in which the exchange-correlation functional is systematically expanded in the gradient and higher-order derivatives of the density. One way of deriving the GEA is to start from the UEG, introduce a weak and slowly-varying external potential δv(r), and expand the exchange-correlation energy in terms of the gradients of the density (see, e.g., Refs. [55, 134, 166, 246]). Alternatively, one can perform a semiclassical expansion (i.e., an expansion in powers of the reduced Planck constant ħ) of the exact Exc[ρ] in terms of the gradients of the external potential and use the mapping between the potential and the density to express it in terms of the gradients of the density (see, e.g., Ref. [49]).

The spin-independent gradient expansion of the exchange functional is known up to fourth order (GEA4) [246],

(1.112)

involving the density gradient ∇ρ(r) and Laplacian ∇²ρ(r). Sham [224] obtained the second-order coefficient $$C_{{\mathrm {x}},\text{S}}^{(2)} = -7/(432\pi (3\pi ^2)^{1/3})\approx -0.001667$$ . The calculation was done by starting with the screened Yukawa interaction $$\mathrm{e} ^{-\kappa r_{12}}/r_{12}$$ and taking the limit κ → 0 at the end of the calculation. It was later shown that this calculation contains an order-of-limit problem and that the correct Coulombic second-order coefficient is $$C_{{\mathrm {x}}}^{(2)}=-5/(216\pi (3\pi ^2)^{1/3})\approx -0.002382$$ [55, 134]. The fourth-order coefficients are $$C_{{\mathrm {x}},1}^{(4)}=-73/(64800\pi ^3)\approx -0.000036$$ , and $$C_{{\mathrm {x}},2}^{(4)}\approx 0.00009$$ , where the last one has been numerically estimated [246]. Note that each term in Eq. (1.112) correctly fulfills the scaling relation of Eq. (1.83). The spin-dependent gradient exchange expansion is simply obtained from the spin-scaling relation [Eq. (1.36)].

Similarly, Ma and Brueckner [166] obtained the spin-independent second-order gradient expansion (GEA2) of the correlation functional,

(1.113)

with a second-order coefficient in the high-density limit $$C_{{\text{c}},\text{MB}}^{(2)}(r_{{\mathrm {s}}}\to 0) = 0.004235$$ . It is believed [149] that this calculation contains a similar order-of-limit problem as in Sham’s coefficient $$C_{{\mathrm {x}},\text{S}}^{(2)}$$ , in such a way that these two coefficients must be combined to obtain the correct second-order exchange-correlation coefficient in the high-density limit $$C_{\text{xc}}^{(2)}(r_{{\mathrm {s}}}\to 0) = C_{{\mathrm {x}},\text{S}}^{(2)} + C_{{\text{c}},\text{MB}}^{(2)}(r_{{\mathrm {s}}}\to 0)$$ . The correct second-order correlation coefficient in the high-density limit is then $$C_{\text{c}}^{(2)}(r_{{\mathrm {s}}}\to 0) = C_{\text{xc}}^{(2)}(r_{{\mathrm {s}}}\to 0) - C_{{\mathrm {x}}}^{(2)} = 0.004950$$ . Similarly, the second-order correlation coefficient as a function of rs can be obtained by $$C_{\text{c}}^{(2)}(r_{{\mathrm {s}}}) = C_{\text{xc}}^{(2)}(r_{{\mathrm {s}}}) - C_{{\mathrm {x}}}^{(2)}$$ , where $$C_{\text{xc}}^{(2)}(r_{{\mathrm {s}}})$$ has been parametrized in Ref. [206]. The spin-dependent generalization has the form [204, 271]

StartLayout 1st Row 1st Column upper E Subscript c Superscript GEA 2 Baseline left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket 2nd Column equals upper E Subscript c Superscript upper L upper S upper D upper A Baseline left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript sigma comma sigma prime element of StartSet up arrow comma down arrow EndSet Endscripts integral Underscript double struck upper R cubed Endscripts upper C Subscript c Superscript sigma comma sigma prime comma left parenthesis 2 right parenthesis Baseline left parenthesis r Subscript normal s Baseline left parenthesis bold r right parenthesis comma zeta left parenthesis bold r right parenthesis right parenthesis StartFraction nabla rho Subscript sigma Baseline left parenthesis bold r right parenthesis Over rho Subscript sigma Baseline left parenthesis bold r right parenthesis Superscript 2 divided by 3 Baseline EndFraction dot StartFraction nabla rho Subscript sigma Sub Superscript prime Subscript Baseline left parenthesis bold r right parenthesis Over rho Subscript sigma Sub Superscript prime Subscript Baseline left parenthesis bold r right parenthesis Superscript 2 divided by 3 Baseline EndFraction normal d bold r comma EndLayout

(1.114)

where the functions $$C_{{\text{c}}}^{\sigma ,\sigma ',(2)}(r_{{\mathrm {s}}},\zeta )$$ have been numerically calculated in the high-density limit [204, 205].

The GEA should improve over the LDA for sufficiently slowly varying densities. Since the spin-independent GEA2 exchange energy per particle has the form

where x = |∇ρ|∕ρ4∕3 is a dimensionless reduced density gradient, the precise condition for exchange is x ≪ 1. Unfortunately, for real systems like atoms and molecules, the reduced density gradient x can be large in some regions of space. In particular, in the exponential density tail, , the reduced density gradient diverges $$x({\mathbf {r}}) \xrightarrow [|{\mathbf {r}}|\to \infty ]{} \infty $$ . But this is not as bad as it seems since $$\rho \; \varepsilon _{{\mathrm {x}}}^{\text{GEA2}}$$ goes to zero anyway in this limit. The situation is more catastrophic for correlation. Indeed, in the high-density limit, the spin-independent GEA2 correlation energy per particle behaves as

StartLayout 1st Row 1st Column upper E Subscript c Superscript GEA 2 Baseline left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket 2nd Column equals upper E Subscript c Superscript upper L upper S upper D upper A Baseline left bracket rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right bracket 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript sigma comma sigma prime element of StartSet up arrow comma down arrow EndSet Endscripts integral Underscript double struck upper R cubed Endscripts upper C Subscript c Superscript sigma comma sigma prime comma left parenthesis 2 right parenthesis Baseline left parenthesis r Subscript normal s Baseline left parenthesis bold r right parenthesis comma zeta left parenthesis bold r right parenthesis right parenthesis StartFraction nabla rho Subscript sigma Baseline left parenthesis bold r right parenthesis Over rho Subscript sigma Baseline left parenthesis bold r right parenthesis Superscript 2 divided by 3 Baseline EndFraction dot StartFraction nabla rho Subscript sigma Sub Superscript prime Subscript Baseline left parenthesis bold r right parenthesis Over rho Subscript sigma Sub Superscript prime Subscript Baseline left parenthesis bold r right parenthesis Superscript 2 divided by 3 Baseline EndFraction normal d bold r comma EndLayout

where y = |∇ρ|∕ρ7∕6 is another reduced density gradient adapted to correlation. Therefore, in this limit, the GEA2 correlation correction diverges to + ∞ even faster than the LDA diverges to −∞.

Another aspect of the deficiency of the GEA is that the corresponding GEA exchange and correlation holes have unphysical long-range parts which break the negativity [Eq. (1.52)] and sum-rule conditions [Eqs. (1.50) and (1.57)].

In practice, the GEA tends to deteriorate the results obtained at the LDA level. Truncated gradient expansions should not be directly used but need to be resummed.

1.3.3 Generalized-Gradient Approximations

The failures of the GEA led to the development, which really started in the 1980s, of generalized-gradient approximations (GGAs) with the generic form

(1.115)

where $$e_{\text{xc}}^{\text{GGA}}$$ is some function. The GGAs are often called semilocal approximations in the sense that $$e_{\text{xc}}^{\text{GGA}}$$ does not only use the local value of the spin densities ρ↑(r) and ρ↓(r) but also “semilocal information” through its gradients⁶ ∇ρ↑(r) and ∇ρ↓(r).

Many GGA functionals have been proposed. They generally provide a big improvement over LDA for molecular systems. However, their accuracy is still limited, in particular by self-interaction and static-correlation errors. We review here some of the most widely used GGA functionals.

B88 Exchange Functional

In the Becke 88 (B88 or B) exchange functional [18], the exchange energy density is written as

(1.116)

where $$x_\sigma =|\nabla \rho _\sigma |/\rho _\sigma ^{4/3}$$ . The fact that $$e_{{\mathrm {x}}}^{\text{B88}}$$ depends linearly on $$\rho _\sigma ^{4/3}$$ and nonlinearly only on the dimensionless reduced density gradient xσ(r) guarantees the scaling relation of Eq. (1.83). Using the exponential decay of the ground-state spin densities of Coulombic systems, x left parenthesis bold r right parenthesis right arrow Underscript StartAbsoluteValue bold r EndAbsoluteValue right arrow normal infinity Overscript Endscripts normal infinity ), it can be verified that the chosen form for $$e_{{\mathrm {x}}}^{\text{B88}}$$ satisfies the exact asymptotic behavior of the exchange energy density per particle [Eq. (1.56)], although the corresponding exchange potential does not satisfy the exact − 1∕r asymptotic behavior [Eq. (1.27)] [54]. For small xσ, $$e_{{\mathrm {x}}}^{\text{B88}}$$ is correctly quadratic in xσ. The parameter β = 0.0042 was found by fitting to HF exchange energies of rare-gas atoms. A very similar value of β can also be found by imposing the coefficient Bx of the approximate large-Z asymptotic expansion of the exchange energy of neutral atoms [Eq. (1.96)] [51]. It turns out that imposing the coefficient of the second-order gradient expansion [Eq. (1.112)] would lead to a value of β about two times smaller and would greatly deteriorate the accuracy of the functional for atoms and molecules.

LYP Correlation Functional

The Lee–Yang–Parr (LYP) [150] correlation functional is one of the rare functionals which have not been constructed starting from LDA. It originates from the Colle–Salvetti [38] correlation-energy approximation depending on the curvature of the HF hole. By using a gradient-expansion approximation of the curvature of the HF hole, LYP turned the Colle–Salvetti expression into a density functional depending on the density, the density gradient, and the Laplacian of the density. The dependence on the Laplacian of the density can be exactly eliminated by an integration by parts [174], giving the following correlation energy density

StartLayout 1st Row 1st Column e Subscript c Superscript LYP Baseline left parenthesis rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline comma nabla rho Subscript up arrow Baseline comma nabla rho Subscript down arrow Baseline right parenthesis 2nd Column equals minus a StartFraction 4 Over 1 plus d rho Superscript negative 1 divided by 3 Baseline EndFraction StartFraction rho Subscript up arrow Baseline rho Subscript down arrow Baseline Over rho EndFraction minus a b omega left parenthesis rho right parenthesis left brace rho Subscript up arrow Baseline rho Subscript down arrow Baseline 2nd Row 1st Column Blank 2nd Column times left bracket sigma summation Underscript sigma element of StartSet up arrow comma down arrow EndSet Endscripts left parenthesis 2 Superscript 11 divided by 3 Baseline upper C Subscript upper F Baseline rho Subscript sigma Superscript 8 divided by 3 Baseline minus left parenthesis five halves minus StartFraction delta left parenthesis rho right parenthesis Over 18 EndFraction right parenthesis StartAbsoluteValue nabla rho Subscript sigma Baseline EndAbsoluteValue squared 3rd Row 1st Column Blank 2nd Column minus StartFraction delta left parenthesis rho right parenthesis minus 11 Over 9 EndFraction StartFraction rho Subscript sigma Baseline Over rho EndFraction StartAbsoluteValue nabla rho Subscript sigma Baseline EndAbsoluteValue squared right parenthesis 4th Row 1st Column Blank 2nd Column plus left parenthesis StartFraction 47 Over 18 EndFraction minus StartFraction 7 delta left parenthesis rho right parenthesis Over 18 EndFraction right parenthesis StartAbsoluteValue nabla rho EndAbsoluteValue squared right bracket minus two thirds rho squared StartAbsoluteValue nabla rho EndAbsoluteValue squared 5th Row 1st Column Blank 2nd Column plus left parenthesis two thirds rho squared minus rho Subscript up arrow Superscript 2 Baseline right parenthesis StartAbsoluteValue nabla rho Subscript down arrow Baseline EndAbsoluteValue squared plus left parenthesis two thirds rho squared minus rho Subscript down arrow Superscript 2 Baseline right parenthesis StartAbsoluteValue nabla rho Subscript up arrow Baseline EndAbsoluteValue squared right brace comma EndLayout

where $$\omega (\rho ) = \rho ^{-11/3} \exp (-c \rho ^{-1/3})/(1+d\rho ^{-1/3})$$ , δ(ρ) = cρ−1∕3 + dρ−1∕3∕(1 + dρ−1∕3), and CF = (3∕10)(3π²)2∕3. The parameters a = 0.04918, b = 0.132, c = 0.2533, and d = 0.349 were obtained in the original Colle–Salvetti expression by a fit to Helium data. Note that the LYP correlation energy vanishes for fully spin-polarized densities (ρ↑ = 0 or ρ↓ = 0) and therefore correctly vanishes for one-electron systems [Eq. (1.75)].

PW91 Exchange-Correlation Functional

The Perdew–Wang 91 (PW91) (see Refs. [29, 187, 189]) exchange-correlation functional is based on a model of the exchange hole hx(r₁, r₂) in Eq. (1.55) and of the coupling-constant-integrated correlation hole $$\bar {h}_{\text{c}}({\mathbf {r}}_1,{\mathbf {r}}_2)$$ in Eq. (1.69). The idea is to start from the GEA model of these holes given as gradient expansions and remove the unrealistic long-range parts of these holes to restore important constraints satisfied by the LDA. Specifically, the spurious positive parts of the GEA exchange hole are removed to enforce the negativity condition of Eq. (1.52) and a cutoff in |r₁ −r₂| is applied to enforce the normalization condition of Eq. (1.50). Similarly, a cutoff is applied on the GEA correlation hole to enforce the condition that the hole integrates to zero [Eq. (1.70)]. The exchange and correlation energies per particle calculated from these numerical holes are then fitted to functions of the density and density gradient chosen to satisfy a number of exact constraints.

The spin-independent PW91 exchange energy density is written as

(1.117)

where the so-called enhancement factor is

(1.118)

with the reduced density gradient s = |∇ρ|∕(2kFρ) = x∕[2(3π²)1∕3] where kF = (3π²ρ)1∕3 is the Fermi wave vector. The spin-dependent PW91 exchange energy density is simply obtained from the spin-scaling relation [Eq. (1.36)]: . The enhancement factor $$F_{{\mathrm {x}}}^{\text{PW91}}(s)$$ satisfies the second-order gradient expansion [Eq. (1.112)], $$F_{{\mathrm {x}}}^{\text{PW91}}(s)= 1 + \mu \; s^2 + O(s^4)$$ with $$\mu =-16\pi (\pi /3)^{2/3} C_{{\mathrm {x}}}^{(2)} = 10/81 \approx 0.1235$$ , the local Lieb–Oxford bound, $$F_{{\mathrm {x}}}^{\text{PW91}}(s) \leqslant -C_{\text{LO}}/(C_{{\mathrm {x}}} 2^{1/3}) \approx 1.804$$ , which is a sufficient and necessary condition for a spin-dependent GGA exchange functional to satisfy the Lieb–Oxford lower bound [Eq. (1.98)] for all densities [194] (note however that 1.804 is not an optimal bound), and the condition $$\lim _{s\to \infty } s^{1/2} F_{{\mathrm {x}}}^{\text{PW91}}(s) &lt; \infty $$ which guarantees the non-uniform scaling finiteness conditions of Eqs. (1.89) and (1.91) [158, 194].

The PW91 correlation energy density is written as

(1.119)

where the gradient correction HPW91(ρ↑, ρ↓, t) = H₀(ρ↑, ρ↓, t) + H₁(ρ↑, ρ↓, t) depends on another reduced density gradient (adapted to correlation) t  = |∇ρ|∕(2ϕ₂(ζ)ksρ) = y∕[4ϕ₂(ζ)(3∕π)1∕6] where $$k_{{\mathrm {s}}}=\sqrt {4 k_{\text{F}}/\pi }$$ is the Thomas–Fermi screening wave vector and the spin-scaling function ϕ₂(ζ) is defined by Eq. (1.103), with

and Cxc(rs) is taken from Ref. [206]. The function H₀(ρ↑, ρ↓, t) was chosen so that it fulfills the second-order gradient expansion [Eq. (1.114)], H₀(ρ↑, ρ↓, t) = βϕ₂(ζ)³t² + O(t⁴), using an approximate ζ dependence [271] and the Ma–Brueckner high-density-limit second-order coefficient [166] $$\beta = 16(3/\pi )^{1/3} C_{{\text{c}},\text{MB}}^{(2)}(r_{{\mathrm {s}}}\to 0) \approx 0.06673$$ , and so that it cancels the LDA correlation in the large-t limit, $$\lim _{t\to \infty } H_0(\rho _\uparrow ,\rho _\downarrow ,t) = - \varepsilon _{\text{c}}^{\text{UEG}}(\rho _\uparrow ,\rho _\downarrow )$$ . The only fitted parameter is α = 0.09. The function H₁(ρ↑, ρ↓, t) only serves to restore the correct second-order gradient expansion, such that HPW91(ρ↑, ρ↓, t) = 16(3∕π)1∕3Cxc(rs)ϕ₂(ζ)³t² + O(t⁴), while keeping the large-t limit unchanged.

PBE Exchange-Correlation Functional

The Perdew–Burke–Ernzerhof (PBE) [188] exchange-correlation functional is a simplification of the PW91 functional with no fitted parameters which gives almost the same energies. The spin-independent PBE exchange energy density is written as

(1.120)

where the enhancement factor is

(1.121)

The function $$F_{{\mathrm {x}}}^{\text{PBE}}(s)$$ has the second-order gradient expansion $$F_{{\mathrm {x}}}^{\text{PBE}}(s)= 1 + \mu \; s^2 + O(s^4)$$ , and the parameter is chosen as $$\mu = 16\pi (\pi /3)^{2/3} C_{{\text{c}},\text{MB}}^{(2)}(r_{{\mathrm {s}}}\to 0) \approx 0.21951$$ so as to cancel the correlation second-order gradient expansion. The second parameter κ is chosen so as to saturate the local Lieb–Oxford bound, i.e. $$\lim _{s\to \infty } F_{{\mathrm {x}}}^{\text{PBE}}(s) = 1 + \kappa = -C_{\text{LO}}/(C_{{\mathrm {x}}} 2^{1/3})\approx 1.804$$ , leading to κ = 0.804. The same exchange functional form was in fact proposed earlier in the Becke 86 (B86) functional [17] with empirical parameters (μ = 0.235, κ = 0.967).

A revised version of the PBE exchange functional, called revPBE, was proposed where the local Lieb–Oxford bound constraint is relaxed and the parameter κ = 1.245 is found instead by fitting to exchange-only total atomic energies for He and Ar, resulting in more accurate atomic total energies and molecular atomization energies [276]. Another revised version of the PBE exchange functional, called RPBE, was also proposed to achieve a similar improvement, while still enforcing the local Lieb–Oxford bound, by changing the form of the enhancement factor to $$F_{{\mathrm {x}}}^{\text{RPBE}}(s) = 1+\kappa (1-\exp (-\mu s^2/\kappa ))$$ with the same parameters as in the original PBE [103].

The PBE correlation energy density is written as

(1.122)

with the gradient correction

and

As in the PW91 correlation functional, the function HPBE(ρ↑, ρ↓, t) has the second-order gradient expansion HPBE(ρ↑, ρ↓, t) = βϕ₂(ζ)³t² + O(t⁴) where $$\beta = 16(3/\pi )^{1/3} C_{{\text{c}},\text{MB}}^{(2)}(r_{{\mathrm {s}}} \to 0) \approx 0.06673$$ , and it cancels the LDA correlation in the large-t limit, $$\lim _{t\to \infty } H^{\text{PBE}}(\rho _\uparrow ,\rho _\downarrow ,t) = - \varepsilon _{\text{c}}^{\text{UEG}}(\rho _\uparrow ,\rho _\downarrow )$$ . In contrast with the PW91 correlation functional, under uniform coordinate scaling to the high-density limit, the PBE correlation functional correctly cancels out the logarithm divergence of the LDA correlation functional [Eq. (1.109)], i.e. , where A(0)ϕ₂(ζ)³ is a good approximation to the coefficient A(ζ) [271].

A variant of the PBE exchange-correlation functional, called PBEsol [196], targeted for solid-state systems, was proposed where the correct second-order exchange gradient-expansion coefficient is restored, i.e. $$\mu _{\text{PBEsol}} = 16\pi (\pi /3)^{2/3} C_{{\mathrm {x}}}^{(2)} = 10/81 \approx 0.1235$$ , and the second-order correlation gradient-expansion coefficient βPBEsol = 0.046 is found by fitting to jellium surface exchange-correlation energies.

B97-GGA Exchange-Correlation Functional

The Becke 97 GGA (B97-GGA) exchange-correlation functional is the GGA part of the B97 hybrid functional [22] (see Sect. 1.4.1). The B97-GGA exchange energy density is

(1.123)

where $$e_{{\mathrm {x}},\sigma }^{\text{UEG}}(\rho _\sigma ) = e_{{\mathrm {x}}}^{\text{UEG}}(\rho _\sigma ,0)$$ is the spin-σ contribution to the UEG exchange energy density and the gradient correction gx(xσ) is a function of $$x_\sigma = |\nabla \rho _\sigma |/\rho _\sigma ^{4/3}$$ ,

(1.124)

with $$u_{{\mathrm {x}}}(x_\sigma ) = \gamma _{{\mathrm {x}}} x_\sigma ^2/(1+ \gamma _{{\mathrm {x}}} x_\sigma ^2)$$ . The B97-GGA correlation energy density is written as the sum of opposite- and same-spin contributions

StartLayout 1st Row 1st Column e Subscript c Superscript upper B 97 hyphen GGA Baseline left parenthesis rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline comma nabla rho Subscript up arrow Baseline comma nabla rho Subscript down arrow Baseline right parenthesis 2nd Column equals e Subscript c comma up arrow down arrow Superscript upper B 97 hyphen GGA Baseline left parenthesis rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline comma nabla rho Subscript up arrow Baseline comma nabla rho Subscript down arrow Baseline right parenthesis 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript sigma element of StartSet up arrow comma down arrow EndSet Endscripts e Subscript c comma sigma sigma Superscript upper B 97 hyphen GGA Baseline left parenthesis rho Subscript sigma Baseline comma nabla rho Subscript sigma Baseline right parenthesis comma EndLayout

(1.125)

where

(1.126)

and

(1.127)

In these expressions, and $$e_{{\text{c}},\sigma \sigma }^{\text{UEG}}(\rho _\alpha ) = e_{{\text{c}}}^{\text{UEG}}(\rho _\sigma ,0)$$ are estimations of the opposite- and same-spin contributions to the UEG correlation energy density [238, 239]. The opposite-spin gradient correction is taken as a function of $$x_{\uparrow \downarrow } = \sqrt {(x_\uparrow ^2 + x_\downarrow ^2)/2}$$ ,

(1.128)

with , and the same-spin gradient correction is

(1.129)

with $$u_{{\text{c}}}^{\sigma \sigma }(x_\sigma ) = \gamma _{{\text{c}}}^{\sigma \sigma } x_\sigma ^2/(1+ \gamma _{{\text{c}}}^{\sigma \sigma } x_\sigma ^2)$$ . The parameters γx = 0.004, $$\gamma _{{\text{c}}}^{\uparrow \downarrow }=0.006$$ , and $$\gamma _{{\text{c}}}^{\sigma \sigma }=0.2$$ , were roughly optimized on atomic exchange and correlation energies. The other parameters cx,i, $$c_{{\text{c}},i}^{\uparrow \downarrow }$$ , $$c_{{\text{c}},i}^{\sigma \sigma }$$ for a polynomial degree m = 2 in Eqs. (1.124), (1.128), and (1.129) were optimized in the B97 hybrid functional in the presence of a fraction of HF exchange energy (see Sect. 1.4.1).

The Hamprecht–Cohen–Tozer–Handy (HCTC) [104] exchange-correlation functional uses the same form as the B97-GGA exchange-correlation functional but with a polynomial degree m = 4 and the parameters cx,i, $$c_{{\text{c}},i}^{\uparrow \downarrow }$$ , $$c_{{\text{c}},i}^{\sigma \sigma }$$ were optimized without HF exchange a set of energetic properties (atomic total energies, ionization energies, atomization energies), nuclear gradients, and accurate exchange-correlation potentials.

1.3.4 Meta-Generalized-Gradient Approximations

The meta-generalized-gradient approximations (meta-GGAs or mGGAs) are of the generic form, in their spin-independent version,

(1.130)

i.e., they use more ingredients than the GGAs, namely the Laplacian of the density ∇²ρ(r) and/or the non-interacting positive kinetic energy density τ(r) associated with a single-determinant wave function Φ,

(1.131)

where {φi}i=1,…,N are the orbitals occupied in Φ. The meta-GGAs are considered as part of the family of semilocal approximations, in the sense that τ(r) contains semilocal information with respect to the orbitals.

Meta-GGAs can be viewed as implicit functionals of the density only, i.e. $$E_{\text{xc}}^{{\mathrm {mGGA}}}[\rho ,\tau _{\Phi [\rho ]}]$$ , since τ(r) can be considered itself as an implicit functional of the density via the KS single-determinant wave function Φ[ρ]. This view in which $$E_{\text{xc}}^{{\mathrm {mGGA}}}[\rho ]$$ is a proper approximation to the exchange-correlation density functional Exc[ρ] of the KS scheme is normally adopted when constructing meta-GGAs approximations. However, the calculation of the functional derivative of $$E_{\text{xc}}^{{\mathrm {mGGA}}}[\rho ]$$ with respect to the density then requires the use of the complicated optimized-effective-potential method (see Sect. 1.7). Therefore, in practical calculations, meta-GGAs are usually reinterpreted as explicit functionals of a single-determinant wave function Φ, i.e. $$E_{\text{xc}}^{{\mathrm {mGGA}}}[\rho _{\Phi },\tau _{\Phi }]$$ , [2, 9, 73, 180, 181, 232, 242, 275] or, in other words, approximations to an exact GKS exchange-correlation functional (see Sect. 1.1.4).

In the latter approach, which we will here refer to as the meta-Kohn–Sham (mKS) scheme, we introduce a functional $$E_{\text{xc}}^{{\mathrm {mKS}}}[\rho ,\tau ]$$ (to which meta-GGAs are approximations) defined for ρ and τ simultaneously representable by a single-determinant wave function $$\Phi \in \mathcal {S}^N$$ and which defines the GKS functional $$E_{\text{xc}}^{S}[\Phi ]=E_{\text{xc}}^{{\mathrm {mKS}}}[\rho _{\Phi },\tau _{\Phi }]$$ [see Eq. (1.38)] giving the exact ground-state energy via Eq. (1.39),

StartLayout 1st Row upper E 0 equals inf Underscript upper Phi element of script upper S Superscript upper N Baseline Endscripts left brace left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript n e Baseline vertical bar upper Phi right angle bracket plus upper E Subscript normal upper H Baseline left bracket rho Subscript upper Phi Baseline right bracket plus upper E Subscript xc Superscript m upper K upper S Baseline left bracket rho Subscript upper Phi Baseline comma tau Subscript upper Phi Baseline right bracket right brace comma EndLayout

(1.132)

which, by taking variations with respect to the orbitals, gives the mKS equations:

(1.133)

Here, $$v_{\text{xc}}^{{\mathrm {mKS}}}({\mathbf {r}})=v_{{\mathrm {xc}},1}^{{\mathrm {mKS}}}({\mathbf {r}})+v_{{\mathrm {xc}},2}^{{\mathrm {mKS}}}({\mathbf {r}})$$ contains a usual local potential

and a non-multiplicative operator [2, 9, 73, 242, 275]

(1.134)

evaluated with $$\rho ({\mathbf {r}}) =\sum _{i=1}^N |\varphi _i({\mathbf {r}})|{ }^2$$ and $$\tau ({\mathbf {r}})=(1/2) \sum _{i=1}^{N} | \nabla \varphi _{i} ({\mathbf {r}}) |{ }^2$$ . Interestingly, the mKS equations can be rewritten as a Schrödinger-like equation with a position-dependent mass m(r) [50],

(1.135)

where $$m({\mathbf {r}}) = \left (1+\delta E_{\text{xc}}^{{\mathrm {mKS}}}[\rho ,\tau ]/\delta \tau ({\mathbf {r}})\right )^{-1}$$ . As in the KS scheme, the functional $$E_{\text{xc}}^{{\mathrm {mKS}}}[\rho ,\tau ]$$ is decomposed into exchange and correlation contributions: $$E_{\text{xc}}^{{\mathrm {mKS}}}[\rho ,\tau ] = E_{{\mathrm {x}}}^{{\mathrm {mKS}}}[\rho ,\tau ] + E_{\text{c}}^{{\mathrm {mKS}}}[\rho ,\tau ]$$ . In the spin-dependent version of the mKS scheme, we consider a similar functional of the spin-resolved densities and non-interacting positive kinetic energy densities $$E_{\text{xc}}^{{\mathrm {mKS}}}[\rho _\uparrow ,\rho _\downarrow ,\tau _\uparrow ,\tau _\downarrow ]$$ and the spin-scaling relation of Eq. (1.36) is generalized to

(1.136)

Correspondingly, the spin-dependent versions of the meta-GGAs are formulated in terms of the spin-resolved quantities ρ↑, ρ↓, ∇ρ↑, ∇ρ↓, ∇²ρ↑, ∇²ρ↓, τ↑, and τ↓.

One motivation for the introduction of the variable τ(r) is that it appears in the expansion of the spherically averaged exchange hole [entering in Eq. (1.55)] for small interelectronic distances r₁₂ [16], which for the case of a closed-shell system is

StartLayout 1st Row StartFraction 1 Over 4 pi r 12 squared EndFraction integral Underscript upper S left parenthesis bold 0 comma r 12 right parenthesis Endscripts h Subscript normal x Baseline left parenthesis bold r 1 comma bold r 1 plus bold r 12 right parenthesis normal d bold r 12 equals minus StartFraction rho left parenthesis bold r 1 right parenthesis Over 2 EndFraction 2nd Row minus one third left parenthesis one fourth nabla squared rho left parenthesis bold r 1 right parenthesis minus tau left parenthesis bold r 1 right parenthesis plus StartFraction StartAbsoluteValue nabla rho left parenthesis bold r 1 right parenthesis EndAbsoluteValue squared Over 8 rho left parenthesis bold r 1 right parenthesis EndFraction right parenthesis r 12 squared plus upper O left parenthesis r 12 Superscript 4 Baseline right parenthesis comma EndLayout

(1.137)

where S(0, r₁₂) designates the sphere centered at 0 and of radius r₁₂ = |r₁₂|. Thus τ(r) is needed to describe the curvature of the exchange hole.

Another important motivation is that τ(r) is useful for identifying different types of spatial regions of electronic systems [245]. This is done by comparing τ(r) with the von Weizsäcker kinetic energy density,

(1.138)

which is the exact non-interacting kinetic energy density for one-electron systems and two-electron spin-unpolarized systems, and, more generally, for one-orbital regions as introduced in Sect. 1.2.3. For example, the indicator

(1.139)

which takes its values in the range [0, 1] [141], identifies one-orbital regions (z = 1). A better indicator is

(1.140)

where τUEG(r) = (3∕10)(3π²)2∕3ρ(r)5∕3 is the non-interacting kinetic energy density of the UEG. This indicator α(r) distinguishes one-orbital regions (α = 0), slowly varying density regions (α ≈ 1), and regions of density overlap between closed shells that characterize noncovalent bonds (α ≫ 1).

Nowadays, ∇²ρ(r) is rarely used to construct meta-GGAs because it contains similar information to τ(r), which can be seen by the second-order gradient expansion of τ(r) [27]:

(1.141)

In comparison to GGAs, meta-GGAs are more versatile and generally constitute an improvement. Significantly, thanks to the use of τ, self-interaction errors in the correlation functional can be essentially eliminated with meta-GGAs. They still suffer however from self-interaction errors in the exchange functional. We now describe some of the most used meta-GGA functionals.

TPSS Exchange-Correlation Functional

In the Tao–Perdew–Staroverov–Scuseria (TPSS) [198, 249] functional, the exchange energy density is written as

(1.142)

where the enhancement factor is a function of s = |∇ρ|∕(2kFρ) and z = τW∕τ,

(1.143)

with κ = 0.804 so as to saturate the local Lieb–Oxford bound (just like in the PBE exchange functional) and

(1.144)

and $$\widetilde {q}_b = (9/20) (\alpha -1)/[1 + b \alpha (\alpha -1)]^{1/2} + 2s^2/3$$ (where α = (τ − τW)∕τUEG = (5s²∕3)(z⁻¹ − 1)) is a quantity that tends to the reduced density Laplacian $$q=\nabla ^2 \rho /(4 k_{\text{F}}^2 \rho )$$ in the slowly varying density limit [using Eq. (1.141)]. The function xTPSS(s, z) is chosen so as to satisfy the fourth-order gradient expansion [Eq. (1.112)] which can be written in the form of the enhancement factor $$F_{{\mathrm {x}}}^{\text{GEA4}}(s,z) = 1+ (10/81) s^2 +(146/2025) q^2 -(73/405) s^2 q$$ . The constant μ = 0.21951 is chosen to retain the same large-s behavior of the PBE exchange functional, i.e. . The constants c = 1.59096 and e = 1.537 are chosen so as to eliminate the divergence of the potential at the nucleus for a two-electron exponential density and to yield the correct exchange energy (− 0.3125 hartree) for the exact ground-state density of the hydrogen atom. Finally, the constant b = 0.40 is chosen, quite arbitrarily, as the smallest value that makes $$F_{{\mathrm {x}}}^{\text{TPSS}}(s,z)$$ a monotonically increasing function of s.

The TPSS correlation functional is constructed by making minor refinements to the previously developed Perdew–Kurth–Zupan–Blaha (PKZB) [192] meta-GGA correlation functional,

(1.145)

where the revised PKZB correlation energy per particle is

(1.146)

with

where $$\varepsilon _{{\text{c}}}^{\text{PBE}}(\rho _\uparrow ,\rho _\downarrow ,\nabla \rho _\uparrow , \nabla \rho _\downarrow )$$ is the PBE correlation energy per particle. Equation (1.146) constitutes a one-electron self-interaction correction on the PBE correlation functional. Indeed, for one-electron densities we have z = 1 and ζ = ±1, and the TPSS correlation energy correctly vanishes [Eqs. (1.75)]. The TPSS correlation functional preserves many properties of the PBE correlation functional: it has correct uniform coordinate scaling in the high- and low-density limits, vanishing correlation energy in the large density-gradient limit, and the same second-order gradient expansion (since the additional terms beyond PBE are at least in z² and thus only change the fourth-order terms of the gradient expansion). The parameters d = 2.8 hartree⁻¹ and C(0, 0) = 0.53 are chosen so as to recover the PBE surface correlation energy of jellium [145] over the range of valence-electron bulk densities. The rest of the function is taken as

where ξ = |∇ζ|∕(2kF) is a reduced spin-polarization gradient. The function C(ζ, ξ) is chosen so as to make the exchange-correlation energy independent of the spin polarization ζ in the low-density limit [Eq. (1.86)] and to avoid that the self-interaction correction introduces additional correlation energy density in the core-valence overlap region of monovalent atoms such as Li.

M06-L Exchange-Correlation Functional

In the Minnesota 06 local (M06-L) exchange-correlation functional [278], the exchange energy density is written as

(1.147)

The first term in Eq. (1.147), which has the same form as in the previously developed M05 exchange functional [277], contains the spin-σ PBE exchange energy density $$e_{{\mathrm {x}},\sigma }^{\text{PBE}}(\rho _\sigma ,\nabla \rho _\sigma ) = e_{{\mathrm {x}}}^{\text{PBE}}(\rho _\sigma ,0,\nabla \rho _\sigma ,0)$$ and the kinetic-energy density correction factor

(1.148)

where $$w_\sigma =(\tau _\sigma ^{\text{UEG}}/\tau _\sigma -1)/(\tau _\sigma ^{\text{UEG}}/\tau _\sigma +1)$$ with $$\tau _\sigma ^{\text{UEG}}=(3/10) (6\pi ^2)^{2/3} \rho _\sigma ^{5/3}$$ is an indicator of the delocalization of the exchange hole [24]. The second term in Eq. (1.147), which has the same form as in the VS98 exchange functional [264], contains the spin-σ UEG exchange energy density $$e_{{\mathrm {x}},\sigma }^{\text{UEG}}(\rho _\sigma ) = e_{{\mathrm {x}}}^{\text{UEG}}(\rho _\sigma ,0)$$ and the correction factor

(1.149)

where $$x_\sigma =|\nabla \rho _\sigma |/\rho _\sigma ^{4/3}$$ and $$Z_\sigma = 2(\tau _\sigma -\tau _\sigma ^{\text{UEG}})/\rho _\sigma ^{5/3}$$ and h is the parametrized function

with γ(x, Z, α) = 1 + α(x² + Z).

The M06-L correlation energy is written as the sum of opposite- and same-spin contributions, similarly to the B97-GGA correlation functional [Eq. (1.125)],

where

and

where the spin-decomposed UEG correlation energies $$e_{{\text{c}},\uparrow \downarrow }^{\text{UEG}}(\rho _\uparrow ,\rho _\downarrow )$$ and $$e_{{\text{c}},\sigma \sigma }^{\text{UEG}}(\rho _\sigma )$$ were already defined after Eq. (1.127), and the gradient corrections gc,↑↓(x↑↓) and gc,σσ(xσ) are given in Eqs. (1.128) and (1.129). The additional correction factors are

(1.150)

where $$x_{\uparrow \downarrow } = \sqrt {(x_\uparrow ^2 + x_\downarrow ^2)/2}$$ , Z↑↓ = Z↑ + Z↓, and

(1.151)

The factor Dσ(zσ) = 1 − zσ, where $$z_\sigma =\tau ^{\text{W}}_\sigma /\tau _\sigma $$ and $$\tau ^{\text{W}}_\sigma =|\nabla \rho _\sigma |{ }^2/(8\rho _\sigma )$$ , ensures that the correlation energy correctly vanishes for one-electron systems [23].

The parameters $$\gamma _{{\text{c}}}^{\uparrow \downarrow }=0.0031$$ , and $$\gamma _{{\text{c}}}^{\sigma \sigma }=0.06$$ , were optimized on the correlation energies of He and Ne. The parameters αx = 0.001867, $$\alpha _{{\text{c}}}^{\uparrow \downarrow }=0.003050$$ , and $$\alpha _{{\text{c}}}^{\sigma \sigma }=0.005151$$ were taken from Ref. [264]. The constraints a₀ + dx,0 = 1, $$c_{{\text{c}},0}^{\uparrow \downarrow }+d_{{\text{c}},0}^{\uparrow \downarrow }=1$$ , and $$c_{{\text{c}},0}^{\sigma \sigma }+d_{{\text{c}},0}^{\sigma \sigma }=1$$ are enforced to obtain the correct UEG limit. The remaining 34 free parameters ai, $$c_{{\text{c}},i}^{\uparrow \downarrow }$$ , $$c_{{\text{c}},i}^{\sigma \sigma }$$ for a polynomial degree m = 4 in Eqs. (1.148), (1.128), and (1.129), and dx,i, $$d_{{\text{c}},i}^{\uparrow \downarrow }$$ , $$d_{{\text{c}},i}^{\sigma \sigma }$$ in Eqs. (1.149), (1.150), and (1.151) were optimized on a large set of diverse physicochemical properties concerning main-group thermochemistry, reaction barrier heights, noncovalent interactions, electronic spectroscopy, and transition metal bonding.

SCAN Exchange-Correlation Functional

In the SCAN (strongly constrained and appropriately normed) [244] exchange-correlation functional, the exchange energy density is written as

(1.152)

where the enhancement factor is a function of s = |∇ρ|∕(2kFρ) and α  = (τ − τW)∕τUEG,

(1.153)

which interpolates between α = 0 and α ≈ 1, and extrapolates to α →∞ using the function

where θ is the Heaviside step function. The function $$g_{{\mathrm {x}}}(s) = 1 - \exp (-a_1 s^{-1/2})$$ is chosen to make $$F_{{\mathrm {x}}}^{\text{SCAN}}(s,\alpha )$$ vanish like s−1∕2 as s →∞, which guarantees the non-uniform scaling finiteness conditions [Eqs. (1.89) and (1.91)] [158, 194], and a₁ = 4.9479 is taken to recover the exact exchange energy of the hydrogen atom. For α ≈ 1 (slowly varying density regions), $$F_{{\mathrm {x}}}^{\text{SCAN}}(s,\alpha ) \approx h_{{\mathrm {x}}}^1(s,\alpha ) g_{{\mathrm {x}}}(s)$$ , where $$h_{{\mathrm {x}}}^1(s,\alpha )$$ is a PBE-like resummation of the fourth-order gradient expansion [Eq. (1.112)],

where

with μ = 10∕81, b₂ = (5913∕405000)1∕2, b₁ = (511∕13500)∕(2b₂), b₃ = 0.5, and $$b_4=\mu ^2/k_1-1606/18225-b_1^2$$ . For α = 0 (one-orbital regions), $$F_{{\mathrm {x}}}^{\text{SCAN}}(s,\alpha =0) = h_{{\mathrm {x}}}^0 g_{{\mathrm {x}}}(s)$$ where $$h_{{\mathrm {x}}}^0=1.174$$ is chosen to saturate the local two-electron tight bound $$F_{{\mathrm {x}}}^{\text{SCAN}}(s,\alpha =0) \leqslant 1.174$$ , which is a sufficient and necessary condition for a meta-GGA exchange functional to satisfy the global tight bound of Eq. (1.101) for all two-electron spin-unpolarized densities [194].

The SCAN correlation energy density is written as

(1.154)

which is again an interpolation between α = 0 and α = 1, and an extrapolation to α →∞ using the function

For α = 1, the correlation energy par particle is taken as a revised version of the PBE correlation energy per particle,

(1.155)

where

(1.156)

with t = |∇ρ|∕(2ϕ₂(ζ)ksρ), $$w_1 = \exp [-\varepsilon _{\text{c}}^{\text{UEG}}(\rho _\uparrow ,\rho _\downarrow )/(A(0)\phi _2(\zeta )^3)]-1$$ , $$\mathcal {A} = \beta (r_{{\mathrm {s}}})/(A(0)w_1)$$ , and $$g(\mathcal {A} t^2)=1/(1+4\mathcal {A} t^2)^{1/4}$$ . The function has a second-order gradient expansion $$H_1^{\text{SCAN}}(\rho _\uparrow ,\rho _\downarrow ,t) = \beta (r_{{\mathrm {s}}}) \phi _2(\zeta ) t^2 + O(t^4)$$ , where the coefficient β(rs) = 0.066725(1 + 0.1rs)∕(1 + 0.1778rs) is a rough fit of the density dependence of the second-order gradient expansion correlation coefficient beyond the Ma–Brueckner high-density-limit value and designed so that for rs →∞ the second-order gradient expansion terms for exchange and correlation cancel each other [195]. For α = 0, the correlation energy par particle is constructed to be accurate for one- and two-electron systems and is written as

(1.157)

The spin function Gc(ζ) = [1 − 2.3631(ϕ₄(ζ) − 1)](1 − ζ¹²) is designed to make the correlation energy vanish for one-electron densities (α = 0 and ζ = ±1) and to make the exchange-correlation energy independent of ζ in the low-density limit [Eq. (1.86)]. Equation (1.157) includes a LDA-type term [243]

and a gradient correction

with $$w_0 = \exp (-\varepsilon _{\text{c}}^{\text{LDA0}}(\rho )/b_{1{\text{c}}}) -1$$ and $$g_\infty (\zeta =0,s)= \lim _{\zeta \to 0} \lim _{r_{{\mathrm {s}}} \to \infty } g(\mathcal {A} t) = 1/(1+0.512104 s^2)^{1/4}$$ . The parameter b1c = 0.0285764 is determined so that the high-density limit of $$\varepsilon _{\text{c}}^{0}(\rho _\uparrow ,\rho _\downarrow ,s)$$ reproduces the exact correlation energy of the Helium isoelectronic series in the large-nuclear charge limit, i.e. $$\lim _{Z\to \infty } E_{\text{c}}[\rho _{N=2,Z}] = E_{\text{c}}^{{\mathrm {GL2}}}[\rho _{N=2,Z=1}^{\text{H}}]=-0.0467$$ hartree [Eq. (1.94)]. The parameter b3c = 0.125541 is determined to saturate the lower bound on the exchange-correlation energies of two-electron densities [Eq. (1.99)]. The parameter b2c = 0.0889 is determined to reproduce the exact exchange-correlation energy of the He atom.

The remaining seven parameters (k₁ = 0.065, c1x = 0.667, c2x = 0.8, dx = 1.24, c1c = 0.64, c2c = 1.5, and dc = 0.7) are determined by fitting to the approximate asymptotic expansions of the exchange and correlation energies of neutral atoms in large nuclear charge limit [Eqs. (1.96) and (1.97)], the binding energy curve of compressed Ar₂, and jellium surface exchange-correlation energies.

1.4 Single-Determinant Hybrid Approximations

1.4.1 Hybrid Approximations

Based on arguments relying on the adiabatic-connection formalism, in 1993 Becke [19] proposed to mix a fraction of the exact or Hartree–Fock (HF) exchange energy $$E_{{\mathrm {x}}}^{{\mathrm {HF}}}$$ with GGA functionals. In particular, he proposed a three-parameter hybrid (3H) approximation [20] of the form, written here in its spin-independent version,

(1.158)

with empirical parameters a, b, and c. The functional $$E_{\text{xc}}^{\text{3H}}[\Phi ]$$ is thought of as a functional of a single-determinant wave function $$\Phi \in \mathcal {S}^N$$ since $$E_{{\mathrm {x}}}^{{\mathrm {HF}}}[\Phi ]$$ is itself a functional of Φ,

StartLayout 1st Row 1st Column upper E Subscript normal x Superscript upper H upper F Baseline left bracket upper Phi right bracket 2nd Column equals 3rd Column left angle bracket upper Phi vertical bar ModifyingAbove upper W With caret Subscript ee Baseline vertical bar upper Phi right angle bracket minus upper E Subscript normal upper H Baseline left bracket rho Subscript upper Phi Baseline right bracket 2nd Row 1st Column Blank 2nd Column equals 3rd Column minus one half sigma summation Underscript sigma element of StartSet up arrow comma down arrow EndSet Endscripts sigma summation Underscript i equals 1 Overscript upper N Subscript sigma Baseline Endscripts sigma summation Underscript j equals 1 Overscript upper N Subscript sigma Baseline Endscripts integral Underscript double struck upper R cubed times double struck upper R cubed Endscripts StartFraction phi Subscript i sigma Superscript asterisk Baseline left parenthesis bold r 1 right parenthesis phi Subscript j sigma Baseline left parenthesis bold r 1 right parenthesis phi Subscript j sigma Superscript asterisk Baseline left parenthesis bold r 2 right parenthesis phi Subscript i sigma Baseline left parenthesis bold r 2 right parenthesis Over StartAbsoluteValue bold r 1 minus bold r 2 EndAbsoluteValue EndFraction normal d bold r 1 normal d bold r 2 comma EndLayout

(1.159)

where $$\{\varphi _{i\sigma }\}_{i=1,\ldots ,N_\sigma }$$ are the orbitals occupied in Φ. In 1996, Becke proposed a simpler one-parameter hybrid (1H) approximation [21],

(1.160)

where the fraction a of HF exchange has to be determined. For simplicity, we considered GGA functionals $$E_{{\mathrm {x}}}^{\text{GGA}}[\rho _\Phi ]$$ and $$E_{\text{c}}^{\text{GGA}}[\rho _\Phi ]$$ in Eq. (1.160) but we can more generally use meta-GGA functionals $$E_{{\mathrm {x}}}^{\text{mGGA}}[\rho _\Phi ,\tau _\Phi ]$$ and $$E_{\text{c}}^{\text{mGGA}}[\rho _\Phi ,\tau _\Phi ]$$ .

These hybrid approximations should be considered as approximations of the GKS exchange-correlation functional $$E_{\text{xc}}^{S}[\Phi ]$$ in Eq. (1.38) with $$S[\Phi ]=a \; E_{{\mathrm {x}}}^{{\mathrm {HF}}}[\Phi ]$$ . The corresponding GKS equations [Eq. (1.1.4)] then include the term

(1.161)

where $$v_{{\mathrm {x}},\sigma }^{{\mathrm {HF}}} ({\mathbf {r}},{\mathbf {r}}')$$ is the nonlocal HF exchange potential⁷

The main benefit of adding a fraction of HF exchange is to decrease the self-interaction error (see Sect. 1.2.3) introduced by semilocal exchange functionals which tends to favor too much delocalized electron densities over localized electron densities. The fraction of HF exchange should however be small enough to keep the compensation of errors usually occurring between the approximate semilocal exchange and correlation functionals. First, Becke used the value a = 0.5 in the so-called Becke Half-and-Half functional [19], but then fits to various experimental data often repeatedly gave an optimal parameter a around 0.20–0.25. A rationale has been proposed in favor of the value 0.25 [191]. By decreasing self-interaction errors in the exchange energy, hybrid approximations are often a big improvement over semilocal approximations for molecular systems with sufficiently large electronic gaps. However, for systems with small HOMO-LUMO gaps, such as systems with stretched chemical bonds or with transition metal elements, they tend to increase static-correlation errors.

An interesting extension of the hybrid approximations are the so-called local hybrids, which use a position-dependent fraction a(r) of a (non-uniquely defined) HF exchange energy density $$e_{{\mathrm {x}}}^{{\mathrm {HF}}}({\mathbf {r}})$$ [125] (see, Ref. [167] for a recent review), and which belong to the wider family of hyper-GGA functionals in which the correlation energy can also be expressed as a function of $$e_{{\mathrm {x}}}^{{\mathrm {HF}}}({\mathbf {r}})$$ [197]. The local-hybrid approximations are much more flexible than the global hybrid approach exposed in this section but require more complicated and computationally expensive implementations. For this reason, they have not often been used and we will not consider them any further here.

We now describe some of the most used hybrid approximations.

B3LYP Exchange-Correlation Functional

The B3LYP exchange-correlation functional [237] is the most famous and widely used three-parameter hybrid approximation [Eq. (1.158)]. It uses the B88 exchange functional and the LYP correlation functional,

(1.162)

and the parameters a = 0.20, b = 0.72, and c = 0.81 were found by optimizing on a set of atomization energies, ionization energies, proton affinities of small molecules and first-row total atomic energies [20]. A caveat is that the VWN parametrization of the RPA correlation energy (sometimes referred to as VWN3) of the UEG was actually used for $$E_{\text{c}}^{{\mathrm {LSDA}}}[\rho _{\uparrow },\rho _{\downarrow }]$$ instead of the VWN parametrization of the accurate correlation energy (sometimes referred to as VWN5) of the UEG [265].

B97 Exchange-Correlation Functional

The Becke 97 (B97) exchange-correlation functional [22] is a GGA hybrid of the form

(1.163)

where the form of the B97-GGA exchange and correlation functionals were given in Eqs. (1.123) and (1.125). The fraction of HF exchange a = 0.1943 and the remaining parameters cx,0 = 1.00459, cx,1 = 0.629639, cx,2 = 0.928509, $$c_{{\text{c}},0}^{\uparrow \downarrow }=0.9454$$ , $$c_{{\text{c}},1}^{\uparrow \downarrow }=0.7471$$ , $$c_{{\text{c}},2}^{\uparrow \downarrow }=-4.5961$$ , $$c_{{\text{c}},0}^{\sigma \sigma }=0.1737$$ , $$c_{{\text{c}},1}^{\sigma \sigma }=2.3487$$ , and $$c_{{\text{c}},2}^{\sigma \sigma }=-2.4868$$ for a polynomial degree m = 2 in Eqs. (1.124), (1.128), and (1.129) were optimized on a set of total energies, atomization energies, ionization energies, and proton affinities. Note that, for xσ = 0, the UEG limit is not imposed, which would require the parameters cx,0, $$c_{{\text{c}},0}^{\uparrow \downarrow }$$ , and $$c_{{\text{c}},0}^{\sigma \sigma }$$ to be all strictly equal to 1. With the above optimized parameters, we see that it is nearly satisfied for the exchange energy and the opposite-spin correlation energy, but very far from it for the same-spin correlation energy, which is drastically reduced compared to the LDA.

PBE0 Exchange-Correlation Functional

The PBE0 exchange-correlation functional [1, 60] is a GGA hybrid using the PBE exchange and correlation functionals,

(1.164)

and the fraction of the HF exchange is fixed at a = 0.25 according to the rationale of Ref. [191]. This functional is also known under the name PBE1PBE. The “1” in the latter name emphasizes that there is one parameter, a, while the “0” in the more common name PBE0 emphasizes that this parameter is not found by fitting.

TPSSh Exchange-Correlation Functional

The TPSSh exchange-correlation functional [234] is a meta-GGA hybrid using the TPSS exchange and correlation functionals,

(1.165)

and the fraction of the HF exchange a = 0.10 was determined by optimizing on a large set of atomization energies.

M06 and M06-2X Exchange-Correlation Functionals

The M06 exchange-correlation functional [279] is a meta-GGA hybrid using the M06-L exchange and correlation functionals,

(1.166)

and the parameters in the M06-L exchange and correlation functionals were reoptimized together with the fraction of HF exchange a = 0.27 on the same large set of diverse physicochemical properties used for the M06-L functional. In the M06-2X exchange-correlation functional the fraction of HF exchange is doubled, i.e. a = 0.54, and the parameters were reoptimized with the function hx(xσ, Zσ) in Eq. (1.149) set to zero and excluding transition metal properties in the training set. With this large fraction of HF exchange, the M06-2X functional is designed for systems without transition metal elements.

1.4.2 Range-Separated Hybrid Approximations

Based on earlier ideas of Savin [216] (exposed in detail in Sect. 1.5.2), in 2001, Iikura et al. [121] proposed a long-range correction (LC) scheme in which the exchange-correlation energy is written as, in its spin-independent version,

(1.167)

This scheme has also been referred to as the range-separated hybrid exchange (RSHX) scheme [77]. In Eq. (1.167), $$E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu ,{\mathrm {HF}}}[\Phi ]$$ is the HF exchange energy for a long-range electron-electron interaction $$w_{{\mathrm {ee}}}^{{\mathrm {lr}},\mu }(r_{12}) = \operatorname {\mathrm {erf}}(\mu r_{12})/r_{12}$$ (where $$ \operatorname {\mathrm {erf}}$$ is the error function and the parameter μ ∈ [0, +∞) controls the range of the interaction),

(1.168)

and $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu ,{\mathrm {GGA}}}[\rho ]$$ is a GGA exchange energy functional for the complementary short-range interaction $$w_{{\mathrm {ee}}}^{{\mathrm {sr}},\mu }(r_{12}) = 1/r_{12} - w_{{\mathrm {ee}}}^{{\mathrm {lr}}}(r_{12})$$ . This latter functional can be thought of as an approximation to the short-range exchange functional

(1.169)

where hx(r₁, r₂) is the KS exchange hole of Sect. 1.2.1. For μ = 0, the long-range HF exchange energy vanishes, i.e. $$E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu =0,{\mathrm {HF}}}[\Phi ]=0$$ , and the short-range exchange functional reduces to the standard KS exchange functional, i.e. $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu =0}[\rho ] = E_{{\mathrm {x}}}[\rho ]$$ . Reversely, for μ →∞, the long-range HF exchange energy reduces to the full-range HF exchange energy, i.e. $$E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu \to \infty ,{\mathrm {HF}}}[\Phi ]=E_{{\mathrm {x}}}^{{\mathrm {HF}}}[\Phi ]$$ , and the short-range exchange functional vanishes, i.e. $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu \to \infty }[\rho ] = 0$$ . Significantly, for large μ, the short-range exchange functional becomes a local functional of the density [81, 255]:

StartLayout 1st Row 1st Column upper E Subscript normal x Superscript upper H upper F Baseline left bracket upper Phi right bracket 2nd Column equals 3rd Column left angle bracket upper Phi vertical bar ModifyingAbove upper W With caret Subscript ee Baseline vertical bar upper Phi right angle bracket minus upper E Subscript normal upper H Baseline left bracket rho Subscript upper Phi Baseline right bracket 2nd Row 1st Column Blank 2nd Column equals 3rd Column minus one half sigma summation Underscript sigma element of StartSet up arrow comma down arrow EndSet Endscripts sigma summation Underscript i equals 1 Overscript upper N Subscript sigma Baseline Endscripts sigma summation Underscript j equals 1 Overscript upper N Subscript sigma Baseline Endscripts integral Underscript double struck upper R cubed times double struck upper R cubed Endscripts StartFraction phi Subscript i sigma Superscript asterisk Baseline left parenthesis bold r 1 right parenthesis phi Subscript j sigma Baseline left parenthesis bold r 1 right parenthesis phi Subscript j sigma Superscript asterisk Baseline left parenthesis bold r 2 right parenthesis phi Subscript i sigma Baseline left parenthesis bold r 2 right parenthesis Over StartAbsoluteValue bold r 1 minus bold r 2 EndAbsoluteValue EndFraction normal d bold r 1 normal d bold r 2 comma EndLayout

(1.170)

Like the hybrid approximations of Sect. 1.4.1, Eq. (1.167) should be considered as an approximation of the GKS exchange-correlation functional $$E_{\text{xc}}^{S}[\Phi ]$$ in Eq. (1.38) with $$S[\Phi ]= E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu ,{\mathrm {HF}}}[\Phi ]$$ , and the corresponding GKS equations [Eq. (1.1.4)] then includes a long-range nonlocal HF exchange potential . Similarly to the hybrid approximations, the introduction of a fraction of long-range HF exchange reduces the self-interaction error (see, e.g., Ref. [179]). In addition, the short-range exchange part is easier to approximate with semilocal density-functional approximations, as Eq. (1.170) strongly suggests. In particular, the − 1∕r asymptotic behavior of the exchange potential [Eq. (1.27)], which is difficult to satisfy with semilocal approximations, does not apply anymore to the short-range exchange potential.

In 2004, Yanai et al. [273], introduced a more flexible scheme called the Coulomb-attenuating method (CAM) [273] in which fractions of HF exchange are added at both short range and long range,

(1.171)

where $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu ,{\mathrm {HF}}}[\Phi ]=E_{{\mathrm {x}}}^{{\mathrm {HF}}}[\Phi ] - E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu ,{\mathrm {HF}}}[\Phi ]$$ is the short-range HF exchange energy and $$E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu ,{\mathrm {GGA}}}=E_{{\mathrm {x}}}^{{\mathrm {GGA}}} - E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu ,{\mathrm {GGA}}}$$ is a long-range GGA exchange energy. The reintroduction of HF exchange at short range further reduces the self-interaction error and improves thermodynamic properties such as atomization energies. Again, Eq. (1.171) should be considered as an approximation of the GKS exchange-correlation functional $$E_{\text{xc}}^{S}[\Phi ]$$ in Eq. (1.38) with $$S[\Phi ]= a \; E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu ,{\mathrm {HF}}}[\Phi ] + b \; E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu ,{\mathrm {HF}}}[\Phi ]$$ . Other forms of modified electron-electron interactions are also possible (see, e.g., Refs. [113, 217, 255]).

The approximations in Eqs. (1.167) and (1.171) are usually collectively referred to as range-separated hybrid approximations. Range-separated hybrids in the form of Eq. (1.171) are more flexible than the hybrid approximations of Sect. 1.4.1, and consequently are potentially more accurate, in particular for long-range electronic excitations. However, like the hybrid approximations, the presence of HF exchange tends to induce static-correlation errors for systems with small HOMO-LUMO gaps.

The range-separation parameter μ (also sometimes denoted as ω) is generally chosen empirically, e.g. by fitting to experimental data. In practice, a value around μ ≈ 0.3 − 0.5 bohr⁻¹, fixed for all systems, is often found to be optimal. It has also been proposed to adjust the value of μ in each system, e.g. by requiring that the opposite of the HOMO energy be equal to the ionization energy calculated by total energy differences [12, 235, 236]. These so-called optimally tuned range-separated hybrids are well suited for the calculation of charge-transfer electronic excitations but have the disadvantage of not being size consistent [130].

A natural idea is to use a position-dependent range-separation parameter μ(r) which allows the range of the modified interaction to adapt to the local average electron-electron distance in the diverse spatial regions of the system. These locally range-separated hybrids [11, 133, 139] are promising but they induced computational complications and are still in the early stages of development. We will thus not consider them any further here.

We now describe some of the most used approximations in the context of the range-separated hybrids.

Short-Range LDA Exchange Functional

The short-range LDA exchange functional [81, 216] can be obtained by using in Eq. (1.169) the LDA exchange hole [Eq. (1.110)], which leads to

(1.172)

with the short-range UEG exchange energy density

(1.173)

where $${\widetilde {\mu }}=\mu /(2k_{\text{F}})$$ is a dimensionless range-separation parameter. The spin-dependent version is obtained from the same spin-scaling relation as in the standard case [Eq. (1.36)]. The short-range LDA exchange functional becomes exact for large μ [Eq. (1.170)] and is the first building block for constructing short-range exchange GGA functionals.

CAM-B3LYP Exchange-Correlation Functional

The CAM-B3LYP exchange-correlation functional [273] uses Eq. (1.171) with short- and long-range versions of the B88 exchange functional and the same correlation functional used in B3LYP (i.e., $$0.81 \; E_{\text{c}}^{\text{LYP}} + 0.19 \; E_{\text{c}}^{\text{LSDA}}$$ ),

StartLayout 1st Row 1st Column upper E Subscript xc Superscript CAM hyphen upper B 3 LYP Baseline left bracket upper Phi right bracket 2nd Column equals a upper E Subscript normal x Superscript s r comma mu comma upper H upper F Baseline left bracket upper Phi right bracket plus b upper E Subscript normal x Superscript l r comma mu comma upper H upper F Baseline left bracket upper Phi right bracket plus left parenthesis 1 minus a right parenthesis upper E Subscript normal x Superscript s r comma mu comma upper B 88 Baseline left bracket rho Subscript up arrow comma upper Phi Baseline comma rho Subscript down arrow comma upper Phi Baseline right bracket 2nd Row 1st Column Blank 2nd Column plus left parenthesis 1 minus b right parenthesis upper E Subscript normal x Superscript l r comma mu comma upper B 88 Baseline left bracket rho Subscript up arrow comma upper Phi Baseline comma rho Subscript down arrow comma upper Phi Baseline right bracket plus 0.81 upper E Subscript c Superscript LYP Baseline left bracket rho Subscript up arrow comma upper Phi Baseline comma rho Subscript down arrow comma upper Phi Baseline right bracket 3rd Row 1st Column Blank 2nd Column plus 0.19 upper E Subscript c Superscript LSDA Baseline left bracket rho Subscript up arrow comma upper Phi Baseline comma rho Subscript down arrow comma upper Phi Baseline right bracket comma EndLayout

(1.174)

where the parameters a = 0.19 and b = 0.65 were optimized on atomization energies and the range-separation parameter μ = 0.33 bohr⁻¹ was taken from Ref. [250], where it was optimized on equilibrium distances of diatomic molecules. In this expression, the short-range B88 exchange functional $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu ,\text{B88}}$$ is defined by using in Eq. (1.169) the following generic GGA model for the exchange hole [121] (given here in its spin-independent version)

(1.175)

with $$k_{\text{GGA}} = k_{\text{F}}/\sqrt {e_{{\mathrm {x}}}^{\text{GGA}}(\rho ,\nabla \rho )/e_{{\mathrm {x}}}^{\text{UEG}}(\rho )}$$ . The exchange-hole model of Eq. (1.175) properly yields the GGA exchange energy density $$e_{{\mathrm {x}}}^{\text{GGA}}(\rho ,\nabla \rho )$$ for μ = 0 and thus allows one to extend any standard GGA exchange functional to a short-range GGA exchange functional. Note however that it does not fulfill the sum rule [Eq. (1.50)]. The long-range B88 exchange functional is then simply $$E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu ,\text{B88}} = E_{{\mathrm {x}}}^{\text{B88}} - E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu ,\text{B88}}$$ .

LC-ωPBE Exchange-Correlation Functional

The LC-ωPBE exchange-correlation functional [266, 267] uses a short-range version of the PBE exchange functional as well as the standard PBE correlation functional,

(1.176)

The short-range PBE exchange functional is obtained by using in Eq. (1.169) the following GGA exchange hole model constructed to yield the PBE exchange energy [59],

(1.177)

where s = |∇ρ|∕(2kFρ) and

(1.178)

Here, $$\mathcal {A}$$ , $$\mathcal {B}$$ , $$\mathcal {C}$$ , $$\mathcal {D}$$ , and $$\mathcal {E}$$ are constants chosen to obtain an oscillation-averaged UEG exchange hole for s = 0, and $$\mathcal {F}(s)$$ , $$\mathcal {G}(s)$$ and $$\mathcal {H}(s)$$ are functions determined so that the hole yields the PBE exchange density for μ = 0, and satisfies the sum rule [Eq. (1.50)] and the small-r₁₂ expansion [Eq. (1.137)] using the gradient expansion of τ of Eq. (1.141). The range-separation parameter is fixed at μ = ω = 0.4 bohr⁻¹ which has been found to be close to optimal for atomization energies, reaction barrier heights, and ionization energies [266].

ωB97X Exchange-Correlation Functional

The ωB97X exchange-correlation functional [34] has the form of Eq. (1.171) with b = 1:

(1.179)

The short-range B97-GGA exchange density is defined as

where $$e_{{\mathrm {x}},\sigma }^{{\mathrm {sr}},\mu ,\text{UEG}}(\rho _\sigma ) = e_{{\mathrm {x}}}^{{\mathrm {sr}},\mu ,\text{UEG}}(\rho _\sigma ,0)$$ is the spin-σ contribution to the short-range UEG exchange energy density [Eq. (1.173)] and the gradient correction gx(xσ) where $$x_\sigma = |\nabla \rho _\sigma |/\rho _\sigma ^{4/3}$$ has the same form as in Eq. (1.124) with polynomial degree m = 4. In Eq. (1.179), the correlation functional has the same form as the B97-GGA correlation functional but again with polynomial degree m = 4 in Eqs. (1.128) and (1.129). The fraction of short-range HF exchange a ≈ 0.16, the range-separation parameter μ = ω = 0.3 bohr⁻¹, and the linear coefficients in Eqs. (1.124), (1.128), and (1.129) were optimized on sets of atomic energies, atomization energies, ionization energies, electron and proton affinities, reaction barrier heights, and noncovalent interactions, with the constraints a + cx,0 = 1, $$c_{{\text{c}},0}^{\uparrow \downarrow }=1$$ , and $$c_{{\text{c}},0}^{\sigma \sigma }=1$$ to enforce the correct UEG limit.

HSE Exchange-Correlation Functional

The Heyd–Scuseria–Ernzerhof (HSE) exchange-correlation functional [117] is of the form of Eq. (1.171) with b = 0 (i.e., no long-range HF exchange),

(1.180)

and involves the long-range PBE exchange functional $$E_{{\mathrm {x}}}^{{\mathrm {lr}},\mu ,\text{PBE}}= E_{{\mathrm {x}}}^{\text{PBE}}- E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu ,\text{PBE}}$$ complementary to the short-range PBE exchange functional constructed from the PBE exchange hole model [Eqs. (1.177) and (1.178)]. In order to reproduce reliable values for the band gap in semiconducting solids, the range-separation parameter is fixed at μ = 0.15 bohr⁻¹, which is a very small value compared to the other range-separated hybrids. It means that the range of electron-electron distances covered by HF exchange is large, and the HSE functional could be thought of as a regular hybrid approximation but with the very long-range contribution of the HF exchange removed. This is particularly appropriate for solids since in these systems the very long-range HF exchange is effectively balanced by the correlation effects (a phenomenon known as screening). The fraction of (short-range) HF exchange is fixed at a = 0.25 like in the PBE0 hybrid functional.

1.5 Multideterminant Hybrid Approximations

1.5.1 Double-Hybrid Approximations

In 2006, Grimme [98] introduced a two-parameter double-hybrid (2DH) approximation, written here in its spin-independent version,

(1.181)

mixing a fraction ax of the HF exchange energy with a GGA exchange functional, and a fraction ac of the second-order Møller–Plesset (MP2) correlation energy $$E_{\text{c}}^{{\mathrm {MP2}}}$$ with a GGA correlation functional. In Eq. (1.181), the first three terms are first calculated in a self-consistent manner, and then the last term $$E_{\text{c}}^{{\mathrm {MP2}}}$$ is added perturbatively using the orbitals determined in the first step. The expression of $$E_{\text{c}}^{{\mathrm {MP2}}}$$ is [247]

upper E Subscript c Superscript MP 2 Baseline equals minus one fourth sigma summation Underscript i equals 1 Overscript upper N Endscripts sigma summation Underscript j equals 1 Overscript upper N Endscripts sigma summation Underscript a greater than or slanted equals upper N plus 1 Endscripts sigma summation Underscript b greater than or slanted equals upper N plus 1 Endscripts StartFraction StartAbsoluteValue left angle bracket phi Subscript i Baseline phi Subscript j Baseline vertical bar phi Subscript a Baseline phi Subscript b Baseline right angle bracket EndAbsoluteValue squared Over epsilon Subscript a Baseline plus epsilon Subscript b Baseline minus epsilon Subscript i Baseline minus epsilon Subscript j Baseline EndFraction comma

(1.182)

where i, j and a, b run over occupied and virtual spin orbitals, respectively, εk are spin orbital energies, and left angle bracket phi Subscript i Baseline phi Subscript j Baseline vertical bar phi Subscript a Baseline phi Subscript b Baseline right angle bracket equals left angle bracket phi Subscript i Baseline phi Subscript j Baseline vertical bar phi Subscript a Baseline phi Subscript b Baseline right angle bracket minus left angle bracket phi Subscript i Baseline phi Subscript j Baseline vertical bar phi Subscript b Baseline phi Subscript a Baseline right angle bracket ) are antisymmetrized two-electron integrals with (in physicists’ notation)

(1.183)

Note that the notation in Eq. (1.182) assumes that the one-electron wave-function space is spanned by a discrete set of spin orbitals. In the exact theory, the continuum limit of the set of virtual spin orbitals is implied.

The rigorous framework underlying these double-hybrid approximations was established by Sharkas et al. [226]. The idea is to decompose the universal density functional of Eq. (1.7) as

StartLayout 1st Row upper F left bracket rho right bracket equals min Underscript upper Psi element of script upper W Subscript rho Superscript upper N Baseline Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript lamda Baseline left bracket rho right bracket comma EndLayout

(1.184)

where λ ∈ [0, 1] is a coupling constant and $$\bar {E}_{{\mathrm {Hxc}}}^\lambda [\rho ]$$ is a complementary density functional defined to make Eq. (1.184) exact. From Eqs. (1.10) and (1.62), we see that $$\bar {E}_{{\mathrm {Hxc}}}^\lambda [\rho ] = E_{{\mathrm {Hxc}}}[\rho ] - E_{{\mathrm {Hxc}}}^\lambda [\rho ]$$ , where EHxc[ρ] is the standard Hartree-exchange-correlation functional of the KS scheme and $$E_{{\mathrm {Hxc}}}^\lambda [\rho ]$$ is the Hartree-exchange-correlation functional along the adiabatic connection. The Hartree and exchange contributions are simply linear in λ,

(1.185)

(1.186)

where EH[ρ] and Ex[ρ] are the standard Hartree and exchange functionals of the KS scheme. Moreover, from the uniform coordinate scaling relation of Eq. (1.84), we have

(1.187)

where Ec[ρ] is the standard correlation functional of the KS scheme and ρ1∕λ(r) = (1∕λ)³ρ(r∕λ) is the scaled density. The decomposition in Eq. (1.184) leads to the following expression of the exact ground-state energy

StartLayout 1st Row upper E 0 equals inf Underscript upper Psi element of script upper W Superscript upper N Baseline Endscripts left brace left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript lamda Baseline left bracket rho Subscript upper Psi Baseline right bracket right brace comma EndLayout

(1.188)

where the infimum is over general multideterminant wave functions $$\Psi \in \mathcal {W}^{N}$$ . This constitutes a multideterminant extension of the KS scheme. Note that this multideterminant KS scheme can trivially be extended to spin-dependent density functionals and functionals depending on the kinetic-energy density [232].

The double-hybrid ansatz can be seen as a particular approximation within this multideterminant KS scheme [226]. To see this, we define a density-scaled one-parameter hybrid (DS1H) approximation by restricting the minimization in Eq. (1.188) to single-determinant wave functions $$\Phi \in \mathcal {S}^{N}$$ ,

StartLayout 1st Row 1st Column upper E 0 Superscript DS 1 upper H comma lamda 2nd Column equals 3rd Column inf Underscript upper Phi element of script upper S Superscript upper N Baseline Endscripts left brace left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript lamda Baseline left bracket rho Subscript upper Phi Baseline right bracket right brace comma EndLayout

(1.189)

obtaining an energy which necessarily depends on λ. A minimizing single-determinant wave function Φλ must satisfy the self-consistent eigenvalue equation

StartLayout 1st Row left parenthesis ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus lamda ModifyingAbove upper V With caret Subscript upper H x Superscript upper H upper F Baseline left bracket upper Phi Superscript lamda Baseline right bracket plus ModifyingAbove Above ModifyingAbove upper V With bar With caret Subscript upper H x c Superscript lamda Baseline left bracket rho Subscript upper Phi Sub Superscript lamda Subscript Baseline right bracket right parenthesis vertical bar upper Phi Superscript lamda Baseline right angle bracket equals script upper E 0 Superscript lamda Baseline vertical bar upper Phi Superscript lamda Baseline right angle bracket comma EndLayout

(1.190)

where is the nonlocal HF potential operator evaluated with the DS1H wave function Φλ and is the local Hartree-exchange-correlation potential operator generated by the energy functional $$\bar {E}_{{\mathrm {Hxc}}}^{\lambda }[\rho ]$$ and evaluated at the DS1H density $$\rho _{\Phi ^\lambda }$$ . If written explicitly in terms of spin orbitals, Eq. (1.190) would have the form of the GKS equations [Eq. (1.40)]. The DS1H ground-state energy can be finally written as

StartLayout 1st Row 1st Column upper E 0 Superscript DS 1 upper H comma lamda Baseline equals left angle bracket upper Phi Superscript lamda Baseline vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline vertical bar upper Phi Superscript lamda Baseline right angle bracket 2nd Column plus upper E Subscript normal upper H Baseline left bracket rho Subscript upper Phi Sub Superscript lamda Subscript Baseline right bracket plus lamda upper E Subscript normal x Superscript upper H upper F Baseline left bracket upper Phi Superscript lamda Baseline right bracket 2nd Row 1st Column Blank 2nd Column plus left parenthesis 1 minus lamda right parenthesis upper E Subscript normal x Baseline left bracket rho Subscript upper Phi Sub Superscript lamda Subscript Baseline right bracket plus ModifyingAbove upper E With bar Subscript c Superscript lamda Baseline left bracket rho Subscript upper Phi Sub Superscript lamda Subscript Baseline right bracket comma EndLayout

(1.191)

where the full Hartree energy EH[ρ] has been recomposed. The exchange-correlation energy in Eq. (1.191) is of the form of a hybrid approximation [Eq. (1.160)].

All that is missing in Eq. (1.191) is the correlation energy associated with the scaled interaction . It can be calculated by a nonlinear Rayleigh–Schrödinger perturbation theory [5, 6, 69] starting from the DS1H reference. Consider the following energy expression with the perturbation parameter α ∈ [0, 1],

StartLayout 1st Row 1st Column upper E 0 Superscript lamda comma alpha 2nd Column equals 3rd Column inf Underscript upper Psi element of script upper W Superscript upper N Baseline Endscripts left brace left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus lamda ModifyingAbove upper V With caret Subscript upper H x Superscript upper H upper F Baseline left bracket upper Phi Superscript lamda Baseline right bracket plus alpha lamda ModifyingAbove upper W With caret vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript lamda Baseline left bracket rho Subscript upper Psi Baseline right bracket right brace comma EndLayout

(1.192)

where is the scaled Møller–Plesset perturbation operator. For α = 0, the stationary equation associated with Eq. (1.192) reduces to the DS1H eigenvalue equation [Eq. (1.190)]. For α = 1, Eq. (1.192) reduces to Eq. (1.188), so $$E^{\lambda ,\alpha =1}_0$$ is the exact energy, independently of λ. The sum of the zeroth-order energy and first-order energy correction gives simply the DS1H energy, $$E^{\text{DS1H},\lambda }_0=E^{\lambda ,(0)}_0+E^{\lambda ,(1)}_0$$ . Thanks to the existence of a Brillouin theorem just like in standard Møller–Plesset perturbation theory (see Refs. [5, 6, 69]), only double excitations contribute to the first-order wave-function correction Ψλ, (1) and the second-order energy correction has a standard MP2 form

StartLayout 1st Row upper E 0 Superscript lamda comma left parenthesis 2 right parenthesis Baseline equals lamda squared left angle bracket upper Phi Superscript lamda Baseline vertical bar ModifyingAbove upper W With caret vertical bar upper Psi Superscript lamda comma left parenthesis 1 right parenthesis Baseline right angle bracket equals lamda squared upper E Subscript c Superscript MP 2 Baseline comma EndLayout

where $$E_{\text{c}}^{\text{MP2}}$$ has the expression in Eq. (1.182) with DS1H spin orbitals and associated orbital eigenvalues (which implicitly depend on λ). This second-order perturbation theory defines a density-scaled one-parameter double-hybrid (DS1DH) approximation

(1.193)

which contains the exchange-correlation energy contribution

(1.194)

To make connection with the double-hybrid ansatz of Eq. (1.181), we can also define a one-parameter double-hybrid (1DH) approximation, obtained by neglecting the density scaling in the correlation functional, i.e. Ec[ρ1∕λ] ≈ Ec[ρ] in Eq. (1.187),

(1.195)

which, after using semilocal approximations for Ex[ρ] and Ec[ρ], has the form of Eq. (1.181) with parameters ax = λ and ac = λ². In this rigorous formulation of the double-hybrid approximations, the fraction of HF exchange is thus connected to the fraction of MP2 correlation. Taking into account approximately the scaling of the density in Ec[ρ1∕λ], it has also been proposed to use ac = λ³ [260]. Fromager [67] also proposed an extension of this rigorous formulation in order to justify the use of double-hybrid approximations with two parameters such that $$a_{\text{c}} \leqslant a_{{\mathrm {x}}}^2=\lambda ^2$$ .

An essential advantage of double-hybrid approximations is that the presence of nonlocal MP2 correlation allows one to use a larger fraction of nonlocal HF exchange, which helps decreasing the self-interaction error. This usually provides an improvement over hybrid approximations for molecular systems with sufficiently large electronic gaps. However, a large fraction of HF exchange and a fraction of MP2 correlation also generally means large static-correlation errors in systems with small HOMO-LUMO gaps.

The first and still best known double-hybrid approximation is B2PLYP [98], which is based on the B88 exchange functional and the LYP correlation functional,

and the parameters ax = 0.53 and ac = 0.27 have been optimized on a set of atomization energies. Interestingly, even though the two parameters have been optimized without any constraint, we have $$a_{\text{c}} \approx a_{{\mathrm {x}}}^2 = 0.28$$ as predicted by Eq. (1.195).

It has also been proposed to use the spin-component-scaled (SCS) version of MP2 [95] to construct spin-component-scaled double-hybrid approximations of the form [136, 137]

(1.196)

which contains four empirical parameters ax, ac, cOS, and cSS. In this expression, $$E^{\text{MP2}}_{{\text{c}},\text{OS}}$$ and $$E^{\text{MP2}}_{{\text{c}},\text{SS}}$$ are the opposite-spin (OS) and same-spin (SS) contributions to the MP2 correlation energy obtained by restricting the sums over i and j in Eq. (1.182) to spin orbitals of opposite and same spins, respectively. Since in MP2 the same-spin component is usually overestimated relative to the opposite-spin component, this SCS variant is a simple way to achieve higher accuracy without increasing computational cost.

For reviews on different flavors of double hybrids and their assessments, the reader may consult Refs. [82, 171, 212, 241]. It has also been proposed to construct double-hybrid approximations where the MP2 correlation term is extended to a higher-order correlation method such as RPA [3, 100, 172, 173, 211] or coupled-cluster [35, 76]. More generally, the multideterminant extension of the KS scheme of Eq. (1.188) allows one to define hybrids combining any wave-function method with density functionals. For example, a multiconfiguration hybrid approximation based on Eq. (1.188) which combines a multiconfiguration self-consistent-field (MCSCF) wave function with density functionals has been proposed in the goal of tackling strongly correlated systems [225]. This approach has also been used to combine valence-bond (VB) theory [274] or variational two-electron reduced-density-matrix theory [176] with DFT.

1.5.2 Range-Separated Double-Hybrid Approximations

1.5.2.1 Range-Separated One-Parameter Double-Hybrid Approximations

In 2005, Ángyán et al. [6] introduced what could be called the first range-separated one-parameter double-hybrid approximation, i.e. combining HF exchange and MP2 correlation with density functionals using a one-parameter decomposition of the electron-electron interaction. This is based on the range-separated multideterminant extension of the KS scheme introduced earlier by Savin [216] (see, also, Refs. [215, 217, 255]) and which actually predates and inspired the multideterminant extension of the KS scheme of Eq. (1.188).

The idea is to decompose the universal density functional of Eq. (1.7) as

StartLayout 1st Row upper F left bracket rho right bracket equals min Underscript upper Psi element of script upper W Subscript rho Superscript upper N Baseline Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper W With caret Subscript e e Superscript l r comma mu Baseline vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript s r comma mu Baseline left bracket rho right bracket comma EndLayout

(1.197)

where is the long-range electron-electron interaction operator (associated with the pair potential $$w_{{\mathrm {ee}}}^{{\mathrm {lr}},\mu }(r_{12}) = \operatorname {\mathrm {erf}}(\mu r_{12})/r_{12}$$ as already used in the range-separated hybrids of Sect. 1.4.2) and $$\bar {E}_{{\mathrm {Hxc}}}^{{\mathrm {sr}},\mu }[\rho ]$$ is the complementary short-range density functional defined to make Eq. (1.197) exact. As before, the parameter μ ∈ [0, +∞) controls the range of the separation. The complementary short-range functional can be written as $$\bar {E}_{{\mathrm {Hxc}}}^{{\mathrm {sr}},\mu }[\rho ] = E_{{\mathrm {Hxc}}}[\rho ] - E_{{\mathrm {Hxc}}}^{{\mathrm {lr}},\mu }[\rho ]$$ , where EHxc[ρ] is the standard Hartree-exchange-correlation functional of the KS scheme and $$E_{{\mathrm {Hxc}}}^{{\mathrm {lr}},\mu }[\rho ]$$ is the Hartree-exchange-correlation functional associated with the long-range interaction $$w_{{\mathrm {ee}}}^{{\mathrm {lr}},\mu }(r_{12})$$ . It is often convenient to decompose the short-range functional as (see Refs. [240, 254, 258] for an alternative decomposition)

where $$E_{{\mathrm {H}}}^{{\mathrm {sr}},\mu }[\rho ]$$ is the short-range Hartree functional,

with the short-range interaction $$w_{{\mathrm {ee}}}^{{\mathrm {sr}},\mu }(r_{12})=1/r_{12} - w_{{\mathrm {ee}}}^{{\mathrm {lr}},\mu }(r_{12})$$ , $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu }[\rho ]$$ is the short-range exchange functional [Eq. (1.169)] which can also be written as

StartLayout 1st Row 1st Column upper E Subscript normal x Superscript s r Baseline left bracket rho right bracket 2nd Column equals 3rd Column left angle bracket upper Phi left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Superscript s r comma mu Baseline vertical bar upper Phi left bracket rho right bracket right angle bracket minus upper E Subscript normal upper H Superscript s r comma mu Baseline left bracket rho right bracket comma EndLayout

with the KS single-determinant wave function Φ[ρ], and $$\bar {E}_{\text{c}}^{{\mathrm {sr}},\mu }[\rho ]$$ is the complementary short-range correlation functional. Just like for Eq. (1.188), the decomposition in Eq. (1.197) leads to the following expression of the exact ground-state energy

StartLayout 1st Row upper E 0 equals inf Underscript upper Psi element of script upper W Superscript upper N Baseline Endscripts left brace left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus ModifyingAbove upper W With caret Subscript e e Superscript l r comma mu Baseline vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript s r comma mu Baseline left bracket rho Subscript upper Psi Baseline right bracket right brace comma EndLayout

(1.198)

where the infimum is over general multideterminant wave functions $$\Psi \in \mathcal {W}^{N}$$ .

To obtain an MP2/DFT hybrid scheme, we proceed analogously to Sect. 1.5.1. First, we define the following range-separated hybrid (RSH) approximation by restricting the minimization in Eq. (1.198) to single-determinant wave functions $$\Phi \in \mathcal {S}^{N}$$ ,

StartLayout 1st Row 1st Column upper E 0 Superscript RSH comma mu 2nd Column equals 3rd Column inf Underscript upper Phi element of script upper S Superscript upper N Baseline Endscripts left brace left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus ModifyingAbove upper W With caret Subscript e e Superscript l r comma mu Baseline vertical bar upper Phi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript s r comma mu Baseline left bracket rho Subscript upper Phi Baseline right bracket right brace comma EndLayout

(1.199)

obtaining an energy which necessarily depends on μ. A minimizing single-determinant wave function Φμ must satisfy the self-consistent eigenvalue equation

StartLayout 1st Row left parenthesis ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus ModifyingAbove upper V With caret Subscript upper H x Superscript l r comma mu comma upper H upper F Baseline left bracket upper Phi Superscript mu Baseline right bracket plus ModifyingAbove Above ModifyingAbove upper V With bar With caret Subscript upper H x c Superscript s r comma mu Baseline left bracket rho Subscript upper Phi Sub Superscript mu Subscript Baseline right bracket right parenthesis vertical bar upper Phi Superscript mu Baseline right angle bracket equals script upper E 0 Superscript mu Baseline vertical bar upper Phi Superscript mu Baseline right angle bracket comma EndLayout

(1.200)

where is the nonlocal long-range HF potential operator evaluated with the RSH wave function Φμ and is the local short-range Hartree-exchange-correlation potential operator generated by the energy functional $$\bar {E}_{{\mathrm {Hxc}}}^{{\mathrm {sr}},\mu }[\rho ]$$ and evaluated at the RSH density $$\rho _{\Phi ^\mu }$$ . The RSH ground-state energy can be finally written as

StartLayout 1st Row 1st Column upper E 0 Superscript RSH comma mu Baseline equals left angle bracket upper Phi Superscript mu Baseline vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline vertical bar upper Phi Superscript mu Baseline right angle bracket 2nd Column plus upper E Subscript normal upper H Baseline left bracket rho Subscript upper Phi Sub Superscript mu Subscript Baseline right bracket plus upper E Subscript normal x Superscript l r comma mu comma upper H upper F Baseline left bracket upper Phi Superscript mu Baseline right bracket 2nd Row 1st Column Blank 2nd Column plus upper E Subscript normal x Superscript s r comma mu Baseline left bracket rho Subscript upper Phi Sub Superscript mu Subscript Baseline right bracket plus ModifyingAbove upper E With bar Subscript c Superscript s r comma mu Baseline left bracket rho Subscript upper Phi Sub Superscript mu Subscript Baseline right bracket comma EndLayout

(1.201)

where the full Hartree energy EH[ρ] has been recomposed. The exchange-correlation energy in Eq. (1.201) has a similar form as in the LC scheme of Eq. (1.167).

To calculate the missing long-range correlation energy in Eq. (1.201), we can define a nonlinear Rayleigh–Schrödinger perturbation theory [5, 6, 69] starting from the RSH reference. We start from the following energy expression with the perturbation parameter α ∈ [0, 1],

upper E 0 Superscript mu comma alpha Baseline equals inf Underscript upper Psi element of script upper W Superscript upper N Baseline Endscripts left brace left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus ModifyingAbove upper V With caret Subscript upper H x Superscript l r comma mu comma upper H upper F Baseline left bracket upper Phi Superscript mu Baseline right bracket plus alpha ModifyingAbove upper W With caret Superscript l r comma mu Baseline vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript l r comma mu Baseline left bracket rho Subscript upper Psi Baseline right bracket right brace comma

(1.202)

where is the long-range Møller–Plesset perturbation operator. For α = 0, the stationary equation associated with Eq. (1.202) reduces to the RSH eigenvalue equation [Eq. (1.200)]. For α = 1, Eq. (1.202) reduces to Eq. (1.198), so $$E^{\mu ,\alpha =1}_0$$ is the exact energy, independently of μ. The sum of the zeroth-order energy and first-order energy correction gives simply the RSH energy, $$E^{\text{RSH},\mu }_0=E^{\mu ,(0)}_0+E^{\mu ,(1)}_0$$ . As in Sect. 1.5.1, only double excitations contribute to the first-order wave-function correction Ψμ, (1) and the second-order energy correction has a standard MP2 form

StartLayout 1st Row upper E 0 Superscript mu comma left parenthesis 2 right parenthesis Baseline equals left angle bracket upper Phi Superscript mu Baseline vertical bar ModifyingAbove upper W With caret Superscript l r comma mu Baseline vertical bar upper Psi Superscript mu comma left parenthesis 1 right parenthesis Baseline right angle bracket equals upper E Subscript c Superscript l r comma mu comma MP 2 Baseline comma EndLayout

where $$E_{\text{c}}^{{\mathrm {lr}},\mu ,\text{MP2}}$$ has the same expression as in Eq. (1.182) with RSH spin orbitals and associated orbital eigenvalues (which implicitly depend on μ) but using the long-range two-electron integrals

(1.203)

instead of the standard two-electron integrals of Eq. (1.183). This second-order perturbation theory defines a RSH+MP2 approximation,

(1.204)

which contains the exchange-correlation energy contribution

(1.205)

When using semilocal density-functional approximations for the short-range functionals $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu }[\rho ]$$ and $$\bar {E}^{{\mathrm {sr}},\mu }_{\text{c}}[\rho ]$$ , the RSH+MP2 exchange-correlation energy expression of Eq. (1.205) thus constitutes range-separated double-hybrid approximations similar to the double hybrids of Sect. 1.5.1. The optimal value for the range-separation parameter is often around μ ≈ 0.5 bohr⁻¹ [77, 177]. This scheme has the advantage of dropping the long-range part of both the exchange and correlation density functionals, which are usually not well described by semilocal density-functional approximations. Moreover, using a long-range MP2 correlation energy has the advantage of leading to a correct qualitative description of London dispersion interaction energies [6, 78, 79, 251], while displaying a fast convergence with the one-electron basis size [63]. Similar to the SCS double hybrids [Eq. (1.196)], a SCS variant of the RSH+MP2 scheme has also been proposed [213].

The range-separated multideterminant extension of the KS scheme of Eq. (1.198) allows one to define various hybrid schemes combining any wave-function method with density functionals. For example, one can go beyond second order by using long-range coupled-cluster [75, 83, 84, 262] or random-phase approximations [124, 185, 257, 261, 262]. To describe strongly correlated systems, one can also use for the long-range part wave-function methods such as configuration interaction (CI) [31, 62, 152, 202], MCSCF [70, 71, 108], density-matrix renormalization group (DMRG) [107], or multireference perturbation theory [68]. Density-matrix functional theory (DMFT) [201, 209, 210] and Green-function methods [128, 207] have also been used for the long-range part.

We now consider the approximations used for $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu }[\rho ]$$ and $$\bar {E}^{{\mathrm {sr}},\mu }_{\text{c}}[\rho ]$$ . In Sect. 1.4.2, we have already described the short-range exchange LDA [Eq. (1.172)] and some short-range exchange GGAs for $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu }[\rho ]$$ . Here, we describe the short-range LDA correlation functional and another short-range GGA exchange-correlation functional.

Short-Range LDA Correlation Functional

The complementary short-range LDA (or LSDA) correlation functional is

(1.206)

where is the complementary short-range UEG correlation energy density. In this expression, $$\bar {\varepsilon }_{\text{c}}^{{\mathrm {sr}},\mu ,{\mathrm {UEG}}}(\rho _\uparrow ,\rho _\downarrow )$$ is defined by

(1.207)

where $$\varepsilon _{\text{c}}^{{\mathrm {UEG}}}(\rho _\uparrow ,\rho _\downarrow )$$ and $$\varepsilon _{\text{c}}^{{\mathrm {lr}},\mu ,{\mathrm {UEG}}}(\rho _\uparrow ,\rho _\downarrow )$$ are the correlation energies per particle of the UEG with the standard Coulomb and long-range electron-electron interactions, respectively. A simple spin-independent parametrization of $$\bar {\varepsilon }_{\text{c}}^{{\mathrm {sr}},\mu ,{\mathrm {UEG}}}$$ was given in Ref. [259]. A better spin-dependent parametrization was constructed in Ref. [186] which uses the PW92 parametrization for $$\varepsilon _{\text{c}}^{{\mathrm {UEG}}}(\rho _\uparrow ,\rho _\downarrow )$$ [Eq. (1.106)] and the following parametrization for $$\varepsilon _{\text{c}}^{{\mathrm {lr}},\mu ,{\mathrm {UEG}}}(\rho _\uparrow ,\rho _\downarrow )$$ in terms of rs = (3∕(4πρ))1∕3 and ζ = (ρ↑− ρ↓)∕ρ:

StartLayout 1st Row 1st Column Blank 2nd Column epsilon Subscript c Superscript l r comma mu comma upper U upper E upper G Baseline left parenthesis rho Subscript up arrow Baseline comma rho Subscript down arrow Baseline right parenthesis equals 2nd Row 1st Column Blank 2nd Column StartFraction Start 2 By 2 Matrix 1st Row 1st Column phi 2 left parenthesis zeta right parenthesis cubed upper Q left parenthesis StartFraction mu StartRoot r Subscript normal s Baseline EndRoot Over phi 2 left parenthesis zeta right parenthesis EndFraction right parenthesis 2nd Column plus a 1 left parenthesis r Subscript normal s Baseline comma zeta right parenthesis mu cubed plus a 2 left parenthesis r Subscript normal s Baseline comma zeta right parenthesis mu Superscript 4 Baseline plus a 3 left parenthesis r Subscript normal s Baseline comma zeta right parenthesis mu Superscript 5 Baseline plus a 4 left parenthesis r Subscript normal s Baseline comma zeta right parenthesis mu Superscript 6 Baseline 2nd Row 1st Column Blank 2nd Column plus a 5 left parenthesis r Subscript normal s Baseline comma zeta right parenthesis mu Superscript 8 Baseline EndMatrix Over left parenthesis 1 plus b 0 left parenthesis r Subscript normal s Baseline right parenthesis squared mu squared right parenthesis Superscript 4 Baseline EndFraction period EndLayout

In this expression, ϕ₂(ζ) is a spin-scaling function defined by Eq. (1.103), Q(x) is a function determined from the small-μ and/or small-rs limit,

with a = 5.84605, c = 3.91744, d = 3.44851, $$b=d-3 \pi \alpha /[4 \ln (2) - 4]$$ , α = 4∕(9π)1∕3, and the functions ai(rs, ζ) are

where $$\varepsilon _{\text{c}}^{\text{PW92}}(r_{{\mathrm {s}}},\zeta )$$ is the PW92 parametrization of the UEG correlation energy per particle. The functions Ci(rs, ζ) are determined from the large-μ limit,

where g(0, rs, ζ) is the on-top pair-distribution function⁸ of the Coulombic UEG and g″(0, rs, ζ) is its second-order derivative with respect to r₁₂ at r₁₂ = 0, and similarly for their correlation parts gc(0, rs, ζ) = g(0, rs, ζ) − (1 − ζ²)∕2 and $$g_{\text{c}}^{\prime \prime }(0,r_{{\mathrm {s}}},\zeta )=g''(0,r_{{\mathrm {s}}},\zeta )-\phi _8(\zeta )/(5 \alpha ^2 r_{{\mathrm {s}}}^2)$$ with ϕ₈(ζ) defined by Eq. (1.103). The ζ-dependence of the latter quantities is assumed to be exchange-like, i.e. g(0, rs, ζ) ≈ (1 − ζ²)g(0, rs, ζ = 0) and $$g''(0,r_{{\mathrm {s}}},\zeta ) \approx \zeta _+^2 g''(0,r_{{\mathrm {s}}}/\zeta _+^{1/3},\zeta =1) + \zeta _-^2 g''(0,r_{{\mathrm {s}}}/\zeta _-^{1/3},\zeta =1)$$ where ζ± = (1 ± ζ)∕2. The on-top pair-distribution function has been parametrized in Ref. [85] as

with B = 0.7317 − d, C = 0.08193, D = −0.01277, E = 0.001859, and F = 0.7524. The remaining functions were determined by fitting to QMC data:

Short-Range PBE(GWS) Exchange-Correlation Functional

The Goll–Werner–Stoll (GWS) variant of the short-range PBE exchange-correlation functional [83, 84] is a slight modification of the short-range PBE functional developed in Ref. [256]. The exchange energy density is

(1.211)

with an enhancement factor of the same form as in the standard PBE exchange functional,

(1.212)

with s = |∇ρ|∕(2kFρ) and $${\widetilde {\mu }}=\mu /(2k_{\text{F}})$$ . In this expression, κ = 0.840, as in the standard PBE exchange functional, to saturate the local Lieb–Oxford bound (for μ = 0) and $$b({\widetilde {\mu }})=b^{\text{PBE}} [b^{\text{T}}({\widetilde {\mu }})/b^{\text{T}}(0)] \mathrm{e} ^{-\alpha _{\text{x}} {\widetilde {\mu }}^2}$$ where bPBE = 0.21951 is the second-order gradient-expansion coefficient of the standard PBE exchange functional, and $$b^{\text{T}}({\widetilde {\mu }})$$ is a function coming from the second-order GEA of the short-range exchange energy [254, 256],

(1.213)

with $$c_1({\widetilde {\mu }})=1 +22{\widetilde {\mu }}^2 +144{\widetilde {\mu }}^4$$ , $$c_2({\widetilde {\mu }})=2{\widetilde {\mu }}^2(-7+72{\widetilde {\mu }}^2)$$ , $$c_3({\widetilde {\mu }})=-864{\widetilde {\mu }}^4(-1+2{\widetilde {\mu }}^2)$$ , and $$c_4({\widetilde {\mu }})={\widetilde {\mu }}^2[-3 -24{\widetilde {\mu }}^2 +32{\widetilde {\mu }}^4 +8{\widetilde {\mu }}\sqrt {\pi } \operatorname {\mathrm {erf}}(1/(2{\widetilde {\mu }}))]$$ . Finally, αx = 19.0 is a damping parameter optimized for the He atom.

Similarly, the correlation energy density has the same form as the standard PBE correlation functional,

with t = |∇ρ|∕(2ϕ₂(ζ)ksρ) and the gradient correction

where

and

(1.214)

and the value of A(0) is given after Eq. (1.104). In Eq. (1.214), β = 0.066725 is the second-order gradient coefficient of the standard PBE correlation functional and αc = 2.78 is a damping parameter optimized for the He atom.

For μ = 0, this short-range PBE exchange-correlation functional reduces to the standard PBE exchange-correlation functional and for large μ it reduces to the short-range LDA exchange-correlation functional.

1.5.2.2 Range-Separated Two-Parameter Double-Hybrid Approximations

In 2018, Kalai and Toulouse [127] introduced what we will call range-separated two-parameter double-hybrid approximations, combining HF exchange and MP2 correlation with density functionals using a two-parameter decomposition of the electron-electron in a way reminiscent of the CAM decomposition [Eq. (1.171)] (see, also, Refs. [40, 75]). This is based on a multideterminant extension of the KS scheme which generalizes the schemes of Sects. 1.5.1 and 1.5.2.1.

We first decompose the universal density functional of Eq. (1.7) as

StartLayout 1st Row upper F left bracket rho right bracket equals min Underscript upper Psi element of script upper W Subscript rho Superscript upper N Baseline Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper W With caret Subscript e e Superscript l r comma mu Baseline plus lamda ModifyingAbove upper W With caret Subscript e e Superscript s r comma mu Baseline vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript s r comma mu comma lamda Baseline left bracket rho right bracket comma EndLayout

(1.215)

where the parameter μ ∈ [0, +∞) controls the range of the separation as always, the parameter λ ∈ [0, 1] corresponds to the fraction of the short-range electron-electron interaction in the wave-function part, and $$\bar {E}_{{\mathrm {Hxc}}}^{{\mathrm {sr}},\mu ,\lambda }[\rho ]$$ is the complementary short-range density functional defined to make this decomposition exact. As before, the latter functional can be decomposed as

The Hartree and exchange contributions are linear in λ,

(1.216)

where $$E_{{\mathrm {H}}}^{{\mathrm {sr}},\mu }[\rho ]$$ and $$E_{{\mathrm {x}}}^{{\mathrm {sr}},\mu }[\rho ]$$ are the short-range Hartree and exchange functionals introduced in Sect. 1.5.2.1, and the correlation contribution can be written as

where Ec[ρ] is the standard KS correlation functional and $$E_{\text{c}}^{\mu ,\lambda }[\rho ]$$ is the correlation functional associated with the interaction $$w_{\text{ee}}^{{\mathrm {lr}},\mu }(r_{12}) + \lambda w_{\text{ee}}^{{\mathrm {sr}},\mu }(r_{12})$$ . The exact ground-state energy can then be expressed as

StartLayout 1st Row upper E 0 equals inf Underscript upper Psi element of script upper W Superscript upper N Baseline Endscripts left brace left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus ModifyingAbove upper W With caret Subscript e e Superscript l r comma mu Baseline plus lamda ModifyingAbove upper W With caret Subscript e e Superscript s r comma mu Baseline vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript s r comma mu comma lamda Baseline left bracket rho Subscript upper Psi Baseline right bracket right brace comma EndLayout

(1.217)

which constitutes a generalization of Eqs. (1.188) and (1.198).

To obtain a MP2/DFT hybrid scheme, we proceed in full analogy to Sects. 1.5.1 and 1.5.2.1. First, we define the following single-determinant range-separated two-parameter hybrid (RS2H) approximation,

upper E 0 Superscript RS 2 upper H comma mu comma lamda Baseline equals inf Underscript upper Phi element of script upper S Superscript upper N Baseline Endscripts left brace left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret plus ModifyingAbove upper V With caret Subscript ne Baseline plus ModifyingAbove upper W With caret Subscript e e Superscript l r comma mu Baseline plus lamda ModifyingAbove upper W With caret Subscript e e Superscript s r comma mu Baseline vertical bar upper Phi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Superscript s r comma mu comma lamda Baseline left bracket rho Subscript upper Phi Baseline right bracket right brace comma

(1.218)

and use it as a reference for defining a perturbation theory similarly to Eqs. (1.192) and (1.202). At second order, we obtain

(1.219)

where $$E^{\mu ,\lambda ,{\mathrm {MP2}}}_{\text{c}}$$ is the MP2 correlation energy expression evaluated with RS2H spin orbitals and orbital eigenvalues, and the two-electron integrals associated with the interaction $$w_{\text{ee}}^{{\mathrm {lr}},\mu }(r_{12}) + \lambda w_{\text{ee}}^{{\mathrm {sr}},\mu }(r_{12})$$ . This RS2H+MP2 scheme thus contains the exchange-correlation energy contribution

(1.220)

where Φμ, λ is a minimizing single-determinant wave function in Eq. (1.218).

A good approximation for the λ-dependence of the complementary correlation functional $$\bar {E}^{{\mathrm {sr}},\mu ,\lambda }_{\text{c}}[\rho ]$$ is [127]

(1.221)

where $$\bar {E}_{\text{c}}^{{\mathrm {sr}},\mu }[\rho ]$$ is the short-range correlation functional introduced in Sect. 1.5.2.1. In particular, the λ-dependence in Eq. (1.221) is correct both in the high-density limit, for a non-degenerate KS system, and in the low-density limit. Thanks to Eqs. (1.216) and (1.221), the semilocal density-functional approximations for $$E^{{\mathrm {sr}},\mu }_{{\mathrm {x}}}[\rho ]$$ and $$\bar {E}^{{\mathrm {sr}},\mu }_{\text{c}}[\rho ]$$ of Sect. 1.5.2.1 can be reused here without developing new ones. In Ref. [127], the short-range PBE(GWS) exchange and correlation functionals were used, and the optimal parameters μ = 0.46 bohr⁻¹ and λ = 0.58 were found on small sets of atomization energies and reaction barrier heights, i.e. values similar to the ones usually used separately in range-separated hybrids and double hybrids.

The RS2H+MP2 scheme improves a bit over the RSH+MP2 scheme of Sect. 1.5.2.1, particularly for interaction energies of hydrogen-bonded systems. Even if the presence of short-range MP2 correlation deteriorates in principle the convergence rate with the one-electron basis size, in practice the fraction of pure short-range MP2 correlation (λ² ≈ 0.34) is small enough to keep a fast basis convergence. Accuracy can be improved, particularly for dispersion interactions, by supplanting the MP2 term by coupled-cluster or random-phase approximations [126]. Like for the approach of Sect. 1.5.2.1, many wave-function methods could be used in the general scheme of Eq. (1.217).

1.6 Semiempirical Dispersion Corrections and Nonlocal van der Waals Density Functionals

Among the previously considered exchange-correlation approximations, only the range-separated double hybrids of Sect. 1.5.2, thanks to their long-range nonlocal correlation component, are capable of fully describing London dispersion interactions, crucial for describing weakly bonded systems. To improve the other approximations (semilocal functionals, single-determinant hybrids, double hybrids without range separation) for weakly bonded systems, it has been proposed to add to them a semiempirical dispersion correction or a nonlocal van der Waals density functional. We now describe these approaches.

1.6.1 Semiempirical Dispersion Corrections

To explicitly account for London dispersion interactions, it has been proposed in the 2000s to add to the standard approximate functionals a semiempirical dispersion correction of the form [52, 96, 97, 272]

(1.222)

where Rαβ is the distance between each pair of atoms and $$C_{6}^{\alpha \beta }$$ is the London dispersion coefficient between these atoms. Here, f(Rαβ) is a damping function which tends to 1 at large Rαβ and tends to zero at small Rαβ, e.g.

(1.223)

with the sum of tabulated atomic van der Waals radii $$R_{\alpha \beta }^{\text{vdW}} = R_{\alpha }^{\text{vdW}} + R_{\beta }^{\text{vdW}}$$ and a constant d, and s is a scaling parameter that can be adjusted for each approximate functional. The dispersion coefficient $$C_{6}^{\alpha \beta }$$ for any pair of atoms is empirically calculated from tabulated same-atom dispersion coefficients $$C_{6}^{\alpha \alpha }$$ and/or atomic polarizabilities. This approach was named “DFT-D” by Grimme [96].

The last version of DFT-D (referred to as DFT-D3) also includes $$C_8^{\alpha \beta }$$ two-body terms and $$C_9^{\alpha \beta \gamma }$$ three-body terms [99]. There have also been various proposals to make the determination of dispersion coefficients less empirical, such as the scheme of Becke and Johnson [25] based on the exchange-hole dipole moment, the scheme of Tkatchenko and Scheffler [253] based on a Hirshfeld atomic partitioning, or the scheme of Sato and Nakai [214] based on the local-response approximation [45].

The “DFT-D” approach provides a big and inexpensive improvement for the description of weakly bonded systems. One limitation is that the semiempirical dispersion correction, being just a force field in its simplest variant, affects only the molecular geometry of the system but not directly its electronic structure. Some of the most used DFT-D functionals are:

The PBE-D exchange-correlation functional [97], based on the PBE functional with a scaling parameter s = 0.75;

The B97-D exchange-correlation functional [97], based on the B97-GGA functional with a scaling parameter s = 1.25 and reoptimized linear coefficients in Eqs. (1.124), (1.128), and (1.129) in the presence of the semiempirical dispersion correction;

The B3LYP-D exchange-correlation functional [97], based on the B3LYP hybrid functional with a scaling parameter s = 1.05;

The ωB97X-D exchange-correlation functional [33], based on the ωB97X range-separated hybrid functional with a scaling parameter s = 1, a modified damping function, and reoptimized parameters in ωB97X in the presence of the semiempirical dispersion correction.

The semiempirical dispersion correction can also be added to double-hybrid approximations. For example, B2PLYP-D [218] is based on the B2PLYP double hybrid with a scaling parameter s = 0.55. The scaling parameter is small since the fraction of MP2 correlation in B2PLYP already partially takes into account dispersion interactions. It has also been proposed to add a semiempirical dispersion correction to the SCS version of the double hybrids [Eq. (1.196)], resulting in a family of dispersion-corrected spin-component-scaled double-hybrid (DSD) approximations [136–138]. An example of double hybrid is this latter family is DSD-BLYP [136], which uses the B88 exchange functional and the LYP correlation functional.

1.6.2 Nonlocal van der Waals Density Functionals

Another approach to describe dispersion interactions is to add to the standard approximate functionals a so-called nonlocal van der Waals density functional of the form [43, 151, 268–270]

(1.224)

where ϕ(r₁, r₂) is a correlation kernel. Two main families of such nonlocal correlation functionals exist: the “van der Waals density functionals” (vdW-DF) [43, 151] and the Vydrov–Van Voorhis (VV) functionals [268–270].

We will only describe the last version of the VV functionals, i.e. the VV10 nonlocal correlation functional [270]. In this functional, the correlation kernel is taken as

where r₁₂ = |r₂ −r₁| is the interelectronic distance, β is a constant determining the local (delta-distribution) part of the kernel, and the function g is defined as

(1.225)

In Eq. (1.225), $$\omega _0({\mathbf {r}}) = \sqrt {\omega _{\text{g}}({\mathbf {r}})^2 + \frac {\omega _{\text{p}}({\mathbf {r}})^2}{3}}$$ involves the square of the local plasma frequency ωp(r)² = 4πρ(r) and the square of the local band gap ωg(r)² = C|∇ρ(r)|⁴∕ρ(r)⁴, where C is an adjustable parameter controlling the large-r₁₂ asymptotic dispersion coefficients, and κ(r) = bkF(r)²∕ωp(r), where kF(r) = (3π²ρ(r))1∕3 is the local Fermi wave vector and b is an adjustable parameter controlling the short-range damping of the large-r₁₂ asymptote. As expected for dispersion interactions, in the large-r₁₂ limit, ϕVV10(r₁, r₂) behaves as $$1/r_{12}^6$$ :

StartLayout 1st Row upper E 0 Superscript RS 2 upper H plus MP 2 comma mu comma lamda Baseline equals upper E 0 Superscript RS 2 upper H comma mu comma lamda Baseline plus upper E Subscript c Superscript mu comma lamda comma upper M upper P 2 Baseline comma EndLayout

The constant β = (3∕b²)3∕4∕16 is chosen to make $$E^{\text{nl}}_{\text{c}}[\rho ]$$ vanish in the uniform density limit, thus leaving this limit unchanged when $$E^{\text{nl}}_{\text{c}}[\rho ]$$ is added to another density functional. The adjustable parameters C ≈ 0.009 and b ≈ 6 are found by optimization of C₆ dispersion coefficients and of weak intermolecular interaction energies, respectively, the precise values depending on which exchange-correlation functional the VV10 correction is added to.

Nonlocal van der Waals density functionals are necessarily more computationally expensive than semiempirical dispersion corrections. However, they have the advantage of being less empirical and, since they are functionals of the density, of impacting directly on the electronic structure of the system. The VV10 nonlocal functional has been incorporated in a number of recently developed exchange-correlation functionals, for example:

The ωB97X-V exchange-correlation functional [169], based on the ωB97X range-separated hybrid [Eq. (1.179)] with reoptimized linear coefficients in Eq. (1.124) with polynomial degree m = 2 and in Eqs. (1.128) and (1.129) with polynomial degree m = 1, as well as reoptimized VV10 parameters C = 0.01 and b = 6.0;

The ωB97M-V exchange-correlation functional [170], based on the ωB97X range-separated hybrid [Eq. (1.179)] but with more general and combinatorially optimized meta-GGA exchange and correlation enhancement factors and the same VV10 parameters C = 0.01 and b = 6.0 as in ωB97X-V.

1.7 Orbital-Dependent Exchange-Correlation Density Functionals

We discuss here some exchange-correlation density functionals explicitly depending on the KS orbitals (for a review, see Ref. [140]). Since the KS orbitals are themselves functionals of the density, these exchange-correlation expressions are thus implicit functionals of the density (for notational simplicity, this dependence on the density of the orbitals and other intermediate quantities will not be explicitly indicated). In fact, the single-determinant and multideterminant hybrid approximations of Sects. 1.4 and 1.5 already belong to this family, with the caveat that the orbitals are obtained with a nonlocal potential. In this section, we are concerned with orbital-dependent exchange-correlation energy functionals with orbitals obtained with a local potential, i.e. staying within the KS scheme.⁹ These approximations tend to be more computationally involved than the approximations previously seen and have thus been much less used so far.

1.7.1 Exact Exchange

The exact exchange (EXX) energy functional [Eq. (1.16)] can be expressed in terms of the KS orbitals,

(1.226)

and has exactly the same form as the HF exchange [Eq. (1.159)], but the orbitals used in both expressions are in general different.

Since the exact exchange energy in Eq. (1.226) is not an explicit functional of the density, the corresponding exchange potential vx(r) = δEx[ρ]∕δρ(r) cannot be calculated directly. We can however find an workable equation for vx(r) by first considering the functional derivative of Ex[ρ] with respect to the KS potential vs(r) and then applying the chain rule:

(1.227)

Introducing the non-interacting KS static linear-response function χ₀(r′, r) = δρ(r′)∕δvs(r), we can rewrite Eq. (1.227) as

which is known as the optimized-effective-potential (OEP) equation for the exact-exchange potential [91, 92, 248].

Using first-order perturbation theory on the KS system, explicit expressions in terms of the orbitals can be derived for χ₀(r′, r) and δEx[ρ]∕δvs(r). The expression of χ₀(r′, r) is

where i and a run over occupied and virtual spatial orbitals, respectively, and c.c. stands for the complex conjugate. The expression of δEx[ρ]∕δvs(r) is

where 〈φaσφjσ|φjσφiσ〉 are two-electron integrals over KS spatial orbitals:

(1.228)

Applying this OEP method with the EXX energy (and no correlation energy functional) is an old idea [227, 248], but reasonably efficient calculations for molecules have been possible only relatively recently [87, 122]. The EXX occupied orbitals turn out to be very similar to the HF occupied orbitals, and thus the EXX ground-state properties are also similar to the HF ones. However, the EXX virtual orbitals (which see a − 1∕r asymptotic potential for a neutral system) tend to be much less diffuse than the HF virtual orbitals (which see an exponentially decaying potential for a neutral system), and may be more adapted for calculating excited-state properties.

1.7.2 Second-Order Görling–Levy Perturbation Theory

In 1993, Görling and Levy [90, 91] developed a perturbation theory in terms of the coupling constant λ of the adiabatic connection (Sect. 1.2.2) which provides an explicit orbital-dependent second-order approximation for the correlation energy functional. The Hamiltonian along the adiabatic connection [Eq. (1.60)] can be written as

StartLayout 1st Row upper E 0 Superscript RS 2 upper H plus MP 2 comma mu comma lamda Baseline equals upper E 0 Superscript RS 2 upper H comma mu comma lamda Baseline plus upper E Subscript c Superscript mu comma lamda comma upper M upper P 2 Baseline comma EndLayout

(1.229)

where is the KS non-interacting reference Hamiltonian (which will be assumed to have a nondegenerate ground state). Equation (1.229) was obtained by decomposing the potential operator keeping the density constant as , where is the KS potential operator, is the Hartree-exchange potential operator which is linear in λ, and is the correlation potential which starts at second order in λ, i.e. . Using a complete set of orthonormal eigenfunctions Φn and eigenvalues $$\mathcal {E}_n$$ of the KS Hamiltonian, , the normalized ground-state wave function of the Hamiltonian can be expanded as Ψλ =  Φ + λ Ψ⁽¹⁾ + ⋯ where Φ = Φ₀ is the ground-state KS single-determinant wave function and Ψ⁽¹⁾ is its first-order correction given by

StartLayout 1st Row vertical bar upper Psi Superscript left parenthesis 1 right parenthesis Baseline right angle bracket equals minus sigma summation Underscript n not equals 0 Endscripts StartFraction left angle bracket upper Phi Subscript n Baseline vertical bar ModifyingAbove upper W With caret Subscript e e Baseline minus ModifyingAbove upper V With caret Subscript normal upper H normal x Baseline vertical bar upper Phi right angle bracket Over script upper E Subscript n Baseline minus script upper E 0 EndFraction vertical bar upper Phi Subscript n Baseline right angle bracket period EndLayout

Using the expression in Eq. (1.63), the correlation energy functional can also be expanded in powers of λ:

StartLayout 1st Row 1st Column upper E Subscript c Superscript lamda Baseline left bracket rho right bracket 2nd Column equals 3rd Column left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove upper T With caret plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline right angle bracket minus left angle bracket upper Phi vertical bar ModifyingAbove upper T With caret plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi right angle bracket period 2nd Row 1st Column Blank 2nd Column equals 3rd Column upper E Subscript c Superscript left parenthesis 0 right parenthesis Baseline plus lamda upper E Subscript c Superscript left parenthesis 1 right parenthesis Baseline plus lamda squared upper E Subscript c Superscript left parenthesis 2 right parenthesis Baseline plus midline horizontal ellipsis period EndLayout

(1.230)

Since Ψλ=0 =  Φ, the zeroth-order term vanishes: $$E_{{\text{c}}}^{(0)} =0$$ . Using the expression of the first-order derivative of $$E_{{\text{c}}}^{\lambda }$$ with respect to λ in Eq. (1.64), i.e. partial differential upper E Subscript c Superscript lamda Baseline divided by partial differential lamda equals left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline right angle bracket minus left angle bracket upper Phi vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi right angle bracket ), we find that the first-order term vanishes as well: $$E_{{\text{c}}}^{(1)}=0$$ . The second-order term corresponds to the second-order Görling–Levy (GL2) correlation energy and is given by

upper E Subscript c Superscript upper G upper L 2 Baseline left bracket rho right bracket equals upper E Subscript c Superscript left parenthesis 2 right parenthesis Baseline equals left angle bracket upper Phi vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript left parenthesis 1 right parenthesis Baseline right angle bracket equals left angle bracket upper Phi vertical bar ModifyingAbove upper W With caret Subscript e e Baseline minus upper V Subscript normal upper H normal x Baseline vertical bar upper Psi Superscript left parenthesis 1 right parenthesis Baseline right angle bracket comma

(1.231)

where the second equality comes from the fact that left angle bracket upper Phi vertical bar ModifyingAbove upper V With caret Subscript normal upper H normal x Baseline vertical bar upper Psi Superscript left parenthesis 1 right parenthesis Baseline right angle bracket equals 0 ) since it is the derivative with respect to λ at λ = 0 of left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove upper V With caret Subscript normal upper H normal x Baseline vertical bar upper Psi Superscript lamda Baseline right angle bracket equals integral Underscript double struck upper R cubed Endscripts v Subscript normal upper H normal x Baseline left parenthesis bold r right parenthesis rho left parenthesis bold r right parenthesis normal d bold r ), which does not depend on λ by virtue of the fact that the density ρ(r) is constant along the adiabatic connection. Using the last expression in Eq. (1.231) allows one to express the GL2 correlation energy as

upper E Subscript c Superscript upper G upper L 2 Baseline left bracket rho right bracket equals minus sigma summation Underscript n not equals 0 Endscripts StartFraction StartAbsoluteValue left angle bracket upper Phi vertical bar ModifyingAbove upper W With caret Subscript e e Baseline minus ModifyingAbove upper V With caret Subscript normal upper H normal x Baseline vertical bar upper Phi Subscript n Baseline right angle bracket EndAbsoluteValue squared Over script upper E Subscript n Baseline minus script upper E 0 EndFraction period

(1.232)

It is instructive to decompose the GL2 correlation energy as

(1.233)

where $$E_{\text{c}}^{{\mathrm {MP2}}}$$ is a MP2-like correlation energy evaluated with KS spin orbitals,

upper E Subscript c Superscript MP 2 Baseline equals minus one fourth sigma summation Underscript i equals 1 Overscript upper N Endscripts sigma summation Underscript j equals 1 Overscript upper N Endscripts sigma summation Underscript a greater than or slanted equals upper N plus 1 Endscripts sigma summation Underscript b greater than or slanted equals upper N plus 1 Endscripts StartFraction StartAbsoluteValue left angle bracket phi Subscript i Baseline phi Subscript j Baseline vertical bar phi Subscript a Baseline phi Subscript b Baseline right angle bracket EndAbsoluteValue squared Over epsilon Subscript a Baseline plus epsilon Subscript b Baseline minus epsilon Subscript i Baseline minus epsilon Subscript j Baseline EndFraction comma

(1.234)

and $$E_{\text{c}}^{\text{S}}$$ is the contribution coming from the single excitations (which does not vanish here, contrary to HF-based MP2 perturbation theory),

upper E Subscript c Superscript upper S Baseline equals minus sigma summation Underscript i equals 1 Overscript upper N Endscripts sigma summation Underscript a greater than or slanted equals upper N plus 1 Endscripts StartFraction StartAbsoluteValue left angle bracket phi Subscript i Baseline vertical bar ModifyingAbove upper V With caret Subscript normal x Superscript upper H upper F Baseline minus ModifyingAbove upper V With caret Subscript normal x Baseline vertical bar phi Subscript a Baseline right angle bracket EndAbsoluteValue squared Over epsilon Subscript a Baseline minus epsilon Subscript i Baseline EndFraction comma

(1.235)

involving the difference between the integrals over the nonlocal HF exchange potential left angle bracket phi Subscript i Baseline vertical bar ModifyingAbove upper V With caret Subscript normal x Superscript upper H upper F Baseline vertical bar phi Subscript a Baseline right angle bracket equals minus sigma summation Underscript j equals 1 Overscript upper N Endscripts left angle bracket phi Subscript i Baseline phi Subscript j Baseline vertical bar phi Subscript j Baseline phi Subscript a Baseline right angle bracket ) and over the local KS exchange potential left angle bracket phi Subscript i Baseline vertical bar ModifyingAbove upper V With caret Subscript normal x Baseline vertical bar phi Subscript a Baseline right angle bracket equals integral Underscript double struck upper R cubed times StartSet up arrow comma down arrow EndSet Endscripts phi Subscript i Superscript asterisk Baseline left parenthesis bold x right parenthesis v Subscript normal x Baseline left parenthesis bold r right parenthesis phi Subscript a Baseline left parenthesis bold x right parenthesis normal d bold x ).

Calculations of the GL2 correlation energy using either a non-self-consistent post-EXX implementation or a more complicated OEP self-consistent procedure have been tested (see, e.g., Refs. [53, 94, 175]) but the results are often disappointing. It is preferable to go beyond second order with random-phase approximations in the adiabatic-connection fluctuation-dissipation approach.

1.7.3 Adiabatic-Connection Fluctuation-Dissipation Approach

1.7.3.1 Exact Adiabatic-Connection Fluctuation-Dissipation Expression

Using the adiabatic-connection formula of Eq. (1.65), the correlation energy functional can be written as

StartLayout 1st Row 1st Column upper E Subscript c Baseline left bracket rho right bracket 2nd Column equals integral Subscript 0 Superscript 1 Baseline normal d lamda left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline right angle bracket minus left angle bracket upper Phi vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Phi right angle bracket 2nd Row 1st Column Blank 2nd Column equals one half integral Subscript 0 Superscript 1 Baseline normal d lamda integral Underscript double struck upper R cubed times double struck upper R cubed Endscripts StartFraction rho Subscript 2 comma c Superscript lamda Baseline left parenthesis bold r 1 comma bold r 2 right parenthesis Over StartAbsoluteValue bold r 1 minus bold r 2 EndAbsoluteValue EndFraction normal d bold r 1 normal d bold r 2 comma EndLayout

(1.236)

where $$\rho _{2,{\text{c}}}^{\lambda }({\mathbf {r}}_1,{\mathbf {r}}_2) = \rho _{2}^{\lambda }({\mathbf {r}}_1,{\mathbf {r}}_2) - \rho _{2,{\mathrm {KS}}}({\mathbf {r}}_1,{\mathbf {r}}_2)$$ is the correlation part of the pair density along the adiabatic connection. The pair density $$\rho _{2}^{\lambda }({\mathbf {r}}_1,{\mathbf {r}}_2)$$ can be expressed with the pair-density operator where is the density operator,

StartLayout 1st Row 1st Column rho 2 Superscript lamda Baseline left parenthesis bold r 1 comma bold r 2 right parenthesis 2nd Column equals left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove rho With caret Subscript 2 Baseline left parenthesis bold r 1 comma bold r 2 right parenthesis vertical bar upper Psi Superscript lamda Baseline right angle bracket 2nd Row 1st Column Blank 2nd Column equals left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove rho With caret left parenthesis bold r 1 right parenthesis ModifyingAbove rho With caret left parenthesis bold r 2 right parenthesis vertical bar upper Psi Superscript lamda Baseline right angle bracket minus delta left parenthesis bold r 1 minus bold r 2 right parenthesis left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove rho With caret left parenthesis bold r 1 right parenthesis vertical bar upper Psi Superscript lamda Baseline right angle bracket comma EndLayout

and the KS pair density ρ2,KS(r₁, r₂) simply corresponds to the case λ = 0,

StartLayout 1st Row 1st Column rho Subscript 2 comma upper K upper S Baseline left parenthesis bold r 1 comma bold r 2 right parenthesis 2nd Column equals rho 2 Superscript lamda equals 0 Baseline left parenthesis bold r 1 comma bold r 2 right parenthesis 2nd Row 1st Column Blank 2nd Column equals left angle bracket upper Phi vertical bar ModifyingAbove rho With caret left parenthesis bold r 1 right parenthesis ModifyingAbove rho With caret left parenthesis bold r 2 right parenthesis vertical bar upper Phi right angle bracket minus delta left parenthesis bold r 1 minus bold r 2 right parenthesis left angle bracket upper Phi vertical bar ModifyingAbove rho With caret left parenthesis bold r 1 right parenthesis vertical bar upper Phi right angle bracket period EndLayout

Since the density does not change with λ, i.e. left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove rho With caret left parenthesis bold r right parenthesis vertical bar upper Psi Superscript lamda Baseline right angle bracket equals left angle bracket upper Phi vertical bar ModifyingAbove rho With caret left parenthesis bold r right parenthesis vertical bar upper Phi right angle bracket equals rho left parenthesis bold r right parenthesis ), the correlation pair density needed in Eq. (1.236) can thus be expressed as

StartLayout 1st Row rho Subscript 2 comma c Superscript lamda Baseline left parenthesis bold r 1 comma bold r 2 right parenthesis equals left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove rho With caret left parenthesis bold r 1 right parenthesis ModifyingAbove rho With caret left parenthesis bold r 2 right parenthesis vertical bar upper Psi Superscript lamda Baseline right angle bracket minus left angle bracket upper Phi vertical bar ModifyingAbove rho With caret left parenthesis bold r 1 right parenthesis ModifyingAbove rho With caret left parenthesis bold r 2 right parenthesis vertical bar upper Phi right angle bracket period EndLayout

(1.237)

We would like to calculate $$\rho _{2,{\text{c}}}^{\lambda }({\mathbf {r}}_1,{\mathbf {r}}_2)$$ without having to calculate the complicated many-body wave function Ψλ. For this, we consider the retarded dynamic linear-response function along the adiabatic connection in frequency space (or the so-called Lehmann representation)

StartLayout 1st Row chi Subscript lamda Baseline left parenthesis bold r 1 comma bold r 2 semicolon omega right parenthesis 2nd Row equals sigma summation Underscript n not equals 0 Endscripts StartFraction left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove rho With caret left parenthesis bold r 1 right parenthesis vertical bar upper Psi Subscript n Superscript lamda Baseline right angle bracket left angle bracket upper Psi Subscript n Superscript lamda Baseline vertical bar ModifyingAbove rho With caret left parenthesis bold r 2 right parenthesis vertical bar upper Psi Superscript lamda Baseline right angle bracket Over omega minus omega Subscript n Superscript lamda Baseline plus i 0 Superscript plus Baseline EndFraction minus StartFraction left angle bracket upper Psi Superscript lamda Baseline vertical bar ModifyingAbove rho With caret left parenthesis bold r 2 right parenthesis vertical bar upper Psi Subscript n Superscript lamda Baseline right angle bracket left angle bracket upper Psi Subscript n Superscript lamda Baseline vertical bar ModifyingAbove rho With caret left parenthesis bold r 1 right parenthesis vertical bar upper Psi Superscript lamda Baseline right angle bracket Over omega plus omega Subscript n Superscript lamda Baseline plus i 0 Superscript plus Baseline EndFraction comma EndLayout

(1.238)

where the sums are over all eigenstates $$\Psi _n^\lambda $$ of the Hamiltonian , i.e. , except the ground state $$\Psi ^\lambda =\Psi ^\lambda _0$$ , and $$\omega _n^\lambda = E_n^\lambda - E_0^\lambda $$ are the corresponding excitation energies. By contour integrating χλ(r₁, r₂, ω) around the right half ω-complex plane, we arrive at the (zero-temperature) fluctuation-dissipation theorem,

(1.239)

which relates ground-state correlations in the time-independent system $$\rho _{2,{\text{c}}}^{\lambda }({\mathbf {r}}_1,{\mathbf {r}}_2)$$ to the linear response of the system due to a time-dependent external perturbation χλ(r₁, r₂, ω).

Combining Eqs. (1.236) and (1.239), we finally obtain the exact adiabatic-connection fluctuation-dissipation (ACFD) formula for the correlation energy [146, 147] (see, also, Ref. [106]):

(1.240)

The usefulness of the ACFD formula is due to the fact that there are practical ways of directly calculating χλ(r₁, r₂;ω) without having to calculate the many-body wave function Ψλ. In linear-response time-dependent density-functional theory (TDDFT), one can find a Dyson-like equation for χλ(r₁, r₂;ω),

StartLayout 1st Row 1st Column chi Subscript lamda Baseline left parenthesis bold r 1 comma bold r 2 semicolon omega right parenthesis 2nd Column equals chi 0 left parenthesis bold r 1 comma bold r 2 semicolon omega right parenthesis 2nd Row 1st Column Blank 2nd Column plus integral Underscript double struck upper R cubed times double struck upper R cubed Endscripts chi 0 left parenthesis bold r 1 comma bold r 3 semicolon omega right parenthesis f Subscript upper H x c Superscript lamda Baseline left parenthesis bold r 3 comma bold r 4 semicolon omega right parenthesis chi Subscript lamda Baseline left parenthesis bold r 4 comma bold r 2 semicolon omega right parenthesis normal d bold r 3 normal d bold r 4 comma EndLayout

(1.241)

where $$f_{{\mathrm {Hxc}}}^\lambda ({\mathbf {r}}_3,{\mathbf {r}}_4;\omega )$$ is the Hartree-exchange-correlation kernel associated to the Hamiltonian Hλ. Here, Eq. (1.241) will be considered as the definition for $$f_{{\mathrm {Hxc}}}^\lambda $$ . In principle, the exact correlation energy can be obtained with Eqs. (1.240) and (1.241). In practice, however, we need to use an approximation for $$f_{{\mathrm {Hxc}}}^\lambda $$ .

1.7.3.2 Random-Phase Approximations

In the direct random-phase approximation (dRPA, also just referred to as RPA, or sometimes as time-dependent Hartree), only the Hartree part of the kernel, which is linear in λ and independent from ω, is retained [146, 148],

(1.242)

where wee(r₁, r₂) = 1∕|r₁ −r₂| is the Coulomb interaction, and the corresponding dRPA linear-response function then satisfies the equation

StartLayout 1st Row 1st Column chi Subscript lamda Superscript d upper R upper P upper A Baseline left parenthesis bold r 1 comma bold r 2 semicolon omega right parenthesis 2nd Column equals chi 0 left parenthesis bold r 1 comma bold r 2 semicolon omega right parenthesis 2nd Row 1st Column Blank 2nd Column plus lamda integral Underscript double struck upper R cubed times double struck upper R cubed Endscripts chi 0 left parenthesis bold r 1 comma bold r 3 semicolon omega right parenthesis w Subscript e e Baseline left parenthesis bold r 3 comma bold r 4 right parenthesis chi Subscript lamda Superscript d upper R upper P upper A Baseline left parenthesis bold r 4 comma bold r 2 semicolon omega right parenthesis normal d bold r 3 normal d bold r 4 period EndLayout

(1.243)

The physical contents of this approximation can be seen by iterating Eq. (1.243), which generates an infinite series,

which, after plugging it into Eq. (1.240), leads to the dRPA correlation energy as the following perturbation expansion¹⁰

StartLayout 1st Row 1st Column Blank 2nd Column upper E Subscript c Superscript d upper R upper P upper A Baseline left bracket rho right bracket 2nd Row 1st Column Blank 2nd Column equals minus one half integral Subscript 0 Superscript 1 Baseline normal d lamda integral Subscript negative normal infinity Superscript plus normal infinity Baseline StartFraction normal d omega Over 2 pi EndFraction left bracket lamda integral Underscript double struck upper R cubed times double struck upper R cubed times double struck upper R cubed times double struck upper R cubed Endscripts StartFraction chi 0 left parenthesis bold r 1 comma bold r 3 semicolon i omega right parenthesis chi 0 left parenthesis bold r 4 comma bold r 2 semicolon i omega right parenthesis Over StartAbsoluteValue bold r 1 minus bold r 2 EndAbsoluteValue StartAbsoluteValue bold r 3 minus bold r 4 EndAbsoluteValue EndFraction normal d bold r 1 normal d bold r 2 normal d bold r 3 normal d bold r 4 3rd Row 1st Column Blank 2nd Column plus lamda squared integral Underscript double struck upper R cubed times double struck upper R cubed times double struck upper R cubed times double struck upper R cubed times double struck upper R cubed times double struck upper R cubed Endscripts StartFraction chi 0 left parenthesis bold r 1 comma bold r 3 semicolon i omega right parenthesis chi 0 left parenthesis bold r 4 comma bold r 5 semicolon i omega right parenthesis chi 0 left parenthesis bold r 6 comma bold r 2 semicolon i omega right parenthesis Over StartAbsoluteValue bold r 1 minus bold r 2 EndAbsoluteValue StartAbsoluteValue bold r 3 minus bold r 4 EndAbsoluteValue StartAbsoluteValue bold r 5 minus bold r 6 EndAbsoluteValue EndFraction 4th Row 1st Column Blank 2nd Column normal d bold r 1 normal d bold r 2 normal d bold r 3 normal d bold r 4 normal d bold r 5 normal d bold r 6 plus midline horizontal ellipsis right bracket period EndLayout

(1.244)

Using now the Lehmann representation [Eq. (1.238)] of the KS dynamic linear-response function in terms of the KS orbitals and their energies,

StartLayout 1st Row 1st Column chi 0 left parenthesis bold r 1 comma bold r 2 semicolon omega right parenthesis 2nd Column equals sigma summation Underscript sigma element of StartSet up arrow comma down arrow EndSet Endscripts sigma summation Underscript i equals 1 Overscript upper N Subscript sigma Baseline Endscripts sigma summation Underscript a greater than or slanted equals upper N Subscript sigma Baseline plus 1 Endscripts left bracket StartFraction phi Subscript i sigma Superscript asterisk Baseline left parenthesis bold r 1 right parenthesis phi Subscript a sigma Baseline left parenthesis bold r 1 right parenthesis phi Subscript a sigma Superscript asterisk Baseline left parenthesis bold r 2 right parenthesis phi Subscript i sigma Baseline left parenthesis bold r 2 right parenthesis Over omega minus left parenthesis epsilon Subscript a sigma Baseline minus epsilon Subscript i sigma Baseline right parenthesis plus i 0 Superscript plus Baseline EndFraction 2nd Row 1st Column Blank 2nd Column minus StartFraction phi Subscript i sigma Superscript asterisk Baseline left parenthesis bold r 2 right parenthesis phi Subscript a sigma Baseline left parenthesis bold r 2 right parenthesis phi Subscript a sigma Superscript asterisk Baseline left parenthesis bold r 1 right parenthesis phi Subscript i sigma Baseline left parenthesis bold r 1 right parenthesis Over omega plus left parenthesis epsilon Subscript a sigma Baseline minus epsilon Subscript i sigma Baseline right parenthesis plus i 0 Superscript plus Baseline EndFraction right bracket comma 3rd Row 1st Column Blank 2nd Column Blank EndLayout

(1.245)

one can obtain, after quite some work,

(1.246)

The dRPA correlation energy is the sum of all the direct terms (i.e., no exchange terms) of the perturbation expansion up to infinite order. In the language of diagrammatic perturbation theory, we say that the dRPA correlation energy is the sum of all direct ring diagrams. Of course, Eq. (1.246) is not the way to calculate the dRPA correlation energy in practice. This is done by solving the Dyson equation [Eq. (1.243)] without explicitly expanding in powers of λ, e.g. using matrix equations from linear-response TDDFT [72, 261] or coupled-cluster theory [219, 262].

Most dRPA correlation energy (combined with the EXX energy) calculations are done in a non-self-consistent way, but self-consistent OEP dRPA calculations have also been performed [26, 112]. One of the main advantage of dRPA is that it accounts for long-range dispersion interactions [46–48]. However, it shows large self-interaction errors. To overcome the latter drawback and improve the general accuracy, one can add exchange and beyond terms in various ways (see, e.g., Refs. [7, 14, 15, 39, 58, 88, 101, 110, 111, 114–116, 120, 178, 257, 261]). This remains an active area of research. For reviews on random-phase approximations, the reader may consult Refs. [37, 61, 208].

Acknowledgements

I thank Éric Cancès and Gero Friesecke for discussions and comments on the manuscript. This review chapter grew out of my lecture notes for DFT courses given in several summer schools (ISTPC summer schools in June 2015 and 2017 in France, ICS summer school in July 2015 in France, and ESQC summer school in September 2019 in Italy). This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement EMC2 No 810367).

Suggested Further Reading

Books

R. M. Dreizler and E. K. U. Gross, Density Functional Theory. Springer-Verlag, Berlin (1990).E. Engel and R. M. Dreizler. Density Functional Theory: An Advanced Course, Theoretical and Mathematical Physics, Springer-Verlag, Berlin Heidelberg (2011).C. Fiolhais, F. Nogueira and M. A. L. Marques, eds., A Primer in Density Functional Theory, Vol. 620 of Lecture Notes in Physics, Springer, Berlin (2003).R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York (1989).T. Tsuneda, Density Functional Theory in Quantum Chemistry, Springer, Tokyo (2014).

Review Articles

A. D. Becke. Perspective: Fifty years of density-functional theory in chemical physics. J. Chem. Phys. 140, 18A301 (2014).K. Burke. Perspective on density functional theory. J. Chem. Phys. 136, 150901 (2012).K. Capelle. A bird’s-eye view of density-functional theory. Braz. J. Phys. 36, 1318 (2006).A. J. Cohen. P. Mori-Sánchez and W. Yang. Challenges for density functional theory. Chem. Rev. 112, 289–320 (2012).R. O. Jones. Density functional theory: Its origins, rise to prominence, and future. Rev. Mod. Phys. 87, 897 (2015).N. Mardirossian and M. Head-Gordon. Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals. Mol. Phys. 115, 2315–2372 (2017).J. P. Perdew and K. Schmidt, Jacob’s ladder of density functional approximations for the exchange-correlation energy. AIP Conf. Proc. 577, 1 (2001).G.E. Scuseria and V.N. Staroverov. Development of approximate exchange-correlation functionals. In: Theory and Applications of Computational Chemistry: The First 40 years, edited by C.E. Dykstra, G. Frenking, K.S. Kim and G.E. Scuseria, pp. 669–724, Elsevier (2005).V.N. Staroverov. Density-functional approximations for exchange and correlation. In: A Matter of Density: Exploring the Electron Density Concept in the Chemical, Biological, and Materials Sciences, edited by N. Sukumar, pp. 125–156, John Wiley & Sons, Hoboken, NJ (2013).H. S. Yu, S. L. Li and D. G. Truhlar. Perspective: Kohn-Sham density functional theory descending a staircase. J. Chem. Phys. 145, 130901 (2016).

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Footnotes

1

An integration over a spin coordinate σ just means a sum over the two values σ ∈{↑, ↓}, i.e. ∫{↑,↓}dσ =∑σ ∈{↑,↓} and $$\int _{\mathbb {R}^3\times \{ \uparrow ,\downarrow \}} \mathrm{d} {\mathbf {x}} = \sum _{\sigma \in \{ \uparrow ,\downarrow \}} \int _{\mathbb {R}^3} \mathrm{d} {\mathbf {r}}$$ .

2

It is also possible to define the non-interacting kinetic-energy functional analogously to the Levy–Lieb functional in Eq. (1.7) by minimizing over wave functions $$\Psi \in \mathcal {W}^N_\rho $$ , i.e. upper T Subscript normal s comma upper L upper L Baseline left bracket rho right bracket equals min Underscript upper Psi element of script upper W Subscript rho Superscript upper N Baseline Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper T With caret vertical bar upper Psi right angle bracket ) [160]. In this case, the corresponding minimizing KS wave function can generally be a linear combination of Slater determinants. However, we often have Ts,LL[ρ] = Ts[ρ], in particular for densities ρ that come from a non-interacting ground-state wave function which is not degenerate. In this chapter, we will usually assume this nondegeneracy condition.

3

Therefore, the correlation hole is really a “hole” only in some region of space, and a “bump” in other regions.

4

In this context, the Hellmann–Feynman theorem states that in the derivative

StartFraction partial differential upper F Superscript lamda Baseline left bracket rho right bracket Over partial differential lamda EndFraction equals left angle bracket StartFraction partial differential upper Psi Superscript lamda Baseline left bracket rho right bracket Over partial differential lamda EndFraction vertical bar ModifyingAbove upper T With caret plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline left bracket rho right bracket right angle bracket plus left angle bracket upper Psi Superscript lamda Baseline left bracket rho right bracket vertical bar ModifyingAbove upper W With caret Subscript e e Baseline vertical bar upper Psi Superscript lamda Baseline left bracket rho right bracket right angle bracket plus left angle bracket upper Psi Superscript lamda Baseline left bracket rho right bracket vertical bar ModifyingAbove upper T With caret plus lamda ModifyingAbove upper W With caret Subscript e e Baseline vertical bar StartFraction partial differential upper Psi Superscript lamda Baseline left bracket rho right bracket Over partial differential lamda EndFraction right angle bracket

the first and third terms involving the derivative of Ψλ[ρ] vanish. This is due to the fact that Ψλ[ρ] is obtained via the minimization of Eq. (1.61) and thus any variation of Ψλ[ρ] which keeps the density constant (which is the case for a variation with respect to λ) gives a vanishing variation of Fλ[ρ].

5

In the definition of Eq. (1.79) we exclude the point λ = 0 in order to allow for the possibility of a discontinuity in λ there due to a degeneracy.

6

For generality and simplicity, we consider here that the GGAs depend on the spin density gradients ∇ρ↑ and ∇ρ↓, but due to rotational invariance GGAs actually depend only on the scalar quantities (∇ρ↑)², (∇ρ↓)², and ∇ρ↑⋅∇ρ↓.

7

The possibility of combining a nonlocal HF potential with a local correlation potential was mentioned already in 1965 in the paper by Kohn and Sham [135].

8

For a general system, the pair-distribution function g(r₁, r₂) is defined from the pair density ρ₂(r₁, r₂) [Eq. (1.42)] as ρ₂(r₁, r₂) = ρ(r₁)ρ(r₂)g(r₁, r₂). The on-top value is the value at electron coalescence, i.e. for r₁ = r₂.

9

The boundary between the various single-determinant and multideterminant hybrids of Sects. 1.4 and 1.5 and the orbital-dependent functionals of the present section is however thin. For example, it is possible to optimize the orbitals using a local potential in hybrids or range-separated hybrids [8, 131, 229], and in double hybrids or range-separated double hybrids [230, 231].

10

Using the operator viewpoint, the series in Eq. (1.244) can be formally summed in the form (see, e.g., Ref. [178]).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_2

2. On Connecting Density Functional Approximations to Theory

Andreas Savin1  

(1)

Laboratoire de Chimie Théorique, CNRS and Sorbonne University, Paris, France

Andreas Savin

Email: andreas.savin@lct.jussieu.fr

Dedicated to Jean-Paul Malrieu on his 80th birthday

Abstract

Usually, density functional models are considered approximations to density functional theory, However, there is no systematic connection between the two, and this can make us doubt about a linkage. This attitude can be further enforced by the vagueness of the argumentation for using spin densities. Questioning the foundations of density functional models leads to a search for alternative explanations. Seeing them as using models for pair densities is one of them. Another is considering density functional approximations as a way to extrapolate results obtained in a model system to those of a corresponding physical one.

2.1 Introduction

2.1.1 On approximations in DFT

Density functional theory is here. It has changed the way the computation of electronic systems is seen by the scientific community. It has a sound theoretical foundation. However, following exact theory is more complicated than solving the Schrödinger equation. Furthermore, it does not tell us how to systematically produce approximations. Usual approximations are convenient and (to a large degree) successful, but how to improve them?

2.1.2 Excuses

This is not a review. References are erratic and biased. My own publications dominate, not because they are more important, but because they are only given to complement argumentation.

2.1.3 Summary

After giving the notations, and recalling the Hohenberg–Kohn theorem, we review some practical solutions, such as the decomposition of the universal functional, in order to comply with different physical requirements. It is argued that this does not necessarily solve the problem. Refinements, such as using the spin density as a supplementary variable, are discussed. It is argued that the need for these refinements may hide a different foundation for the approximations. In order to introduce a “systematic” way to get closer to the physical Hamiltonian, model Hamiltonians are defined that via an adjustable parameter. It can bring the model Hamiltonian arbitrarily close to the physical Hamiltonian. Finally, examples show that simple mathematical recipes provide a quality similar to that of density functional approximations.

2.2 Schrödinger Equation and Notations

We start with a Schrödinger equation:

(2.1)

The wave function Ψ depends on the coordinates of the electrons r₁, r₂, …, rN and their spins. We will be mainly concerned with ground state eigenvalues, E = E₀. We consider Hamiltonians of the form

(2.2)

where T is the operator for the kinetic energy,

(2.3)

and V  is a local one-particle potential

(2.4)

The density operator, $$\widehat {\rho }(\mathbf {r})$$ , can be written using Dirac’s δ function,

(2.5)

Its expectation value is the density

rho left parenthesis bold r right parenthesis equals left angle bracket upper Psi vertical bar ModifyingAbove rho With caret left parenthesis bold r right parenthesis vertical bar upper Psi right angle bracket period

(2.6)

Notice that it integrates to N,

(2.7)

W is a two-particle local potential,

(2.8)

(2.9)

where $$\widehat {P}$$ is the pair density operator,

(2.10)

The pair density is

upper P left parenthesis bold r comma bold r Superscript prime Baseline right parenthesis equals left angle bracket upper Psi vertical bar ModifyingAbove upper P With caret left parenthesis bold r comma bold r Superscript prime Baseline right parenthesis vertical bar upper Psi right angle bracket period

(2.11)

As the interaction depends only on the distance between particles, often the dependence of P on r is reduced to that on u = |r −r′|, using instead of $$\widehat {P}$$ the spherically averaged operator

(2.12)

yielding

upper P Subscript s p h Baseline left parenthesis bold r comma u right parenthesis equals left angle bracket upper Psi vertical bar ModifyingAbove upper P With caret Subscript s p h Baseline left parenthesis bold r comma u right parenthesis vertical bar upper Psi right angle bracket period

(2.13)

Going one step further, one can also integrate over r, to obtain the system-average

upper P Subscript s y s Baseline left parenthesis u right parenthesis equals integral Underscript double struck upper R cubed Endscripts left angle bracket upper Psi vertical bar ModifyingAbove upper P With caret Subscript s p h Baseline left parenthesis bold r comma u right parenthesis vertical bar upper Psi right angle bracket normal d bold r period

(2.14)

For the electronic systems, V = Vne, or v = vne, describes the Coulomb interaction between the nuclei and the electrons,

(2.15)

A is an index for the nuclei, ZA their nuclear charge, and RA their position. Also, W = Vee, or w = vee, describes the Coulomb interaction between electrons

(2.16)

To characterize a given electronic system, one has only to specify N and vne.

Model systems are considered below, where v≠vne and w≠vee. Of course, in this case the energies and wave functions depend also on the choice of v and w. No change of the non-local one-particle operator T is considered in this chapter, but such modifications can be found in the literature (see, e.g., [10] for a density functional context). The ground state energy can also be obtained using the variational principle,

upper E left bracket v comma w comma upper N right bracket equals min Underscript upper Psi Endscripts left angle bracket upper Psi vertical bar upper H vertical bar upper Psi right angle bracket equals min Underscript upper Psi Endscripts left angle bracket upper Psi vertical bar upper T plus upper V plus upper W vertical bar upper Psi right angle bracket period

(2.17)

2.3 The Density Functional Viewpoint

2.3.1 The Hohenberg–Kohn Theorem

In order to see how density functional theory can be useful, one generally argues using the Hohenberg–Kohn theorem [12] (cf. Chap. 3).

One of its formulations: “ρ yields vne and N, and thus everything” is useless, as we do not need to know the density to know the potential of the system under study. This formulation of the theorem is never used in practice.

However, the variational formulation of the Hohenberg–Kohn theorem is consequential. It states that

(2.18)

where for F one uses either the Legendre transform form [18],

(2.19)

or, equivalently, a constrained search for ensembles. For the sake of simplicity, in this chapter its pure state form [16, 18, 20]

upper F left bracket rho comma w right bracket equals min Underscript upper Psi right arrow rho Endscripts left angle bracket upper Psi vertical bar upper T plus upper W vertical bar upper Psi right angle bracket

(2.20)

is used. As F does not depend on v (which specifies the system) the functional is called universal. The dependence on N appears through that of ρ (Eq. (2.7)). As above for the energy functional, the dependence on the operator T is not included in the notation for F. For the physical system, w = vee is implicitly assumed; we write:

(2.21)

The hope raised by Eq. (2.18) is that it can be used with some simple approximation for F[ρ].

2.3.2 Difficulty of Producing F[ρ]

Obtaining F for a given ρ(r) is possible, but still difficult: a constrained minimization, as required by Eq. (2.20), is more demanding than a minimization with the single constraint of normalizing the wave function, Eq. (2.17). The Legendre transformed form of F, Eq. (2.19), first requires us to compute E for all v, but then no F[ρ] is needed.

Up to now, there has been no systematic way to construct approximations for F[ρ]. In practice, F is replaced by some model, $$\widetilde {F}$$ : one speaks about a density functional approximation (DFA).

Note that when using Eq. (2.19), due to the variational principle, the errors will be of second order in F for first-order errors in v. Stated differently: there are many v that give values of F that are close. For example, adding to the potential a very rapidly oscillating function yields essentially the same value F. (For this, and other examples, see, e.g., [31, 32].) Again, it appears to be of little practical importance to follow the line ρgivesvand thus everything. However, obtaining E₀ from Eq. (2.18) is not necessarily affected by this problem once F is known or can be approximated. One can even wonder if the existence of many density functional approximations with a similar quality may be due to the indifference of F to changes in the approximation of an optimizing v in Eq. (2.19).

2.4 Practical Solutions for Density Functional Approximations

In order to create models, two main lines have emerged within density functional theory.

1.

Using a simple ansatz for the density functional.

2.

Considering DFT as an inspiration to develop other methods that do not require an explicit construction of a density functional.

The first approach is a cutting the Gordian knot type of solution. The second approach is in line with methods developed for wave functions, sometimes nothing but such a method.

2.4.1 Ansatz

2.4.1.1 Choice of the Ansatz

Most DFAs start with the so-called local density approximation (LDA). Within this model, a general functional G[ρ] is replaced by the ansatz

(2.22)

The function g has to be defined in some way. Traditionally, it is fixed in the uniform electron gas, a system with an infinite number of particles, and where ρ does not depend on the position (cf. Chap. 3). Typically g is either obtained analytically as a function of ρ, or computed for a series of values of ρ, and fitted to them satisfying asymptotic conditions.

LDA has the important advantage of being (to a certain extent) size-consistent, i.e., satisfying

(2.23)

where EA…B is the system composed of two parts, A and B, at infinite separation, while EA and EB are the energies of these parts computed independently. For the violations of size-consistency by LDA, see, e.g., [21, 27]. Another, major, advantage is its computational simplicity (just a numerical integration to obtain $$\widetilde {G}$$ ), and its linear scaling with system size. Both result from the local character of g: if ρ can be decomposed into contributions from two spatial parts,

(2.24)

then so can g; $$\widetilde {G}$$ becomes the sum of the two contributions.

LDA can be extended by making g depend on other local quantities such as derivatives of the density, giving generalized gradient approximations (GGAs), etc. (cf. Chap. 1).

2.4.1.2 Finding the Right Functional to Approximate by Partitioning

Applying the LDA, Eq. (2.22), to F[ρ], Eq. (2.21), does not provide the accuracy needed in most electronic structure calculations. The strategy chosen is to define some density functional Fd[ρ], and approximate only the remaining part, $$\bar {F}_d[\rho ]=F[\rho ]-F_d[\rho ]$$ .

In the following, some choices for the partitioning of F will be presented.

2.4.1.3 Satisfying Electrostatics

In the classical limit, the electrostatic interaction is given by the nuclear repulsion,

the interaction between the electron cloud and the positive charges of the nuclei,

and the repulsion inside the electron clouds, the Hartree energy,

(2.25)

There is a balance between these contributions. For example, between distant neutral atoms these compensate (there is no |RA −RB|⁻¹ term in the limit $$\vert {\mathbf {R}}_A - {\mathbf {R}}_B \vert ^{-1} \rightarrow \infty $$ ). This balance is destroyed if EH is approximated, for example, by using LDA, Eq. (2.22). An excess or deficit of repulsion produces an unphysical repulsion, or attraction of neutral atoms. Furthermore, even if this balance is enforced by parametrization for a given system, it is not kept for another, even closely related system (see, e.g., [29]). The solution to this problem was already proposed in the original Hohenberg–Kohn paper [12]: EH is treated exactly, and only the remaining part approximated.

Finding good models for F[ρ] − EH[ρ] is still an active field of research; there are already approximations that work well for classes of systems, but we do not have yet a universally applicable model.

2.4.1.4 The Kohn–Sham Method: Imposing the Pauli Principle

The Pauli principle is hidden in the wave function used to define F[ρ], Eq. (2.21). A way to impose it is to use a model system, with F[ρ, w≠vee], where the Pauli principle is imposed, and use approximations for the remaining part,

upper E 0 equals min Underscript upper Psi Endscripts left parenthesis left angle bracket upper Psi vertical bar upper T plus upper V Subscript n e Baseline plus upper W vertical bar upper Psi right angle bracket plus ModifyingAbove upper E With bar Subscript upper H x c Baseline left bracket rho Subscript upper Psi Baseline comma w right bracket right parenthesis comma

(2.26)

where the subscript Ψ indicates that ρ is obtained from this wave function, and

(2.27)

This expression is derived using Eqs. (2.17), (2.18), and (2.20). In general, one takes into account the remark made above about EH[ρ], and defines

(2.28)

The remaining part,

(2.29)

is called the exchange-correlation energy.

With Eq. (2.26) one is back to an unconstrained variation of a wave function that is chosen to be anti-symmetric, thus satisfying the Pauli principle.

The problem is made simpler by a proper choice of w. For the Kohn–Sham model, one chooses the simplest one, namely w = 0.

The Kohn–Sham model is usually presented as a modified Schrödinger equation that is obtained by the variation of Ψ in Eq. (2.26),

(2.30)

where

(2.31)

(2.32)

(2.33)

Observe that this step (with the extra problem of the existence of the functional derivative) is not needed to obtain E₀. Furthermore, $$\mathcal {E}(w)=E[v_{\mathrm {ne}}+v_{\mathrm {Hxc}},w,N]$$ , so that

(2.34)

where ρ₀ is a minimizing density.¹

2.4.1.5 Using the Model Wave Function

One can also use the minimizing model wave function, Ψ(w), and choose to approximate the correlation density functional

ModifyingAbove upper E With bar Subscript normal c Baseline left bracket rho comma w right bracket equals upper F left bracket rho comma w right bracket minus left angle bracket upper Psi left parenthesis w right parenthesis vertical bar upper T plus upper W vertical bar upper Psi left parenthesis w right parenthesis right angle bracket period

(2.35)

2.4.1.6 Problems that Remain After Splitting F

Separating F into a defined part, Fd, and a remainder to be approximated, $$\bar {F}_d$$ , does not necessarily guarantee that an approximation, like that given in Eq. (2.22), works better.

Separating the Hartree part, EH, analogously to what was done in Eq. (2.29), removes a problem, but introduces a new one. Take the limiting case of one-electron systems. There is no contribution of the interaction between electrons: EHxc = 0. Thus, calculating the Hartree part exactly means that the remaining part has to cancel EH exactly. But obtaining approximations for − EH is as difficult as obtaining them for EH, and this was considered not to be reachable with approximations of LDA-type. This is known as the self-interaction problem.

Another (not unrelated) problem is due to degeneracy. For example, this appears when we consider two parts of the system far apart, and this even in the simplest molecules like H₂, or H $$_2^+$$ when they are stretched (the internuclear distance goes to infinity). Then, something related to the Einstein–Podolsky–Rosen effect shows up: an infinitesimal perturbation can produce a drastic change in the wave function, the density, etc., but not in the energy. Unfortunately, this conflicts with the general philosophy of constructing DFAs that are aimed to produce significant changes in the energies for small changes in the density.

One could imagine detecting degeneracy. However, the model systems, in particular the mean-field models (such as Kohn–Sham), do not necessarily have ground states presenting the same degeneracy as the physical system: while one can present some degeneracy, the other may not. While the physical wave functions have the symmetry of the Hamiltonian, the model wave function often breaks symmetry to reduce the energy. (Well-known is the breaking of spin symmetry which shows up when bonds are stretched.) The opposite can occur, too: the Kohn–Sham system can produce degeneracy, but not when Coulomb interaction is present (see, e.g., Fig. 11 in [32]).

Even more difficult is the case of near-degeneracy, i.e., when a small change in the parameters characterizing H can produce degeneracy. In this case, detecting degeneracy is not a trivial problem, and it exists in many-body calculations, too. The standard approach in such situations is to stop using a single Slater determinant as a reference.

2.4.1.7 Problems of the Model Systems

By construction, the minimizing model Ψ(w) gives an exact ground state density. Some other properties can be reproduced, too. Trivially, all the expectation values of local one-particle operators can be found, as they need only the density to compute them. Surprisingly at first, the exact ionization potential can also be obtained. However, this can be easily understood, as it can be related to the asymptotic decay of the density (see, e.g., [6, 17]).

Often, quantities that are not proven to be reproduced exactly by the model system are nevertheless expected to be good approximations. However, there is a danger of over-stretching this analogy. For example, it is fashionable to judge DFAs by their ability to reproduce fundamental gaps (differences between the ionization potentials and the electron affinities) from differences between orbital energies (of the lowest unoccupied and highest occupied ones). However, this is wrong [22, 34]. Let us consider, for example, a system with zero electron affinity. For a neutral system the Kohn–Sham potential, vne + vHxc, Eq. (2.31), decays at large distances as − 1∕r (see, e.g.,[17]), and we know that it supports Rydberg series. Thus, its gap (ionization potential) is necessarily larger than its first excitation energy. In fact, accurate Kohn–Sham orbital energy differences give good approximations to excitation energies. Let us take the He atom as an example [33]. An extremely accurate Kohn–Sham potential can be obtained from an extremely accurate density. The Kohn–Sham one-particle Hamiltonian lowest eigenvalue corresponds to the doubly occupied state (1s). However, higher eigenvalues exist. The next eigenvalue (2s) is ≈ 0.75 hartree above the lowest one. It can be compared to the excitation energies of the triplet and singlet (≈ 0.73, and 0.76 hartree, respectively). However, the fundamental gap of the He atom is of ≈ 0.90 hartree. (This comparison should not to be confused with potentials produced by DFAs, as LDA for Exc that generates a potential that does not support excited states, and has a ionization potential of ≈ 0.55 hartree.) Thus, in general, a DFA that produces an orbital energy difference that reproduces the exact fundamental gap can be expected not to be a good approximation to the exact Kohn–Sham system.

The preceding discussion leads to slippery ground. Could it be that the Kohn–Sham approximations are used because they produce convenient mean-field models? Could it be that (for specific purposes) they may be better than the exact Kohn–Sham system would be?

2.4.2 Refining the Approximations

2.4.2.1 Spin Densities

The quality of approximations improves considerably when spin densities are introduced, i.e., when the functional $$\widetilde {G}$$ is made to depend not only on ρ, but on its components, the spin-up, ρ↑(r), and the spin-down ρ↓(r) densities,

Equivalently, one may add to the dependence on ρ that on the spin polarization

(2.36)

A justification is brought by the fact that the exchange term acts only for electrons of the same spin, and that correlation is not the same for a pair of electrons of different spins as for that between two electrons of the same spin (which are kept apart by the Pauli principle).

An example of the importance of making the functional depend on ρ↑ and ρ↓ is shown in Fig. 2.1. According to the Hohenberg–Kohn theorem, neither the energy nor the value of F for the hydrogen atom should depend on ζ. However, for LDA where a dependence on ζ is introduced by adjusting the exchange-correlation of the spin-polarized uniform electron gas, there is a clear dependence on ζ, the best value being obtained when ζ = ±1, i.e., for maximal spin polarization.

A line graph of functional F for spin-up and spin-down densities versus spin polarization Zeta. The graph plots a downward parabola of around (0.52, negative 1.0), (0.56, 0), and (0.52, 1.0). A dotted line is at F equals 0.50.

Fig. 2.1

Dependence of the local density approximation of the functional F on the spin polarization ζ, Eq. (2.36) for the exact density of the hydrogen atom; the exact value of F is 0.5 hartree (dotted line)

In spite of contributing in a decisive way to the success of DFAs (most achievements in thermo-chemistry would be non-existent without using spin-densities), there is a problem: the theoretical foundation of this approach has never been established. There are several arguments in support of this affirmation. Here we give a few. One hears that the spin-density shows up in a weak magnetic field, and wrongly assumes that

1.

a weak magnetic field should not affect the result,

2.

a linear magnetic field should be sufficient, because the field is weak,

3.

it is sufficient to take into account the interaction between the magnetic field and the spins (i.e., only a term BzSz),

4.

the magnetic fields used for spin-polarized systems are weak.

The first point is wrong, because lifting degeneracy by a magnetic field can produce a different ground state. For example, putting a stretched H₂ molecule in a weak uniform magnetic field produces a triplet ground state, while in the absence of a magnetic field, it is a singlet. The second point is wrong, because it ignores a general problem: “a small perturbation parameter does not mean a small perturbation” [24]. For the specific case we are considering, we notice that even a weak linear magnetic field stabilizes states with high angular momentum below the ground state in the absence of the magnetic field. The variational principle cannot be applied, and the Hohenberg–Kohn theorem cannot be proven [28]. The third point is wrong, as we know from the elementary treatment of the Zeeman effect: the orbital momentum is as important as the spin, but if we introduce a dependence on it, we have a dependence on the external potential, and this is not allowed for a universal density functional. Finally, the fourth point is wrong, because in order to produce a spin-polarized electron gas (for densities of chemical interest, ρ ≈ 3∕4π, i.e., rs = 1) a strong electronic excitation is needed, and this can be produced only by a huge magnetic field (see Fig. 2.2).

A line graph of the strength of magnetic field B versus polarization zeta. The graph plots a positive growth slope with r subscript s equals 1. A horizontal dashed line at B subscript z equals zero.

Fig. 2.2

Strength of the magnetic field, B, needed to stabilize the uniform electron gas with polarization ζ with respect to the unpolarized electron gas with density ρ = 3∕4π, i.e., rs = 1. The strongest magnetic field ever produced on earth is indicated by a horizontal dotted line

There is, however, a different viewpoint: the spin-density is a proxy for another quantity that can be related to the spin-density. It was noticed long ago, for unrestricted Hartree–Fock calculations [38], that spin-densities can be connected to on-top pair density, P(r, r), cf. Eq. (2.11). Starting from

an alternative interpretation of the spin-density in DFT is obtained [2, 23]:

(2.37)

Could it be that the theory behind DFAs is not DFT?

Let us mention that a relationship can also be found between spin-densities and first-order reduced density matrices ([35, 36]).

2.4.2.2 The Adiabatic Connection

The adiabatic connection was invoked in order to understand what a density approximation should do [9, 11, 15, 39]. The basic idea is that one constructs a model Hamiltonian depending continuously on a parameter, H(μ). The corresponding Schrödinger equation has an eigenvalue E(μ) and an eigenfunction Ψ(μ). We require that for a certain value of this parameter the model Hamiltonian becomes the physical one. Let us now choose μ = ∞ for it. Furthermore, we assume that the Hellmann–Feynman theorem (or first-order perturbation theory) can be applied to this model:

StartFraction normal d Over normal d mu EndFraction upper E left parenthesis mu right parenthesis equals left angle bracket upper Psi left parenthesis mu right parenthesis vertical bar partial differential Subscript mu Baseline upper H left parenthesis mu right parenthesis vertical bar upper Psi left parenthesis mu right parenthesis right angle bracket period

(2.38)

Suppose that the model system, say at μ₀, is accessible (for example, it is a Kohn–Sham calculation). We want to know how to correct the model energy, E(μ₀), to obtain E = E(μ = ∞). For the missing part, we use the notation, $$\bar {E}(\mu _0)$$ :

(2.39)

By integrating Eq. (2.38) we get what is also called the integrated Hellmann–Feynman formula [4]

ModifyingAbove upper E With bar left parenthesis mu 0 right parenthesis equals upper E minus upper E left parenthesis mu 0 right parenthesis equals integral Subscript mu 0 Superscript normal infinity Baseline left angle bracket upper Psi left parenthesis mu right parenthesis vertical bar partial differential Subscript mu Baseline upper H left parenthesis mu right parenthesis vertical bar upper Psi left parenthesis mu right parenthesis right angle bracket normal d mu period

(2.40)

If we consider (as above) that the model only changes V  and W, we also write

upper E minus upper E left parenthesis mu 0 right parenthesis equals integral Subscript mu 0 Superscript normal infinity Baseline left angle bracket upper Psi left parenthesis mu right parenthesis vertical bar partial differential Subscript mu Baseline left parenthesis upper V left parenthesis mu right parenthesis plus upper W left parenthesis mu right parenthesis right parenthesis vertical bar upper Psi left parenthesis mu right parenthesis right angle bracket normal d mu period

(2.41)

In density functional theory, one furthermore assumes that one can choose V (μ) such that the density does not change with μ. Using Eqs. (2.4) and (2.6), and the convention used here that $$v \rightarrow v_{\mathrm {ne}}$$ when $$\mu \rightarrow \infty $$ , we can write

StartLayout 1st Row upper E minus left angle bracket upper Psi left parenthesis mu 0 right parenthesis vertical bar upper T plus upper W left parenthesis mu 0 right parenthesis plus upper V Subscript n e Baseline vertical bar upper Psi left parenthesis mu 0 right parenthesis right angle bracket equals integral Subscript mu 0 Superscript normal infinity Baseline left angle bracket upper Psi left parenthesis mu right parenthesis vertical bar partial differential Subscript mu Baseline upper W left parenthesis mu right parenthesis vertical bar upper Psi left parenthesis mu right parenthesis right angle bracket normal d mu period EndLayout

(2.42)

Using a relationship analogous to Eqs. (2.9), and (2.11),

StartLayout 1st Row left angle bracket upper Psi left parenthesis mu right parenthesis vertical bar partial differential Subscript mu Baseline upper W left parenthesis mu right parenthesis vertical bar upper Psi left parenthesis mu right parenthesis right angle bracket equals one half integral Underscript double struck upper R cubed Endscripts integral Underscript double struck upper R cubed Endscripts upper P Subscript mu Baseline left parenthesis bold r 1 comma bold r 2 comma mu right parenthesis partial differential Subscript mu Baseline w left parenthesis StartAbsoluteValue bold r 1 minus bold r 2 EndAbsoluteValue comma mu right parenthesis normal d bold r 1 normal d bold r 2 period EndLayout

(2.43)

Notice that as Ψ depends on μ, so does P. A comparison with Eq. (2.26) (where the dependence on w is replaced by that on μ) gives the correction to E(μ₀):

(2.44)

The integrand e(r₁) shows a superficial similarity with the function g appearing in LDA, Eq. (2.22). However, unlike LDA, the connection with ρ is not evident.

One can eliminate a known term from $$\bar {E}_{\mathrm {Hxc}}$$ , and correct correspondingly the r.h.s. For example, if we would like to have $$\bar {E}_{\mathrm {xc}}$$ , Eq. (2.29), we eliminate the contribution of $$\bar {E}_H$$ , by taking the derivative w.r.t. μ in Eq. (2.28), i.e., by subtracting ρ(r)ρ(r′) from P on the r.h.s. of Eq. (2.44).

2.4.2.3 Density or Pair-Density Functional Theory?

Starting from Eq. (2.44), we may ask whether we should construct functionals of the pair density, P(r, r′), instead of one that depends on ρ(r). We first notice that the pair density, P(r, r′), yields, by integration over r′, the density ρ(r), up to a factor N − 1. The already mentioned relationship between spin-densities and the on-top pair density, Eq. (2.37), presents itself as a further argument. However, the conditions to be imposed on P such that it is fermionic are difficult, while those to be imposed on ρ are simple (ρ should be non-negative, and integrate to N).

In fact, LDA can be seen as replacing, in each point of space r₁, P(r₁, r₂) in Eq. (2.44) by that obtained in the uniform electron gas with density ρ(r₁) (see, e.g., [9]). This idea can be extended beyond LDA: many successful functionals have been constructed starting from this perspective (among them those developed by A.D. Becke, and by J.P. Perdew and co-workers, see, e.g., [1]).

Some people consider the random phase approximation (RPA) as a density functional model. It can also be seen as constructing a simplified form of P to be used in Eq. (2.44) (see, e.g., [5]).

Recently, new approximations using the pair density showed up (see, e.g., [37]).

2.4.3 Approaching the Exact Result

2.4.3.1 Limitations of the Mean Field Model

Even if by miracle we had the exact Kohn–Sham determinant (and potential), we would still miss information about the exact system (with Coulomb interaction). For example, we still would not have the exact energy. Unfortunately, the task of obtaining simple functionals capable of dealing with cases when a single Slater determinant is not a good approximation is not solved.

Sometimes ensembles of Kohn–Sham states are discussed. A formula expressing the correlation energy in terms of weighted Kohn–Sham orbital energies exists [25]. However, we do not know a simple expression for obtaining the weights, and it does not seem that they follow a Boltzmann distribution [30].

Long experience in quantum chemistry shows that a single Slater determinant is often a bad starting point for obtaining many properties such as the energy. There, it seems natural to consider multi-reference methods, i.e., wave functions where more than one determinant deserves a preferential treatment. The selection of determinants is an art, unless selective methods are used, such as CIPSI (configuration interaction by perturbation with multiconfigurational zeroth-order wave function selected by iterative process) [13]. In the following, a special way of generating a multi-determinant wave function will be discussed, namely using some (ideally) weak interaction operator W. Degenerate (and near-degenerate) states are detected by such operators, and this automatically introduces more than one Slater determinant if needed. Using more complicated wave functions is a price to pay for getting forms that make existing DFAs closer to a theoretically justifiable form.

2.4.3.2 Choosing w

Equation (2.44) suggests that it may be more easy to obtain approximations for EHxc when w≠0, i.e., μ > 0. Indeed, if ∂μw is short-ranged, we can use some approximation of P(r, r′) that is valid only when r′ is close to r, and use an expansion around r. In the limit of zero-range (δ-function) we obtain the on-top pair density P(r, r) that for a single Slater determinant produces a connection to the spin-density (Eq. (2.37)), i.e., a form that resembles LDA with spin-dependence. Furthermore, expanding P in r′ around r produces semi-local terms such as density derivatives [7]. Finally, we can expect a better transferability between systems when electrons are close, justifying the transferability from other systems like the uniform electron gas, in other words, expecting “universality”.

Also, it seems advantageous to avoid using w that posses a singularity (like the Coulomb interaction), because this induces a strong dependence on the basis set used, and hence a very slow convergence to the exact results (cf. the difficulty of converging left angle bracket upper Psi vertical bar delta left parenthesis r 12 right parenthesis vertical bar upper Psi right angle bracket ) with a finite basis set [3]).

A simple and computationally convenient form for w satisfying the requirements above is given by

(2.45)

Its derivative is short-ranged,

(2.46)

and, when μ is very large

partial differential Subscript mu Baseline w left parenthesis bold r comma mu right parenthesis right arrow StartFraction 2 pi Over mu cubed EndFraction delta left parenthesis bold r right parenthesis comma f o r mu right arrow normal infinity period

(2.47)

The interaction in Eq. (2.45) also has the properties:

$$w \rightarrow v_{\mathrm {ee}}$$ when $$\mu \rightarrow \infty $$ ,

w = 0 when μ = 0

i.e., by changing μ it is possible to switch between the Kohn–Sham and the physical system. This allows us to consider this method to be systematically improvable, in the sense that increasing μ brings the model closer to the physical Hamiltonian.

However, we do not know how far we have to get away from w = 0 to get reliable approximations. This can be explored numerically.

2.4.3.3 Errors of DFAs for w > 0

Below are results obtained with w given by Eq. (2.45) and the dependence on μ is analyzed.

First, to construct a density functional approximations to $$\bar {E}_{\mathrm {xc}}$$ , Eq. (2.29), uniform electron gas calculations are used [19, 26]. Now, the LDA, Eq. (2.22), can be applied to $$\widetilde {G}=\bar {E}_{\mathrm {xc}}$$ for any value of μ.

The numerical results given below are for the 2-electron harmonium, a system with the Hamiltonian

(2.48)

The variables can be separated, and the solutions can be found for real values of μ and ω by solving numerically a one-dimensional differential equation (see, e.g., [14]). For ω = 1∕2, which is chosen below, analytical solutions are known for the non-interacting (μ = 0) and the fully interacting (μ = ∞) system.

Figure 2.3 shows the errors made for the harmonium as a function of the choice of the parameter μ. At μ = 0, the error is that given by the usual LDA. It decreases steadily, and around μ = 0.5…1 a change of behavior occurs, quickly reaching chemical accuracy (1 kcal/mol ≈ 2 mhartree).

A line graph of delta E versus mu. It plots a negative exponential curve of around (0.025, 0), (0.002, 0.6), and (0.000, 3.0). A dashed line is at delta E equals 0.002.

Fig. 2.3

Errors made by the local density approximation for the ground state energy of harmonium (Eq. (2.48)) as a function of the range separation parameter of the model, μ (Eq. (2.45))

As having w≠0 requires having more than one Slater determinant, the time required for computing the wave function rapidly increases with μ. However, as the convergence with the basis set is faster if w has no singularity, less computational effort is needed to obtain the wave function. Figure 2.4 shows the error that can be achieved in a given time. Calculations were done first for spherically symmetric basis functions to saturation (s-limit). Next a new value was obtained for the p-limit (l = 1), next for the d-limit, (l = 2), etc. For such a small system, there is no gain in computing the integrals. However, one can see that a high accuracy is reached much faster when μ = 1 than when μ = ∞.

A line graph of delta E versus C P U time at omega equals 1 over 2. Delta E drops from about (0.04, 0.4), (0.0008, 0.4) through (0, 1) at mu equals one. Mu equals infinite line is from (0.08, 0.4), then (0.004, 0.4), (0.001, 1) to (0.0008, 9.8).

Fig. 2.4

Harmonium energy errors obtained by saturating the basis set with l = 0, 1, 2, 3, in a calculation with μ = 1, blue, and for the Coulomb interaction, red

For this system, choosing a value of μ between 0.5 and 1 seems to provide a good compromise between the supplementary effort needed to have w≠0, and having a good density functional approximation.

2.4.3.4 Approaching the Exact Result Analytically

Instead of using universal models for P in Eq. (2.44), one can construct corrections for a given model Hamiltonian whose energies are determined by some v and w, $$\bar {E}=E[v_{\mathrm {ne}},v_{\mathrm {ee}},N]-E[v,w,N]$$ , Eq. (2.39). The role of the approximation is to correct for the difference between the energy of the exact system and that of the model system. We explore whether standard techniques from numerical analysis could compete with density functional approximations in estimating these corrections.

Notice that as $$v \rightarrow v_{\mathrm {ne}}$$ and $$w \rightarrow v_{\mathrm {ee}}$$ , the correction vanishes: $$\bar {E} \rightarrow 0$$ . One can also try to improve the result by using a set of model Hamiltonians for which obtaining the model energy is simpler than finding E[vne, vee, N].

In a density functional context it is tempting to use v as given by some density functional approximation, or even to use the potential that yields the exact density ρ (to show the principle of the procedure). Below the simplest expression for the external potential is chosen, v = vne. Of course, this brings the model system very far from the physical system when the interaction w is weak: the errors of the model at w = 0 are a very important part of the total energy. For example, for the harmonium studied above, at μ = 0 the error is of 0.5 hartree, as shown in Fig. 2.5.

A multi-line graph of delta E versus mu. 4 lines for order x equals mu superscript negative 2 and order x equals mu superscript negative 2 plus k times mu superscript negative 3 are plotted. All lines have the same trend of a positive exponential curve. The upper curves are dragged at the peaks. 2 dashed lines are marked at delta E equals positive and negative 0.0016, approximately.

Fig. 2.5

Errors of different approximations for the ground state energy of harmonium: E(μ) (thin, black), and the Taylor series around μ to order 1 (blue), and to order 2 (red); the dashed curves correspond to a transformation to x(μ) = μ⁻², the others to x(μ) = μ⁻² + κmu⁻³. The horizontal dotted lines indicate chemical accuracy (± 1 kcal/mol). The inset shows a zoom on the same curves

First, we analyze how the energy of the model system, E(μ), approaches that of the Coulomb system, i.e., how E(μ) approaches E(μ = ∞). From the large μ behavior of the interaction w of Eqs. (2.45) and (2.47), we derive

(2.49)

The coefficient a−k in the equation above is proportional to

The coefficient a−k−1 is proportional to a−k, and given by the cusp condition, as Ψ(μ) has to approach Ψ =  Ψ(μ = ∞) when μ gets large [8], and k is equal to 2l + 2, 2l being the power of the expansion of P(r, r + u) in |u| around zero. In particular, for a pair of singlet coupled electrons (of anti-parallel spin), we have k = 2 and

(2.50)

yielding

2.4.4 Taylor Series Truncation Error

We consider a Taylor series for large μ. First, we make a change of variable to x(μ) such that x monotonously approaches 0 as $$\mu \rightarrow \infty $$ ,

(2.51)

Using the chain rule, we go back to the μ variable,

(2.52)

Obtaining the first derivative of E with respect to the energy is not expensive, because it does not require the computation of a new wave function. Of course, the cost increases with higher derivatives.

A choice for a change of variable that makes the expansion correct at large μ, is x = μ⁻², cf. Eq. (2.49). As we know the next term in this expansion, we can also choose x = μ⁻² + κμ⁻³. Not surprisingly, the latter choice is more reliable than the first (cf. Fig. 2.5). However, it is a surprise that the approximation works very well up to μ ≈ 1, while the expansion was derived in the limit $$\mu \rightarrow \infty $$ .

Comparing these results with Fig. 2.3, one notices that the range of models for which the density functional approximation works well is comparable to that for which the Taylor series works well.

2.5 Outlook

Most applications of density functional theory rely on the simplicity of using a single Slater determinant. This chapter does not intent to discourage the traditional search of density functionals. They are successful in practice, and there still is room for improvement. However, using the simple mathematical techniques discussed in the preceding section lead to good quality approximations, and this is encouraging. There are many paths that could be followed. One, of course, is to improve the mathematical techniques. Another is to change the interaction w to a form for which the extrapolations considered here would work better. Finally, one can envisage using density functional models in combination with the extrapolation approach presented here.

Acknowledgements

Eric Cancès’ comments on the first draft on the manuscript are gratefully acknowledged.

References

1.

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Footnotes

1

For the standard Kohn–Sham model, $$\mathcal {E}(w=0)$$ is a sum of orbital energies.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_3

3. Universal Functionals in Density Functional Theory

Mathieu Lewin1  , Elliott H. Lieb2   and Robert Seiringer3  

(1)

CNRS & CEREMADE, Université Paris-Dauphine, PSL University, Paris, France

(2)

Departments of Mathematics and Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA

(3)

IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria

Mathieu Lewin (Corresponding author)

Email: mathieu.lewin@math.cnrs.fr

Elliott H. Lieb

Email: lieb@princeton.edu

Robert Seiringer

Email: robert.seiringer@ist.ac.at

Abstract

In this chapter we first review the Levy–Lieb functional, which gives the lowest kinetic and interaction energy that can be reached with all possible quantum states having a given density. We discuss two possible convex generalizations of this functional, corresponding to using mixed canonical and grand-canonical states, respectively. We present some recent works about the local density approximation, in which the functionals get replaced by purely local functionals constructed using the uniform electron gas energy per unit volume. We then review the known upper and lower bounds on the Levy–Lieb functionals. We start with the kinetic energy alone, then turn to the classical interaction alone, before we are able to put everything together. A later section is devoted to the Hohenberg–Kohn theorem and the role of many-body unique continuation in its proof.

3.1 Introduction

Density Functional Theory (DFT) attempts to describe all the relevant information about a many-body quantum system at or near its ground state in terms of its one-body density ρ(r). In its orbital-free variational formulation, due to Levy [99] and Lieb [116], DFT relies completely on a universal functional ρ↦FLL[ρ] which gives the lowest (kinetic plus interaction) energy that can be reached with all possible quantum states having a given density function r↦ρ(r). This functional is exact and it is able to describe interacting quantum Coulomb systems in their ground states. It is of course not known explicitly and one of the main purpose of DFT is to find suitable approximations. In this chapter we review known upper and lower bounds on this functional and discuss some regimes in which it simplifies. In particular we focus on the Local Density Approximation (LDA) which becomes exact in the regime where the density is very flat on sufficiently large regions of space, as was recently proved in [106, 108]. A simpler older approximation is the Thomas–Fermi functional [53, 169], which is surprisingly accurate for heavy atoms [122, 123] and is reviewed in [114].

It turns out that there are several possible Levy–Lieb-type functionals. Instead of considering N-particle wavefunctions one can also work with mixed states [116], or even with grand-canonical states for which only the average number of particles is fixed. The latter has not been thoroughly discussed in the literature. We consider the three possibilities in this chapter. When we discuss bounds, it is useful to consider the kinetic and interaction energies separately. This naturally brings in Lieb–Thirring [125, 126] and Lieb–Oxford [113, 120] inequalities, which provide lower bounds on these two functionals.

In Sect. 3.7 we recall the Hohenberg–Kohn theorem, which is another important abstract result in DFT. It turns out that its proof relies on some unique continuation problems for N-particle systems, which are not yet completely understood.

Notation

Everywhere in this chapter x = (r, σ) denotes both the space variable $$\mathbf {r}\in \mathbb {R}^d$$ and the spin variable $$\sigma \in \mathbb {Z}_q$$ . Although the physical case corresponds to d = 3 and q = 2, it is sometimes useful to keep d and q general to better emphasize the role of the dimension and spin. To simplify our notation we use the convention that $$\int _{\mathbb {R}^d\times \mathbb {Z}_q}f(\mathbf {x})\,\mathrm {d}\mathbf {x}=\sum _{\sigma \in \mathbb {Z}_q}\int _{\mathbb {R}^d}f(\mathbf {r},\sigma )\,\mathrm {d}\mathbf {r}$$ . For N particles we use the notation $$\mathbf {X}=({\mathbf {x}}_1,\ldots ,{\mathbf {x}}_N)\in (\mathbb {R}^d\times \mathbb {Z}_q)^N$$ with a similar convention for the integral $$\int _{(\mathbb {R}^d\times \mathbb {Z}_q)^N}$$ .

We recall that the one-body density ρΨ of an N-particle fermionic (normalized) wavefunction

is defined by

The interpretation of ρΨ is that it provides the average number of particles in space, without taking their spin into account.

3.2 Universal Functionals in Density Functional Theory

Following [99, 116] we express the ground state energy as a variational problem involving only the density, and we discuss some (known and unknown) mathematical properties of the associated functionals.

3.2.1 The Levy–Lieb Universal Functional

For completeness we consider general interaction potentials w in any space dimension d. This is useful to understand the particularities of the physical case at work in DFT, namely the Coulomb interaction w(r) = |r|⁻¹ in dimension d = 3.

Let

with w even, with $$v_\pm \geqslant 0$$ , with v := v₊ − v₋ and with

(3.1)

Under the assumption (3.1), the N-body potential

is infinitesimally (− Δ)–form bounded from below, which means that

for all Ψ and all ε > 0, with CN,ε a constant depending only on ε and N. From this we can deduce that the quadratic form associated with the symmetric N-body operator

(3.2)

is closed on the energy space

(3.3)

This allows us to work with the associated Friedrichs self-adjoint realization of $$H_N^{v,w}$$ , see [149, Sec. VIII.6] and [150, Sec. X.3]. Everywhere we work with fermions, that is, the wavefunction Ψ is assumed to be anti-symmetric with respect to exchanges of its variables. Note that the bosonic case is obtained when the number of spin states q is equal to N, see [121, Sec. 3.1.2–3.1.3].

The Hamiltonian (3.2) describes N fermionic particles evolving in $$\mathbb {R}^d$$ , with q spin states, submitted to an external potential v and interacting via the pair potential w. An important physical quantity is the ground state energy of the system, which is obtained by minimizing the corresponding energy

StartEnclose box upper E Subscript upper N Baseline left bracket v right bracket colon equals inf Underscript StartLayout 1st Row upper Psi element of script upper Q left parenthesis upper H Subscript upper N Superscript v comma w Baseline right parenthesis 2nd Row integral Underscript left parenthesis double struck upper R Superscript d Baseline times double struck upper Z Subscript q Baseline right parenthesis Superscript upper N Baseline Endscripts StartAbsoluteValue upper Psi EndAbsoluteValue squared equals 1 EndLayout Endscripts left angle bracket upper Psi comma upper H Subscript upper N Superscript v comma w Baseline upper Psi right angle bracket EndEnclose

where $${{\left \langle \Psi ,H^{v,w}_N\Psi \right \rangle }}$$ is understood in the sense of quadratic forms. By the variational principle, this is just the bottom of the spectrum of the operator $$H^{v,w}_N$$ :

in the fermionic subspace. We do not emphasize the interaction w in our notation since it will usually be fixed. When needed we will instead use the notation $$E_N^w[v]$$ .

The main idea of [99, 116] is to replace the infimum over Ψ by a two-step minimization

where we first minimize over the density ρ and then over all the wavefunctions having this prescribed density. This procedure requires us to identify the set of N-representable densities, that is, those arising from a Ψ in the form domain of $$H_N^{v,w}$$ , a question that we now address.

The Hoffmann-Ostenhof inequality [81] states that

(3.4)

for all (bosonic or fermionic) wavefunctions (see also Theorem 3.11 below). This inequality implies that we should restrict ourselves to densities such that

This turns out to be the optimal condition.

Theorem 3.1 (Representability of the One-Particle Density [116])

Let $$\rho \in L^1(\mathbb {R}^d,\mathbb {R}_+)$$ be such that $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ and $$\int _{\mathbb {R}^d}\rho (\mathbf {r})\,\mathrm {d}\mathbf {r}=N\in \mathbb {N}$$ . Then there exists one normalized antisymmetric wavefunction $$\Psi \in \bigwedge _1^NL^2(\mathbb {R}^d\times \mathbb {Z}_q,\mathbb {C})$$ of finite kinetic energy, $$\int _{(\mathbb {R}^d\times \mathbb {Z}_q)^N}|\nabla \Psi |{ }^2&lt;\infty $$ , such that ρ = ρΨ.

The proof of the theorem is much easier for $$q\geqslant N$$ , where the antisymmetry can be put entirely in the spin variables. One can just take

When q < N the proof in [116], inspired by March and Young [129] and Harriman [79], consists of taking a Slater determinant

with the orbitals

(3.5)

where the phases θj are chosen to make the φj orthonormal. Although we know that there exist such phases (an explicit example will be given later in (3.44)), we have very bad control over their behavior in N. This will be discussed later in Sect. 3.4 when we consider the kinetic energy cost of introducing such phases.

At this point we have found the set of N-representable densities

(3.6)

Note that this is a convex set since $$\rho \mapsto \int _{\mathbb {R}^d}|\nabla \sqrt {\rho }|{ }^2$$ is convex [118, Thm. 7.8]. With Theorem 3.1 at hand, we can rewrite the ground state energy as a minimization principle over ρ in this convex set:

StartEnclose box upper E Subscript upper N Baseline left bracket v right bracket colon equals inf Underscript StartLayout 1st Row StartRoot rho EndRoot element of upper H Superscript 1 Baseline left parenthesis double struck upper R Superscript d Baseline right parenthesis 2nd Row integral Underscript double struck upper R Superscript d Baseline Endscripts rho equals upper N 3rd Row integral Underscript double struck upper R Superscript d Baseline Endscripts v Subscript plus Baseline rho less than normal infinity EndLayout Endscripts left brace upper F Subscript upper L upper L Baseline left bracket rho right bracket plus integral Underscript double struck upper R Superscript d Baseline Endscripts v left parenthesis bold r right parenthesis rho left parenthesis bold r right parenthesis normal d bold r right brace EndEnclose

(3.7)

where

(3.8)

is called the Levy–Lieb functional. It is the lowest possible (kinetic plus interaction) energy of a quantum system having the prescribed density ρ. This universal functional is the central object of DFT, since knowing it would allow us to compute the ground state energy of a system with any external potential v, by (3.7). In this chapter we will review what is known about FLL. But first we need to introduce two other universal functionals, which are obtained after convexifying FLL in two different ways.

3.2.2 Lieb’s Universal Functional

Note that v↦EN[v] is concave (as seen from the variational principle, it is a minimization over Ψ of affine functions in v). More precisely, from (3.7) we see that EN is the Legendre transform of FLL on the convex set of N-representable densities. This naturally raises the question of whether FLL is, conversely, the Legendre transform of EN. This turns out to be wrong since FLL is not convex [116]. It is therefore convenient to look at the convex hull (also called lower convex envelope)

which is the Legendre transform of EN. Here the convex hull means that it is the largest convex function below FLL. We always assume that $$\int _{\mathbb {R}^d}\rho =N$$ . Another kind of convex hull with respect to N will be considered later. As proved in [116], the function FL is explicit and given by a similar definition as in (3.8) but with mixed states instead of pure states:

StartEnclose box upper F Subscript normal upper L Baseline left bracket rho right bracket colon equals inf Underscript StartLayout 1st Row upper Gamma equals upper Gamma Superscript asterisk Baseline greater than or slanted equals 0 2nd Row trace upper Gamma equals 1 3rd Row trace left parenthesis negative upper Delta right parenthesis upper Gamma less than normal infinity 4th Row rho Subscript upper Gamma Baseline equals rho EndLayout Endscripts trace left parenthesis upper H Subscript upper N Superscript 0 comma w Baseline upper Gamma right parenthesis period EndEnclose

(3.9)

We recall that the density ρΓ of a mixed state Γ (a non-negative self-adjoint operator satisfying Tr ( Γ) = 1), diagonalized in the form

with αj⩾0 and ∑jαj = 1, is defined by

It is useful to know that the infimum is attained in (3.9), as well as for the Levy–Lieb functional in (3.8).

Theorem 3.2 (Existence of Optimal (Pure and Mixed) States [116])

Let $$\rho \in L^1(\mathbb {R}^d,\mathbb {R}_+)$$ be such that $$\int _{\mathbb {R}^d}\rho =N\in \mathbb {N}$$ and $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ . Then the infima in (3.8) and (3.9) are attained.

The proof uses the fact that a minimizing sequence Ψj (resp. Γj) is necessarily compact in $$L^2(\mathbb {R}^d\times \mathbb {Z}_q,\mathbb {C})^N$$ (resp. in the trace class), since the density is fixed, hence the sequence is tight.

Let us consider a density ρ and a corresponding minimizing N-particle mixed state Γ. Diagonalizing Γ in the form Γ =∑jαj| Ψj〉〈 Ψj|, we see that

and since the upper bound is obvious, we conclude that (3.9) can also be written in the form

(3.10)

This is the claimed convex hull of FLL.

Since an affine function always attains its minimum at an extreme point of a convex set, the ground state energy EN[v] is given by the same formula

as we had in (3.7) for pure states. From this discussion it seems more natural to work with the convex Lieb functional FL, instead of FLL. The following duality principle holds.

Theorem 3.3 (Duality [116])

We have, with p as in (3.1),

(3.11)

In the second line the constant EN[v] has been included in v, hence the constraint that $$H^{v,w}_N\geqslant 0$$ in the operator sense.

We have seen in Theorem 3.2 that (3.8) and (3.9) are attained. On the other hand, the supremum in (3.11) will not be attained for most densities. Indeed, ρ would then be the density of a mixed ground state for the corresponding EN[v] but this set is believed to be very small. For instance, if ρ vanishes on a set of positive measure, the supremum cannot be attained with a v for which unique continuation holds on the whole space. This is discussed in Sect. 3.7.

The importance in applications of the convex formulation of DFT based on the functionals FLL and FL is reviewed in [80].

3.2.3 The Grand Canonical Universal Functional

At this step we have defined two universal functionals FLL and FL of the density ρ, which provide the same ground state energy EN[v] in the presence of an external potential v. Since Lieb’s functional FL is convex, it is to be preferred over FLL. The convexity implies the dual formula stated in Theorem 3.3.

In spite of its convexity, the functional FL does not behave well with respect to the weak topology of $$H^1(\mathbb {R}^d)$$ . This might lead to some difficulties in processes where some electrons are lost, e.g. for scattering. Indeed, if we have a sequence ρn such that $$\sqrt {\rho _n}\rightharpoonup \sqrt {\rho }$$ weakly but not strongly, then we may have $$\int _{\mathbb {R}^d}\rho (\mathbf {r})\,\mathrm {d}\mathbf {r}&lt;N$$ . The integral does not even need to be an integer, in which case FL[ρ] would not make any sense. For this reason it is natural to introduce a functional allowing for non-integer values of the particle number. The following grand-canonical version did not appear in [116], but was mentioned in [141] and was recently studied in [106, 108].

For us, a grand-canonical state (commuting with the particle number) will be a collection Γ = ( Γn)n⩾0 of non-negative self-adjoint operators, each of them acting on the n-particle space $$\bigwedge _1^nL^2(\mathbb {R}^d\times \mathbb {Z}_q)$$ , and such that

Here Γ₀ is a number in [0, 1] which gives the probability that there is no particle at all. The corresponding density is the sum

so that the average number of particles in the system is given by $$\int _{\mathbb {R}^d}\rho _\Gamma =\sum _{n\geqslant 1}n\mathrm {Tr}\,(\Gamma _n)$$ . We define the grand-canonical universal functional as

StartEnclose box upper F Subscript upper G upper C Baseline left bracket rho right bracket colon equals inf Underscript StartLayout 1st Row sigma summation Underscript n greater than or slanted equals 1 Endscripts trace upper Gamma Subscript n Baseline less than or slanted equals 1 2nd Row sigma summation Underscript n greater than or slanted equals 1 Endscripts trace left parenthesis negative upper Delta right parenthesis upper Gamma Subscript n Baseline less than normal infinity 3rd Row sigma summation Underscript n greater than or slanted equals 1 Endscripts rho Subscript upper Gamma Sub Subscript n Subscript Baseline equals rho EndLayout Endscripts sigma summation Underscript n greater than or slanted equals 1 Endscripts trace left parenthesis upper H Subscript n Superscript 0 comma w Baseline upper Gamma Subscript n Baseline right parenthesis period EndEnclose

(3.12)

In order to guarantee that the problem is well posed, we need some more assumptions on the interaction potential w. We assume that the system is stable of the second kind [152], that is, there exists a constant C such that

(3.13)

Inserting this in (3.12) implies

hence the infimum in (3.12) is finite. A typical example is that of a non-negative interaction potential w⩾0 such as Coulomb, or more generally a potential w which is classically stable of the second-kind [152], that is, which satisfies the same assumption as (3.13) with the kinetic energy removed:

(3.14)

For a density with integer particle number $$\int _{\mathbb {R}^d}\rho (\mathbf {r})\,\mathrm {d}\mathbf {r}=N\in \mathbb {N}$$ the grand canonical functional is the lowest of the three universal functionals:

The following can be shown similarly as for Theorem 3.2, using the stability assumption (3.13).

Theorem 3.4 (Existence of Optimal Grand-Canonical States)

Assume that the system is stable of the second kind as in (3.13). Let $$\rho \in L^1(\mathbb {R}^d,\mathbb {R}_+)$$ be such that $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ . Then the infimum in (3.12) is attained.

From the existence of a minimizer we deduce as before that

In other words, the grand canonical functional is also a convex hull of the original Levy–Lieb functional FLL, but convex combinations ρ =∑jβjρj are considered with the ρj having an arbitrary number of particles. It is not required that all the ρj have the fixed number N of particles like for FL.

An interesting question is to determine whether an optimal grand canonical state ( Γn)n⩾0 corresponding to a given ρ always satisfies Γn ≡ 0 for $$n\geqslant n_{\mathrm {max}}$$ . In this case we say that Γ = ( Γn)n⩾0 has a compact support in n. No result of this sort seems to have appeared in the literature up to now.

The following theorem asserts that FGC is the weak-∗ lower semi-continuous envelope of FL, in an appropriate sense.

Theorem 3.5 (Weak Lower Semi-Continuity)

We assume that

with p as in (3.1) and that w tends to 0 at infinity. The functional FGCis the weak-∗ lower semi-continuous closure of FL, in the following sense:

(i)

For any sequence $$(\sqrt {\rho _j})_{j\geqslant 1}\subset H^1(\mathbb {R}^d)$$ converging weakly in $$\dot {H}^1(\mathbb {R}^d)$$ to $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ , we have

(3.15)

(ii)

For any $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ , there exists a sequence $$(\sqrt {\rho _j})_{j\geqslant 1}\subset H^1(\mathbb {R}^d)$$ converging strongly to $$\sqrt {\rho }$$ in $$\dot {H}^1(\mathbb {R}^d)\cap L^{p}(\mathbb {R}^d)$$ for all 2 < p < p∗, such that

(3.16)

Here

is the critical Sobolev exponent. Since we are not aware that a proof of Theorem 3.5 has been explicitly written anywhere, we provide the full argument later in Sect. 3.8. It is inspired by [102].

When the liminf on the right of (3.15) is finite (which we can always assume, otherwise the statement is void), then the Hoffmann-Ostenhof inequality (3.4) implies that $$\sqrt {\rho _j}$$ is bounded in the homogeneous Sobolev space $$\dot {H}^1(\mathbb {R}^d)$$ . However $$\int _{\mathbb {R}^d}\rho _j$$ need not be bounded in general and this is why only the weak convergence in $$\dot {H}^1(\mathbb {R}^d)$$ was assumed. This plays an important role in (ii). Consider a ρ and an associated optimal grand-canonical state Γ = ( Γn)n⩾0 for FGC[ρ]. If we have $$\Gamma _{n_k}\neq 0$$ for a sequence nk →∞, then this means that infinitely many particles are needed to properly represent ρ grand-canonically. Although this is very unlikely to happen in practical situations, this can probably not be avoided for a general interaction w and a general ρ. Then we need a diverging number of particles

in our canonical state associated with ρj, even if it has a bounded energy. On the other hand, if there exists one minimizer ( Γn)n⩾0 for FGC[ρ] which has a compact support in n, then (ii) holds with a sequence converging weakly in $$H^1(\mathbb {R}^d)$$ , as will be clear from our proof. This is one reason why it is important to understand whether optimal states always have a compact support in n, as we have mentioned previously.

Next we discuss the dual formulation of FGC. For v = v₊ − v₋ with $$v_-\in L^p(\mathbb {R}^d)+L^\infty (\mathbb {R}^d)$$ and $$v_+\in L^1_{\mathrm {loc}}(\mathbb {R}^d)$$ , we find that the Legendre transform of FGC is given by

(3.17)

For $$\lambda =N\in \mathbb {N}$$ , we have $$E^{\mathrm {GC}}_N[v]\leqslant E_N[v]$$ but equality will in general not hold. If the function n↦En[v] is convex in the discrete sense, that is,

(3.18)

then it follows that

for all $$N\in \mathbb {N}$$ and θ ∈ [0, 1) [141]. In particular $$E_N^{\mathrm {GC}}[v]=E_N[v]$$ . Therefore, if (3.18) holds, then the grand canonical functional FGC[ρ] provides the same ground state energy in an external potential v as the canonical ones FLL[ρ] and FL[ρ]. In physical terms the condition (3.18) means that the electron ionization energy is greater than or equal to the electron affinity. It is a famous conjecture that (3.18) holds for the Coulomb potential in dimension d = 3, for atomic or molecular external potentials v [5, 141]. A counterexample is provided in [116] for a different w. Note that (3.18) always holds for w ≡ 0.

If (3.18) does not hold, then $$E^{\mathrm {GC}}_\lambda [v]$$ is equal to the convex hull of n↦En[v]. This amounts to considering the set $$\mathcal {K}\subset \mathbb {N}$$ of the points k such that

One obtains

where n₁ < N < n₂ are the two closest points in $$\mathcal {K}$$ on the left and right of N.

3.2.4 Kohn–Sham Exchange Correlation

In the previous sections we have explained the Levy–Lieb variational formulation of the ground state energy of the N-particle problem in terms of the density only, which is really in the spirit of DFT. Practitioners prefer to use an auxiliary set of N orthonormal functions Φ = (φ₁, …, φN), which describe N fictitious uncorrelated electrons, to build the desired density through the formula

with the Slater determinant

(3.19)

This method provides a better representation of the kinetic energy, but it is much more costly from a computational point of view. We quickly explain this approach due to Kohn–Sham [93] here.

For a density ρ with $$\int _{\mathbb {R}^d}\rho (\mathbf {r})\mathrm {d}\mathbf {r}=N\in \mathbb {N}$$ , we introduce the lowest kinetic energy of Slater determinants

(3.20)

(the min is attained for the same reason as in Theorem 3.2). We then add and subtract TS from FLL, which allows us to rewrite the N-particle ground state using N orbitals as

(3.21)

where

(3.22)

is called the exchange-correlation energy. From a mathematical point of view, the Kohn–Sham approach a priori requires the study of both FLL[ρ] and TS[ρ] as separated functionals. It is an interesting question to find a way to study Exc[ρ] directly, without interpreting it as a difference. In chemistry one often relies on the adiabatic connection formula (see Remark 3.1 below), which however involves another kinetic energy functional T[ρ] discussed later in Sect. 3.4.

Instead of using N uncorrelated electrons as the main variable, one may also use a one-particle density matrix γ, which is often called the Kohn–Sham method with fractional occupations. The method is similar but one has to subtract the lowest kinetic energy TGC[ρ] of all possible one-particle density matrices, which is defined later in Sect. 3.4.

3.3 The Uniform Electron Gas and the Local Density Approximation

The universal functionals FLL, FL and FGC defined in the previous section allow us in principle to describe any fermionic system interacting via the potential w. But these functionals are of course not known exactly and finding them is essentially the same as solving the N-particle problem. One of the main purposes of DFT is to find reliable and efficient approximations. Here we discuss the most widely used of these approximations, called the Local Density Approximation (LDA) [46, 82, 93, 128, 136, 140], where they are replaced by purely local ones. The LDA is often considered as “the mother of all approximations” [142] and it yields surprisingly good results, even in cases where the density is not at all slowly varying [128, 136]. Its successors involving gradient corrections are even better and have become the standard in DFT calculations. In this section we only consider the Coulomb case in dimension d = 3 but we expect similar results for other potentials in all dimensions.

Of course, the functionals FLL, FL and FGC are not local at all. Two electrons at different places are always entangled and, furthermore, the Coulomb potential has a very long range so that electrons interact even when they are far apart. In the LDA one makes the assumption that the only non-local part is the Hartree term (the classical Coulomb energy of the density ρ) and one approximates the rest by a local function of ρ, that is, the integral of a function f depending only on the value ρ(r) at r:

(3.23)

The function f is chosen to be the energy per unit volume of an infinite gas of constant density ρ, called the Uniform Electron Gas (UEG), so that the approximation becomes exact when ρ is constant over a very large domain. Because the UEG is an infinite system it should not depend on whether it is defined canonically or grand-canonically. Hence the function f must be the same for the three functionals FLL, FL and FGC.

The idea behind the LDA is as depicted in Fig. 3.1. After subtraction of the Hartree term, one splits the space into small boxes (of volume dr) and assumes that the remaining energy is the sum of the local energies. In each little box, one replaces the density by a constant. One does not use the energy of the constant function in the small box, but rather the energy per unit volume of an infinite system having the corresponding uniform density, multiplied by the volume dr of the small box.

Fig. 3.1

Main idea of the Local Density Approximation in DFT. The Levy–Lieb energy (with the Hartree term subtracted) is replaced by the sum of the energies per unit volume of an infinite uniform gas with the local density ρ(r₀), times the volume dr

In the LDA the complicated Levy–Lieb functionals therefore get replaced by a new universal function $$f:\mathbb {R}_+\to \mathbb {R}$$ , which is much simpler since it only depends on one real parameter. But the function f is also not known exactly, and we will see that it displays a very rich structure.

In this section we report on the results in [108] where the LDA was rigorously justified for the first time. The proper regime is that of slowly varying densities, that is, densities ρ which are very flat on sufficiently large domains such that f(ρ) becomes a good local approximation. To this end we start by defining the function f.

3.3.1 The Uniform Electron Gas

The uniform electron gas was rigorously defined in [106, 108] and it is obtained by assuming that the density is exactly constant over a large domain which grows so as to cover the whole space. The result is the following.

Theorem 3.6 (Uniform Electron Gas [106, 108])

Let w(r) = |r|⁻¹in dimension d = 3 and q⩾1 be the number of spin states. Let ρ₀ > 0. Let Ω be a fixed open convex set of unit volume | Ω| = 1. Let $$\chi \in L^1(\mathbb {R}^3)$$ be a radial non-negative function of compact support such that $$\int _{\mathbb {R}^3}\chi =1$$ and $$\int _{\mathbb {R}^3}|\nabla \sqrt {\chi }|{ }^2&lt;\infty $$ . Then the following thermodynamic limit exists

(3.24)

and does not depend on Ω and χ.

Electrons have q = 2 spin states but we have written the result with a general q for convenience. It is expected that the exact same result holds for the other two functionals FLL and FL, with the same limit f(ρ₀). This has not yet been proved, except in the classical case where the kinetic energy is dropped (see Sect. 3.5).

The number f(ρ₀) is the energy per unit volume (with the Hartree energy subtracted) of an infinite gas submitted to the constraint that its density is exactly constant over $$\mathbb {R}^3$$ , ρ(r) ≡ ρ₀. In the literature, f(ρ₀) is often confused with the corresponding Jellium energy. In Jellium there is no constraint on ρ but one adds instead a uniform background of density ρ₀ which compensates the long range of the Coulomb potential [104, 119]. That the two models coincide has only been shown in the classical case [36, 107] so far, as we will mention in Sect. 3.5 below. In the quantum case the same result should hold, but the proof has not been written yet.

Here are some rigorously known properties of the function f.

Theorem 3.7 (Properties of f [72, 106, 108, 119])

The function f is locally Lipschitz: There exists a constant C such that

(3.25)

for all ρ₁, ρ₂⩾0. The function f satisfies the uniform bound

(3.26)

for all ρ⩾0, where cTF(3) = (3∕10)(6π²)2∕3and $$c_{\mathrm {D}}(1,3)=(3/4)\left (6/\pi \right )^{1/3}$$ are respectively the Thomas–Fermi and Dirac constants, discussed later in Sects.3.4and3.5. It behaves at small densities like

(3.27)

where

is the classical UEG energy discussed later in Sect.3.5, and at large densities like

(3.28)

The statement involves several constants that will be introduced in the next sections. It is believed that f is smooth except at finitely many points corresponding to phase transitions. In the case of spin-1∕2 particles like electrons (q = 2), numerical simulations in [3, 20, 24, 47, 84, 88, 146, 175] indicate that there might be one or two such points. For a long time it was believed that the system can be a ferromagnetic Wigner crystal, a ferromagnetic fluid and a paramagnetic fluid. Recent results indicate that the ferromagnetic fluid phase might not exist [3, 84], however. More transitions could occur in the solid phase (for instance an anti-ferromagnetic crystal). In spite of the clear numerical evidence that there are phase transitions, proving it remains a very challenging open problem [10]. Several approximate formulas for the function f are used in DFT, including for instance the celebrated Perdew–Wang (PW92) functional [144].

3.3.2 The Local Density Approximation of FGC

We now state a result from [108], where the LDA was justified for the first time in the quantum case.

Theorem 3.8 (LDA for FGC [108])

Let w(r) = |r|⁻¹in dimension d = 3 and q⩾1 be the number of spin states. Then there exists a constant C = C(q) such that

(3.29)

for every ε > 0 and every non-negative density $$\rho \in L^1(\mathbb {R}^3)\cap L^2(\mathbb {R}^3)$$ such that $$\nabla \sqrt {\rho }\in L^2\cap L^4(\mathbb {R}^3)$$ . Here f is the function defined in Theorem3.6.

It is expected that the exact same result holds for the canonical functionals FLL and FL, with of course the additional constraint that $$\int _{\mathbb {R}^3}\rho \in \mathbb {N}$$ .

The last gradient term $$\varepsilon ^{-15}|\nabla \sqrt {\rho }|{ }^4$$ was chosen for simplicity but the same result actually holds with ε1−4p|∇ρθ|p instead, under the conditions that p > 3, 0 < θ < 1 and $$2\leqslant p\theta \leqslant 1+{p}/{2}$$ . The constant C then depends on the chosen p and θ.

In addition to the large power of ε, which is an artifact of the proof in [108], the form of the error term is probably not optimal. It is reasonable to expect that the right side of (3.29) should only involve quantities like ρ5∕3, ρ4∕3, $$|\nabla \sqrt {\rho }|{ }^2$$ and |∇ρ1∕3|² or perhaps |∇ρ|, which have the same scaling as the kinetic and Coulomb terms.

The inequality (3.29) holds for every density but it is useful only when the two gradient terms are much smaller than the first term,

so that after optimizing over ε one gets a small term on the right side of (3.29). One interesting case is when the density is given in terms of a fixed function ρ with $$\int _{\mathbb {R}^3}\rho =1$$ , which is rescaled in the manner

After optimizing over ε we obtain the following expansion of the grand-canonical Levy–Lieb energy:

(3.30)

The first term is the trivial non-local Coulomb term, whereas the next term in the expansion is the LDA. It is an interesting open question to determine the next order correction, which is believed to be also local, of order N1∕3, and to involve gradients. The exact same result as (3.30) is expected for FLL and FL.

Remark 3.9 (LDA for the Exchange-Correlation Energy)

Theorem 3.15 below is a result similar to Theorem 3.8 for the grand-canonical kinetic energy TGC alone, and implies a corresponding bound for the difference of the two functionals. These two bounds justify the LDA for the (grand-canonical) exchange-correlation energy, as was defined in Sect. 3.2.4.

In the next two sections we study separately the kinetic energy functional and the classical interaction functional. We discuss known upper and lower bounds and derive the LDA for these functionals in a similar (but simpler) manner as for the full Levy–Lieb functional FGC. Although the minimum of a sum is in general not the sum of the two minima, understanding the kinetic and interaction energies separately will give us useful information on the full functional, as explained in Sect. 3.6 below.

3.4 Kinetic Energy and Lieb–Thirring Inequalities

3.4.1 Three Kinetic Energy Functionals

We have introduced in (3.20) the lowest kinetic energy TS[ρ] that can be reached with Slater determinants, for a given density ρ with $$\int _{\mathbb {R}^3}\rho (\mathbf {r})\,\mathrm {d}\mathbf {r}=N$$ . We can define in a similar manner the lowest kinetic energy that can be reached with all possible wave functions

StartEnclose box upper T left bracket rho right bracket colon equals min Underscript StartLayout 1st Row upper Psi element of and Underscript 1 Overscript upper N Endscripts upper H Superscript 1 Baseline left parenthesis double struck upper R Superscript d Baseline times double struck upper Z Subscript q Baseline comma double struck upper C right parenthesis 2nd Row double vertical bar upper Psi double vertical bar Subscript upper L squared Baseline equals 1 3rd Row rho Subscript upper Psi Baseline equals rho EndLayout Endscripts one half sigma summation Underscript j equals 1 Overscript upper N Endscripts integral Underscript left parenthesis double struck upper R Superscript d Baseline times double struck upper Z Subscript q Baseline right parenthesis Superscript upper N Baseline Endscripts StartAbsoluteValue nabla Subscript j Baseline upper Psi left parenthesis bold upper X right parenthesis EndAbsoluteValue squared normal d bold upper X period EndEnclose

(3.31)

This is nothing but $$F_{\mathrm {LL}}^0[\rho ]$$ , the Levy–Lieb functional with interaction w ≡ 0. Recall that T and TS depend on the number of spin states q.

Since this is a non-interacting problem, one may think at first sight that minimizers will always be Slater determinants, that is, T[ρ] and TS[ρ] should coincide. But this is not true in general [116] and the best one can say for a general ρ is that $$T_{\mathrm {S}}[\rho ]\geqslant T[\rho ]$$ .

There are two other natural kinetic functionals corresponding to $$F_{\mathrm {L}}^0[\rho ]$$ and $$F_{\mathrm {GC}}^0[\rho ]$$ , respectively. For the first one the minimization is extended to mixed canonical states and for the second one to grand-canonical states. It turns out that these two are equal:

The reason is that the kinetic energy can be expressed in terms of the one-particle density matrix γ and that the set of such matrices which are N-representable by a mixed state coincides with those which are representable by a grand-canonical state [27, 28]. By duality, this also follows from the fact that the inequality (3.18) always holds in the non-interacting case.

In order to explain all this in detail, we first recall that the one-particle density matrix γΨ of a wavefunction Ψ is the self-adjoint operator acting on the one-particle space $$L^2(\mathbb {R}^d\times \mathbb {Z}_q)$$ with integral kernel

This gives

with the trace interpreted in the quadratic form sense. Every density matrix of an antisymmetric Ψ satisfies $$0\leqslant \gamma _\Psi =(\gamma _\Psi )^*\leqslant 1$$ and Tr (γΨ) = N. When we consider mixed N-particle states we obtain the convex hull of the set of N-representable density matrices. This convex hull is definitely contained in the convex set

But the extreme points of this set are the rank-N orthogonal projections with finite kinetic energy. Those are exactly the one-particle density matrices of the Slater determinants. Hence we must have equality of the two convex sets. By considering grand-canonical states the set will not increase further. See Theorem 3.26 for a related result.

This discussion leads us towards introducing the following kinetic energy functional

StartEnclose box upper T Subscript upper G upper C Baseline left bracket rho right bracket colon equals min Underscript StartLayout 1st Row 0 less than or slanted equals gamma equals gamma Superscript asterisk Baseline less than or slanted equals 1 2nd Row trace left parenthesis negative upper Delta right parenthesis gamma less than normal infinity 3rd Row rho Subscript gamma Baseline equals rho EndLayout Endscripts trace left parenthesis StartFraction negative upper Delta Over 2 EndFraction right parenthesis gamma period EndEnclose

(3.32)

We call it “grand-canonical” since $$\int _{\mathbb {R}^d}\rho $$ can now take any positive value. But for $$\int _{\mathbb {R}^d}\rho \in \mathbb {N}$$ , this is just Lieb’s canonical energy $$T_{\mathrm {GC}}[\rho ]=F_{\mathrm {L}}^0[\rho ]$$ . In this case we also have

The functional TGC[ρ] is convex and it is the convex hull and the weak-∗ semi-continuous closure of both T[ρ] and TS[ρ], similarly as in Theorem 3.5. Minimizers exist for these three functionals, as in Theorem 3.2. It suffices to take w ≡ 0 in all those theorems.

Remark 3.10 (N-Representability of the One-Particle Density Matrix)

There are complicated constraints on a one-particle density matrix γ to ensure that it arises from an N-particle wavefunction Ψ. For instance, when N = 2 then all the eigenvalues of γ must be of even multiplicity. Another example is that no γ of rank N + 1 is N-representable. See [28, 62, 101] for these two examples and [2, 13, 91, 154] for more advanced results when N⩾3.

Now we mention some known upper and lower bounds on the kinetic energy functionals. Lower bounds naturally involve the lowest functional TGC[ρ]. Upper bound should ideally involve TS[ρ] but we will see that much more is known on TGC[ρ].

3.4.2 Lower Bounds: Hoffmann-Ostenhof and Lieb–Thirring Inequalities

The first lower bound is the Hoffmann-Ostenhof inequality mentioned previously in (3.4) and which holds for the grand-canonical kinetic energy as well.

Theorem 3.11 (Hoffmann-Ostenhof Inequality [81])

For every ρ⩾0 such that $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ , we have

(3.33)

Using the Gagliardo–Nirenberg inequality [118]

(3.34)

for $$u=\sqrt {\rho }$$ , we obtain

and this is optimal for bosons. But for fermions this is not optimal at all. The Lieb–Thirring inequality states that one can replace the N-dependent prefactor by an N-independent one (or, rather, by a q-dependent constant where q is the number of spin states).

Theorem 3.12 (Lieb–Thirring [121, 125, 126])

There exists a positive constant cLT(d) > 0 such that

(3.35)

for all ρ⩾0 such that $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ .

Note the spin dependence in q−2∕d, which is compatible with the fact that the bosonic case is recovered when q = N. For particles like electrons we have q = 2 and the constant is N-independent.

For large fermionic systems the Lieb–Thirring inequality is an advantageous replacement for the Gagliardo–Nirenberg inequality, to which it reduces in the case N = 1 (in particular we always have $$c_{\mathrm {LT}}(d)\leqslant c_{\mathrm {GN}}(d)/2$$ ). Since its invention, the Lieb–Thirring inequality (3.35) has played a central role in the mathematical understanding of large fermionic systems. It was originally used to give a proof of stability of matter [112, 117, 121, 125] that is much shorter than the original proof of Dyson and Lenard [48]. Later the Lieb–Thirring inequality was generalized to systems at positive density [57, 58] where ρ is a local perturbation of a constant, and to the dynamic case where it extends Strichartz’s inequality [59, 60].

The right side of (3.35) is related to the kinetic energy of the free Fermi gas. Indeed, we recall that the translation-invariant orthogonal projector

(3.36)

has the constant density $$\rho _{P_{\rho _0}}=\rho _0$$ and the constant kinetic energy density $$c_{\mathrm {TF}}(d)\,q^{-2/d}\rho _0^{1+2/d}$$ , where

(3.37)

is called the Thomas–Fermi constant.

The best constant cLT(d) in (3.35) is unknown but it is definitely less than or equal to cTF(d). This is seen by using the trial state $$\gamma _R=\chi (\cdot /R)P_{\rho _0}\chi (\cdot /R)$$ and taking the limit R →∞. The famous Lieb–Thirring conjecture [159] states that

(3.38)

In other words, the conjecture states that the best constant is obtained either for the infinite non-interacting uniform electron gas, or for one isolated electron. This conjecture was investigated numerically in [98]. The proof of the conjecture (3.38) in dimension d = 3 would have a great impact since it would mean that the Thomas–Fermi–Dirac (TFD) energy is an exact lower bound to the many-particle problem [114], as we will mention later in Sect. 3.6. The Thomas–Fermi energy is the simplest functional in Density Functional Theory and knowing that it is an exact lower bound would simplify drastically many mathematical results, in addition to increasing its physical significance.

The best known estimate on cLT(d) was recently proved in [56] and reads

It improves upon the previously best known result where 1.456 was replaced by 1.814 and which was proved in d = 1 in 1991 by Eden and Foias [49] and in d⩾2 by Dolbeault et al. [45] in 2008. We refer to [54–56, 159] for a recent overview of other important results on the Lieb–Thirring inequality.

By duality, the Lieb–Thirring inequality implies a bound on the sum of the negative eigenvalues of a one-particle Schrödinger operator in an external potential v, denoted by functional calculus as −Tr (− Δ∕2 + v)₋. Namely, we have by (3.7)

(3.39)

where $$E_N^0$$ indicates that we take w ≡ 0. Since $$T_{\mathrm {GC}}[\rho ]=F^0_{\mathrm {GC}}[\rho ]$$ is the Legendre transform of $$E_N^0[v]$$ , the inequality (3.39) is actually equivalent to the Lieb–Thirring inequality (3.35). The original proof of Lieb and Thirring [125, 126] actually showed (3.39) and it was only much later that Rumin [153] found a direct proof of (3.35).

The semi-classical constant (3.37) naturally occurs for slowly varying densities in the LDA regime, as we will see. Nam proved in [133] that one can replace the Lieb–Thirring (unknown) constant cLT(d) by cTF(d) at the expense of a gradient correction.

Theorem 3.13 (Nam’s Lieb–Thirring Inequality with Gradient Correction [133])

Let q, d⩾1. There exists a universal constant κ(d) (independent of the number of spin states q) such that

(3.40)

for all 0 < ε < 1 and all d⩾1.

This was the first step towards a proof of the validity of the LDA for the kinetic energy, to which we will come back soon.

Li and Yau proved in [111] a lower bound involving the optimal Thomas–Fermi constant in a bounded domain $$\Omega \subset \mathbb {R}^d$$ :

(3.41)

(see also [118, Thm. 12.3] and [75, Lem. 9]). The bound is particularly useful for densities ρ which are (almost) constant over a domain Ω. For instance for $$\rho (\mathbf {r})=\rho _0{\mathbbm 1}_\Omega \ast \chi (\mathbf {r})$$ with supp(χ) ⊂ B₁ we find the exact lower bound

whereas (3.40) yields a worse error term.

3.4.3 Upper Bounds

For upper bounds one should ideally consider the larger functional TS[ρ]. In dimension d = 1, March and Young [129, Eq. (9)] gave the proof of an estimate similar to (3.40) without the parameter ε in front of the gradient correction

(3.42)

where π²∕6 = cTF(1). In the same paper they also state a result in 3D (for a constant c > cTF(3)) but the proof has a mistake [116, Sec. 5.B]. The bound (3.42) is proved by using as trial state the orbitals

(3.43)

(we take q = 1 for simplicity), where $$n\in \mathbb {Z}$$ and the phases are seen to make the φn orthonormal. Computing the kinetic energy of this trial state, one obtains

Taking all the integers n less than or equal to N∕2 in absolute value and using the precise behavior of the series gives the result for q = 1.

The method can be generalized to higher dimensions using a similar method, but the estimate has a bad behavior in N. The orbitals

(3.44)

were considered in [79, 116] and these are the phases which we already mentioned in (3.5). Using this trial state one obtains [116] for q = 1

(3.45)

An upper bound on TS[ρ] involving only $$\int _{\mathbb {R}^d}|\nabla \sqrt {\rho }|{ }^2$$ has to have a constant diverging at least as fast as N2∕d, due to the Lieb–Thirring inequality. In [11, 176, 177] the optimal upper bound of this form was shown:

(3.46)

The idea of the proof is to apply a deformation of the space in order to map ρ onto the constant density in a box, which is then represented by a usual Slater determinant made of plane waves. One would expect an upper bound on TS[ρ] involving both $$\int _{\mathbb {R}^d}\rho ^{1+2/d}$$ and $$\int _{\mathbb {R}^d}|\nabla \sqrt {\rho }|{ }^2$$ , with coefficients independent of N as in (3.42), but this seems unknown at present. The periodic case was studied in [12].

Recently, an upper bound similar to (3.40) was proved in [108] for the grand-canonical functional TGC[ρ].

Theorem 3.14 (Upper Bound on TGC[ρ] [108])

Let d, q⩾1. There exists a constant κ′(d) such that

(3.47)

for all ε > 0 and all ρ⩾0 with $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ .

The main difficulty in the proof of (3.47) is the constraint that the one-particle density matrix must have the exact density ρ. One can provide rather good upper bounds if we allow the density to vary a bit. For instance, by using coherent states [114] the density ρ is replaced by ρ ∗|f|² where f is the profile used to build the coherent states (typically a Gaussian).

The proof of (3.47) instead relies on the following trial one-particle density matrix

(3.48)

Here the two functions $$\sqrt {\eta (t/\rho (\mathbf {r}))}$$ are interpreted as multiplication operators, whereas the operator in the middle is the Fourier multiplier Pt introduced before in (3.36). The non-negative function η is chosen such that

(3.49)

The main idea is to represent the density ρ by using the smooth “layer cake principle” [118, Thm. 1.13]

where we think of η as very concentrated around 1, and to then take the free Fermi gas Pt as in (3.36) on the support of η(t∕ρ), where ρ is very close to t. The measure dt∕t in (3.48) ensures that ργ = ρ exactly. On the other hand the condition $$\int _0^\infty \eta (t)\,{\mathrm {d} t}/{t}\leqslant 1$$ ensures that $$0\leqslant \gamma \leqslant 1$$ and means that η must put slightly more weight on the right of 1 than on the left. Computing the kinetic energy of the trial state (3.48) and optimizing over η, one obtains (3.47).

We have explained the construction of the trial state (3.48) to emphasize how much easier it is to work in the grand-canonical setting. It is an important open problem to obtain a bound similar to (3.47) on TS[ρ] or T[ρ]. For TS[ρ] this amounts to understanding how to build N orthogonal orbitals with the prescribed density, and to obtain the lowest possible energy. This problem is somewhat related to the smooth Hobby–Rice problem. There one considers N arbitrary L²-normalized functions $$\varphi _1,\ldots ,\varphi _N\in H^1(\mathbb {R}^d,\mathbb {C})$$ and looks for the minimal kinetic energy cost to orthonormalize them using only phases: $$\varphi _j^{\prime }=\varphi _j \mathrm{e} ^{i\theta _j}$$ . It was proved in [61, 97, 155] that such phases θj always exist, but known bounds involve $$ \left \| \nabla \theta _j \right \|{ }_{L^1}$$ which are not enough to deduce anything on the H¹ norm of the orbitals $$\varphi _j^{\prime }$$ . In view of (3.45), one would suspect that

but this seems unknown at present. In (3.43) the reference orbitals are all equal to $$\sqrt {\rho /N}$$ but this is probably not the optimal choice for TS[ρ] in dimension d⩾2.

3.4.4 Local Density Approximation for the Kinetic Energy

From the lower bound (3.40) and the upper bound (3.47) we obtain the following result, which is similar to Theorem 3.8 but involves only quantities that all scale the same, namely like inverse-length squared.

Theorem 3.15 (Local Density Approximation of the Kinetic Energy [108, 133])

Let d, q⩾1. There exists a universal constant C(d) such that

(3.50)

for all ρ⩾0 with $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ and all ε > 0.

In the regime where

the optimization over ε gives a right side which is negligible compared to the left side. In this regime we can approximate the kinetic energy functional in the manner

(3.51)

The right side is called the Thomas–Fermi kinetic energy and it is the simplest approximation to TGC[ρ]. If we fix a density ρ with $$\int _{\mathbb {R}^d}\rho =1$$ and take ρN(r) = ρ(rN−1∕d), then we find from (3.50) that

From semi-classical analysis it is expected that for a sufficiently regular ρ the next term should be equal to

(3.52)

which is called the von Weizsäcker correction, see [136, Sec. 6.7] and [128, pp. 89–90]. This is in reference to the historical work [171] for atoms in dimension d = 3 where however von Weizsäcker chose the coefficient 1∕2 instead of 1∕18.¹ The value of the prefactor in (3.52) was predicted in [83, 90, 94, 156, 170]. That the coefficient is negative in dimension d = 1 is related to the non-optimality of the Thomas-Fermi constant in the Lieb–Thirring inequality (3.35) and is well known in one-dimensional semi-classical analysis [21].

Even without having a clean upper bound like (3.47), it is reasonable to believe that

but this does not seem to be known at present. If the fixed density ρ is replaced by a well chosen locally constant density ρN converging to ρ, then this was proved in [71, Thm. 4].

3.4.5 Derivation from Levy–Lieb at Large Densities

In this section we show that our kinetic energy functionals can be obtained from the corresponding Levy–Lieb functionals in a proper limit of large densities. For completeness, we consider a rather arbitrary interaction potential w in any dimension.

Theorem 3.16 (Convergence at High Density)

Let $$w\in L^p(\mathbb {R}^d)+L^\infty (\mathbb {R}^d)$$ with p as in (3.1) and ρ⩾0 such that $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ . If $$\int _{\mathbb {R}^d}\rho \in \mathbb {N}$$ we have

(3.53)

If $$\int _{\mathbb {R}^d}\rho \in \mathbb {R}_+$$ and the additional classical stability assumption (3.14) holds, we have

(3.54)

One can also prove the convergence of optimal states or even write the theorem in the form of Gamma convergence. In a similar manner, TS[ρ] arises from the Hartree–Fock-type Levy–Lieb functional where one only minimizes over Slater determinants.

Since we have not found the proof in the literature, we provide it here for completeness.

Proof

We start with FLL. By scaling we see that

with the new interaction potential wλ(r) = λ⁻²w(r∕λ). Our assumptions on w imply that w is infinitesimally (− Δ)-form bounded, that is, $$|w|\leqslant \varepsilon (-\Delta )+C_\varepsilon $$ for all ε > 0. After scaling this implies

For the two-particle operator this gives

and we thus obtain

(3.55)

This yields the bound

The limit (3.53) follows after taking first λ →∞ and then ε → 0. For an explicit potential such as Coulomb we know how Cε depends on ε and one can then give a quantitative bound.

For FL the argument is exactly the same, with the same bound and T[ρ] replaced by TGC[ρ]. For FGC the above argument does not work due to the bad behavior in N. Instead, we rescale the stability assumption (3.14) on w and obtain

which provides the lower bound

For the upper bound we consider a fixed grand-canonical state Γ = ( Γn)n⩾0 such that

for some small η > 0. From the proof in Sect. 3.8, we can assume that Γ has compact support: Γn ≡ 0 for $$n\geqslant K$$ . Using the previous bound (3.55) in the canonical case, we obtain the bound

The limit now follows after taking λ →∞, ε → 0 and finally η → 0. □

Remark 3.1 (Adiabatic Connection)

For homogeneous potentials such as Coulomb, scaling ρ is the same as changing the strength of the interaction. This is the spirit of the adiabatic connection formula, which is often used in quantum chemistry to interpolate between the non-interacting and interacting problems [80]. Let us for instance discuss $$F^{w}_{\mathrm {L}}[\rho ]$$ and the corresponding kinetic energy $$T_{\mathrm {GC}}[\rho ]=F^{0}_{\mathrm {L}}[\rho ]$$ . We introduce a coupling constant t in front of w and look at the function $$t\mapsto F^{tw}_{\mathrm {L}}[\rho ]$$ . It is concave on [0, 1] (and increasing if w⩾0). It has left and right derivatives everywhere, which are given by the minimal and maximal values of the interaction energy among all the possible minimizers Γt of $$F^{tw}_{\mathrm {L}}[\rho ]$$ , by the Feynman-Hellmann theorem. These two derivatives are equal, except possibly on a countable subset of [0, 1]. We can express

where Γt is any minimizer for $$F^{tw}_{\mathrm {L}}[\rho ]$$ . This is a formula for the direct plus exchange-correlation energy in Kohn-Sham theory with fractional occupations. It is sometimes useful to consider a general path t ∈ [0, 1]↦wt in place of the simple linear switching, see [174] and [80, Sec. 2.4].

3.5 The Classical Interaction Energy and Lieb–Oxford Inequalities

In this section we study the Levy–Lieb functional with the kinetic energy dropped, which then becomes a purely classical problem.

3.5.1 A Multi-Marginal Optimal Transport Problem

In the classical problem there is no difference between fermions and bosons. In the canonical setting, the main variable is a symmetric probability density $$\mathbb {P}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N)$$ over $$(\mathbb {R}^d)^N$$ , which in the quantum case corresponds to

for pure states and to an average of such quantities for mixed states. The sum over the spin variables occurs since the interaction potential has been assumed to be spin-independent. In general, $$\mathbb {P}$$ will not be absolutely continuous with respect to the Lebesgue measure, however. The problem is therefore better stated in the form

StartEnclose box upper F Subscript upper S upper C upper E Baseline left bracket rho right bracket equals inf Underscript StartLayout 1st Row double struck upper P colon 2nd Row rho Subscript double struck upper P Baseline equals rho EndLayout Endscripts integral Underscript left parenthesis double struck upper R Superscript d Baseline right parenthesis Superscript upper N Baseline Endscripts sigma summation Underscript 1 less than or slanted equals j less than k less than or slanted equals upper N Endscripts w left parenthesis bold r Subscript j Baseline minus bold r Subscript k Baseline right parenthesis normal d double struck upper P left parenthesis bold r 1 comma ellipsis comma bold r Subscript upper N Baseline right parenthesis EndEnclose

(3.56)

with the density

The acronym SCE means Strictly Correlated Electrons [69, 160–164] since, as we will explain, the minimizing solution $$\mathbb {P}$$ is typically supported on a set of small dimension where the positions of the particles are highly dependent on each other. In general ρ can be a singular measure. In the worst case ρ is the sum of N Dirac deltas, in which case $$\mathbb {P}$$ has to be the symmetrized tensor product of these N deltas so that the locations of the particles are then completely fixed. For simplicity we will always assume that $$\rho \in L^1(\mathbb {R}^d)$$ . Nevertheless, the minimizing $$\mathbb {P}$$ need not be a function.

There is a grand-canonical version of FSCE which is stated in the form

StartLayout 1st Row StartEnclose box upper F Subscript upper G upper S upper C upper E Baseline left bracket rho right bracket equals inf Underscript StartLayout 1st Row double struck upper P equals left parenthesis double struck upper P Subscript n Baseline right parenthesis Subscript n greater than or slanted equals 0 Baseline 2nd Row sigma summation Underscript n greater than or slanted equals 0 Endscripts double struck upper P Subscript n Baseline left parenthesis left parenthesis double struck upper R Superscript d Baseline right parenthesis Superscript n Baseline right parenthesis equals 1 3rd Row sigma summation Underscript n greater than or slanted equals 1 Endscripts rho Subscript double struck upper P Sub Subscript n Subscript Baseline equals rho EndLayout Endscripts sigma summation Underscript n greater than or slanted equals 2 Endscripts integral Underscript left parenthesis double struck upper R Superscript d Baseline right parenthesis Superscript n Baseline Endscripts sigma summation Underscript 1 less than or slanted equals j less than k less than or slanted equals n Endscripts w left parenthesis bold r Subscript j Baseline minus bold r Subscript k Baseline right parenthesis normal d double struck upper P Subscript n Baseline left parenthesis bold r 1 comma ellipsis comma bold r Subscript n Baseline right parenthesis period EndEnclose EndLayout

(3.57)

This was introduced in [106] and further studied in [42, 107, 108]. For the problem to be well posed for all densities, w needs to satisfy the stability condition (3.14).

The two classical problems (3.56) and (3.57) belong to the class of multi-marginal optimal transport problems [33, 35, 41, 137, 162]. We only mention here a few striking results. The existence of a minimizing $$\mathbb {P}$$ for (3.56) follows by compactness arguments similar to Theorem 3.2, for a large class of interaction potentials including the Coulomb potential. The argument is the same in the grand-canonical case (3.57). It was proved in [30] that the infimum can be restricted to Monge states, which are the most correlated N-particle probability densities with one-particle density ρ and take the form

(3.58)

where $$T:\mathbb {R}^d\to \mathbb {R}^d$$ is a transport map such that T#ρ = ρ and TN = Id and Sym denotes symmetrization. The formula means that the position y = r₁ of the first particle completely determines the positions r₂ = Tr₁, …, rN = TN−1r₁ of the other N − 1 particles through the transport map T (and the picture is symmetrized with respect to the indices of the particles at the end). When moving the first particle (at the appropriate speed so as to build the desired density ρ) the other particles follow in a ‘strictly correlated’ way.

Even if the infimum in (3.56) is the same when restricted to Monge states, there might exist no Monge minimizer [32, 162]. Only when N = 2, or in one dimension for all N⩾2 [29] one can be sure that Monge minimizers exist. In fact, in dimension d = 1 and for a positive interaction w⩾0, the problem admits a minimizer $$\mathbb {P}$$ which does not depend on w at all! It is the Monge state with increasing transport map $$T\#(\rho {\mathbbm 1}_{(r_{k-1},r_k)})=\rho {\mathbbm 1}_{(r_{k},r_{k+1})}$$ where r₀ = −∞ < r₁ < r₂ < ⋯ < rN−1 < rN = +∞ are chosen such that $$\int _{r_i}^{r_{i+1}}\rho (r)\,\mathrm {d} r=1$$ [29]. The corresponding N-particle probability can also be expressed in the manner

(3.59)

where rk(s) : [0, 1] → [rk−1, rk] is the inverse of the increasing function $$r\mapsto s_k(r)=\int _{r_{k-1}}^r\rho (t)\,\mathrm {d} t$$ . This is displayed in Fig. 3.2. For instance, for the uniform density $$\rho (r)={\mathbbm 1}(0\leqslant r\leqslant N)$$ we have simply T(y) = y. The N points are placed on the lattice $$(y+\mathbb {Z})\cap [0,N)$$ and their position is averaged over s ∈ [0, 1]:

(3.60)

This is called a floating Wigner crystal in Physics and Chemistry [9, 47, 107, 130], since the particles are exactly located on a lattice, whose position is varied. We will come back to this special state later in Sect. 3.5.4.

Fig. 3.2

Form (3.59) of the optimal Monge-type probability $$ \mathbb {P}$$ in one dimension. The positions of all the particles are fixed by the position of the first particle and they are moved to the right at a proper speed so as to reproduce the desired density ρ

In [22] some interesting properties of the exact (Monge or not Monge) minimizer $$\mathbb {P}$$ of (3.56) were established. This includes the fact that the N particles have a positive distance to each other on the support of $$\mathbb {P}$$ , for a repulsive interaction such as Coulomb. The dual formulation is similar to (3.11) and takes the form

(3.61)

One important feature of the SCE problem is that there exists an optimal potential vSCE solving the supremum in (3.61), under rather weak assumptions on the interaction potential w. The optimal vSCE is called a Kantorovich potential. This is in stark contrast to the quantum case (3.11), where unique continuation drastically reduces the set of densities ρ for which the supremum is attained (see Sect. 3.7 below). Under the sole assumption that w is radial decreasing, diverges at the origin and is C¹ outside of the origin (like for the Coulomb potential w(r) = |r|⁻¹ in dimension d = 3), it was proved in [22] that there exists an optimal Kantorovich potential vSCE which is bounded and Lipschitz. An optimal N-particle probability $$\mathbb {P}$$ must then be supported on the set

In other words, the N particles should minimize the associated N-particle classical problem with the external potential vSCE. All densities $$\rho \in L^1(\mathbb {R}^3)$$ are v-representable in the classical case.

3.5.2 Convergence of the Levy–Lieb Functional at Low Density

We have seen above in Theorem 3.16 that the kinetic energy functional becomes dominant at large densities. Similarly, the interaction becomes dominant at low densities, provided that w has the right scaling at large distances. To simplify our exposition, from now on we restrict our discussion to power-law (Riesz) potentials

The main result is the following.

Theorem 3.17 (Convergence at Low Density)

Let w(r) = |r|−swith $$0&lt;s&lt;\min (2,d)$$ . Let ρ⩾0 such that $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ . If $$\int _{\mathbb {R}^d}\rho \in \mathbb {N}$$ we have

(3.62)

If $$\int _{\mathbb {R}^d}\rho \in \mathbb {R}_+$$ we have

(3.63)

Note that the stability condition (3.13) is always satisfied for the positive potential w(r) = |r|−s, hence the grand-canonical energies are well defined.

The proof is much more complicated than Theorem 3.16, since an optimizer $$\mathbb {P}$$ for FSCE[ρ] or FGSCE[ρ] will never have a finite kinetic energy, even under the assumption that $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ . The limit for FLL was shown for $$\int _{\mathbb {R}^d}\rho =2$$ with spin first by Cotar et al. in [33] and later extended to $$\int _{\mathbb {R}^d}\rho =3$$ by Bindini and de Pascale in [8]. The limit for FL[ρ] and all $$\int _{\mathbb {R}^d}\rho \in \mathbb {N}$$ was solved in [103] whereas the case of FLL was finally treated in [34]. The proof for FGC follows along the lines of [103]. The next order in λ was predicted in [70] and proved in some cases in [31].

To summarize, at large densities the Levy–Lieb functional behaves like the kinetic energy of non-interacting quantum particles, whereas at low density the particles tend to be very correlated and solve the corresponding classical problem.

3.5.3 Lieb–Oxford Inequality

We discuss here upper and lower bounds on the interaction energy, with an emphasis on lower bounds (Lieb–Oxford inequality).

The easiest upper bound is obtained by taking the decorrelated trial state $$\mathbb {P}=(\rho /N)^{\otimes N}$$ , that is, independent particles distributed according to the density ρ. This gives the bound

(3.64)

The right side is, up to the constant 1 − 1∕N, the classical energy of the density distribution ρ and it is a non-local term. The factor 1∕N can be dropped for repulsive potentials.

It is relatively easy to prove a similar lower bound, under the additional assumption that w is continuous and has a non-negative Fourier transform, ModifyingAbove w With caret greater than or slanted equals 0 ). In this case we have

double integral Underscript double struck upper R Superscript 2 d Baseline Endscripts w left parenthesis bold r minus bold r Superscript prime Baseline right parenthesis normal d eta left parenthesis bold r right parenthesis normal d eta left parenthesis bold r Superscript prime Baseline right parenthesis equals left parenthesis 2 pi right parenthesis Superscript d divided by 2 Baseline integral Underscript double struck upper R Superscript d Baseline Endscripts ModifyingAbove w With caret left parenthesis k right parenthesis StartAbsoluteValue ModifyingAbove eta With caret left parenthesis k right parenthesis EndAbsoluteValue squared normal d k greater than or slanted equals 0

for every signed measure η. Taking $$\eta =\sum _{j=1}^N\delta _{{\mathbf {r}}_j}-f$$ and expanding we find the pointwise inequality on $$(\mathbb {R}^d)^N$$

This is valid for all f and the last error term comes from the case j = k. Integrating against any state $$\mathbb {P}$$ with density ρ and taking f = ρ, we obtain the following lower bound:

StartLayout 1st Row upper F Subscript upper S upper C upper E Baseline left bracket rho right bracket greater than or slanted equals one half double integral Underscript double struck upper R Superscript 2 d Baseline Endscripts w left parenthesis bold r minus bold r Superscript prime Baseline right parenthesis rho left parenthesis bold r right parenthesis rho left parenthesis bold r Superscript prime Baseline right parenthesis normal d bold r normal d bold r Superscript prime Baseline minus StartFraction w left parenthesis 0 right parenthesis Over 2 EndFraction integral Underscript double struck upper R Superscript d Baseline Endscripts rho left parenthesis bold r right parenthesis normal d bold r comma when ModifyingAbove w With caret greater than or slanted equals 0 period EndLayout

(3.65)

For a long-range potential the first term grows faster than N for most densities, hence the last error term is often much lower than the classical energy.

For Coulomb or other power-law potentials, the previous argument does not work since w(0) = +∞. One solution is to regularize the potential at the origin but this also modifies the classical interaction energy. One can estimate the error under appropriate regularity assumptions on ρ. But Lieb [113] and then Lieb–Oxford [120] have proved a universal bound which has the right scaling behavior and does not require the potential to be smeared out. We state it for power-law potentials but the inequality is slightly more general.

Theorem 3.18 (Lieb–Oxford Inequality [4, 72, 113, 120, 121, 124, 127])

Assume that w(r) = |r|−swith 0 < s < d in dimension d⩾1. Then there exists a universal constant cLO(s, d) > 0 such that

(3.66)

for every $$\rho \in (L^1\cap L^{1+\frac {s}{d}})(\mathbb {R}^d,\mathbb {R}_+)$$ .

From now on we always call cLO(s, d) the smallest constant for which the inequality (3.66) is valid for all ρ. Note that cLO(s, d) works for every particle number $$\int _{\mathbb {R}^d}\rho $$ . If one adds the constraint that $$\int _{\mathbb {R}^d}\rho =\lambda $$ then the optimal constant depends on λ but it is non-decreasing and has the limit cLO(s, d) when λ →∞.

Although only the case s = 1 and d = 3 was considered in the original papers [113, 120], the proof for s = 1 and d = 2 given in [4, 72, 124] extends to any 0 < s < d in any dimension, see [127, Lemma 16]. This proof involves the Hardy–Littlewood estimate for the maximal function Mρ [73],

and, consequently, the best known estimate on cLO(s, d) involves the unknown constant cHL(s, d). A Lieb-Oxford bound was shown for $$w(\mathbf {r})=-\log |\mathbf {r}|$$ in two dimensions in [110, Prop. 3.8]. In dimension d = 1, optimal Lieb-Oxford bounds are studied in [40].

In the 3D Coulomb case, d = 3 and s = 1, the best estimate known so far on the optimal Lieb-Oxford constant is

StartEnclose box 1.4442 less than or slanted equals c Subscript upper L upper O Baseline left parenthesis 1 comma 3 right parenthesis less than or slanted equals 1.5765 period EndEnclose

(3.67)

The upper constant was equal to 8.52 in [113], to 1.68 in [120] and later improved to 1.64 in [89]. The better value 1.58 was obtained very recently in [109]. The lower bound has been claimed in [100, 138] and only shown recently in [36, 108]. It will be discussed in the next section. It was conjectured in [100, 135, 148] that the best Lieb-Oxford constant is indeed about 1.44. It remains an important challenge to find the optimal constant in (3.66). Several of the most prominent functionals used in Density Functional Theory make use of the value of the Lieb-Oxford constant for calibration [100, 138, 139, 143, 166–168].

A different Lieb-Oxford inequality was recently proved in the 3D Coulomb case in [109]. It reads

(3.68)

under the additional assumption that $$\mathbb {P}$$ has negative correlations, which means

(3.69)

This condition is satisfied when $$\mathbb {P}$$ is the square of a Slater determinant (3.19), in which case the left side of (3.68) is called the exchange energy and the best constant is believed to be 1.09 [143]. But many other states satisfy the condition (3.69). In statistical mechanics, this is typical of gas phases [152] at high temperature. Since 1.25 < 1.44 < cLO(1, 3), this means that such states cannot provide the optimal Lieb-Oxford constant. In fact, we explain below how to obtain the lower bound 1.44 from a solid (periodic) phase.

The indirect energy is the equivalent of the exchange-correlation energy defined in the quantum case in Sect. 3.2.4:

For power-law interactions it is always negative and bounded from below by a constant times $$\int _{\mathbb {R}^d}\rho ^{1+s/d}$$ .

3.5.4 Constant Densities and the Classical Uniform Electron Gas

We discuss here the special case of densities which are constant over a finite set and the limit when this set fills the whole space. This is the classical equivalent of the Uniform Electron Gas discussed in the quantum case in Sect. 3.3.1 above. This special case will give us some lower bounds on the Lieb–Oxford constant cLO(s, d), including the bound 1.44 in dimension d = 3 stated in (3.67).

The classical equivalent of Theorem 3.6 was proved in [106].

Theorem 3.19 (The Classical Uniform Electron Gas Energy [106])

Assume that w(r) = |r|−swith 0 < s < d in dimension d⩾1. Let ρ₀ > 0. Let Ω be a fixed open convex set of unit volume | Ω| = 1. Then there exists a universal constant cUEG(s, d) > 0 such that

(3.70)

In particular we obtain $$c_{\mathrm {LO}}(s,d)\geqslant -c_{\mathrm {UEG}}(s,d)$$ .

The constant cUEG(1, 3) is the one which has appeared before in Theorem 3.7. At low density, the quantum UEG behaves like a classical gas by an equivalent of Theorem 3.17 for infinite systems [106].

Note that the classical canonical and grand-canonical functionals are known to give the same thermodynamic limit. In the quantum case this is not yet known. We have stated the theorem for a fixed domain Ω which is scaled but the same result holds for a general sequence ΩL that has a regular boundary in the sense of Fisher [106].

Except in dimensions d ∈{1, 8, 24} and $$\max (0,d-2)\leqslant s&lt;d$$ , some special cases to which we will come back, the constant cUEG(s, d) is unknown. In order to get upper bounds on cUEG(s, d), we need to construct trial states. The idea is to use a floating crystal similar to (3.60), that is, to place the particles on a lattice and then average over translations to obtain a constant density.

Let $$\mathscr {L}\subset \mathbb {R}^d$$ be a lattice of normalized unit cell Q. We then only retain the points of the lattice intersecting the large cube CL = (−L∕2, L∕2)d and average over the translations of this finite lattice over Q. This way we obtain a trial state which is constant over the union of the corresponding translates of Q. In general this is only an approximation of CL but since the limit (3.70) is insensitive to the type of domains, this will not create any difficulty. The trial state is, therefore, given by

(3.71)

Then $$\mathbb {P}_{\mathscr {L},L}$$ has the constant density

(3.72)

The state is as displayed in Fig. 3.3. Note that the energy of the probability measure $$\mathbb {P}_{\mathscr {L},L}$$ is simply the interaction of the lattice points, since the interaction potential is translation-invariant:

Fig. 3.3

A two-dimensional picture of the Jellium model. The dots represent the point particles, which are placed on a finite subset of a lattice $$\mathscr {L}$$ . The colored set is the union of the corresponding unit cells and it represents a uniform background charge distribution of opposite charge. The indirect energy of the floating crystal is obtained after integrating the position of the lattice over the unit cell Q as in (3.79). When the lattice is not centered, this results in an excess of point charges on one side and an excess of background charge on the other side, indicated by the two rectangles. These large boundary charge fluctuations are responsible for the shift in Theorem 3.20 at s = d − 2

It is instructive to see first what happens in the short range case s > d. Then the energy per unit volume converges to

(3.73)

The function on the right side is called the Epstein Zeta function [10, 17, 51] and it is the natural generalization to $$\mathbb {R}^d$$ of the usual Riemann Zeta function, which it coincides with when d = 1 (hence $$\mathscr {L}=\mathbb {Z}$$ ). It turns out that the limit in the long range case can be expressed with the (analytic extension) of $$\zeta _{\mathscr {L}}$$ , for potentials decaying to zero at infinity faster than Coulomb. Something special is happening at s = d − 2.

Theorem 3.20 (Indirect Energy of the Floating Wigner Crystal [14–16, 96, 104, 105])

Let $$d-2\leqslant s&lt;d$$ in dimension d⩾3 and 0 < s < d in dimensions d = 1, 2. Let $$\mathscr {L}\subset \mathbb {R}^d$$ be a lattice with a normalized unit cell Q having no dipole and no quadrupole moment:

Then the indirect energy per unit volume of the floating Wigner crystal (3.71) converges to

(3.74)

where $$\zeta _{\mathscr {L}}(s)$$ is the analytic continuation to $$\mathbb {C}\setminus \{d\}$$ of the Epstein Zeta function on the right of (3.73), initially defined for $$\Re (s)&gt;d$$ .

A similar result holds for $$-1\leqslant s\leqslant 0$$ in d = 1 and s = 0 in d = 2 [96, 104].

The first divergent term in the lattice sum is the classical energy, which behaves like N2−s∕d and depends on the shape of the chosen large domain CL:

(3.75)

This is because the lattice sum is a Riemann sum for the corresponding integral at that scale. Replacing CL by another set changes this macroscopic term. Note that the analytic extension of (3.75) is a o(Ld) for s > d. This term probably exists in the short range case too, but it is lower order and it was not seen in the limit (3.73).

The theorem provides the next order term in the long range case d − 2 < s < d. This is an extensive quantity (of the order of the volume) which has a limit independent of the shape CL. This limit is simply the analytic extension of the short range energy. This is compatible with our picture that the classical energy (3.75) is the leading term for s < d but once it is removed, we are essentially back to (3.73).

At s = d − 2 the picture changes. Another term of the order L2(d−1)−s, which was lower order for all s > d − 2, becomes relevant for the energy per unit volume at s = d − 2 and dominates for s < d − 2. As we will explain later, this is a kind of surface term.

In dimension d = 1 we know from [29] that the floating crystal is optimal and provides the minimal classical energy at constant density. Therefore we deduce from Theorem 3.20 that

(3.76)

In particular, $$-c_{\mathrm {LO}}(s,1)\leqslant \zeta (s)$$ .

In higher dimensions, the floating crystal is not known to be an exact minimum, and furthermore there are several possible crystals. Therefore we only obtain the upper bound

(3.77)

The minimum is over all lattices of normalized unit cell. It is expected that the last inequality should be an equality for some values of the dimension d including d = 1, 2, 3. So far this is only known in dimensions d = 8 and d = 24 [25, 145]. In dimension d = 2 the minimum on the right of (3.77) is known to be achieved by the triangular lattice [23, 43, 50, 131, 147] whereas in dimension d = 3, numerics indicates that it is achieved by the Body-Centered Cubic lattice (BCC) for 0 < s⩽3∕2 and the Face-Centered Cubic lattice (FCC) for $$3/2\leqslant s&lt;3$$ [10, 17, 68, 157].

The surprising jump of the energy per unit volume (3.74) in the Coulomb case s = d − 2 was first discovered in 1979 by Hall [76] based on an unpublished remark by Plaskett in 1959. The conendrum raised by Hall was discussed in several papers in the 80s, see for instance [1, 39, 77, 78, 86, 134]. It was rediscovered in 1988 by Borwein et al. [15] and was recently revived and reformulated in [105, App. B]. It has indeed always been assumed in the Physics and Chemistry literature that the floating crystal is a good trial state for the UEG, and that cUEG(1, 3) should even be equal to the BCC lattice energy, whose value is ζBCC(1) ≃−1.4442 (see [26] and [68, p. 43]). This value is used in most DFT functionals based on the Uniform Electron Gas. But the Coulomb potential is exactly the one for which the floating crystal behaves badly, by Theorem 3.20.

Note that the jump exists and is unavoidable in 1D, where the floating crystal is known to be optimal, but it happens at the negative value s = −1. Indeed, for w(r) = −|r| we have the expansion similar to Theorem 3.20

(3.78)

with 1∕6 > −ζ(−1) = 1∕12.

In order to better understand what is going on, it is useful to reinterpret the result in terms of the Jellium model [104, 119]. In this model there is no constraint on the electronic density but the particles interact with a compensating uniform background of opposite charge. At density one, the corresponding energy is defined by

where Ω is any measurable set of volume | Ω| = N, representing the uniform background. A short calculation shows that the indirect energy of the floating crystal can be written in the form

(3.79)

In other words, it is the average of Jellium energies where the lattice points are moved over the fixed background ΩL. In this interpretation it becomes clear why the averaging over y induces the shift in (3.74): moving the particles away from the center of the unit cells is not at all energetically favorable. When the particles are moved in one direction this creates a large excess of negative charges on one side and a corresponding excess of background charge on the opposite side (Fig. 3.3). These two opposite boundary charges have an interaction energy proportional to (Ld−1)²∕Ls = L2(d−1)−s which is exactly of the order of the volume in the Coulomb case s = d − 2 and grows faster for s < d − 2. On the other hand, when the particles are placed exactly at the center of the unit cells, one recovers the analytic extension of the short range energy for all s > d − 4.

Theorem 3.21 (Jellium Energy of the Clamped Wigner Crystal [10, 14–16, 96, 104, 105, 119])

Assume that d⩾1. Let $$\mathscr {L}\subset \mathbb {R}^d$$ be a lattice satisfying the same assumptions as in Theorem3.20. Then the Jellium energy per unit volume of the Wigner crystal clamped at the center of the unit cells converges to

(3.80)

for all $$\max (0,d-4)&lt;s&lt;d$$ .

The floating crystal is really not a good trial state for the UEG. The conundrum raised in Theorem 3.20 was recently resolved in [36, 107]. Cotar and Petrache managed to prove in [36] that the (unknown) UEG energy cUEG(s, d) is always continuous for 0 < s < d and that it is equal to the (also unknown) Jellium energy for $$d-2\leqslant s&lt;d$$ . The proof of continuity in s is very delicate and requires the use of advanced analytical techniques due to Fefferman and collaborators [52, 74, 85]. A short time later, the same result was obtained in [107] with a different and much simpler argument. Here we only explain this argument for the special case of the floating crystal, that is, we show how to modify the trial state (3.71) in order to cancel the shift appearing in (3.74).

The main idea of [107] is to immerse the crystal in a thin layer of fluid. In other words, the floating crystal is melted close to the boundary in order to reduce the large charge fluctuations. The fluid gets displaced with the crystal when the latter is averaged over translations. Think of a block of ice completely filling a container. In order to move the ice it is necessary to melt it close to the container walls.

To describe this procedure, let us denote by $$C^{\prime }_L$$ a slightly larger cubic container such that $$\Omega _L+Q\subset C^{\prime }_L$$ , where we recall that ΩL is the union of the unit cells Q + ℓ with $$\ell \in \mathscr {L}\cap C_L$$ . We can take $$C^{\prime }_L=C_{L+\lambda }$$ , where λ is any fixed distance larger than the diameter of Q. We assume that the volume of the fluid $$|C^{\prime }_L\setminus \Omega _L|=M$$ is an integer. It satisfies $$M\leqslant CL^{d-1}\ll L^d$$ . The new trial state has the N = | ΩL| particles on the floating crystal, translated by y ∈ Q as before in (3.71), together with M other particles forming an uncorrelated fluid in $$C^{\prime }_L\setminus (\Omega _L+\mathbf {y})$$ , the set remaining after we have subtracted the union of all the cells centered at the particle positions (Fig. 3.4):

(3.81)

Note that the state of the fluid is correlated with the position y of the crystal.

Fig. 3.4

A two-dimensional picture of the modified floating crystal (3.81) from [107]. The dots represent the point particles which are at the centers of hexagons of volume one. As the whole crystal block is translated by y, the incompressible fluid gets displaced to fill the remaining space $$C^{\prime }_L\setminus (\Omega _L+\mathbf {y})$$ . The resulting density is only constant well inside the container

Theorem 3.22 (Indirect Energy of the Modified Floating Crystal [107])

Let $$\max (0,d-4)&lt; s&lt;d$$ in dimension d⩾1 and $$\mathscr {L}\subset \mathbb {R}^d$$ a lattice satisfying the same assumptions as in Theorem3.20. Then the indirect energy per unit volume of the modified floating Wigner crystal (3.81) converges to

(3.82)

In particular, we obtain

For the BCC lattice in dimension d = 3, one finds the claimed lower bound

It is reasonable to conjecture that

for d = 1, 2, 3 and all 0 < s < d, which amounts to saying that the uniform electron gas is always crystallized at zero temperature [10, 104]. The modified trial state (3.81) suggests that the system can only be a solid in the bulk. In a neighborhood of the boundary, it is probably a fluid because the particles have to be able to move sufficiently far away around the crystal to compensate the large charge fluctuations.

In this section we have explained some major difficulties encountered when trying to construct good trial states for the Uniform Electron Gas. Those are entirely due to the boundary, namely to the fact that we work with a finite piece of material in the physical space $$\mathbb {R}^d$$ . If we set up the model on the torus, as is often done in practical calculations, these difficulties disappear [104, Sec. IV.C].

It is worth mentioning an estimate due to Lieb and Narnhofer [119] in the Coulomb case s = 1 in dimension d = 3 which states that

(3.83)

for any open set Ω of integer volume. The constant (3∕5)(9π∕2)1∕3 ≃ 1.4508 is surprisingly close to the expected optimal value − ζBCC(1) and it implies in any case that

StartEnclose box negative 1.4508 less than or slanted equals c Subscript upper U upper E upper G Baseline left parenthesis 1 comma 3 right parenthesis less than or slanted equals negative 1.4442 period EndEnclose

This is the bound that appeared in Theorem 3.7. For negatively-correlated states as in (3.69), the constant can be replaced by (3∕2)(π∕6)1∕3 ≃ 1.2090 [109].

In dimension d = 3 for s = 1 it was conjectured in [135, 148] that the classical Uniform Electron Gas gives the optimal Lieb–Oxford constant, i.e., cLO(1, 3) = −cUEG(1, 3).

Remark 3.23 (Determinantal Processes and the Dirac Constant)

Let $$\rho =\rho _0{\mathbbm 1}_{C_L}$$ with $$N=\rho _0 L^d\in \mathbb {N}$$ . Instead of the floating crystal (3.71) and its modified version (3.81), one can consider the square of a Slater determinant as a trial state:

where k₁, …, kN are N distinct points in $$\mathbb {Z}^d$$ (our trial state contains no spin). We find

As for the free Fermi gas we choose all the points ki in a given ball centered at the origin. For w(r) = |r|−s the second term behaves like $$L^d\rho _0^{1+s/d}c_{\mathrm {D}}(s,d)$$ where the Dirac constant [44]

c Subscript normal upper D Baseline left parenthesis s comma d right parenthesis equals StartFraction 1 Over 2 left parenthesis 2 pi right parenthesis Superscript d Baseline EndFraction integral Underscript double struck upper R Superscript d Baseline Endscripts StartFraction StartAbsoluteValue ModifyingAbove double struck 1 Subscript upper B Sub Subscript k Sub Sub Subscript normal upper F Sub Subscript Subscript Baseline With caret left parenthesis bold r right parenthesis EndAbsoluteValue squared Over StartAbsoluteValue bold r EndAbsoluteValue Superscript s Baseline EndFraction normal d bold r

is the exchange energy per unit volume of a free Fermi gas. Here the Fermi radius is $$k_{\mathrm {F}}=\sqrt {2(d+2)c_{\mathrm {TF}}(d)/d}$$ . This proves that

but this bound is worse than the floating crystal. The particles are not correlated enough. In dimension d = 3 with s = 1 one finds $$c_{\mathrm {D}}(1,3)=(3/4)\left (6/\pi \right )^{1/3}\simeq 0.9305$$ . Recall that for an arbitrary determinantal point process, we have the better Lieb-Oxford inequality (3.68) from [109].

3.5.5 Local Density Approximation for the Classical Interaction Energy

We have discussed in the previous section the case of exactly constant densities and their limit of infinite volume. We now consider the case of slowly varying densities, which are assumed to be essentially constant over large sets.

First, we mention that the Lieb–Narnhofer bound (3.83) from [119] was generalized to arbitrary densities, in the form of lower bounds involving gradient-type corrections [7, 105]. For instance, the bound

(3.84)

was shown to hold in [105] for any ε > 0.

The following result gives a quantitative estimate on the grand-canonical classical energy for slowly varying densities.

Theorem 3.24 (Local Density Approximation of the Classical Coulomb Energy [108])

Consider the case w(r) = |r|⁻¹in dimension d = 3. There exists a constant C such that

(3.85)

for every ε > 0 and every non-negative density $$\rho \in L^1(\mathbb {R}^3)\cap L^{4/3}(\mathbb {R}^3)$$ such that $$\nabla \rho ^{1/3}\in L^4(\mathbb {R}^3)$$ .

The gradient term can be replaced by $$\varepsilon ^{-b}\int _{\mathbb {R}^3}|\nabla \rho ^\theta (\mathbf {r})|{ }^p\,\mathrm {d}\mathbf {r}$$ for any p > 3 and 0 < θ < 1 such that θp⩾4∕3, with $$b=\max \{2p-1,(1+3\theta )p-4\}$$ .

If we fix a density ρ with $$\int _{\mathbb {R}^3}\rho =1$$ and take ρN(r) = ρ(rN−1∕3), then we find that

(3.86)

Compare this expansion with the quantum case (3.30).

It is an open problem to prove an estimate similar to (3.85) for the canonical SCE functional FSCE. However, a non-quantitative convergence similar to (3.86) is known even for w(r) = |r|−s in all dimensions d > s > 0:

(3.87)

In the Coulomb case for d = 3, (3.87) was proved in [106] whereas general power-law potentials were covered in [37].

3.6 Upper and Lower Bounds on the Levy–Lieb Functionals

In the previous sections we have studied the kinetic and interaction energies separately and reviewed several known upper and lower bounds. Since the minimum of a sum is always greater than or equal to the sum of the minima, we easily obtain lower bounds on the full Levy–Lieb functional. For instance, putting together the Lieb–Thirring and Lieb–Oxford inequalities, we find

for the interaction w(r) = |r|−s in dimension d > s > 0. The right side takes the same form as the Thomas–Fermi–Dirac functional, except for the values of the two constants in front of the terms ρ1+2∕d and ρ1+s∕d. One would obtain the right constant in front of the term ρ1+2∕d in dimension d⩾3 if the Lieb–Thirring conjecture mentioned in Sect. 3.4.2 had been proved. We refer to [114] for a review of results on Thomas–Fermi-type functionals. From Nam’s bound (3.40) one can replace the Lieb–Thirring constant by (1 − ε)cTF(d) at the expense of a (negative) gradient correction.

Upper bounds are more complicated because a trial state that works for T[ρ] could be very bad for the classical energy and conversely. We have seen in Sect. 3.4 that the set of one-particle density matrices that are N-representable by a mixed states is exactly given by the operators γ = γ∗ such that $$0\leqslant \gamma \leqslant 1$$ and Tr (γ) = N. This is because any such operator is the convex combination of rank-N projections which correspond to Slater determinants. But for estimating the interaction energy we need some more information on the two-particle density.

An explicit convex combination which provided a bound on the two-particle density matrix was derived in [115]. The idea is the following. Assume for simplicity that $$\gamma =\sum _{i=1}^Kn_i|u_i\rangle \langle u_i|$$ has finite rank K. By Horn’s lemma [115] there exists a set of N orthonormal vectors V¹, …, VN in $$\mathbb {C}^K$$ such that $$\sum _{k=1}^N|V^k_i|{ }^2=n_i$$ for all i. Define then the new orbitals $$f^k_\theta =\sum _{j=1}^K\mathrm{e} ^{i\theta _j}V^k_ju_j$$ where θ = (θ₁, …, θK) ∈ (0, 2π)K and the associated trial mixed state

A computation shows that its one-particle density matrix is exactly γ, whereas its two-particle density matrix is

(3.88)

where $$\mathscr {A}$$ is the orthogonal projection onto the anti-symmetric two-particle subspace $$L^2(\mathbb {R}^d\times \mathbb {Z}_q,\mathbb {C})\wedge L^2(\mathbb {R}^d\times \mathbb {Z}_q,\mathbb {C})$$ . The first operator has the integral kernel

(3.89)

The operator $$\mathscr {A} (\gamma \otimes \gamma ) \mathscr {A}$$ is exactly the two-particle density matrix of the unique quasi-free state over the Fock space that has the one-particle density γ [6]. Integrating against the potential w, the first term in (3.89) gives the classical energy (Hartree term) whereas the second gives the exchange term, which is non-positive for w⩾0.

Since the last term in (3.88) is a non-positive operator, the following was obtained in [115] after using the density of finite rank operators.

Lemma 3.25 (Mixed Canonical States and Quasi-Free States [115])

Let $$0\leqslant \gamma =\gamma ^*\leqslant 1$$ be a one-particle density matrix such that $$\mathrm {Tr}\,(\gamma )=N\in \mathbb {N}$$ . Then there exists a mixed state Γ over the fermionic N-particle space $$\bigwedge _1^NL^2(\mathbb {R}^d\times \mathbb {Z}_q,\mathbb {C})$$ such that its one-particle density matrix is γ and its two-particle density matrix Γ⁽²⁾satisfies

(3.90)

in the operator sense.

Using the lemma for the trial state (3.48) employed in the proof of Theorem 3.14 and neglecting the exchange term, the following was derived in [108].

Theorem 3.26 (Upper Bound on FL[ρ] [108])

For w(r) = |r|−sin dimension d > s > 0, we have

(3.91)

for any ε > 0.

This time, the right side of (3.91) involves an energy functional that looks like the Thomas–Fermi–von Weizäcker energy [114]. It is an open problem to derive a similar upper bound for FLL.

3.7 The Hohenberg–Kohn Theorem

In this chapter we have mainly discussed the convex formulation of Density Functional Theory [80, 116] based on the universal functionals of Levy and Lieb. Another important result is the Hohenberg-Kohn theorem [82] which, in spite of its rather abstract character, is often cited as the main justification for the use of the density to replace the N-particle wavefunction. As we will explain, the necessary assumptions for the validity of this theorem are not yet fully understood mathematically. In fact, this result relies on the unique continuation principle which is not completely settled for N-particle Hamiltonians. Before stating the Hohenberg-Kohn theorem, we therefore start by discussing unique continuation in detail.

For simplicity we assume throughout the whole section that there is no spin:

Adding q does not change anything in the following results but it makes the notation a bit heavier.

3.7.1 Many-Body Unique Continuation

We refer for instance to [95] for a discussion on the importance of the unique continuation for the Hohenberg–Kohn theorem. Because unique continuation is a purely local property we allow here external potentials v whose positive part $$v_+=\max (v,0)$$ is only locally integrable. For simplicity, we assume that its negative part $$v_-=\max (-v,0)$$ and the interaction potential w are infinitesimally (− Δ)-form-bounded, as was done in the body of the chapter.

Definition 3.27 (Many-Body Unique Continuation)

Let

with v±⩾0 and p satisfying (3.1). We say that the potentials v = v₊ − v₋ and w satisfy the many-body unique continuation principle if, for every integer N⩾1, (the Friedrichs realization of) $$H_N^{v,w}$$ satisfies the unique continuation principle in its form domain: if we have $$H_N^{v,w}\Psi =\lambda \Psi $$ for some $$\lambda \in \mathbb {R}$$ and $$\Psi \in \mathcal {Q}(H_N^{v,w})$$ with |{ Ψ = 0}| > 0, then Ψ ≡ 0.

The equation $$H_N^{v,w}\Psi =0$$ is understood in $$\mathcal {Q}(H_N^{v,w})'$$ , that is,

for every $$\Phi \in \mathcal {Q}(H_N^{v,w})$$ or, equivalently, in the sense of distributions. Recall that the full N-body potential is defined by

Our formulation of unique continuation is one of the strongest possible, in that it only requires Ψ to vanish on a set of positive measure in order to deduce that Ψ ≡ 0. This is the property which is needed in the proof of the Hohenberg–Kohn theorem, as we will see. This is sometimes called “unique continuation on sets of positive measures”. For $$v_+\in L^p_{\mathrm {loc}}(\mathbb {R}^d,\mathbb {R}^+)$$ with p as in (3.1) it is shown in [38] that any $$\Psi \in H^1_{\mathrm {loc}}(\mathbb {R}^{dN})$$ vanishing on a set of positive measure and solving $$H_N^{v,w}\Psi =\lambda \Psi $$ must have a point $${\mathbf {X}}_0\in \mathbb {R}^{dN}$$ where it vanishes to infinite order, that is, such that

Unique continuation for functions vanishing to infinite order at one point is usually called “strong unique continuation”. Many authors consider instead the “weak unique continuation” problem where Ψ is instead assumed to vanish on an open set, but this is not sufficient for the Hohenberg–Kohn theorem.

Unique continuation is a very well studied question. Note first that if we restrict our attention to the potentials for N electrons in a molecule, where

then any eigenfunction of $$H_N^{v,w}$$ is analytic outside of the singularities of the potential [132], which form a set of zero measure. Therefore it satisfies the unique continuation principle. However, restricting the theory to this very special, though physically relevant, class of potentials is not appropriate in density functional theory. In order to fully understand the density, it is necessary to allow the largest possible class of potentials.

In a famous work [87], Jerison and Kenig have proved that the (strong) unique continuation principle holds for − Δ + W in $$\mathbb {R}^D$$ under the sole assumption that $$W\in L^p_{\mathrm {loc}}(\mathbb {R}^D)$$ with p satisfying (3.1) and d replaced by D. This was then generalized by Koch and Tataru in [92] and many other authors. These results apply to the N-particle setting under the condition that

and this is valid for all N⩾2 when

This is not far from asking that the potentials are locally bounded (in which case the result would follow for instance from the singular Carleman-type estimate proved in [151]). We see that Lp conditions are not well adapted to the N-particle problem, since they yield N-dependent constraints on v and w. More natural assumptions on v and w involve relative bounds with respect to the Laplacian, since such properties are easily propagated to all N. For instance if v and w are infinitesimally (− Δ)-form bounded in $$\mathbb {R}^d$$ ,

(3.92)

then so is the N-particle potential $$W_N^{v,w}$$ in $$\mathbb {R}^{dN}$$ for every N. It seems reasonable to conjecture that the many-body unique continuation principle holds under the sole assumption (3.92), but this is not known, even for N = 1. See [165] for a similar conjecture in the Kato class, which involves L¹ norms instead of L² norms.

Georgescu [67] and Schechter–Simon [158] have provided one of the first results for N-body systems with N-independent assumptions on v and w, but they required the wavefunction to vanish on an open set (weak unique continuation). Recently [63, 64], Garrigue has extended their result to cover the case of functions vanishing on a set of positive measure. His main assumption is that

for some δ > 0 and all ε, R > 0. This condition in the one-body space $$\mathbb {R}^d$$ is inherited by $$W_N^{v,w}$$ in $$\mathbb {R}^{dN}$$ for every N⩾1. After using the Sobolev inequality, the following result was shown in [64].

Theorem 3.28 (Unique Continuation for Lp Potentials [64])

Any potentials v, w with $$v_+\in L^p_{\mathrm {loc}}(\mathbb {R}^d)$$ and $$v_-,w\in L^p(\mathbb {R}^d)+L^\infty (\mathbb {R}^d)$$ with $$p&gt;\max (2,2d/3)$$ satisfy the many-body unique continuation property of Definition3.27.

Theorem 3.28 now covers Coulomb-type potentials. This is the best result known at the moment for many-body unique continuation. It is an important problem to generalize it to more singular potentials v.

3.7.2 Main Theorem and Some Open Problems

Let us now state the Hohenberg–Kohn theorem, which says that the density uniquely determines the potential, under the condition that unique continuation holds.

Theorem 3.29 (Hohenberg–Kohn)

Let $$(v_1)_-,(v_2)_-,w\in L^p(\mathbb {R}^d)+L^\infty (\mathbb {R}^d)$$ and $$(v_1)_+,(v_2)_+\in L^1_{\mathrm {loc}}(\mathbb {R}^d)$$ with p as in (3.1) and assume that (v₁, w) or (v₂, w) satisfies the many-body unique continuation property of Definition3.27. If there are two ground states Ψ₁and Ψ₂of, respectively, $$H_N^{v_1,w}$$ and $$H_N^{v_2,w}$$ so that $$\rho _{\Psi _1}=\rho _{\Psi _2}$$ , then we have v₁ = v₂ + C for some constant C.

The following proof is essentially the one given in [63, 116].

Proof

Changing v₂ into v₂ − (EN[v₂] − EN[v₁])∕N we can assume that the two ground state energies are equal. Note that the assumption $$\rho _{\Psi _1}=\rho _{\Psi _2}$$ implies that $$\Psi _1\in \mathcal {Q}(H^{v_2,w}_N)$$ and $$\Psi _2\in \mathcal {Q}(H^{v_1,w}_N)$$ , the form domains of the two operators defined in (3.3). We can write

Exchanging the two indices and using that the two densities are equal, we obtain that there is equality everywhere. In particular, Ψ₁ is a ground state for $$H_N^{v_2,w}$$ , hence belongs to its operator domain and solves the equation

in the sense of distributions hence also almost everywhere. Due to the unique continuation principle (for either v₁ or v₂), we know that |{ Ψ₁ = 0}| = 0 hence this implies that

for almost every $${\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N\in \mathbb {R}^d$$ . Integrating against f⊗N with $$f\in C^\infty _c(\mathbb {R}^d)$$ , we deduce that

We obtain v₁ = v₂ a.e., as we wanted. □

Remark 3.30 (Mixed States)

There is a similar Hohenberg–Kohn theorem for mixed states. That is, if we have two N-particle mixed states Γ₁ and Γ₂ supported on the ground state eigenspaces of $$H_N^{v_1,w}$$ and $$H_N^{v_2,w}$$ respectively, such that $$\rho _{\Gamma _1}=\rho _{\Gamma _2}$$ , then v₁ = v₂ + C. The proof is similar.

From Theorem 3.28, we know that the Hohenberg–Kohn theorem holds when $$p&gt;\max (2,2d/3)$$ . We now discuss some consequences of this result.

Let us consider a fixed interaction $$w\in L^p(\mathbb {R}^d)+L^\infty (\mathbb {R}^d)$$ (in DFT w is usually the Coulomb potential in dimension d = 3). We introduce the set of v-representable densities

The Hohenberg–Kohn theorem states that any $$\rho \in \mathcal {R}_w$$ arises from a unique potential v, up to a constant. We remark that the set $$\mathcal {R}_w$$ might be quite small. In fact, all the densities $$\rho \in \mathcal {R}_w$$ are positive in the following sense. If we had |{ρ = 0}| > 0 then the associated Ψ would vanish on a set of infinite measure, which contradicts the unique continuation principle.² It is an interesting question [65, 66] to determine how small $$\mathcal {R}_w$$ is.

That the set $$\mathcal {R}_w$$ might be quite small is not really a problem in density functional theory, since the sought-after ground state density of course always belongs to $$\mathcal {R}_w$$ . However, the smallness of $$\mathcal {R}_w$$ creates some technical difficulties. For instance if we have $$\rho \in \mathcal {R}_w$$ with associated potential v then one cannot use the implicit function theorem to determine a potential for other densities in the neighborhood of ρ.

In DFT, one important question is to understand the dependence of $$\mathcal {R}_w$$ on the interaction potential w. In fact, in Kohn–Sham theory the goal is to replace the many-body problem with interaction w by a non-interacting eigenvalue problem. If we have $$\rho \in \mathcal {R}_w\cap \mathcal {R}_0$$ with ρ = ρΨ and $$H_N^{v,w}\Psi =0$$ , then we conclude that there exists a unique potential vKS, called the Kohn–Sham potential, and a normalized ground state Ψ′ such that $$\rho _{\Psi '}=\rho $$ and

The spectrum of $$H^{v+v_{\mathrm {KS}},0}$$ is determined in terms of the one-particle Kohn–Sham operator hKS = − Δ∕2 + v + vKS. When its eigenvalues satisfy

then Ψ′ is unique up to a phase and equal to the Slater determinant

where the orbitals φ₁, …, φN are the N first eigenfunctions of hKS. Hence the interacting problem has been mapped onto a non-interacting problem. For this reason, it is desirable that $$\mathcal {R}_w\cap \mathcal {R}_0$$ is as large as possible, perhaps even equal to the whole set $$\mathcal {R}_w$$ . This question does not seem to have been studied in detail. Understanding the Coulomb case is the main goal of the Kohn–Sham formulation of DFT. See [66] for numerical results in this direction.

In the original approach of [82, 93], the Hohenberg–Kohn theorem is used to define universal functionals. Namely, for every $$\rho \in \mathcal {R}_w$$ , we know that there exists a unique potential v (up to constants) and an N-particle wavefunction Ψ such that ρ = ρΨ and Ψ is a ground state for EN[v]. Should Ψ be non-degenerate, this defines a map ρ↦ Ψ[ρ] and therefore one can define the universal energy functional by $$\rho \mapsto \langle \Psi [\rho ],H^{0,w}_N\Psi [\rho ]\rangle $$ , and similarly for the kinetic and interaction energies. This approach is not satisfactory from a mathematical point of view, however, since the set $$\mathcal {R}_w$$ is essentially unknown.

3.8 Proof of Theorem 3.5

Note that since w⩾0, we always have FGC[ρ]⩾0. In particular, there is nothing to prove when ρ = 0. In addition, the Lieb–Thirring inequality (3.35) or the Hoffmann-Ostenhof inequality (3.4) imply that FGC[ρ] > 0 for ρ ≠ 0.

For the proof we need to introduce the k-particle density matrices [102]

of a grand-canonical state Γ = ( Γn)n⩾0, where Trk+1→n means the partial trace in the n − k + 1 last variables. The energy can be expressed in terms of Γ⁽¹⁾ and Γ⁽²⁾ only, as follows

Here w₁₂⩾0 denotes the multiplication operator by w(r₁ −r₂) on the two-particle space $$L^2(\mathbb {R}^d\times \mathbb {Z}_q,\mathbb {C})\wedge L^2(\mathbb {R}^d\times \mathbb {Z}_q,\mathbb {C})$$ .

We will also use the concept of localized states [102]. For a function $$0\leqslant \chi \leqslant 1$$ on $$\mathbb {R}^d$$ , the localized state Γ|χ of a grand-canonical state Γ is characterized by the property that its density matrices are equal to $$\Gamma _{|\chi }^{(k)}=\chi ^{\otimes k}\Gamma ^{(k)}\chi ^{\otimes k}$$ for all k.

Proof of (i)

Let Γj = ( Γj,n)n⩾0 be a grand-canonical minimizer for FGC[ρj], whose existence is guaranteed by Theorem 3.4. After extraction of a subsequence we may assume that FGC[ρj] converges to a finite limit (if the limit is + ∞ there is nothing to show).

Since the second term is non-negative, the kinetic energy $$\mathrm {Tr}\,(-\Delta )\Gamma _j^{(1)}$$ must be uniformly bounded, hence $$\sqrt {-\Delta }\Gamma _j^{(1)}\sqrt {-\Delta }$$ is bounded in the trace-class. For fermions, $$\Gamma _j^{(1)}$$ is in addition bounded in operator norm by 1. Note however that we have no a priori bound on the number of particles $$\mathrm {Tr}\,(\Gamma ^{(1)}_j)=\int _{\mathbb {R}^d}\rho _j$$ , which could diverge. The idea is to use the local trace class topology instead.

By the Hoffmann-Ostenhof and Lieb–Thirring inequalities (3.4) and (3.35), $$\sqrt {\rho _j}$$ is bounded in $$\dot {H}^1(\mathbb {R}^d)\cap L^{2+4/d}(\mathbb {R}^d)$$ . In particular ρj is bounded in L¹ on any finite ball, uniformly with respect to the center of the ball. This means that $$\Gamma _j^{(1)}$$ is locally uniformly bounded in the trace-class. Due to the kinetic energy bound, we can therefore assume, after extraction of a subsequence, that $$\Gamma _j^{(1)}$$ converges strongly locally in the trace class to some operator Γ⁽¹⁾ which is such that $$\rho _{\Gamma ^{(1)}}=\rho $$ , the weak limit of the sequence ρj. Since ρ is integrable by assumption, Γ⁽¹⁾ is indeed trace-class.

The argument is somewhat more complicated for $$\Gamma ^{(2)}_j$$ . Let $$0\leqslant \chi \leqslant 1$$ be a function of compact support. Denoting the localized state by Γj|χ, we know that $$\Gamma ^{(1)}_{j|\chi }=\chi \Gamma ^{(1)}_j\chi $$ is bounded in the trace class. By Yang’s inequality [172, 173], we have $$\|\Upsilon _n^{(2)}\|\leqslant Cn\mathrm {Tr}\,(\Upsilon _n)$$ for any n-particle operator $$\Upsilon _n=\Upsilon _n^*\geqslant 0$$ and a universal constant C. This implies that

Hence $$\Gamma _j^{(2)}$$ is a locally bounded sequence of operators. No local bound on the trace is known, but this does not create any difficulty. Up to extraction of a subsequence, we can therefore assume that $$\Gamma _j^{(2)}\rightharpoonup \Gamma ^{(2)}$$ weakly locally as operators. Since w₁₂⩾0 by assumption, Fatou’s lemma for operators now implies that

(3.93)

We have thus shown that the energy is weakly lower semi-continuous when expressed in terms of the one and two-particle density matrices. Our next task is to go back to states in Fock space.

Following [102, Lemma 3], we know that there exists a state Γ = ( Γn)n⩾0 on the Fock space which has the density matrices Γ⁽¹⁾ and Γ⁽²⁾. The argument uses a different notion of weak convergence and goes as follows. First we extract a subsequence in the sense of weak-∗ convergence over the local algebra of anti-commutation relations. The weak limit is in principle an abstract state over the local CAR but since its density is ρ, which is integrable over $$\mathbb {R}^d$$ , the state is actually normal and arises from a grand-canonical state Γ = ( Γn)n⩾0 [18, 19], which has the above density matrices, as we claimed. In particular, the right side of (3.93) is by definition $$\geqslant F_{\mathrm {GC}}[\rho ]$$ , which concludes the proof of (3.15). □

Proof of (ii)

Let ρ⩾0 be such that $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ and let Γ = ( Γn)n⩾0 be a grand-canonical state such that ρΓ = ρ and

By optimality, we must have

(3.94)

for all n⩾1 such that Γn ≠ 0 (otherwise we could decrease the energy by choosing another state, without changing the density). We split the rest of the argument into several steps. □

Step 1. Approximation by a State with $$\rho _{\Gamma _{n_0}}\geqslant \alpha \rho $$

In this first step we slightly modify Γ in order to guarantee that $$\rho _{\Gamma _{n_0}}\geqslant \alpha \rho $$ for some α > 0 and some n₀, a property which will play a role in the next step. To this end we remark that there exists a state $$\widetilde \Gamma $$ with density $$\rho _{\widetilde \Gamma }=\rho $$ and which only lives over the N and (N + 1)–particle subspaces, where N is the integer part of $$\int _{\mathbb {R}^d}\rho $$ . If $$\int _{\mathbb {R}^d}\rho \in \mathbb {N}$$ we just take $$\widetilde \Gamma $$ to minimize FLL[ρ]. Otherwise, we can write $$\int _{\mathbb {R}^d}\rho =N+\kappa $$ with κ ∈ (0, 1) and we consider the state

where ΓN optimizes FLL[Nρ∕(N + κ)] and ΓN+1 optimizes FLL[(N + 1)ρ∕(N + κ)]. The total density is then

as desired. With the trial state $$\widetilde \Gamma $$ at hand, we consider the new state $$\widetilde \Gamma _\varepsilon =(1-\varepsilon )\Gamma +\varepsilon \widetilde \Gamma $$ . This state has the exact density ρ and its energy converges to that of Γ when ε → 0. This new state has the desired property that

and an energy very close to that of Γ.

Without loss of generality, we can thus assume for the rest of the proof that we have a state Γ with the exact density ρΓ = ρ and such that $$\rho _{\Gamma _{n_0}}\geqslant \alpha \rho $$ for some n₀ and α > 0.

Step 2. Approximation by a State with Compact Support in n

In case that Γn does not vanish for large n, we replace the state Γ = ( Γn)n⩾0 by a new state $$\Gamma '=(\Gamma ^{\prime }_n)$$ so that $$\Gamma ^{\prime }_n\equiv 0$$ for n large enough, at the expense of a small error in the energy. Although it is possible to keep the exact density, we will here allow the density to vary a little.

First note that since ρ ≠ 0, we have FGC[ρ] > 0 and therefore Γ₀ < 1 (otherwise the energy would vanish). This allows us to define Γ′ by

In other words, we truncate the state and compensate the missing mass in the vacuum. The energy of Γ′ is equal to $$\sum _{n=1}^K\mathrm {Tr}\,(H^{0,w}_n\Gamma _n)$$ and it converges to FGC[ρ] as K →∞. Similarly, its density is equal to $$\sum _{n=1}^K\rho _{\Gamma _n}$$ and it converges to ρ in $$L^{1}(\mathbb {R}^d)$$ . In addition, we have for the one-particle density matrix $$\mathrm {Tr}\,(-\Delta )\gamma '\leqslant \mathrm {Tr}\,(-\Delta )\gamma $$ which proves that

(3.95)

by the Hoffmann-Ostenhof inequality (3.33). This gives the strong convergence of $$\rho _{\Gamma '}$$ to ρ in $$L^p(\mathbb {R}^d)$$ when K →∞ for $$1\leqslant p&lt;p^*/2$$ where p∗ is the critical Sobolev exponent. Finally, we would like to prove the convergence

(3.96)

strongly in $$L^2(\mathbb {R}^d)$$ , and this is where the first step helps. Indeed, we have the following lemma.

Lemma 3.31

Let (ρj) be a sequence such that $$\alpha \rho \leqslant \rho _j\leqslant \rho $$ for some α > 0 and ρj(x) → ρ(x) a.e., where $$\sqrt {\rho }\in H^1(\mathbb {R}^d)$$ . If $$\nabla \sqrt {\rho -\rho _j}\to 0$$ strongly in $$L^2(\mathbb {R}^d)$$ , then $$\nabla \sqrt {\rho _j}\to \nabla \sqrt {\rho }$$ strongly in $$L^2(\mathbb {R}^d)$$ .

Proof of of Lemma3.31

We write $$u_j=\sqrt {\rho _j}/\sqrt {\rho }$$ , which satisfies $$\sqrt {\alpha }\leqslant u_j\leqslant 1$$ by assumption and which converges almost everywhere to 1. Then the assumption means that

strongly in $$L^2(\mathbb {R}^d)$$ . The first term goes to 0 by dominated convergence, hence we conclude that the second term tends to 0 in $$L^2(\mathbb {R}^d)$$ . Then we have

The first term converges again to 0, whereas the second can be estimated by

and the lemma follows. □

In our case, the inequality (3.95) implies that $$\sum _{n\geqslant 1}\int _{\mathbb {R}^d}|\nabla \sqrt {\rho _{\Gamma _n}}|{ }^2$$ is a convergent series. In addition, by the Cauchy–Schwarz inequality for series,

integral Underscript double struck upper R Superscript d Baseline Endscripts StartAbsoluteValue nabla StartRoot sigma summation Underscript n greater than or slanted equals upper K plus 1 Endscripts rho Subscript upper Gamma Sub Subscript n Subscript Baseline EndRoot EndAbsoluteValue squared less than or slanted equals sigma summation Underscript n greater than or slanted equals upper K plus 1 Endscripts integral Underscript double struck upper R Superscript d Baseline Endscripts StartAbsoluteValue nabla StartRoot rho Subscript upper Gamma Sub Subscript n Subscript Baseline EndRoot EndAbsoluteValue squared long right arrow Underscript upper K right arrow normal infinity Endscripts 0 period

Since we have for $$K\geqslant n_0$$

for some α > 0, the lemma implies the convergence (3.96).

At this step we have replaced Γ = ( Γn)n⩾0 by a new state $$\Gamma '=(\Gamma ^{\prime }_n)$$ with compact support in n, and a close energy. In addition, $$ \left \| \rho _{\Gamma '}-\rho _\Gamma \right \|{ }_{L^{1}(\mathbb {R}^d)}$$ and $$ \left \| \nabla \sqrt {\rho _{\Gamma '}}-\nabla \sqrt {\rho _\Gamma } \right \|{ }_{L^{2}(\mathbb {R}^d)}$$ are small. To simplify our exposition we assume henceforth that Γ itself satisfies Γn ≡ 0 for $$n\geqslant K+1$$ .

Step 3. Approximation by a State with Compact Support in Space

Next we localize the state Γ in order to make it have a compact support in space. Let χR := χ(⋅∕R) for some $$\chi \in C^\infty _c(\mathbb {R}^d)$$ satisfying χ(0) = 1 and $$0\leqslant \chi \leqslant 1$$ . We consider the localized state $$\Gamma _{|\chi _R}$$ which has the density $$\chi _R^2\rho $$ and the energy

Note that the space localization does not modify the support in n [102]. That is, the state $$\Gamma _{|\chi _R}$$ satisfies $$(\Gamma _{|\chi _R})_n\equiv 0$$ for $$n\geqslant K+1$$ (the same value as for Γ). The energy and the density converge strongly as R →∞ to that of Γ. Hence we can assume in the following that Γ = ( Γn)n⩾0 has both a compact support in n and in space.

Step 4. Construction of the Canonical Sequence

In the previous approximations we have replaced the initial grand-canonical state by a new state Γ which has an energy close to the minimal energy FGC[ρ] and a density close to the initial density, in all the appropriate function spaces. In this step we finally construct the sequence ρj but it will only be close to ρΓ in the spaces mentioned in the statement of the theorem. We recall that Γn ≡ 0 for $$n\geqslant K+1$$ .

Let $$\varphi _\ell \in C^\infty _c(B_1)$$ be K orthonormal functions in the unit ball $$B_1\subset \mathbb {R}^d$$ and define φj,ℓ(r) = j−d∕2φℓ(r∕j −v) where v ≠ 0 is any fixed vector in $$\mathbb {R}^d$$ . These are K orthonormal functions in the translated and dilated ball jv + jB₁. We then introduce the following K-particle mixed state

where

is an (K − n)-particle Slater determinant. For j large enough the Γn’s and φj,ℓ have disjoint support, hence

as required. After a lengthy but straightforward calculation, one finds that Υj has the energy

Under our assumptions on w the last two terms converge to 0 in the limit j →∞, hence the energy of Υj converges to that of Γ. In particular,

In addition, the density is

and its square root converges strongly to $$\sqrt {\rho _{\Gamma }}$$ in $$\dot {H}^1(\mathbb {R}^d)\cap L^{p}(\mathbb {R}^d)$$ for all 2 < p < p∗, but not for p = 2.

Step 5. Conclusion

Using an ε∕2 argument to justify the approximations made in Steps 1–3, we have managed to construct the sequence ρj mentioned in the statement and proved that it satisfies

(3.97)

Next we notice that the lower bound

follows from (i). Therefore we obtain the stated limit

and this completes the proof of the theorem. $$\square $$

Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements MDFT No 725528 of M.L. and AQUAMS No 694227 of R.S.).

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Footnotes

1

In order to recover Scott’s correction in atoms, the coefficient must actually be taken equal to 0.083 [114].

2

To include densities vanishing on a set of positive measure we have to allow v₊ to be infinite on such sets and rephrase the unique continuation principle accordingly.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_4

4. The Strong-Interaction Limit of Density Functional Theory

Gero Friesecke1  , Augusto Gerolin2, 3   and Paola Gori-Giorgi4, 5  

(1)

Department of Mathematics, Technische Universität München, Garching bei München, Germany

(2)

Department of Chemistry and Biomolecular Sciences, University of Ottawa, Ottawa, Canada

(3)

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada

(4)

Department of Theoretical Chemistry, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

(5)

AI4Science Microsoft Research, Amsterdam, The Netherlands

Gero Friesecke

Email: gf@ma.tum.de

Abstract

This is a comprehensive review of the strong-interaction limit of density functional theory. It covers the derivation of the limiting strictly correlated electrons (SCE) functional from exact Hohenberg–Kohn DFT, basic aspects of SCE physics such as the nonlocal dependence of the SCE potential on the density, equivalent formulations and the mathematical interpretation as optimal transport with Coulomb cost, rigorous results (including exactly soluble cases), approximations, numerical methods, integration into Kohn–Sham DFT (KS SCE), and applications to molecular systems, an example being that KS SCE, unlike the local density approximation or generalized gradient approximations, dissociates H₂ correctly. We have made an effort to make this review accessible to a broad audience of physicists, chemists, and mathematicians.

4.1 Introduction

The strong-interaction limit of DFT is the inhomogeneous low-density limit associated with the uniform coordinate scaling

of the single-particle density at fixed particle number, with γ → 0. In this limit, the Levy–Lieb functional, which gives the minimum kinetic and interaction energy subject to the given density, has the leading order asymptotics

and the corresponding optimal wavefunction Ψγ has the asymptotics

where ρN solves the variational principle of having minimal Coulomb energy subject to the given density ρ and $$\operatorname {V}_{\mathrm {ee}}^{\operatorname {SCE}}[\rho ]$$ denotes the resulting minimal energy.

This appears to be the only case in which one can obtain insight into how to extract information about the interaction energy directly from the density. As it turns out, in this limit none of the ingredients from the traditional “Jacob’s ladder” of DFT approximations (local density, local density gradients, Kohn–Sham kinetic energy density, Hartree–Fock exchange, virtual orbitals) play any role. Instead, maps based on integrals not derivatives of the density appear. These maps are mathematically related to the field of optimal transport, and physically describe strictly correlated electrons (SCE). The SCE functional $$\operatorname {V}_{\mathrm {ee}}^{\operatorname {SCE}}$$ appearing above is the limiting Hartree-exchange-correlation functional.

While the strong-interaction limit is, of course, not reached in nature, it points the way towards the real physics happening in molecular systems containing strong correlations, without having to leave the realm of Kohn–Sham DFT. Two important examples whose physics is missed by Kohn–Sham DFT with semilocal or hybrid exchange-correlation functionals but captured correctly by integrating the SCE functional into Kohn–Sham DFT (KS SCE) are weakly charged nanosystems, see Fig. 4.11, and H₂ near the dissociation limit, see Fig. 4.14.

This chapter provides a self-contained introduction to this limit and its fascinating physics and mathematics which has been unearthed in the past two decades, and reviews the current state of the art.

4.2 The Many-Electron Schrödinger Equation and Universal Density Functional

In this section we quickly introduce the time-independent electronic Schrödinger equation and the exact reformulation of the ground state problem via a universal density functional.

4.2.1 The Many-Electron Schrödinger Equation

We consider a quantum mechanical system of N non-relativistic electrons (of mass me and charge − e), moving around classical nuclei with positions $${\mathbf {R}}_1,\dots ,{\mathbf {R}}_M \in \mathbb {R}^d$$ and charges Z₁e, …, ZMe (Born–Oppenheimer approximation). Our main interest is in the physical space $$\mathbb {R}^3$$ , but we consider the general space dimension d ≥ 1 since it will be instructive to illustrate key properties of the strong interaction limit with lower dimensional examples. The electrons are described by a wave function $$\Psi :(\mathbb {R}^d\times \mathbb {Z}_2)^N\to \mathbb {C}$$ of N positions $${\mathbf {r}}_i \in \mathbb {R}^d$$ and spin coordinates $$s_i\in \lbrace \uparrow ,\downarrow \rbrace =\mathbb {Z}_2$$ .

The Pauli exclusion principle states that the electronic wave function must be antisymmetric with respect to permutations of the electron coordinates,

(4.1)

where $$\mathfrak {S}_N$$ denotes the group of permutations of the indices 1, …, N. The set of square-integrable N-electron wave functions, $$\{\Psi \in L^2((\mathbb {R}^d\times \mathbb {Z}_2)^N;\mathbb {C}) \, : \,~(4.1)\}$$ , will be denoted $$\bigwedge ^N_{i=1}L^2(\mathbb {R}^d\times \mathbb {Z}_2;\mathbb {C})$$ . The one-body density of an electronic wave function $$\Psi \in \bigwedge ^N_{i=1}L^2(\mathbb {R}^d\times \mathbb {Z}_2;\mathbb {C})$$ is defined by

The energy E[ Ψ, v] of a fermionic state Ψ with external potential $$v:\mathbb {R}^d\to \mathbb {R}$$ is given, in atomic units ħ = me = e = 1, by

(4.2)

where T[ Ψ] is the kinetic energy,

Vee[ Ψ] is the electron-electron interaction energy

and Vne[ Ψ, v] is the electron-nuclei interaction energy,

where $$w \, : \, \mathbb {R}^d\to \mathbb {R}$$ is an interaction potential satisfying w(r) = w(−r), so that the total interaction potential

(4.3)

is symmetric.¹ Typically,

(4.4)

is the Coulomb electron repulsion and v is the Coulomb potential generated by M nuclei which are at positions Rν with charges Zν,

(4.5)

If additional fields are present, the external potential v contains extra terms.

The central quantity of interest is the ground state energy of the system. By the Rayleigh–Ritz variational principle, it is given by

(4.6)

where the infimum is taken over the class $$\mathcal {W}^{N}$$ of wavefunctions which are antisymmetric and have finite kinetic energy,

(4.7)

Here H¹ is the usual Sobolev space of square-integrable functions with square-integrable gradient, and || Ψ|| denotes the L² norm of Ψ. The ground state energy (4.6) is well defined whenever the potentials v and w are sufficiently regular so that the functional E is well defined on $$H^1((\mathbb {R}^d\times \mathbb {Z}_2)^N)$$ . A simple sufficient condition in dimension d = 3 which encompasses (4.5), (4.4) is v, $$w\in L^{3/2}(\mathbb {R}^3) + L^\infty (\mathbb {R}^3)$$ .

Whether or not the infimum in (4.6) is actually a minimum, that is, a minimizing Ψ exists, is much more subtle. For neutral or positively charged molecules in dimension d = 3 ((4.4), (4.5) with $$Z =\sum ^M_{i=1}Z_i &gt; N-1$$ ) the answer is yes, as was proved by Zhislin [145] via a careful spectral analysis of the underlying Hamiltonian operator. For an alternative proof based on variational methods see Friesecke [48].

4.2.2 Universal Density Functional

In quantum mechanics, the absolute value squared | Ψ(r₁, s₁, …, rN, sN)|² of a wave function $$\Psi \in \mathcal {W}^{N}$$ corresponds to an N-point probability distribution: it gives the probability density of finding the electrons at positions $${\mathbf {r}}_i\in \mathbb {R}^d$$ with spins $$s_i~\in ~\mathbb {Z}_2, ~i\in \lbrace 1,\dots ,N\rbrace $$ .

By integrating the N-point probability distribution over the spins, we obtain the N-point position density,

(4.8)

The single particle density ρΨ(rj) is then obtained by integrating out all but one electron position $${\mathbf {r}}_j\in \mathbb {R}^d$$ ,

(4.9)

We denote by Ψ↦ρ the relation between Ψ and ρ given by Eqs. (4.8) and (4.9). This means that the wave function Ψ has single-electron density ρ.

Following the work of Hohenberg and Kohn [78], Levy [91] and Lieb [97] showed that the electronic ground state problem (4.6) can be recast as a minimization over single-electron densities ρ instead of many-electron wavefunctions Ψ:

(4.10)

with

(4.11)

where FLL is the Levy–Lieb functional. The above direct definition of FLL by a constrained search replaced an earlier, indirect existence proof of a universal functional satisfying (4.10) [78]. The space $$\mathcal {D}^N$$ is defined as the set of densities ρ coming from a wave function $$\Psi \in \mathcal {W}^{N}$$ (i.e, Ψ↦ρ), i.e., the N-representable one-particle densities. It can be fully characterized [97] and is given by

(4.12)

Also, it is known that the minimum in (4.11) is attained. For more details about these matters see Chap. 3 by Lewin et al.

4.3 The Strictly Correlated Electrons (SCE) Functional

4.3.1 Constrained-Search Definition

From the early days of DFT it has been clear that a useful approximation to the kinetic energy contribution in (4.11) is given by the functional

(4.13)

and by its further approximation TS[ρ] obtained by Kohn and Sham [85] via restricting the above search to Slater determinants built from orthonormal spin orbitals,

(4.14)

The natural analogue of Ts,LL for the interaction energy contribution in (4.11) is the SCE functional

(4.15)

which was introduced by Seidl [122]. The acronym SCE stands for strictly correlated electrons, and will be explained shortly. As detailed in the next section, the functional (4.15) is a rigorous leading-order asymptotic limit of FLL[ρ] in the low-density regime, where interaction dominates, just as the kinetic functional (4.13) is a leading-order asymptotic limit at high density, where the kinetic energy dominates.

What is more, there also exists a natural analogue to TS for interaction, which approximates, in the case of N electrons in the physical space $$\mathbb {R}^3$$ , the high-dimensional minimization over wavefunctions on 3N dimensional space in (4.15) by a minimization over just N maps on $$\mathbb {R}^3$$ ; see Sect. 4.3.4.

4.3.2 Derivation as a Low-Density or Strong-Interaction Limit of the Levy–Lieb Functional

For any given N-particle density ρ on $$\mathbb {R}^d$$ , consider its dilation obtained by uniform coordinate scaling

where γ > 0 is a scaling factor. Note that this scaling preserves the total particle number,

We are interested in the small-γ regime, which corresponds to a low-density limit.

If Ψ is a wavefunction with density ρ, then the scaled wavefunction

has density ργ. But as first noticed by Levy and Perdew [92], scaling does not commute with constrained search. Instead, by an elementary change of variables,

and therefore

(4.16)

where

(4.17)

is a Levy–Lieb functional with coupling constant λ. This suggests, assuming that the minimization in the second line of (4.16) commutes with taking the limit γ → 0,

(4.18)

or equivalently, by starting from the Levy–Lieb functional with coupling constant, Eq. (4.17), as done in [122, 127]

(4.19)

Mathematically, as pointed out in [127] it is not obvious whether the minimization in the second line of (4.16) commutes with passing to the limit γ → 0 since the optimal wavefunction depends on γ. Nevertheless the above leading-order asymptotics can be rigorously justified; see Theorem 4.3 in the next section.

Repeating the calculation in (4.16) without the kinetic energy and replacing “min” by “inf” shows that

(4.20)

whence the asymptotic result (4.18) can also be re-written as

(4.21)

Off the low-density limit, we remark that $$V_{\mathrm {ee}}^{\mathrm {SCE}}$$ still provides a rigorous lower bound for the Levy–Lieb functional,

(4.22)

This is a trivial consequence of the constrained-search Definitions (4.11) and (4.15) and the nonnegativity of the kinetic energy functional T. For typical atomic densities on $$\mathbb {R}^3$$ , this lower bound is a significant improvement over the Lieb–Oxford bound with best known constant.

4.3.3 Enlarging the Constrained Search to Probability Measures

The variational principle underlying the definition of $$V_{\mathrm {ee}}^{\mathrm {SCE}}[\rho ]$$ in (4.15),

(4.23)

with N-point density $$\pi _N^{\Psi }$$ as in (4.8), typically has no minimizer. That is, no minimizing wavefunction $$\Psi \in {\mathcal {W}}^N$$ exists and the infimum in (4.15) is not attained.² Physically, this reflects the phenomenon that if Ψλ[ρ] is a sequence of square-integrable functions depending on a parameter λ > 0 such that 〈 Ψλ|Vee| Ψλ〉 approaches the infimum in (4.15) as λ tends to infinity—prototypical is the Ψλ that minimizes 〈 Ψ|T + λVee| Ψ〉 subject to Ψ↦ρ—then | Ψλ|² integrates to 1 but is typically concentrating on a lower dimensional subset, as depicted in Fig. 4.1.

4 3-D graphs. A to C is color contoured. A has 2 mountain-like structures close together. B has 2 mountain-like structures separated by a small gap. C has 2 sets of 3 separate thin peaks in decreasing order, with the highest peak opposite to each other. In A to C, the structures have unsharpened boundaries at the bottom. D is a clear graph of C with defined boundaries and gaps.

Fig. 4.1

Numerically computed ground state wave functions for Fλ[ρ] in (4.17) for N = 4 and one-body density $$\rho (r) = \frac {1}{2L}(1+\cos {}( \frac {\pi }{L} r)), r\in [-L,L]$$ (L = 5) for different values of λ: λ = 0.1, 1, 10, ∞. Shown: pair density $$ \sum _{s_1,s_2,s_3,s_4} \int \! \mathrm{d} r_3\, \mathrm{d} r_4 |\Psi ^{\lambda }(r_1,s_1,r_2,s_2,r_3,s_3,r_4,s_4)|{ }^2$$ . Picture from [21], see also [109] for a numerical approximation of (4.17) with N = 2. The pair density on the left is governed by exchange effects, whereas the one on the right is governed purely by Coulombic correlations

2-part diagram. A, a square with 2 right slant lines for r subscript 1 and f of r subscript 1 are at the bottom right corner and top left corner respectively. B has a rectangle with 4 different colored dots for r subscript 1 in the center of the baseline and 2 different colored dots for r subscript 2 at either end of the base. The other 3 sides are labeled rho.

Fig. 4.2

SCE state of a two-electron system with homogeneous density in a one-dimensional domain. By Theorems 4.3 and 4.4, this state is an asymptotically exact approximation to the true quantum ground state at low density. Left: optimal co-motion function or transport map f. Right: position of the two electrons in the one-dimensional domain. The position of the second electron, r₂, is determined by that of the first electron, r₁, through the equation r₂ = f(r₁), with the optimal f keeping the electrons at a constant distance, of half the domain size. The position of the first electron varies over the whole domain according to the density ρ (see Eq. (4.38)). As the first electron (depicted in blue) passes through the mid-point, the position of the second electron (depicted in red) jumps from the right end to the left end, causing a discontinuity of f

A line graph of S C E potentials v of r versus position r plots 3 logarithmically increasing lines for Neon N = 10, Carbon N = 6, and Boron N = 5. They begin from the Y axis at negative 18.75, negative 7.4, and negative 5.4 respectively. Values are approximated. The lines are non-intersecting.

Fig. 4.3

SCE potentials vSCE(r) corresponding to the (radially symmetric) densities of Neon (N = 10), Carbon (N = 6) and Boron (N = 5), Fig. 9 in [127]. Data obtained by the following steps: (i) compute the density ρ(r) by an accurate full CI or quantum Monte Carlo computation; (ii) compute the SGS maps corresponding to ρ as described in Section 4.3.12 below; (iii) obtain the corresponding SCE potentials vSCE via Eqs. (4.59) and (4.64)

This basic shortcoming of (4.23)—that wavefunctions which are closer and closer to being optimal in the constrained search (4.15) do not converge to any proper wavefunction—can be overcome as follows [14, 32]. First, interpret the variational principle (4.23) as a variational principle for the N-point density as suggested by the second expression in (4.23); second, enlarge the space of admissible N-point densities to the space $${\mathcal {P}}(\mathbb {R}^{dN})$$ of probability measures on $$\mathbb {R}^{dN}$$ with density ρ. Then the constrained search becomes well-posed, that is, optimizers exist. See Theorem 4.1 below. This enlargement allows N-point densities to concentrate on lower dimensional subsets as in Fig. 4.1. The condition that a probability measure $$\Pi \in {\mathcal {P}}(\mathbb {R}^{dN})$$ has density ρ now means that Π has marginals equal to the density divided by the particle number, $$\frac {\rho }{N}$$ :

(4.24)

We denote the relation given by Eq. (4.24) by Π↦ρ. This yields the variational principle

(4.25)

and the following enlarged-constrained-search definition of the SCE functional

(4.26)

This alternative definition of $$V_{\mathrm {ee}}^{\mathrm {SCE}}[\rho ]$$ and the underlying enlarged variational problem (4.25) were introduced by Buttazzo et al. [14] and Cotar et al. [32], along with the insight that minimizers now exist (see Theorem 4.1 (1) below) and (4.26) is mathematically an optimal transport problem and can be usefully analyzed with methods from optimal transport theory (see Sect. 4.3.6). We call (4.25) the SIL variational principle, the acronym SIL standing for strong-interaction limit.

The notation in (4.26) (“min” instead of “inf”; using the same notation for the ensuing density functional even though a priori the right-hand side of (4.26) could be lower than that in (4.15) since the minimization is over a larger set) is justified because of:

Theorem 4.1

Let ρ be any N-particle density in the class $${\mathcal {D}}^N$$ (see (4.12)), and let w(r) = |r|⁻¹be the Coulomb interaction.

(1)

The minimum in (4.26) is attained; that is, there exists a minimizing probability measure Π.

(2)

[10, 32, 33] The minimum value in (4.26) is equal to the infimum in (4.15).

Statement (1) is a special case of general existence theorems in optimal transport theory. For a textbook account see [50]. Proofs of such results rely on Prokhorov’s theorem from probability theory as well as on approximation and lower semi-continuity results for functionals of the form $$\Pi \mapsto \int V_{\mathrm {ee}} \mathrm{d} \Pi $$ .

Statement (2), although plausible, is mathematically much more subtle. It rests on the nontrivial result that arbitrary symmetric probability measures $$\Pi \in {\mathcal {P}}(\mathbb {R}^{dN})$$ with marginal ρ can be approximated by N-point densities of quantum wavefunctions $$\Psi \in {\mathcal {W}}^N$$ with the same marginal. Note that such wavefunctions must be antisymmetric and must have a square-integrable gradient; but applying standard smoothing techniques from mathematics—such as mollification—to a given probability measure with marginal ρ does not preserve the marginal, nor does it yield the N-point density of an antisymmetric function. This result, and the ensuing statement (2), was first proved for N = 2 [32], and later extended to N = 3 [10] and general N [33] (see also [95] for a similar extension to general N allowing mixed states).

Remark 4.2 (Symmetrization)

The minimum value in (4.26) is unchanged, and still attained, when the minimization over arbitrary probability measures with marginal ρ∕N, $$\{\Pi \in {\mathcal {P}}(\mathbb {R}^{dN})\, : \, \Pi \mapsto \rho \}$$ , is restricted to symmetric probability measures with marginal ρ∕N, where a probability measure $$\Pi \in {\mathcal {P}}(\mathbb {R}^{dN})$$ is said to be symmetric if

This is because whenever Π is a probability measure in $${\mathcal {P}}(\mathbb {R}^{dN})$$ with marginals ρ, Eq. (4.24), then so is its symmetrization SN Π defined by

(4.27)

the sum being over all permutations of {1, …, N}; and the integral on the r.h.s. of (4.26) for Π agrees with that for SN Π, thanks to the permutation symmetry of Vee.

Next we rigorously justify the asymptotic relations (4.18), (4.19), (4.21) and complement them with an asymptotic result on the associated constrained-search wavefunctions.

Theorem 4.3 ([33]))

For any N-electron density ρ in the class $${\mathcal {D}}^N$$ (see (4.12)), and with w(r) = |r|⁻¹being the Coulomb interaction, the asymptotic results (4.18), (4.19), (4.21) hold. Moreover if Ψλ[ρ] is any minimizer in the constrained-search definition of Fλ[ρ] (see (4.17)), then every limit point³Π of the sequence of N-point densities $$\pi ^{\Psi _\lambda [\rho ]}$$ is a minimizer in the enlarged-search Definition (4.26) of $$V_{\mathrm {ee}}^{\mathrm {SCE}}[\rho ]$$ .

Proof of (4.18), (4.19), and (4.21)

The proof, taken from [33], is easy, so we include it. We show (4.19), the other statements being equivalent. Fix ρ. First, pick any minimizer Ψλ[ρ] in the constrained-search definition of Fλ[ρ], then

(4.28)

that is, the SCE functional is a lower bound of the left-hand side. To show that it is also an asymptotic upper bound for large λ, we fix any positive number 𝜖 and pick a wavefunction $$\widetilde {\Psi }[\rho ]$$ in $${\mathcal {W}}^N$$ such that $$V_{\mathrm {ee}}[\widetilde {\Psi }[\rho ]]\le V_{\mathrm {ee}}^{SCE}[\rho ] + \epsilon $$ . It follows that

Since $$\widetilde {\Psi }$$ belongs to $${\mathcal {W}}^N$$ , its kinetic energy $$T[\widetilde {\Psi }]$$ is finite, and so

Since 𝜖 > 0 was arbitrary,

(4.29)

Combining (4.28) and (4.29) yields (4.19). □

The above simple argument only shows that the asymptotic error in (4.19) is o(1∕λ), but does not give its order, which turns out to be O(1∕λ1∕2), see Sect. 4.3.5.

4.3.4 The SCE Ansatz

The SIL variational principle (4.25) still requires minimization over a high-dimensional space of N-point probability measures.

Seidl [122] (see also [127]) proposed the following low-dimensional ansatz: we restrict minimization over N-point probability measures to minimization over singular probability measures of the special form

(4.30)

where, for any $${\mathbf {r}}_1\in \mathbb {R}^d$$ , δ(rn − fn−1(r₁)) denotes the delta function of rn (alias Dirac measure) centered at fn−1(r₁), and f₁, …, fN−1 are maps from $$\mathbb {R}^d$$ to $$\mathbb {R}^d$$ . The singular densities (4.30) are concentrated on the d-dimensional set

(4.31)

From a physical point of view, such a density describes a state in which the position of one of the electrons, say r₁, can be freely chosen according to the density ρ, but this then uniquely fixes the position of all the other electrons through the functions f₂, …, fN, that is, r₂ = f₁(r₁) etc. Thus states of form (4.30) are called strictly correlated states, or SCE states for short. The fi are called co-motion functions or transport maps.

The marginal constraint that Π must have marginals ρ, Eq. (4.24), turns into the following constraint on the maps fn: the fn must transport the density ρ to itself,

(4.32)

where, for any measurable map $$f\, : \, \mathbb {R}^p\to \mathbb {R}^q$$ and any measure μ on $$\mathbb {R}^p$$ , the push-forward f♯μ is the measure on $$\mathbb {R}^q$$ defined by

(4.33)

More explicitly, if p = q, μ is absolutely continuous with density ρ, f is a diffeomorphism, and the density of the push-forward f♯μ is denoted by f♯ρ, we have

By substituting this formula for the push-forward into (4.32) and changing variables f⁻¹(r′) = r, the constraint (4.32) turns—provided the fn are diffeomorphisms—into the following nonlinear first-order partial differential equation:

Plugging the ansatz (4.30) into the SIL variational principle (4.25) and integrating out the variables r₂, …, rN yields the SCE variational principle

(4.34)

with the minimization being over maps in the admissible class

(4.35)

Thanks to Theorem 4.4 (1) below, this yields a third construction of the SCE functional,

(4.36)

In the Coulomb case, (4.4), and denoting f₀(r) = r, we thus have

(4.37)

Physically, this means that one needs to minimize the mutual Coulomb repulsion of the co-motion functions. This construction of the SCE functional was introduced by Seidl [122]. A priori it is not clear, but was conjectured by Seidl, that it is equivalent to the original construction (4.15). This is now rigorously known (see Corollary 4.6 below).

The construction (4.36) should be considered the analogue for interaction of the classical Kohn–Sham kinetic energy functional TS. Just as TS is determined by N low-dimensional functions (the Kohn–Sham spin orbitals $$\varphi _1,\ldots \varphi _N\, : \, \mathbb {R}^3\times \mathbb {Z}_2\to \mathbb {C}$$ ), $$V_{\mathrm {ee}}^{\mathrm {SCE}}$$ is determined by N − 1 low-dimensional maps (the co-motion functions or transport maps $$f_1,\ldots ,f_{N-1}\, : \, \mathbb {R}^3\to \mathbb {R}^3$$ ), which can be easily stored on a computer. Moreover—like the Kohn–Sham orbitals—the co-motion functions are obtained by just minimizing a 3-dimensional integral.

The reader is warned, however, that the behavior of the SCE variational principle and its relationship to the SIL variational principle is subtle, and open questions remain. In particular, it is not known—except in special cases—whether minimizers in (4.37) exist. The following results have been rigorously proved.

Theorem 4.4

Let $$\rho \, : \, \mathbb {R}^d\to \mathbb {R}$$ be any N-particle density in the class $${\mathcal {D}}^N$$ (see (4.12)), and let w(r) = |r|⁻¹be the Coulomb interaction.

(1)

The infimum in (4.36) is equal to the minimum in (4.26).

(2)

For two electrons (N = 2), and in arbitrary space dimension d, the infimum in (4.36) is attained; that is, there exists a minimizing map f₁. Moreover f₁is unique, and the induced probability measure (4.30) is the unique minimizer of the SIL variational principle (4.25).

(3)

In one space dimension (d = 1), and for arbitrary N, the infimum in (4.36) is attained; that is, there exist minimizing maps f₁, …, fN−1. Moreover the symmetrization (see Remark4.2) of the associated probability measure (4.30) is the unique symmetric minimizer of the SIL variational principle (4.25).

Statement (1) is a consequence, pointed out in [28], of a general theorem of Ambrosio [5] and Pratelli [116] in optimal transport theory. For N = 2 or d = 1, use of the Ambrosio–Pratelli theorem can be avoided since the assertion follows from (2) respectively (3).

The existence of optimal maps in (2) and (3) is subtle and depends on special Coulombic features. For non-Coulombic counterexamples see Remark 4.7 below. In the Coulomb case, it is an open question whether the infimum in (4.36) is attained for general (physically reasonable) densities ρ when d > 1 and N ≥ 3.

Statement (2) completely justifies Seidl’s SCE ansatz for N = 2: the SCE problem

has a unique minimizer and the associated SCE state

(4.38)

is the unique minimizer of the SIL problem

This was proved in [32], by modifying the analysis by Gangbo and McCann [55] of optimal transport with costs w(r, r′) which are convex or concave in the displacement z = r −r′. Note that the Coulomb cost is neither: near any displacement z₀ ≠ 0, it is convex in radial direction and concave in all perpendicular directions. A simpler proof using Kantorovich duality (see Sect. 4.3.7) was suggested in [14], and made rigorous in [42]. The SCE map is given by

(4.39)

for some function $$u:\mathbb {R}^d\to \mathbb {R}$$ (Kantorovich potential). The notion of Kantorovich potential will be explained in Sect. 4.3.7. Equation (4.39) follows by solving Eq. (4.59) for f₁.

Statement (3), together with an explicit construction of the optimal maps given in Sect. 4.3.11, was suggested in the original paper by Seidl [122] on grounds of physical arguments, and was rigorously proved in [27] with the help of cyclical monotonicity methods from optimal transport theory. See Sect. 4.3.11 for more information.

The uniqueness statements in (2) and (3) are somewhat surprising: the optimal N-point densities arising from Levy–Lieb constrained search in the strongly interacting limit are always unique when either N = 2 or d = 1! No analogue holds off the strongly interacting limit.

Example 4.5

Consider a two-electron system with uniform density in a one-dimensional interval [0, L]. The unique minimizer f₁ = f of the SCE variational principle (4.34) can be shown (see Sect. 4.3.11) to be

(4.40)

See Fig. 4.2.

2 diagrams. A, a square with a right diagonal and 4 curves of different curvatures are present on the other 2 corners. The order of the 4 colors of the curves in the top left corner is reversed in the bottom right corner. B plots a bell curve rho. A part of the area to the right of the mean is 1, which ranges from f subscript i of r to f subscript i + 1 of r. Point r is to the left of the mean.

Fig. 4.4

SCE state of N electrons in one dimension. Right: Electron positions. Given that the first electron (depicted in blue) is at r, the positions of the other electrons are completely determined by the requirement that neighboring electrons are separated by an amount of density of 1 (blue area). The co-motion functions or transport maps fi (i = 1, …, N − 1) of SCE theory are defined as the positions of the other electrons as a function of the first position r. The latter is distributed according to the given single-particle density ρ. Left: Graphs of the maps fi, with f₀(r) = r also shown. The figure corresponds to the Lorenzian density (Example 4.18) and N = 5

By combining Theorems 4.4 and 4.1 we obtain:

Corollary 4.6

Let $$\rho \, : \, \mathbb {R}^d\to \mathbb {R}$$ be any N-particle density in the class $${\mathcal {D}}^N$$ (see (4.12)), and let w(r) = |r|⁻¹be the Coulomb interaction. Then the infimum in (4.36) is equal to that in (4.15).

Proof

This follows from the fact that both quantities are equal to the minimum value of the SIL variational principle (4.25), by Theorem 4.1 (2) respectively Theorem 4.4 (1). □

We remark that no proof is known which bypasses the SIL variational principle, even though the corollary was conjectured before the latter was introduced.

We close this introductory section on the SCE ansatz with some remarks.

Remark 4.7 (Nonattainment)

For simple non-Coulombic counterexamples to attainment of the infimum in (4.36) for N = 3 even in one space dimension, see [49, 59]. For instance, one can take the uniform density in the interval [0, 3] and the interaction potential w(r) = r⁴∕4 − r³∕3 [49]. Earlier more intricate counterexamples can be found in [107]. Such a nonattainment has the undesirable consequence that numerically computed optimal maps will necessarily exhibit wilder and wilder oscillations as the mesh is refined or the basis set approaches completeness, and fail to converge in any pointwise sense to actual optimal maps.

Remark 4.8 (Existence of Non-SCE Minimizers)

For a Coulombic example for N = 3 in three space dimensions showing that the SIL variational principle can possess minimizers which are not of SCE form, see [113]. This example exhibits nonuniqueness and it is not known whether it also admits minimizers which are of SCE form.

Remark 4.9 (Alternative Formulations of the SCE Ansatz)

Denoting f₀(r) = r, one can write the SCE ansatz (4.30) in the following form in which all coordinates r₁, …, rN appear on an equal footing:

(4.41)

Also, one can work with the symmetrized form of this ansatz,

(4.42)

(where σ runs over the permutations of the SCE map indices 0, …, N − 1); the symmetrization doesn’t change the energy $$\int V_{\mathrm {ee}}\, \mathrm{d} \Pi $$ , and the symmetrized form (4.42) minimizes the SIL problem (4.26) if and only if the unsymmetrized form (4.41) does, as was explained in Remark 4.2.

Remark 4.10 (Nonsmoothness of Optimal Maps)

The reader might wonder why, in SCE theory, no differentiability and not even continuity is imposed on the competing maps (the maps in the admissible class (4.35) are merely required to be measurable). This is because optimal maps, when they exist, are typically discontinuous. This important effect can be understood intuitively from simple examples as in Fig. 4.2. As the first electron passes through the midpoint of the domain, the position of the second electron jumps from the right end of the domain to the left end, yielding the discontinuous map depicted in the Figure. For a radial density in three dimensions (d = 3), an analogous discontinuity occurs in that spheres near zero are mapped to spheres near infinity [32]. For general densities and general N, the presence of discontinuities across unknown surfaces makes Eq. (4.36) very challenging for numerical computations.

4.3.5 The Next Leading Term

So far we have treated the limit of the Levy–Lieb functional at infinite coupling strength λ (or, equivalently, at extreme low density). One could ask how this limit is approached, or, in other words, what is the next leading term in Eqs. (4.18)–(4.19).

The strategy employed in [68] to compute this next leading term relies on the assumption that the minimizer in (4.25) is of the SCE or Monge type, see the detailed discussion in the previous Sect. 4.3.4. Under this assumption, as shown in Sect. 4.3.8, the classical potential energy

(4.43)

with vSCE(r) defined by Eqs. (4.59) and (4.64), attains its minimum on the manifold Ω₀ parametrized by the co-motion functions,

(4.44)

When λ in Eq. (4.17) is very large but finite, we can expect that the support of the minimizer in Eq. (4.17) be strongly localized around Ω₀, as illustrated by Fig. 4.1 in Sect. 4.3.3. We can then expand Epot around its minimum through second order. The corresponding Hessian matrix $$\mathbb {H}({\mathbf {r}})$$ evaluated on Ω₀, for any fixed r, will have d zero eigenvalues (along the manifold Ω₀) and dN − d positive eigenvalues. By using curvilinear coordinates along the manifold Ω₀ and orthogonal to it, the sought next leading term is determined by adding the kinetic energy to the second-order expansion of Epot, which corresponds to the Hamiltonian of zero-point oscillations in the space orthogonal to Ω₀ [68]. The final result is that Eqs. (4.18)–(4.19) are extended to [66, 68]

(4.45)

(4.46)

where

(4.47)

In [71] this term has been computed explicitly for N = 2 electrons in 1d and it has been compared with accurate numerical calculations for the Levy-Lieb functional at very large λ, finding excellent agreement.

The intuition that the next term of the Levy–Lieb functional at infinite coupling strength λ should be given by zero-point oscillations around the manifold parametrized by the co-motion functions appeared for the first time in Seidl’s seminal work [122]. He also carried out explicit calculations in 3D for the spherically-symmetric case with N = 2 electrons, using the co-motion function introduced in Sect. 4.3.12. This idea was extended to the general many-electron case in [68], where it was also found that Seidl’s original calculation had a wrong factor 2. Very recently, a rigorous proof of Eqs. (4.45)–(4.47) for the many-electron 1d case has been provided by Colombo et al. [29].

4.3.5.1 The Fermionic Statistics

Equations (4.45)–(4.47) are the first-order correction due to kinetic energy in the large-λ (or ħ → 0) limit of the Levy–Lieb functional. This correction is still independent of the particle statistics. A natural question to ask is then at which order will the fermionic antisymmetry enter.

In Refs. [67, 68] it has been conjectured that the particle statistics enters in the λ →∞ limit at orders $$\sim \mathrm{e} ^{-\sqrt {\lambda }}$$ . The physical intuition behind this idea is simply that the effect on the energy of antisymmetrization vanishes as the overlap between gaussians centered at each set of strictly-correlated positions (each r value in Ω₀). The scaling $$\sqrt {\lambda }$$ of such gaussians comes from the zero-point hamiltonian. This conjecture has been confirmed numerically [71] for the case of N = 2 electrons in 1D, again by comparison with accurate numerical calculations of the exact Levy-Lieb functional at large λ.

4.3.6 The Strongly Interacting Limit of DFT from the Point of View of Optimal Transport

We now introduce a fruitful interpretation of the strongly interacting limit of DFT as “optimal transport with Coulomb cost”.

Optimal transport theory (see [50, 117, 119, 136] for textbook accounts) is concerned with the following two problems, introduced in special cases in fundamental work by Kantorovich [80] respectively Monge [108]:

(a)

Kantorovich optimal transport problem: For given probability measures μ₁, …, μN defined on closed subsets X₁, …, XN of $$\mathbb {R}^d$$ , find a joint probability measure Π on the product space $$X=X_1\times \ldots \times X_N\subseteq \mathbb {R}^{Nd}$$ which minimizes a cost functional

subject to the marginal constraints

Here $$c \, : \, X_1\times \ldots \times X_N\to \mathbb {R}\cup \{+\infty \}$$ is some given cost function, and the validity of the above constraint is denoted Π↦μ₁, …, μN.

(b)

Monge optimal transport problem: For given probability measures μ₁, …, μN defined on measurable subsets X₁, …, XN of $$\mathbb {R}^d$$ of positive volume which possess integrable densities p₁, …, pN (i.e. pi ∈ L¹(Xi)), and a cost function c as above, find measurable maps f₁, …, fN−1 with fi  :  X₁ → Xi+1 which minimize

subject to the marginal constraints

This corresponds to making the ansatz

(4.48)

or equivalently—using the notion of push-forward introduced in (4.33)—

(4.49)

in the Kantorovich problem, where id denotes the identity map id(r₁) = r₁.

Example 4.11 (N Equal Marginals, Coulomb Cost)

If we take

the Kantorovich optimal transport problem is precisely the SIL variational problem, (4.25), and the Monge optimal transport problem is precisely the SCE variational problem, (4.34).

Thus the strongly interacting limit of DFT can be viewed as optimal transport with Coulomb cost. This viewpoint, introduced by Buttazzo et al. [14] and Cotar et al. [32], opened the door to much of the current understanding of the strong-interaction limit of DFT.

Example 4.12 (Two Unequal Marginals, Positive Power Cost)

The prototype problem of classical optimal transport theory going back to [80, 108] is to instead take

That is, one considers:

only two marginals;

unequal instead of equal marginals;

a positive instead of a negative power of the Euclidean distance as cost.

Denoting μ₁ = μ, μ₂ = ν, f₁ = T, r₁ = x, r₂ = y, the Kantorovich problem then becomes

(4.50)

and the Monge problem becomes

(4.51)

The analogue of the SCE functional is the optimal cost as a functional of the two prescribed marginals,

Its p-th root, Wp(μ, ν) = (Copt[μ, ν])1∕p, is the celebrated p-Wasserstein distance, which is a metric on the space of probability measures.

Thus the SCE functional can be thought of as the Coulomb analogue of the Wasserstein distance.

We remark that the motivation of Monge and Kantorovich for considering Example 4.12 came from civil engineering, respectively economics, and explains the name optimal transport: Monge thought of moving a given pile of sand on a construction site into a given hole in a way that minimizes the overall distance of transport, with T(x) describing the target position of sand originally located at x and with pile and hole modelled, respectively, by μ and ν. Kantorovich thought of transporting some economic good, say steel, from producers (steel mines) to consumers (factories), at minimal transportation cost; Π(x, y) then describes the density of goods transported from location x to location y, and is called a transport plan. In the latter context it is natural not to make the Monge ansatz

but instead allow one producer located at x to supply several consumers located at different positions y, i.e. consider the general problem (4.50).

The general question for which costs and marginals the Monge and Kantorovich problems are equivalent, i.e. the Kantorovich problem admits minimizers of Monge form, is not well understood. A sufficient condition [119] for N = 2 (and, say, compact convex sets X₁ and X₂ and continuously differentiable costs c) is that the marginal measure μ₁ is absolutely continuous and c satisfies the so-called twist condition that the map $${\mathbf {r}}_2 \mapsto \nabla _{{\mathbf {r}}_1}c({\mathbf {r}}_1,{\mathbf {r}}_2)$$ be injective for every r₁. For N > 2, generalized twist conditions have been studied by Pass [111, 114, 115]; unfortunately these are not satisfied for the Coulomb cost.

4.3.7 Dual Construction of the SCE Functional

We now introduce a fourth—dual—construction of the SCE functional.

A cornerstone principle of optimal transport theory, Kantorovich duality, says that the minimum of a given Kantorovich optimal transport problem (see Sect. 4.3.6) equals the supremum of an associated explicit dual problem. The general form of the dual is recalled in Appendix 4.6. For the SIL problem (4.25), the dual problem is the following (see Appendix 4.6 for a quick derivation from general OT theory): maximize the functional

(4.52)

over potentials $$u \, : \, \mathbb {R}^d\to \mathbb {R}$$ which must satisfy the pointwise constraint

(4.53)

Maximization is over the admissible class

(4.54)

This yields the following alternative definition of the SCE functional:

(4.55)

This construction is due to Buttazzo, DePascale, and Gori-Giorgi [14]. Note that the optimization here is not over N-point densities, but over (suitable) external potentials u. Optimizers are called Kantorovich potentials. Heuristically, they can be thought of as Lagrange multipliers associated with the marginal constraints in the original problem (4.25). This is explained in our discussion of optimality conditions in Sect. 4.3.8.

It can be rigorously shown that the new construction yields, again, the SCE functional, and that optimal potentials exist:

Theorem 4.13

Let $$\rho \, : \, \mathbb {R}^d\to \mathbb {R}$$ be any N-particle density in the class $${\mathcal {D}}^N$$ (see (4.12)), and let w(r) = |r|⁻¹be the Coulomb interaction. Then:

(1)

[14] The supremum in (4.55) is equal to the minimum in (4.26).

(2)

[14,38] The supremum in (4.55) is attained; that is, there exists a maximizing potential u in the class (4.54).

(3)

[13,42] If, in addition, ρ > 0 everywhere, there exists a maximizing potential which is in addition Lipschitz continuous.

Statement (1) follows directly from the general Kantorovich duality theorem of OT theory; see Appendix 4.6. The question of existence and regularity of optimal potentials is more delicate. Note that the Coulomb potential Vee which upper-bounds u(r₁) + ⋯ + u(rN) tends to plus infinity as the distance ri −rj between any two position coordinates goes to zero; so one might a priori think that u’s are favourable which also tend to plus infinity at certain places. But statement (2) in the above theorem says that this does not happen; the existence proof of bounded optimal potentials is due to [14] for N = 2 and to [38] for general N.

Quantum Analogue

We remark that the dual construction of the SCE functional in Eq. (4.55) admits a quantum analogue. In [97], Lieb proposed an extension of the Levy–Lieb functional (4.11) to mixed states, i.e. $$\mathrm {F}_{\mathrm {L}}:\mathcal {D}_N\to \mathbb {R}$$ ,

(4.56)

where Γ is an operator acting on the fermionic Hilbert space and, similarly to (4.9), Γ↦ρ denotes the relation . In [91], M. Levy introduced a similar functional requiring in addition that Γ = |ψ〉〈ψ| be a rank-one operator. An advantage of the Lieb functional FL is that it is convex. Ignoring issues of rigor, (4.56) admits a dual formulation

(4.57)

with the above inequality understood in the sense of self-adjoint operators. For a rigorous discussion of Eq. (4.57) see Chap. 3 by Lewin et al. This equation is the quantum analogue (for mixed states) of the dual construction of the SCE functional. Note that because the right-hand side of the constraint on u now contains an additional positive term, the value of the supremum will be higher than in (4.55), as it should be.

4.3.8 Optimality Conditions

With the help of Kantorovich duality one obtains very interesting necessary conditions for solutions to the SIL variational principle (4.25). In particular, for optimizers of SCE (alias Monge) form one can express the gradient of the Kantorovich potential u in terms of the co-motion functions (alias transport maps).

We follow the rigorous presentation for general OT problems in [50], but specialize throughout to the SIL problem. For the benefit of less mathematically minded readers, we also include a heuristic derivation at the end of this section.

Theorem 4.14 (Optimality Conditions [50])

Let $$\rho \, : \, \mathbb {R}^d\to \mathbb {R}$$ be any N-particle density in the class $${\mathcal {D}}^N$$ (see (4.12)). Let $$V_{\mathrm {ee}} \, : \, \mathbb {R}^{dN}\to \mathbb {R}\cup \{+\infty \}$$ be any interaction potential which is symmetric, bounded from below, lower semi-continuous, and has the property that the minimum in (4.26) is finite. Suppose Π is a solution to the SIL problem (4.26), and u is a solution to the dual problem, i.e. a maximizer of J in the class $${\mathcal {A}}$$ .

(1)

Π is zero outside the set

(2)

Π is an unconstrained minimizer (i.e., a minimizer on $${\mathcal {P}}(\mathbb {R}^{dN})$$ ) of the modified functional

(3)

At any point (r₁, …, rN) in $${\mathcal {M}}$$ where the function in (1) is differentiable with respect tor₁,

(4.58)

In particular, if Π is of SCE form, (4.30), and Vee(r₁, …, rN) is the Coulomb interaction $$\sum _{1\le i&lt;j\le N}\tfrac {1}{|{\mathbf {r}}_i-{\mathbf {r}}_j|}$$ ,

(4.59)

The physical and mathematical meaning of these results is as follows.

(1) says that the classical potential energy

is minimal on the manifold of configurations which occur with nonzero probability under the optimal plan Π. In particular, when Π is of the SCE or Monge type, the classical potential energy is minimal on the manifold (4.31) parametrized by the co-motion functions. Besides being interesting in its own right, this underlies the derivation of the next leading term of the Levy–Lieb functional outlined in Sect. 4.3.5.

(3) says that the Kantorovich potential u is an effective one-body potential emulating the many-body system, in the following sense: its gradient at the point r is precisely the classical repulsive force exerted on an electron at r by the other electrons at positions fi(r). Equation (4.59) is called the force equation.

(2) can be viewed as an infinite-dimensional Lagrange multiplier rule, with any Kantorovich potential (i.e. any optimizer of the dual variational principle (4.55)) playing the role of a Lagrange multiplier associated with the constraint Π↦ρ.

We remark that results of the above form have a long history in OT theory; for the two-marginal problem with interaction potential |r₁ −r₂| respectively |r₁ −r₂|², (1) goes back to Kantorovich himself [80], while the differential version (3) and its usefulness were first realized by Knott and Smith [83].

Proof

The following proof, taken from [50], is simple and illuminating, so we include it. By Kantorovich duality (in the form of Theorem 4.13 (1)),

Since Π has equal marginals $$\frac {\rho }{N}$$ , , and so

But since u satisfies the constraint (4.149) at every point in $$\mathbb {R}^{dN}$$ , the integrand is nonnegative. So the minimum value of the integrand must be zero and attained, and Π must vanish wherever the integrand is positive. This establishes (1) and (2). The elementary calculus fact that the gradient of a differentiable function vanishes at minimum points now yields (4.58). Finally, (4.59) follows since the point $$\bigl ({\mathbf {r}}_1,f_1({\mathbf {r}}_1),\ldots ,f_{N-1}({\mathbf {r}}_1)\bigr )$$ belongs to $${\mathcal {M}}$$ whenever the density ρ is positive at r₁. □

We complete this section with a more heuristic derivation of the optimality conditions.

Heuristic derivation of Theorem 4.14

Let us re-write the SIL variational principle (4.25) in the form

(4.60)

where $$G^{({\mathbf {r}}_1)}[\Pi ]$$ is the functional which assigns to an N-point probability measure Π the value of its single-particle density at the point r₁, and where the minimization is over symmetric probability measures (see Remark 4.2). Let us now postulate the existence of a family of Lagrange multipliers $$(\lambda ({\mathbf {r}}_1))_{{\mathbf {r}}_1\in \mathbb {R}^d}$$ , one for each $$G^{({\mathbf {r}}_1)}$$ , such that minimizers of $${\mathcal {C}}$$ subject to the constraints $$G^{({\mathbf {r}}_1)}[\Pi ]=\rho ({\mathbf {r}}_1)$$ are unconstrained minimizers of the Lagrangian

But since $$G^{({\mathbf {r}}_1)}[\Pi ]$$ is the one-body density of Π, and Π is symmetric,

(4.61)

so the Lagrangian coincides with the functional in (2) with λ = u. It is clear that minimizers of the Lagrangian must be concentrated on the set of pointwise minimizers of the integrand, yielding (1). Statement (3) now follows as in the rigorous proof.

The above argumentation obtains the Kantorovich potential u quickly but non-rigorously as a Lagrange multiplier. In fact, with such a heuristic construction of u, statements (1) and (3) were already derived in [127] before the discovery of the SCE theory/optimal transport connection.

But readers are put on notice that there is no such thing as a general and rigorous Lagrange multiplier rule which would guarantee the existence of Lagrange multipliers for infinite-dimensional non-smooth problems like Levy–Lieb constrained search or its strongly interacting limit (4.25). In DFT (in its original form with both kinetic energy and electron repulsion present), the existence problem for Lagrange multipliers—i.e., the existence of one-body potentials which, when added to the Hamiltonian T + Vee, reduce a constrained search to an unconstrained search—is known as the v-representability problem. This is a longstanding open problem, see e.g. [76, 84, 90, 97, 133]. For variants of the problem at positive temperature respectively quantum lattices see [19, 20]; v-representability for a regularization of the exact Levy–Lieb functional is discussed in Chap. 5 by Kvaal.

4.3.9 Solution of the Purely-Interacting v-Representability Problem

We now show that in the strongly interacting limit the v-representability problem, alias the problem of existence of Lagrange multipliers for density functionals defined by constrained search, can be completely solved. As we will see, this fact follows by combining known results. We assume in this section that w(r) = |r|⁻¹ is the Coulomb interaction.

Recall that a density $$\rho \, : \, \mathbb {R}^d\to \mathbb {R}$$ is called

N-representable if it comes from a wave function $$\Psi \in {\mathcal {W}}^{\mathcal {N}}$$ (i.e. Ψ↦ρ)

v-representable if it comes from a minimizer of 〈 Ψ|T + Vee +∑iv(ri)| Ψ〉 on $${\mathcal {W}}^{\mathcal {N}}$$ for some potential $$v\, : \, \mathbb {R}^d\to \mathbb {R}$$

non-interactingv-representable if it comes from a minimizer of 〈 Ψ|T +∑iv(ri)| Ψ〉 on $${\mathcal {W}}^{\mathcal {N}}$$ for some potential $$v\, : \, \mathbb {R}^d\to \mathbb {R}$$

purely-interactingv-representable if it comes from a minimizer of $$\int _{\mathbb {R}^{dN}} \bigl (V_{\mathrm {ee}} + \sum _i v({\mathbf {r}}_i)\bigr )\mathrm{d} \Pi $$ on $${\mathcal {P}}(\mathbb {R}^{dN})$$ for some potential $$v\, : \, \mathbb {R}^d\to \mathbb {R}$$ .

Theorem 4.15 (N-Representability Implies Purely-Interacting v-Represent -ability)

Any N-representable ρ, i.e. any ρ belonging to the class $${\mathcal {D}}^N$$ (see (4.12)), is purely-interacting v-representable by some bounded measurable potential $$v\, : \, \mathbb {R}^d\to \mathbb {R}$$ . Explicitly, the following choice will do:

(4.62)

where u is any bounded Kantorovich potential for ρ (see Theorem4.13for existence of the latter).

This result is quite remarkable, given that—to our knowledge—not much is known on the rigorous level off the strongly interacting limit.

Proof of Theorem4.15

By Theorem 4.13 (2), there exists a bounded maximizer u of the dual functional, i.e. an associated Kantorovich potential. Let v = −u. By Theorem 4.1 (1), there exists a minimizer Π[ρ] of the SIL variational problem (4.25). By Theorem 4.14 (2), this Π[ρ] is a minimizer of $$\int _{R^{dN}}\bigl (V_{\mathrm {ee}}+\sum _i v({\mathbf {r}}_i)\bigr )\mathrm{d} \Pi $$ on $${\mathcal P}(\mathbb {R}^{dN})$$ . Since by construction Π has density ρ, it follows that v represents ρ. □

If in addition ρ > 0 everywhere, the above proof together with Theorem 4.13 (3) shows that ρ is even purely-interacting v-representable by some Lipschitz continuous potential.

4.3.10 Functional Derivative and SCE Potential

It is not difficult to deduce from Theorem 4.15 that when the density ρ is sufficiently nice (say, continuous and everywhere positive) and the Kantorovich potential u[ρ] (i.e. the maximizer of the dual problem (4.55)) is unique, the SCE functional is functionally differentiable at ρ with functional derivative

(4.63)

where const is an arbitrary additive constant. Here for any functional F on $${\mathcal {D}}^N$$ the functional derivative $$\frac {\delta F}{\delta \rho }[\rho ]$$ at some density ρ (if it exists) is defined by the requirement

for all smooth mass-preserving localized perturbations $$\eta \, : \, \mathbb {R}^d\to \mathbb {R}$$ (mathematically: $$\eta \in C_0^\infty (\mathbb {R}^d)$$ , $$\int \eta = 0$$ ), and is unique up to an additive constant. For an informal derivation of Eq. (4.63) see e.g. [22], and for a rigorous proof under suitable assumptions see [41].

As for any Hartree-exchange-correlation functional, the Hartree-exchange-correlation potential associated to the SCE functional is the functional derivative with additive constant chosen so that the potential vanishes at infinity, in our case

(4.64)

This functional derivative is called the SCE potential.

To summarize: the SCE potential for the strongly correlated limit of DFT agrees up to a shift with the Kantorovich potential from optimal transport theory.

Assume now that the density is everywhere positive, that the ground state of (4.25) is an SCE state, and that

(4.65)

It then follows from (4.59) that the SCE potential has the correct asymptotic behavior

(4.66)

By contrast, Hartree-exchange-correlation potentials for all semilocal functionals (LDA, GGAs) are well known to have the wrong asymptotics on physical (i.e. exponentially decaying) densities,

(4.67)

Open Problem

Rigorously justify (4.65), and hence (4.66), for general densities ρ. Note that for N  =  2 and radial densities, or any N and arbitrary densities in one dimension, assumption (4.65) follows from the explicit formulae for the fi in [32] respectively [27] (Fig. 4.3).⁴

2 graphs. A, the graph plots 2 concentric circles at the origin and 3 clustered dots in the inner circle. There is a path of dots from the outer circle to the inner circle using a dotted curve and arrow. There is another path of dots moving out of the outer circle. B, a line graph plots a line that peaks initially and then falls to the x-axis. The term 4 pi r square, pho of r is mentioned.

Fig. 4.5

SGS state (left panel) for the Lithium atom density (right panel). The extreme angular and radial correlation exhibited by this state is illustrated here while sending one of the electrons (the leftmost) to infinity

Example 4.16

Let N = 2, and let ρ(r) = 2∕π(1 + r²) be the one-dimensional Lorenzian density, normalized so that $$\int \rho = 2.$$ The co-motion function f₁ = f can be computed explicitly and is given by f(r) = −1∕r, see Example 4.18 in Sect. 4.3.11. The SCE potential must satisfy the differential equation (4.59) which in our case reads

The boundary condition vSCE(r) → 0 for r →∞ (Eq. (4.64)) yields the solution

4.3.11 Strictly Correlated Electrons in One Dimension

In one dimension the strong interaction limit (Eq. (4.25)) can be solved exactly. The minimizing probability measure is given by an SCE state (4.30) with explicit co-motion functions alias transport maps. The minimizer was found by Seidl himself in the original paper [122], on grounds of physical intuition. A proof of its optimality was found much later by Colombo, De Pascale and Di Marino [27].

Seidl’s Construction

For a given integrable density $$\rho \, : \, \mathbb {R}\to \mathbb {R}$$ with ρ ≥ 0 and $$\int \rho = N$$ , begin by choosing $$f_1 \, : \, \mathbb {R}\to \mathbb {R}$$ so that the amount of density between r and f₁(r) is 1. Now choose f₂ so that the amount of density between f₁(r) and f₂(r) is again 1, and so on, i.e., denoting f₀(r) = r,

(4.68)

for all i = 0, …, N − 1. For Eq. (4.68) to possess a solution fi+1(r) in $$\mathbb {R}\cup \{+\infty \}$$ we must have $$\int _{f_i(r)}^\infty \rho \ge 1$$ ; otherwise one needs to integrate first up to + ∞ and then onwards from −∞ so as to obtain a total value of 1,

(4.69)

Physically this means that, given that the first electron is at some position x₁ = r, all the other electrons at x₂ = f₁(r), …, xN = fN−1(r) are separated by an equal amount of density between nearest neighbors. See Fig. 4.4, right panel. As always for SCE states, the first electron position is distributed according to the given density ρ.

3 graphs. A has a bell curve with vertical regions for x subscripts 1 to 7 from left to right. The curve forms a crest initially and then decreases gradually to the x-axis. B and C, x 1 and y 1 are the linear lines. x 2 and x 3, x 4 and x 5, and x 6 and x 7 begin from the same point in the y-axis. In B, x 3, x 5, and x 7 increase steeply and reduce gradually. x 2, x 4, and x 6 decrease to the x-axis and increase logarithmically. C, lines x 3, x 5, and x 7 increase parallelly from the y-axis with a difference of 2 units, peak at some point, and decrease uniformly. x 2, x 4, and x 6 decrease parallelly to x axis and increase. They all form a mesh-like structure.

Fig. 4.6

SGS state when N = 7. Top: a radial measure ρrad(r). Bottom left: the maps S, S⁽²⁾, …, S⁽⁶⁾ : [0, ∞) → [0, ∞), plotted with colors green, blue, red, violet, yellow, and brown. Bottom right: the same graphs under a change of variables $$y_1 = \int _0^{x_1} \tfrac {\rho _{\mathrm {rad}}}{N}$$ which transports ρrad to the uniform density on the interval [0, 1]. Picture from [124]

The above construction can be expressed concisely in terms of the cumulative distribution function

(4.70)

and its generalized inverse

(4.71)

(When ρ is continuous and everywhere positive, $$G_{\rho }^{-1}$$ is just the usual inverse function; the above definition has the virtue that it works for any nonnegative integrable ρ with integral N.) Equations (4.68) and (4.69) now take the form Gρ(fi+1(r)) − Gρ(fi(r)) = 1∕N respectively 1 − Gρ(fi(r)) + Gρ(fi+1(r)) = 1∕N, so by solving for fi+1 in terms of fi and using f₀(r) = r

(4.72)

for i ∈{1, …, N − 1}.

Optimality

This construction is indeed optimal:

Theorem 4.17

Let w(r) = |r|⁻¹. For any nonnegative integrable density $$\rho \, : \, \mathbb {R}\to \mathbb {R}$$ with $$\int \rho = N$$ , the SCE state (4.30) with f₁, …, fN−1given by the Seidl construction (4.72) is a minimizer of the SIL problem

Moreover when ρ is everywhere positive, this minimizer is unique for N = 2, and its symmetrization (see Remark4.2) is the unique symmetric minimizer for arbitrary N.

This theorem is due to [32] for N = 2 and to [27] for arbitrary N. Despite the intuitive nature of the optimizer, the proof is not elementary. It is based on a careful analysis of the structure of Vee-cyclically monotone sets in $$\mathbb {R}^N$$ , and strongly relies on both optimal transport theory and the ordering properties of the real line. Note that uniqueness cannot hold for N ≥ 3 unless symmetry is required, as re-labelling the fi then yields another solution. This is purely a mathematical, not a physical effect since solutions to the SIL problem arising as low-density limits of N-point densities of quantum wavefunctions (as described by Theorem 4.3) are always symmetric, corresponding to the symmetrization of the state (4.30) and (4.72).

Group Law

Formula (4.72) implies an interesting group law for the co-motion functions, already noticed in [122]: the ith function is the i-fold composition of the first function with itself,

and the N-fold composition of the first function gives the identity f₀(r) = r.

Explicit Examples

The following examples further illustrate the nonlinear governing Eqs. (4.68)–(4.69), and may serve as useful benchmarks for numerical simulations in the strongly interacting limit (or close to it).

Example 4.5, ctd

Consider a two-electron system with ρ being the uniform density in a one-dimensional interval [0, L]. In this case we have Gρ(r) = r∕L, and formula (4.72) readily yields the co-motion function (4.40). For its graph, see Fig. 4.2. Mathematically this map switches the right and left half of the interval; note that its composition with itself indeed gives the identity, as it must by the group law.

Example 4.18 ([68, 71])

Let ρ(r) = N∕π(1 + r²) be the Lorenzian density, normalized so that $$\int \rho = N$$ . In this case $$G_{\rho }(r)=\tfrac {1}{\pi }\arctan r + \tfrac {1}{2}$$ and so Eq. (4.68) for f₁ in the region $$G_\rho (r)\le \tfrac {N-1}{N}$$ is, recalling the notation f₀(r) = r,

(4.73)

When N = 2 it follows that

(4.74)

(note that then the derivatives of both sides of (4.73) agree, as do their values at r = 0). From now on let us assume N ≥ 3. In this case we can use the addition formula for the tangent, $$\tan {}(x+y) = (\tan x + \tan y)/(1 - \tan x \tan y)$$ for $$x,\, y,\, x\! + \! y\not \in \pi /2 + \mathbb {Z}$$ , and obtain

(4.75)

In the region $$G_\rho (r)&gt;\tfrac {N-1}{N}$$ , or equivalently $$\arctan r &gt; \tfrac {\pi }{2}-\tfrac {\pi }{N}$$ , or equivalently (because $$\tan (\tfrac {\pi }{2}-x)=1/\tan x$$ ) r > 1∕t₁, Eq. (4.69) for f₁ is

that is to say $$\arctan f_1(r) = \arctan r + \tfrac {\pi }{N} - \pi $$ . Using the addition formula for the tangent and $$\tan x = \tan {}(x-\pi )$$ we again find that f₁ is given by (4.75), so this formula describes f₁ on the whole real line. It remains to compute its i-fold composition fi. Here we give a different derivation as compared to [68, 71]. Note that mathematically f₁ is a Moebius map, i.e. a map of the form Ma(r) = (r + a)∕(1 − ar). Using the (elementary to check) composition formula $$M_a\circ M_b = M_{\frac {a+b}{1-ab}}$$ and the addition formula for the tangent we find

(4.76)

Moreover, setting i = N in the above formula we recover the abstract fact that the N-fold composition of f₁ must be the identity. Hence the co-motion functions for the Lorenzian density form a discrete subgroup of the Moebius group. For the graph of these functions when N = 5 see Fig. 4.4.

4.3.12 Radially Symmetric Densities

When the one-body density ρ is radially symmetric, Seidl, Gori-Giorgi and Savin [127] conjectured an explicit minimizing probability measure in (4.25) of a radial-symmetry-preserving SCE form which is related to the explicit SCE state of one-dimensional systems.⁵ Let us describe their conjecture in detail.

The starting point is the following reduction to a 1d problem with effective interaction.

Lemma 4.19 (Reduction to a 1d Problem, [9, 113])

Let $$\rho \, : \, \mathbb {R}^d\to \mathbb {R}$$ be an integrable density with ρ ≥ 0 and $$\int \rho = N$$ which is radially symmetric, that is, ρ(r) = ρ₀(|r|) for some function ρ₀, and let

where ωdis the area of the unit sphere in $$\mathbb {R}^d$$ (for d = 3, ωd = 4π). Then the SCE functional defined by (4.26) reduces to

(4.77)

where $$\operatorname {V}_{\mathrm {ee}}^{\operatorname {rad}}$$ is the reduced Coulomb cost

(4.78)

Moreover, $$\Pi \in {\mathcal P}(\mathbb {R}^{dN})$$ is a minimizer for the full SIL variational principle (4.25) in d dimensions if and only if its radial projection Πrad, defined by

for all intervals A₁, …, AN, is a minimizer for the right-hand side of (4.77) and $$V_{\mathrm {ee}}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N)=V_{\mathrm {ee}}^{\mathrm {rad}}(|{\mathbf {r}}_1|,\ldots ,|{\mathbf {r}}_N|)$$ Π-a.e.

In [122, 127], the following interesting explicit state was conjectured to be optimal for the reduced problem in (4.77):

(4.79)

where S(n) denotes the n-fold composition S ∘⋯ ∘ S and S  :  [0, ∞) → [0, ∞) is defined as follows. Let 0 = a₀ < a₁ < ⋯ < aN−1 < aN = ∞ be such that the intervals An = [an−1, an) between successive an’s carry equal mass, that is, $$\int _{A_n} \rho ^{\mathrm {rad}} = 1$$ for all n, and let $$S|{ }_{A_n}$$ be the unique function such that

(4.80)

(with the convention AN+1 = A₁). In terms of the original SIL problem (4.25), this ansatz corresponds to the SCE ansatz (4.30) with maps satisfying the additional property

for the above explicit S and with suitably chosen angles so that Π↦ρ and $$V_{\mathrm {ee}}({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N)=V_{\mathrm {ee}}^{\mathrm {rad}}(|{\mathbf {r}}_1|,\ldots ,|{\mathbf {r}}_N|)$$ Π-a.e. We call S the SGS map, and the probability measure η given by (4.79), (4.80) the SGS state. See Figs. 4.5 and 4.6.

4-part diagram. A has a shaded square of 1 unit with a diagonal from bottom left to right. B has a 3 by 3 matrix with the same pattern as A in the center cell of the first row, the left cell in the next row, and the right cell in the last row. C has a 3 by 3 matrix with the same pattern as B at the same places as in B. D has a 3 by 3 matrix with the same pattern as C at the same places as in C.

Fig. 4.7

Construction of the optimal map T in Example 4.22. The picture shows the graph (in blue) of the first few iterations of Eq. (4.83); each graph consists of three scaled copies of the previous one. The exact map is reached in the limit of infinitely many iterations

4 3-D graphs, A to D have 2 mountain-like structures side by side with a gap in between and steep slopes on the adjacent side. From A to D, the gaps get wider, the structure gets thinner with sharper edges and their 2 ends bend backward like 2 crowns placed opposite to each other.

Fig. 4.8

Numerically computed optimal density πτ(r₁, r₂) in (4.89) for two electrons in one dimension and the density $$\rho (r)=c_0/\cosh r$$ , r ∈ [−10, 10], for different values of the regularization parameter τ. Here c₀ is a normalization constant, and we have used the effective Coulomb interaction [7] $$V_{\mathrm {ee}}(r_1,r_2)=1.07\, \mathrm{e} ^{-| r_1-r_2|/2.39}$$

The SGS state has been rigorously proved to be optimal in some specific cases.

Example 4.20 (SCE for Radially Symmetric Densities, [32, Theorem 4.10])

Let N = 2, and let ρ be a radially symmetric density on $$\mathbb {R}^d$$ such that ρ(r) > 0 for all r. Then the optimal co-motion function f is given by

for some function $$s:[0,+\infty )\to \mathbb {R}$$ such that $$s\leqslant 0$$ , s is increasing, limr→+∞s(z) = 0, and limr→0+s(r) = −∞.

The function s in the above example corresponds to minus the SGS map, i.e. s = −S in the N = 2 case.

Optimality of the SGS state has also been proved for some special class of densities ρ when N = d = 3 [30, 57, 125] and N = 3 and d = 2 [11].

Recently, counterexamples of radially symmetric probability densities were found for which the SGS state is not optimal. The simplest one is a uniform density on a thin annulus:

Example 4.21 ([30], see also [11, 57, 124, 125] for related examples)

Let N = 3. For sufficiently small ε > 0, and the density

(4.81)

(with the constant cε chosen such that $$\int \rho ^{\mathrm {rad}}=3$$ ), the SGS state ηε defined by (4.79)–(4.80) is not optimal for the variational problem (4.77).

This example illustrates that guessing the optimal SCE states can be a tricky business even for 1d problems, and makes it all the more remarkable that optimality of Seidl’s guess for the 1d Coulomb problem is a rigorous theorem (Theorem 4.17). The proof of nonoptimality relies on a Taylor expansion of the reduced interaction $$\operatorname {V}_{\mathrm {ee}}^{\operatorname {rad}}$$ (defined in Eq. (4.78)) at the point (1, 1, 1) and on cyclical monotonicity methods from optimal transport theory.

While this counterexample disproves optimality of the SGS state in general, the density (4.81) is quite different from typical atomic densities and the following remains an interesting mathematical problem.

Open Problem

Find sufficient conditions on radial densities ρ such that the SGS state is optimal for (4.77).

4.3.13 An Example with Irregular Co-motion Functions for Repulsive Harmonic Interactions

One of the more challenging properties of co-motion functions is that they are typically discontinuous. Here we give an extreme example with modified electron-electron interaction which is discontinuous everywhere, due to Di Marino et al. [42].

Example 4.22

Let d be arbitrary, Vee(r₁, …, rN) = −∑1≤i<j≤N|ri −rj|² (repulsive harmonic interaction), and N = 3. Let $$\rho =3 \, \cdot \, 1_{[0,1]^d}$$ (uniform density on a cube in $$\mathbb {R}^d$$ ). Then there exists a nowhere continuous map T  :  [0, 1]d → [0, 1]d which transports ρ to itself such that

(4.82)

is an optimal probability measure for the SIL problem (4.25).

The map T is an explicit fractal map. For d = 1 it is depicted in Fig. 4.7 and constructed as the unique fixed point of the iteration

(4.83)

starting with f(x) = x. To see what the iteration is doing, divide [0, 1]² into a 3 × 3 grid of squares and put scaled copies of the graph of the original function into the two squares directly above the diagonal and the bottom right square. Optimality of the resulting fractal SCE state (4.82) is easy to see from the following special property of the repulsive harmonic cost which was first observed by Pass [112]: thanks to the identity $$V_{\mathrm {ee}}=|{\mathbf {r}}_1 + \ldots + {\mathbf {r}}_N|{ }^2 - N \sum _{i=1}^N|{\mathbf {r}}_i|{ }^2$$ and the fact that the integral of the second term against a probability measure Π only depends on its marginal, the minimizers of the SIL problem are precisely the probability measures supported on the surface r₁ + ⋯ + rN = 0.

A 4-quadrant graph plots an area covered in the first 2 quadrants between the upper half of a Cassini oval and a semi-ellipse above it, along the X-axis. It ranges symmetrically between negative 2 and 2 on the X-axis. There is another area between 2 U curves of different curvatures, plotted symmetrically along the X-axis. 2 arrows from the area in the first quadrant lead to the area in the third quadrant.

Fig. 4.9

Co-motion function for the H₂ molecule [22]. The blue region—corresponding to the points in a half plane adjacent to the molecular axis with density between 0.04 and 0.08—is mapped to the green region in the opposite half plane. The black dots indicate the positions of the nuclei

The above example and construction works for arbitrary N, see [42].

Open Problem

Do such extreme examples also occur for the Coulomb interaction? Note that the repulsive harmonic interaction arises by locally Taylor-expanding the Coulomb interaction in angular direction.

4.3.14 Minimizers of the Discretized SIL Variational Principle Are Quasi-Monge States

We now come back to the important issue that the SIL variational principle (4.25) still requires minimization over a high-dimensional space of N-point probability measures, whereas the low-dimensional SCE ansatz (4.30) can fail to yield an optimizer of (4.25). One can ask whether some modified low-dimensional ansatz is enough to solve (4.25) exactly. In other words, can one achieve Seidl’s original goal of solving the strongly interacting limit of DFT with a low-dimensional ansatz that can be easily stored on a computer?

For the discretization of (4.25) on a grid, Friesecke and Vögler [53] found a modified ansatz which achieves this, for arbitrary space dimensions, densities and interaction potentials:

(4.84)

where σ runs over all permutations of the indices 0, …, n − 1, α is some (free to choose) probability density on the single-particle space $$\mathbb {R}^d$$ , and the fn are maps from $$\mathbb {R}^d$$ to $$\mathbb {R}^d$$ . States of this form are called quasi-Monge states or quasi-SCE states. With the specific choice $$\alpha =\frac {\rho }{N}$$ , the quasi-Monge ansatz (4.84) reduces precisely to the SCE (alias Monge) ansatz in its symmetric form (4.42). The novelty is the additional freedom of choosing the auxiliary density α. For the quasi-Monge ansatz, the marginal constraint Π↦ρ takes, instead of the conditions fn♯ρ = ρ (n = 0, …, N − 1) (Eq. (4.32)), the form of a single condition,

(4.85)

That is, the average push-forward of the auxiliary density α under the quasi-SCE maps must be the (suitably normalized) physical density.

Plugging the ansatz (4.84) into the SIL variational principle (4.25) and integrating out the variables r₂, …, rN yields the quasi-Monge or quasi-SCE variational principle

(4.86)

with the minimization being subject to the constraint (4.85).

Theorem 4.23 (Justification of the Quasi-Monge Ansatz, [53])

Let ρ be any discrete N-particle density on $$\mathbb {R}^d$$ , that is to say $$\rho ({\mathbf {r}})=\sum _{i=1}^\ell \rho _i \delta ({\mathbf {r}}-{\mathbf {a}}_i)$$ for some distinct discretization points $${\mathbf {a}}_i\in \mathbb {R}^d$$ and some ρi ≥ 0 with ∑iρi = N, and let $$V_{\mathrm {ee}}\, : \, \mathbb {R}^{dN}\to \mathbb {R}\cup \{+\infty \}$$ be any interaction potential which is symmetric in the electron coordinates (e.g., the Coulomb interaction Vee(r₁, …, rN) =∑i<j1∕|ri −rj|). Then the SIL problem (4.25) possesses a minimizer which is a quasi-Monge state (4.84). Equivalently, it possesses a minimizer of the form (4.88), i.e., a superposition of at most ℓ symmetrized Dirac measures.

This result rigorously reduces the number of unknowns from exponential to linear with respect to the number of electrons; more precisely, from ℓN (the dimension of the space of N-point probability measures supported on {a₁, …, aℓ}N) to ℓ ⋅ (N + 1) (ℓ unknowns for each of the N quasi-Monge maps, and another ℓ unknowns for the auxiliary density α).

The above result fails if the class of quasi-Monge states is narrowed to Monge (alias SCE) states, see [49]. For continuous ρ’s, it is an open question whether the SIL problem always (or at least in the Coulomb case) admits minimizers of quasi-Monge form.

Proof of Theorem4.23

(Following [53].) Let us explain the intuition and reasoning behind the quasi-Monge ansatz and Theorem 4.23, which comes from convex geometry. Before passing to a geometric viewpoint, we note that by the symmetry of Vee the minimization in (4.25) can be restricted to symmetric probability measures (see Remark 4.2); moreover any symmetric probability measure $$\Pi \in {\mathcal {P}}(\mathbb {R}^{dN})$$ with Π↦ρ must be of the form

for some symmetric tensor $$(\gamma _{i_1\ldots i_N})\in \mathbb {R}^{\ell \times \ldots \times \ell }$$ with nonnegative entries which sum to 1. Now geometrically, for fixed discretization points a₁, …, aℓ the set of these probability measures is a finite-dimensional convex polytope; let us denote it by $${\mathcal {P}}_{\mathrm {sym}}(\{{\mathbf {a}}_1,\ldots {\mathbf {a}}_\ell \}^N)$$ . The subset satisfying the marginal constraint Π↦ρ, i.e.

(4.87)

is also a convex polytope called the Kantorovich polytope; let us denote it by $${\mathcal {P}}_{\rho }(\{{\mathbf {a}}_1,\ldots {\mathbf {a}}_\ell \}^N)$$ . While general probability measures in these sets possess a huge number of coefficients which increases combinatorially with the number N of particles, the key point is that the extreme points⁶ of these sets are very sparse, with only a small number of nonzero coefficients. The extreme points of $${\mathcal {P}}_\rho (\{{\mathbf {a}}_1,\ldots ,{\mathbf {a}}_\ell \})$$ are easily seen to be symmetrized products of delta functions, $$S_N\delta ({\mathbf {r}}_1-{\mathbf {a}}_{i_1})\cdots \delta ({\mathbf {r}}_N-{\mathbf {a}}_{i_N})$$ , where SN is the symmetrization operator. Now consider a subset of a convex polytope satisfying one linear constraint, geometrically: the intersection of the polytope with a hyperplane. It is geometrically expected (and not difficult to prove) that all extreme points of this new set are convex combinations of just two extreme points of the original polytope. Analogously, by a well-known result in convex geometry the intersection of a convex polytope with k hyperplanes has extreme points given by convex combinations of just k + 1 of the original extreme points. Since the marginal condition (4.87) imposes ℓ − 1 constraints (note that one of the ℓ constraints is redundant due to the sum of the $$\gamma _{i_1\ldots i_N}$$ being 1), the extreme points of the Kantorovich polytope are convex combinations of just ℓ symmetrized delta functions, i.e., probability measures of the form

(4.88)

for some nonnegative coefficients αν. Defining the maps fn by $$f_{n-1}({\mathbf {a}}_\nu ) = {\mathbf {a}}_{i_n^{(\nu )}}$$ yields that all extreme points are quasi-Monge states (4.84). Theorem 4.23 now follows from the general principle that the minimum of a linear functional (such as $$\int V_{\mathrm {ee}} \, \mathrm{d} \Pi $$ ) over a convex polytope is always attained at some extreme point. □

The quasi-Monge (or quasi-SCE) ansatz and Theorem 4.23 underlie the numerical method described in Sect. 4.4.5.3.

4.3.15 Entropic Regularization of the SCE Functional

We have seen in Fig. 4.1 and Sect. 4.3.8 that in the strongly interacting limit, the N-body density concentrates on the lower-dimensional manifold on which the classical effective potential energy Vee(r₁, …, rN) −∑ivSCE(ri) is minimal. A regularization of the SCE functional which has nice mathematical properties and smears out the N-body density is the following:

(4.89)

Here τ > 0 is a small parameter, $${\mathcal {V}}_{\mathrm {ee}}$$ is the usual electron interaction energy, and S is (minus) the Shannon–Von Neumann entropy,

(4.90)

As shown in Lemma 4.27 below, the negative part of the entropy density has finite integral under very mild conditions on ρ (e.g., finite first moment suffices), and so definitions (4.89)–(4.90) make rigorous sense. The existence of a minimizer in (4.89) can be obtained assuming that $$\rho \log \rho \in L^1(\mathbb {R}^d)$$ [60]. Physically, the right-hand side in (4.89) can be viewed as the free energy of N classical particles with interaction potential Vee and density ρ at inverse temperature τ. But our goal here is not to model a physical system at finite temperature, but instead to approximate the SCE functional.

Figure 4.8 illustrates the effect of the entropy term in a two-electron example: the larger the regularization parameter τ, the more the minimizers π are spread out around the support of the SCE state. (Recall that by Theorem 4.4, when N = 2 the SIL variational principle is uniquely minimized by an SCE state.)

7 scatterplots. A graph of density versus x plots dots in a bell curve. The peak is at (50, 0.17). Out of 2 graphs for 500 iterations, one has dots scattered all over the region except along the linear line. In another graph, the dots are scattered in a downward concave trend. Out of 2 graphs for 5000 iterations, 1 graph plots dots in 2 sets of unclear concentric circles at the top left corner and bottom right corner in which the circles repel from the linear line. Another graph plots dots in a bell curve with a vague boundary. The boundaries are defined in 2 graphs for 7000 iterations.

Fig. 4.10

Solution to the SIL problem (4.25) for 10 electrons in 1D with the GenCol algorithm [52]. Top: prescribed density. Left: Evolution under GenCol of the N-electron density from a random initial state, visualized via its two-point marginal (pair density). Gridpoints with nonzero values (i.e., “successful” configurations) are shown in blue, with larger markers indicating higher values. Right: Evolution of the Kantorovich potential. The final N-point density recovers Seidl’s SCE state for the discretized problem with machine precision

The corresponding regularization for the Wasserstein distance squared instead of the SCE functional (Example 4.12 instead of Example 4.11) in fact goes back to Erwin Schrödinger in 1931 [121], and had a completely different motivation: Schrödinger was looking for models for the “most likely” evolution law between two probability distributions of particle positions which have been empirically observed at different times, perhaps hoping to re-discover his—then still controversial—quantum mechanics in a novel way.

An equivalent entropic problem has been considered by Chayes et al. in [19, 20]. In their setting, the integration in the entropy functional S is not against the Lebesgue measure but against the product measure $$\prod _{i=1}^N \rho ({\mathbf {r}}_i)/N$$ , which constitutes a natural model in classical statistical mechanics. The role of the reference measure will be explained below and can be understood via Eq. (4.93).

In optimal transport, entropic regularization became a popular basis for computational methods following an influential paper by Cuturi [34] in machine learning and Galichon and Salanié in economics [54]; see Sect. 4.4 for a computational algorithm. Regularization by entropies other than the Shannon–Von Neumann one is considered in [41, 99].

4.3.15.1 Basic Properties

Let us now informally discuss the basic properties of (4.89).

Unique Minimizer

Assuming that $$\rho \log \rho \in L^1(\mathbb {R}^d)$$ , a minimizer πτ in (4.89) exists [60]. This is expected from the convexity of the functional $${\mathcal {V}}_{\mathrm {ee}}+\tau S$$ . Since the functional is strictly convex on the domain where it is finite, minimizers must be unique.

Euler–Lagrange Equation; Form of Minimizer

Assume a Lagrange multiplier rule as in (4.61). That is, assume the existence of Lagrange multipliers $$(\lambda ({\mathbf {r}}_1))_{{\mathbf {r}}_1\in \mathbb {R}^d}$$ such that πτ is the unconstrained minimizer of the Lagrangian

Thus the function λ(r) has the usual physical interpretation of DFT as minus the potential that enforces the density constraint, and will in the following be denoted uτ(r). It follows that

for all variations η with $$\int \eta = 0$$ and π ± η ≥ 0. That is to say, $$0 = \int [\big ( V_{\mathrm {ee}} - \sum _i u^\tau ({\mathbf {r}}_i)) + \tau \log \pi ]\eta $$ and therefore

By solving for π and adjusting uτ by an additive constant, it follows that

(4.91)

The function aτ can—independently of its construction above with the help of Lagrange multipliers—be interpreted as an entropic weight function which makes the probability density πτ satisfy the constraint πτ↦ρ. Note that by this constraint and Eq. (4.91), aτ must satisfy the following governing equation in which Lagrange multipliers no longer appear:

(4.92)

The above equations constitute the so-called (multi-marginal) Schrödinger system. When the density ρ is Gaussian and the interaction potential w is taken to be the repulsive or attractive harmonic interaction, the entropically regularized problem can be solved exactly, see [58], for the one-dimensional case, and [39, 79, 103] for the general case. In [17], Carlier and Laborde showed the existence of a solution of the system (4.92) via an inverse function theorem argument by assuming that the one-body density ρ belongs to $$L^{\infty }(\mathbb {R}^d)$$ .

Relative Entropy Formulation

The functional $${\mathcal {V}}_{\mathrm {ee}}+\tau S$$ agrees up to an additive constant with the Kullback–Leibler divergence (or minus the relative entropy)⁷ between π and a kernel function $$\mathcal {K}$$ of the electronic interaction Vee [89]:

Thus the optimizer πτ is the density with marginal ρ which has minimal relative entropy with respect to the kernel $$\mathcal {K}$$ .

The Role of the Reference Measure

In the literature, the entropy functionals which are typically studied replace integration against the Lebesgue measure in (4.90) by integration against the product of the marginals $$\mu ^{\otimes N} = \otimes ^N_{i=1}\mu $$ (or any other finite reference measure), where ρ∕N = μ. As shown in Lemma 1.5 in [40] (see also [60] for the Coulomb case), both problems are equivalent since the following identity holds

(4.93)

Therefore, whenever at least one side of the equality above is finite, the original variational problem from the definition of $${\mathrm {V}}_{\mathrm {ee}}^{\tau }[\rho ]$$ (Eq. (4.89)) and the variational problem defined on the right-hand side of (4.93) have the same minimizers.

Dual Formulation

As for the exact (unregularized) strong-interaction limit of DFT, there is a dual variational principle for the Lagrange multiplier and an associated dual construction of $$V_{\mathrm {ee}}^\tau [\rho ]$$ . We have

(4.94)

where

(4.95)

and the supremum in (4.94) is over a suitable class of potentials. The second term in (4.95) can be viewed as a soft version of the inequality constraint Epot(r₁, …, rN) = Vee(r₁, …, rN) − u(r₁) + … + u(rN) ≥ 0 in the unregularized theory (see (4.149)), as it penalizes deviations from this inequality. Indeed, via the Laplace principle we have that, whenever the second term in (4.94) is finite,

In the discrete setting, this is precisely the LogSumExp formula. The existence of an optimizer uτ for the dual problem and the representation formulae (4.91), (4.96) with this uτ were proved in [40, 41] under the assumption that $$\rho \log \rho \in L^1(\mathbb {R}^d)$$ and Vee is measurable and bounded.

Functional Derivative

As in exact SCE theory, the functional derivative of the energy functional is formally given by the optimal potential in the dual problem, that is to say

(4.96)

where uτ is the maximizer of (4.95) (assuming such a maximizer exists and is unique). As in SCE theory, a natural choice of the additive constant is to require $$\lim _{|{\mathbf {r}}|\to \infty } \bigl (u^\tau ({\mathbf {r}}) + \mathit {const}\bigr ) = 0$$ . The ensuing potential vτ = uτ + const is then an approximation to the SCE potential.

4.3.15.2 Relation with the Levy–Lieb Functional

Just like the SCE functional itself, its entropic regularization is a rigorous lower bound of the exact functional, provided the regularization parameter τ is chosen suitably. More precisely:

Theorem 4.24 ([125])

Let Ψ be any N-electron wavefunction in the space $${\mathcal {W}}^N$$ (see (4.7)), or alternatively any bosonic wavefunction in $$H^1(\mathbb {R}^{dN})$$ , and suppose Ψ↦ρ. Let Veebe the Coulomb interaction. Then the scaled Levy–Lieb functional defined in Eq.(4.17) satisfies

(4.97)

In particular, the original Levy–Lieb functional (4.11) satisfies

(4.98)

This result is a consequence of the logarithmic Sobolev inequality (LSI). We include a proof, following Seidl et al. [125]. We begin by recalling a standard version of the LSI.

Theorem 4.25 (LSI, Corollary 7.3 in [70])

Let $$\nu \in \mathcal {P}(\mathbb {R}^n)$$ such that ν(r) = e−V (r)with $${\mathrm {D}}^2 V \geqslant \kappa {\mathrm {Id}}$$ . Then, for every locally integrable function $$f \geqslant 0$$ on $$\mathbb {R}^n$$ such that $$f\nu \in \mathcal {P}(\mathbb {R}^n)$$ we have that $$\int _{\mathbb {R}^n} f\log f \mathrm{d} \nu \leqslant \frac 2{\kappa } \int | \nabla \sqrt {f}|{ }^2 \, d \nu $$ .

This implies the following LSI for the Lebesgue measure:

Corollary 4.26 (LSI for the Lebesgue Measure, [125])

Let $$f\geqslant 0$$ be a function such that $$\sqrt {f} \in H^1(\mathbb {R}^n)$$ and $$f\in \mathcal {P}(\mathbb {R}^n)$$ . Then $$\int _{\mathbb {R}^n} f\log f \mathrm{d} {\mathbf {r}} \leqslant \frac 1{\pi } \int _{\mathbb {R}^n} | \nabla \sqrt {f}|{ }^2d{{\mathbf {r}}}$$ .

Proof of Corollary4.26

1.

In the LSI in Theorem 4.25, the requirement on f that $$\int f \, \mathrm{d} \nu =1$$ can be relaxed to $$0 &lt; \int f \, \mathrm{d} \nu \le 1$$ . This follows by applying the LSI to f∕α, $$\alpha =\int f \, \mathrm{d} \nu $$ , and noting that the extra term $$-(1/\alpha )\int f \log \alpha \, \mathrm{d} \nu $$ on the left-hand side is ≥ 0.

2.

Take $$\nu _{{\mathbf {r}}_2}({\mathbf {r}}_1)=\mathrm{e} ^{-\pi |{\mathbf {r}}_1-{\mathbf {r}}_2|{ }^2}$$ , then ν satisfies the assumption of the LSI with κ = 2π, and moreover $$\int f \mathrm{d} \nu _{{\mathbf {r}}_2}\le 1$$ . Hence by the LSI,

3.

Integrate over r₂ and use that $$\int \mathrm{e} ^{-\pi |{\mathbf {r}}_1-{\mathbf {r}}_2|{ }^2}\mathrm{d} {\mathbf {r}}_2=1$$ . This yields the assertion.

□

Proof of Theorem4.24

Let $$\Psi \in {\mathcal {W}}^N$$ , Ψ↦ρ, and let Π be its N-point position density (4.8). By a version of the Hoffmann-Ostenhof inequality [77],⁸ $$\sqrt {\varPi }\in H^1(\mathbb {R}^{dN})$$ and

This together with the LSI for the Lebesgue measure (Corollary 4.26) applied to π gives

Taking the infimum over $$\Psi \in {\mathcal {W}}^N$$ yields $$F^\lambda [\rho ]/\lambda \ge V_{\mathrm {ee}}^\tau [\rho ] + \tau $$ , with τ as in the theorem. □

Although Theorem 4.24 provides a lower bound for the Levy–Lieb functional (4.11), in practice this bound can be rather loose [58].

4.3.15.3 Well Definedness of Entropy and Convergence to the SCE Functional

We now show that the entropy is well defined under very mild conditions on ρ (e.g., finite first moment suffices), and that the entropically regularized functional $${\mathrm {V}}_{\mathrm {ee}}^{\tau }$$ converges to the SCE functional when the regularization parameter tends to zero.

Note that a priori both the positive and the negative part of the integral (4.90) could be divergent; Lemma 4.27 excludes this for the negative part, and so the integral always has a well defined value in $$\mathbb {R}\cup \{+\infty \}$$ .

Lemma 4.27 (Well-Definedness of Entropy and of the Regularized SCE Functional)

Let $$\rho \in L^1(\mathbb {R}^d)$$ , ρ ≥ 0, $$\int \rho = N$$ , and assume ρ has finite first moment, that is to say $$\int |{\mathbf {r}}|\rho ({\mathbf {r}})\mathrm{d} {\mathbf {r}}&lt;\infty $$ . Let $$\pi \in {\mathcal {P}}(\mathbb {R}^{dN})\cap L^1(\mathbb {R}^{dN})$$ with π↦ρ. Then the negative part $$(\pi \log \pi )_-$$ has finite integral; more precisely, for some constant Aρ > 0 which depends only on ρ but not on π

where $$f_-({\mathbf {r}}) = \min \{f({\mathbf {r}}),0\}$$ denotes the negative part of a function f. Hence S as defined by (4.90) is well defined as a functional

and $$V_{\mathrm {ee}}^\tau [\rho ]$$ as defined by (4.89) is well defined as an element of $$\mathbb {R}\cup \{+\infty \}$$ .

The assumption that ρ has finite first moment cannot be omitted. For instance, for N = 2 and d = 1 the N-body density

continued by zero to $$\mathbb {R}^2$$ and with c₀ chosen such that $$\int \pi = 1$$ , belongs to $$L^1(\mathbb {R}^2)$$ but satisfies $$\int (\pi \log \pi )_- = -\infty $$ , as the interested reader can check using that $$\int _2^\infty \tfrac {1}{z|\log z|{ }^\alpha } \mathrm{d} z = \infty $$ for α = 1 but < ∞ for α > 1. In particular, in such a case the equivalence described in (4.93) does not necessarily hold.

The lemma implies that for any interaction potential Vee on $$\mathbb {R}^{dN}$$ which is symmetric and bounded from below, such as the Coulomb interaction, $$V^{\tau }_{\mathrm {ee}}[\rho ]$$ is well defined as an element of $$\mathbb {R}\cup \{+\infty \}$$ .

Proof of Lemma4.27

$$\pi \log \pi $$ is ≤ 0 precisely in the region $$\Omega =\{{\mathbf {r}}\in \mathbb {R}^{dN} \, : \, \pi ({\mathbf {r}})\le 1\}$$ . Split Ω into $$\Omega _&lt;=\{{\mathbf {r}}\in \Omega \, : \, 0\le \pi ({\mathbf {r}}) &lt; \mathrm{e} ^{-(|{\mathbf {r}}_1|+\ldots +|{\mathbf {r}}_N|)}\}$$ and $$\Omega _&gt;=\{{\mathbf {r}}\in \Omega \, : \, \pi ({\mathbf {r}})\ge \mathrm{e} ^{-(|{\mathbf {r}}_1|+\ldots +|{\mathbf {r}}_N|)}\}$$ . Since $$g(z)=z\log z$$ satisfies $$|g(z)|\le C\sqrt {z}$$ in [0, 1] for some constant C,

StartLayout 1st Row 1st Column Blank 2nd Column integral Underscript double struck upper R Superscript d upper N Baseline Endscripts StartAbsoluteValue left parenthesis pi log pi right parenthesis Subscript minus Baseline EndAbsoluteValue equals integral Underscript upper Omega Subscript left angle bracket Baseline Endscripts StartAbsoluteValue left parenthesis pi log pi right parenthesis Subscript minus Baseline EndAbsoluteValue plus integral Underscript upper Omega Subscript greater than Baseline Endscripts StartAbsoluteValue left parenthesis pi log pi right parenthesis Subscript minus Baseline EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column less than or equals upper C integral Underscript upper Omega Subscript left angle bracket Baseline Endscripts normal e Superscript minus left parenthesis StartAbsoluteValue bold r 1 EndAbsoluteValue divided by 2 plus ellipsis plus StartAbsoluteValue bold r Super Subscript upper N Superscript EndAbsoluteValue divided by 2 right parenthesis Baseline normal d bold r 1 ellipsis normal d bold r Subscript upper N Baseline 3rd Row 1st Column Blank 2nd Column plus integral Underscript upper Omega Subscript greater than Baseline Endscripts pi left parenthesis bold r 1 comma ellipsis comma bold r Subscript upper N Baseline right parenthesis left parenthesis StartAbsoluteValue bold r 1 EndAbsoluteValue plus midline horizontal ellipsis plus StartAbsoluteValue bold r Subscript upper N Baseline EndAbsoluteValue right parenthesis normal d bold r 1 ellipsis normal d bold r Subscript upper N Baseline 4th Row 1st Column Blank 2nd Column less than or equals upper C left parenthesis integral Underscript double struck upper R Superscript d Baseline Endscripts normal e Superscript minus StartAbsoluteValue bold r 1 EndAbsoluteValue divided by 2 Baseline normal d bold r 1 right parenthesis Superscript upper N Baseline plus integral Underscript double struck upper R Superscript d Baseline Endscripts rho left parenthesis bold r 1 right parenthesis StartAbsoluteValue bold r 1 EndAbsoluteValue normal d bold r 1 equals colon upper A Subscript rho Baseline period EndLayout

(4.99)

By the assumption that ρ has finite first moment, the right-hand side is finite, completing the proof of the lemma. □

Finally, we prove that—as intuitively expected—the entropically regularized functional $$V_{\mathrm {ee}}^\tau $$ converges to the exact SCE functional when the regularization parameter tends to zero. The corresponding Γ-convergence result was obtained in [60].

Theorem 4.28

Let ρ be any N-electron density which belongs to the class $${\mathcal {D}}^N$$ (see (4.12)) and has finite first moment, and let Veebe the Coulomb interaction. Then

(4.100)

Proof

We combine the upper bound on $$V_{\mathrm {ee}}^\tau [\rho ]$$ from Theorem 4.24, the asymptotic result on Fλ[ρ]∕λ in Eq. (4.19) (see Theorem 4.3), and the lower bound from Lemma 4.27. By inequality (4.99) we have for any π↦ρ

and hence, by adding $${\mathcal {V}}_{\mathrm {ee}}[\pi ]$$ to both sides and taking the infimum over π

Obviously this lower bound converges to $$V_{\mathrm {ee}}^{\mathrm {SCE}}[\rho ]$$ as τ → 0. On the other hand, by Theorem 4.24 we have $$V_{\mathrm {ee}}^\tau [\rho ] \le F^\lambda [\rho ]/\lambda $$ and by Theorem 4.3 this upper bound also converges to $$V_{\mathrm {ee}}^{\mathrm {SCE}}[\rho ]$$ ; hence so must $$V_{\mathrm {ee}}^\tau [\rho ]$$ . □

4.4 Numerical Methods and Approximations

The SCE functional cannot at the moment be accurately and efficiently computed for general three-dimensional densities and large N. But accurate numerical methods are available for small N or special situations, novel methods aimed at large N are under development, and less accurate approximations can already be computed for large N. We review these methods and approximations in this section, and their use within Kohn–Sham DFT in Sect. 4.5.

4.4.1 Numerical Methods Based on Co-motion Functions

Numerical implementations using co-motion functions were confined to the following cases:

the exact maps are known: general N in one dimension (see Sect. 4.3.11);

an explicit ansatz, able to get very close to the true minimum, exists: spherically symmetric (radial) case (see Sect. 4.3.12).

In addition, co-motion functions can be extracted from optimal plans in the case

N = 2, for which the existence of the map is proven and there are 1-1 correspondences between map, optimal plan, and Kantorovich potential (see Eqs. (4.38) and (4.39)).

We review here and in the following section the implementation for these three classes of problems. Their use in combination with Kohn–Sham DFT is then discussed in Sect. 4.5.

4.4.1.1 One-Dimensional N-Electron Systems

The SCE functional has been implemented for one-dimensional (1D) many-electron systems using the exact co-motion functions (maps) of Seidl [122], which we reported and illustrated in Sect. 4.3.11. These applications typically aim at modeling physical systems in which electrons are confined in elongated traps (quantum wires): the interaction used is thus 3D Coulomb renormalized for small interparticle distances. The idea is that at long range the electrons feel the 1∕|x| interaction, but at short range they can avoid each other due to the finite thickness of the wire, which is mimicked by removing the divergence at x = 0. For example, a widely used effective quasi-1D interaction is obtained by integrating the 3D Coulomb interaction over normalized gaussians in two of the three spatial directions [64], modeling harmonic confinement within a wire of thickness b,

(4.101)

This interaction is finite at x = 0, where it has a cusp, behaves as 1∕|x| for large x and it is convex for x ≥ 0. Other popular quasi-1D interactions are the soft Coulomb and the regularized Coulomb,

(4.102)

(4.103)

Notice, however, that the 1D maps of Seidl [123] are exact only for interactions (costs) that are convex for x ≥ 0 [27]. This means that when using $$v_{\mathrm {ee}}^{\mathrm {soft}}(x)$$ , which is concave for $$x\in [0,a/\sqrt {2}]$$ , the Seidl maps are not guaranteed to yield the true minimizer, as illustrated, for example, in Fig. 2 of Ref. [71].

Numerical realizations of the 1D Seidl maps are reported in Refs. [71, 72, 100–102, 104]. The implementation of the maps directly follows from Sect. 4.3.11: given a density ρ(x) on a grid, the cumulant function Fρ(x) is evaluated on the same grid, and its inverse $$F_\rho ^{-1}(x)$$ is simply obtained by swapping the columns. The grid can be restored by using a spline interpolation for $$F_\rho ^{-1}(x)$$ , and the maps are readily obtained. Numerical issues can appear in regions where the density is close to zero, with $$F_\rho ^{-1}(x)$$ raising extremely steeply. An alternative method to obtain the 1D maps without the need to construct $$F_\rho ^{-1}(x)$$ is discussed in Ref. [72].

4.4.1.2 Spherically Symmetric Densities

For spherically symmetric densities the radial SGS maps (4.79)–(4.80) conjectured in [122, 127] have been implemented in Refs. [63, 127] for the 3D case using numerical densities for atoms from He to Ne, and in Ref. [105] for the 2D case, where the SCE functional has been combined self-consistently with Kohn–Sham DFT to describe electrons confined in a parabolic potential at low density.

The construction of the radial maps is implemented as in the 1D case. However, the computational complexity is now higher due to the evaluation of the reduced radial cost of Eq. (4.77), which requires an angular minimization for given radial distances. For the two-dimensional case treated in Ref. [105], where the number of relative angles to minimize was equal to N − 1, the procedure has been the following. For an initial non-degenerate radial configuration and given initial starting angles, the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm was used to find the closest local minimum. Then the radial position of the “first” electron was changed in small discrete steps, the radial positions of the remaining electrons were computed using the SGS maps, and the angles were optimized using the BFGS algorithm, with starting angles taken from the previous step. This procedure rests on the assumption that the optimal angles change continuously with the radial configuration. The starting angles for the initial radial configuration can be chosen by using simulated annealing as a global optimization strategy. It should be stressed that the angular minimization does not need to be performed for the whole set Ngrid of radial grid points. In fact, the N radial distances are periodic, as each circular shell r ∈ [ai, ai+1] (with $$a_i=F_\rho ^{-1}(i)$$ , $$i\in \mathbb {N}$$ ), corresponds to the same physical situation,[127] simply describing a permutation of the set of distances occurring in the first shell r ∈ [0, a₁]. Thus, by keeping track of the minimizing angles, and by readapting the grid in every circular shell, it is possible to do the angular minimization only Ngrid∕N times rather than Ngrid times.

4.4.2 Methods Based on Linear Programming

Direct discretization of the SIL variational principle (4.25) yields a linear program, which is numerically tractable when N = 2.

4.4.2.1 The N = 2 Case

For two-electron systems in 3D with general density, Chen et al. [22] have implemented a method to directly solve the SIL variational principle via linear programming and extract the co-motion function and the SCE potential from the SIL solution. They used this approach to compute the co-motion function and the KS-SCE binding curve of the H₂ molecule (see Figs. 4.9 and 4.14).

4 line graphs of rho L by 2 versus 2 x by L. 3 graphs for L = 1, 15, and 70 plot 3 curves for K S L D A, K S S C E, and C I. For L = 1, the curves are completely overlapping and forms 2 consecutive crests. For L = 15 and 70, the curves for K S L D A and K S S C E form waves in the center, and C l is a downward curve. Amplitude is higher for L = 70. Graph for L = 70 leads to a graph of rho L by 2 and v subscript K S, L square by 2 versus 2 x by L that plots 2 waves for K S S C E potential and density and 2 parallel horizontal lines for epsilon subscript i.

Fig. 4.11

Left: self-consistent KS SCE densities for N = 4 electrons interacting with $$v_{\mathrm {ee}}^{\mathrm {wire}}(x)$$ of Eq. (4.101) when the external potential is harmonic, $$v_{\mathrm {ne}}(x)= \frac {1}{2}\omega ^2\,x^2$$ , compared with very accurate configuration interaction results (CI) and with KS LDA. Right: the total KS potential at self consistency, vKS = vne + vSCE, for the most correlated case. The horizontal lines are the two eigenvalues of the occupied KS SCE orbitals. Results are in scaled units, where L = 2 ω−1∕2, and are taken from Ref. [101]

One truncates $$\mathbb {R}^3$$ to a bounded domain, discretizes it into ℓ finite regions e₁, …, eℓ, and represents each element by a point aℓ located at its barycenter. The single-particle density becomes a vector in $$\mathbb {R}^\ell $$ with components $$\rho _\ell =\int _{e_\ell }\rho ({\mathbf {r}})\, \mathrm{d} {\mathbf {r}}$$ . The two-particle density π is represented by a matrix $$\gamma =(\gamma _{ij})\in \mathbb {R}^{\ell \times \ell }$$ with $$\gamma _{ij}=\int _{e_i}\int _{e_j}\mathrm{d} \pi ({\mathbf {r}}_1,{\mathbf {r}}_2)$$ , and the interaction Vee(r₁, r₂) becomes a matrix $$(c_{ij})\in \mathbb {R}^{\ell \times \ell }$$ with $$c_{ij}=\tfrac {1}{|{\mathbf {a}}_i-{\mathbf {a}}_j|}$$ . The SIL problem (4.25) then becomes

(4.104)

This is a standard linear programming problem of the form minxfTx subject to Ax = b, xk ≥ 0, where x is the vector containing the entries of γ. The solution can be obtained with a standard linear programming software (in [22], the authors used MOSEK). For a uniform discretization of the density, the number of degrees of freedom in the linear program would still be huge; instead an adaptive mesh was used in which all elements contain roughly the same amount of density, that is to say the mesh is much finer in the high-density region near the nuclei. (For automated generation of such a mesh, the finite element package PHK was used. See Chap. 10 by Dai and Zhou for more information about this package.) The solution to (4.104) entails an approximation to the co-motion function f at the barycenters $$\{{\mathbf {a}}_i\}_{i=1}^\ell $$ , namely the barycenter of the image of ai under the transport plan γ:

(4.105)

where γij can be regarded as the mass transported from ai to aj and the normalization factor ρi∕2 guarantees that the barycentric weights sum to 1. Since, for N = 2, the optimal N-point density π for the continuum problem is unique and of SCE form (see (4.41) and Theorem 4.4), it follows that if the discretization is sufficiently fine, i.e. ℓ is large enough, f(ℓ) is a good approximation to f. The resulting co-motion function for the H₂ molecule is depicted in Fig. 4.9.

4.4.2.2 The N > 2 Case and the Curse of Dimension

Since the above method uses a real-space discretization of the SIL variational principle whose unknown is the N-particle density on $$\mathbb {R}^{3N}$$ , it is limited in practice to N = 2, to keep the number of computational degrees of freedom manageable. Indeed, for general N the N-particle density π must be represented by an order-N tensor $$(\gamma _{i_1\ldots i_N})\in \mathbb {R}^{\ell \times \ldots \times \ell }$$ with entries $$\gamma _{i_1\ldots i_N}=\int _{e_{i_1}\times \ldots \times e_{i_N}} \mathrm{d} \pi ({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N)$$ . Since π can be assumed to be symmetric (see Remark 4.2), γ can be assumed to be symmetric under permutation of indices and Eq. (4.104) becomes

(4.106)

This is still a linear program, but in ℓN (or, using symmetry, $$\binom {N+\ell -1}{\ell +1}$$ ) variables. For rigorous hardness results on problems of the form (4.106) with pairwise interaction see [4, 52].

4.4.3 Methods Based on the Dual Formulation

Mendl and Lin [104] have implemented a method for solving the dual formulation of the SCE functional, Eqs. (4.55) and (4.53), and applied it to the Beryllium atom, a four-electron quantum wire in 1D, and a model trimer in 3D. In the 3D case, they parametrized the (unknown) Kantorovich potential by a pseudocharge,

with m given by a small number of Gaussians and satisfying $$\int m = N-1$$ to account for the asymptotic behavior v(r) ∼ (N − 1)∕|r| for large |r| (see (4.66)). They showed that the constrained maximization in (4.55), (4.53) is equivalent to a nested pair of unconstrained optimizations,

(4.107)

(4.108)

The inner optimization for given v was implemented by a quasi-Newton method and the outer optimization via a gradient-free simplex algorithm. For the Beryllium atom, using just two Gaussians for m resulted in a relative error of the SCE energy of only 1.6% compared to the SCE energy obtained via the SGS co-motion functions for radially symmetric densities [127] as described in Sect. 4.4.1.2. Also, the obtained SCE potential was in good agreement with the one based on the radial co-motion functions.

As the authors point out, this approach is in practice limited to small systems, because the inner optimization is high-dimensional, nonlinear, and highly degenerate for the optimal v (recall that the set of minimizers is typically d-dimensional), and the outer optimization (4.107) is nonlinear and nonsmooth, and hence unsuitable for numerical optimization over a large number of degrees of freedom.

4.4.4 Multi-Marginal Sinkhorn Algorithm

In optimal transport, a standard computational method [35] is to pass to the entropic regularization (in our case, problem (4.89)) and solve the latter via the Sinkhorn algorithm. This is a simple and robust algorithm which goes back to Sinkhorn in the context of estimating Markov transition matrices [131]; it was introduced into two-marginal optimal transport in [34] and generalized to several marginals in [8]. The multi-marginal Sinkhorn algorithm with Coulomb cost was implemented by Benamou et al. [9] (see also [110]) to compute the SCE energy and potential for the He and Li atoms.

The multi-marginal Sinkhorn algorithm goes as follows; we state it here in the continuous setting. One starts from the exact form (4.91) of the optimizer. One now allows the N entropic weight functions aj(rj) = aτ(rj) in this form to be different (so as to be able to update them one by one). One updates them iteratively so as to enforce the j-th marginal constraint, (4.92) for j:

(4.109)

Solving for aj yields an explicit formula for aj in terms of the other ai. Thus a single updating cycle consists of the N steps

(4.110)

One then repeats the cycle until convergence.

Convergence of the Sinkhorn algorithm is rigorously guaranteed under mild conditions on the interaction potential and the density (e.g., bounded potentials and $$\rho \log \rho \in L^1$$ are sufficient); see [131] for the discretized N = 2 case, [118] for the general N = 2 case, and [40] for N ≥ 2. The (linear) rate of convergence for the Sinkhorn algorithm was obtained in [23, 46] in the N = 2 case, and in [15] for the multi-marginal Sinkhorn algorithm. For a two-electron example in dimension one computed with the Sinkhorn algorithm see Fig. 4.8.

In [9], Benamou et al. demonstrated that for the He atom (and the choice τ = 0.02) the algorithm yields an accurate approximation to the SCE energy and the SCE potential compared to the (in this case rigorously justified) SGS map-based solution; the relative error of the potential in the L∞ norm was only 0.4%. Moreover, for the Li atom the numerical Sinkhorn solution exhibited very good qualitative agreement with the SGS solution.

Some regularization is essential for the Sinkhorn approach. As τ approaches zero—so that the entropic regularization $$V^{\tau }_{\mathrm {ee}}[\rho ]$$ from (4.89) approaches the exact SCE functional (4.26)—the convergence speed of the algorithm also goes to zero (see e.g. [35, 50]), and numerical instabilities can appear associated with the extremely small order e−1∕τ of the integrand (see e.g. [8]).

The idea of regularization underlying the algorithm fits well into our DFT context as the optimal N-point density is smeared out anyway off the strongly interacting limit. However, a significant limitation from the point of view of DFT is the high-dimensionality of the integral in (4.109), (4.110). For a discretization of the one-body density by ℓ gridpoint values, the cost of a single integral evaluation for fixed rj is O(ℓN−1), limiting the method to small N.

4.4.5 Towards Large N

Very recently, some promising methods have been proposed which should, at least in principle, be suitable for tackling the case of large N. These have been demonstrated to show good performance on one-dimensional test examples where the Seidl solution from Sect. 4.3.11 is available for comparison. At the time of writing, it has yet to be demonstrated that any of these methods is capable of accurately computing the SCE energy for large N in three dimensions.

4.4.5.1 Semidefinite Convex Relaxation

The starting point of this method, introduced by Khoo and Ying [82], is the fact that the SIL problem 4.25 can, due to the fact that Vee is a two-body potential (4.3), be reformulated as a minimization over N-representable 2-point probability measures:

(4.111)

Here a two-point probability measure on $$\mathbb {R}^d\times \mathbb {R}^d$$ is called N-representable if it is the 2-marginal of a symmetric N-point probability measure on $$\mathbb {R}^{dN}$$ . This two-body formulation of the SCE functional was introduced in [51], and is a direct adaptation of the well-known two-body reduced density matrix formulation [26] of the Rayleigh–Ritz variational principle (4.6) to the strongly correlated limit of DFT.

After discretization as described in Sect. 4.4.2.1, the two-point marginal becomes a matrix $$\Gamma =(\Gamma _{ij})\in \mathbb {R}^{\ell \times \ell }$$ , and N-representability means that Γ is obtained from some symmetric tensor $$(\gamma _{i_1\ldots i_\ell })\in \mathbb {R}^{\ell \times \ldots \times \ell }$$ with nonnegative entries which sum to one by $$\Gamma _{i_1i_2}=\sum _{i_3,\ldots ,i_N}\gamma _{i_1i_2i_3\ldots i_N}$$ .

The extreme points of the set of discrete N-representable 2-marginals have been determined explicitly [53] (see [16, 82] for generalizations to 3-marginals respectively general k-marginals).

Theorem 4.29 ([53])

The set of extreme points of the set $${\mathcal R}_2$$ of discrete N-representable 2-marginals is

In particular, $${\mathcal R}_2$$ is the convex hull of $${\mathcal R}_2^{\mathrm {ext}}$$ .

Here 1 denotes the vector in $$\mathbb {R}^\ell $$ with all components equal to 1. The discretized problem is then

(4.112)

Khoo and Ying [82] introduced the following convex relaxation of this problem in which $${\mathcal R}_2$$ is replaced by a slightly larger but simpler set:

(4.113)

Here Λ ≥ 0 means matrix positivity of Λ.

It is clear that $$\widetilde {{\mathcal R}_2}\supset {\mathcal R}_2$$ , since $$\widetilde {{\mathcal R}_2}$$ is convex and—by inspection—contains the set of extreme points of $${\mathcal R}_2$$ given in Theorem 4.29. A theoretical argument in support of the approximation (4.113) is:

Theorem 4.30 ([82])

The extreme points of the true set $${\mathcal R}_2$$ of discrete N-representable 2-marginals are still extreme points of $$\widetilde {{\mathcal R}_2}$$ .

Intuitively this means that, at least near the extreme points of the exact set $${\mathcal R}_2$$ of N-representable 2-marginals, the relaxation is very tight.

Viewed as a minimization over Λ, (4.112) is a semidefinite program (SDP), i.e. a problem of minimizing a linear cost subject to finitely many linear equalities or inequalities and a matrix positivity constraint. It has been implemented in [82] using a uniform grid and the large-scale SDP solver SDPNAL+. For 1D problems with N = 8, up to ℓ = 1600 gridpoints, and different one-body densities, the solutions reported in [82] are in excellent qualitative agreement with the pair density of the exact Seidl solution. The relative energy error compared to the unapproximated discrete problem (4.111) is estimated to be of the order of 10⁻² to 10⁻⁴, depending on the choice of one-body density. Also, (4.113) is solved for 6 electrons in 2D with a Gaussian density on a 10 × 10 grid.

Khoo and Ying [82] also give a dual formulation of the SDP (4.113) which yields an approximation to the Kantorovich potential. For 1D test problems with 8 electrons and 200 gridpoints, a relative accuracy of 10⁻² to 10⁻³ in the L² norm is reported compared to the exact potential obtained from the Seidl solution and Eq. (4.59).

4.4.5.2 Langevin Dynamics with Moment Constraints

This approach was proposed by Alfonsi et al. [1, 2]. The idea is to only discretize the density constraint, but not the N-point density, and then use a stochastic particle method to simulate the many-electron density. One performs a Galerkin (or “moment”) discretization of the marginal constraint (4.24) by requiring only a fixed number M of integral constraints, of the form

(4.114)

where μ = ρ∕N is the prescribed single-particle density and φ₁, …, φM are suitable single-particle basis functions on $$\mathbb {R}^d$$ . Moreover since the marginal constraint has been relaxed, one introduces a mild additional constraint on the class of admissible N-electron densities γ to prevent mass from escaping to infinity,

(4.115)

for some nonnegative increasing function θ  :  [0, ∞) → [0, ∞) with θ(r) →∞ (r →∞) and some constant A > 0. The SIL problem (4.25) is now approximated by:

(4.116)

Under suitable assumptions on the basis functions, and for A chosen sufficiently large, the minimum value of (4.116) can be shown to converge to the SCE energy $$\operatorname {V}_{\mathrm {ee}}^{\operatorname {SCE}}[\rho ]$$ as the number M of basis functions tends to infinity [2]. The key property of (4.116) opening the door to numerical methods is the following.

Theorem 4.31 ([2])

Assume $$\mu \in {\mathcal P}(\mathbb {R}^d)$$ , and suppose that the basis functions $$\varphi _1,..,\varphi _M\ : \, \mathbb {R}^d\to \mathbb {R}$$ are continuous, belong to L¹(dμ), and satisfy the growth bound |φm(r)|≤ const(1 + θ(|r|))sfor some s ∈ (0, 1). Assume that $$V_{\mathrm {ee}}\, : \, \mathbb {R}^{dN}\to \mathbb {R}\cup \{+\infty \}$$ is nonnegative and $$\int V_{\mathrm {ee}}\mathrm{d} \gamma $$ is finite for some γ satisfying (4.114), and that A is sufficiently large. Then there exists a minimizer of (4.116) of the form for some K ≤ M + 2, some coefficients αν ≥ 0, and some $${\mathbf {a}}_{i}^{(\nu )}\in \mathbb {R}^d$$ .

Thus a sparse ansatz for the many-electron density consisting of K ≤ M + 2 symmetrized Dirac measures (where M is the number of constraints discretizing the marginal condition) is sufficient. This result generalizes Theorem 4.23 from discrete problems to semi-discrete problems with continuous state space and discretized marginal constraint.

In order to numerically solve (4.116), in [1] a stochastic particle method in continuous state space has been implemented. More precisely, the authors use constrained overdamped Langevin dynamics in the potential Vee, which is a natural stochastic evolution equation for minimizing Vee, applied to weighted sums of K symmetrized Dirac measures moving on the constraint manifold (4.114). For 5 electrons in a one-dimensional interval and the regularized Coulomb interaction (4.103) with a = 0.1, up to M = 40 basis functions taken to be Legendre polynomials, and superpositions of up to K = 10, 000 symmetrized Dirac measures, the method achieves good agreement with the Seidl solution described in Sect. 4.3.11. The implementation uses an iterative method to maintain the constraints (which are nonlinear in the particle positions), as well as judicious choices of the time steps, temperature profile, and numbers of symmetrized Diracs to balance accuracy and computational efficiency.

An attractive feature of this method besides its feasibility for large numbers of electrons is the fact that space is not discretized. In [1] simulations are reported for 100 electrons in three dimensions subject to 52 marginal constraints, again using superpositions of 10,000 symmetrized Dirac measures. At the time of writing, it remains an interesting open question to assess, in such situations, the accuracy of the model (4.116) and its numerical solutions.

4.4.5.3 Genetic Column Generation

This method was proposed recently by Friesecke et al. [52]. It directly solves the discretized SIL problem (4.106), by combining the sparse but exact quasi-SCE or quasi-Monge ansatz (see Theorem 4.23), the method of column generation from discrete optimization, and basic ideas from machine learning.

The idea is to alternate between solving the SIL problem on a small but otherwise unconstrained subset of the many-electron configuration space, and updating the subset based on the (primal and dual) SIL solution. Recall that after discretization, the many-electron density becomes a density γ on XN, where X = {a₁, …, aℓ} is a set of discretization points (e.g., a grid) for the single-electron configuration space $$\mathbb {R}^d$$ . One now starts from the quasi-SCE or quasi-Monge ansatz in the form (4.88), which suffices to solve the discrete SIL problem (4.106) exactly (see Theorem 4.23), but—for computational reasons—allows a slightly larger number of delta functions:

(4.117)

Here the $${\mathbf {r}}^{(\nu )}=({\mathbf {r}}_1^{(\nu )},\ldots ,{\mathbf {r}}_N^{(\nu )})$$ are arbitrary N-point configurations in XN and β > 1 is a hyperparameter (taken to be 5 in [52]) which limits the number of N-point configurations to O(ℓ) instead of the naively required O(ℓN). To achieve a unique correspondence between symmetrized Diracs and N-point configurations one restricts the r(ν) to the sector $$X^N_{\mathrm {sym}}=\{({\mathbf {a}}_{i_1},\ldots ,{\mathbf {a}}_{i_N})\in X^N \, : \, i_1\le \ldots \le i_N\}$$ , making the expansion coefficients αν in (4.117) unique.

The ansatz (4.117) involves two sets of degrees of freedom, the subset $$\Omega =\{{\mathbf {r}}^{(1)},\ldots ,{\mathbf {r}}^{(\ell ^{\prime })}\}$$ of the many-electron configuration space and the coefficient vector $$(\alpha _1,\ldots ,\alpha _{\ell ^{\prime }})$$ , which are updated alternatingly. For fixed Ω, the coefficient vector is governed by the SIL problem (4.106) restricted to the ansatz (4.117), which reads, using that $$S_N\delta ({\mathbf {r}}_1-{\mathbf {r}}_1^{(\nu )})\ldots \delta ({\mathbf {r}}_N-{\mathbf {r}}_N^{(\nu )})$$ has single-particle density $$\rho ^{(\nu )}({\mathbf {r}})=\sum _{i=1}^N \delta ({\mathbf {r}}-{\mathbf {r}}_i^{(\nu )})$$ ,

(4.118)

This is just a small linear program with an ℓ × O(ℓ) constraint matrix. Updating the set Ω is done in a simple but subtle manner, as standard methods would incur the curse of dimension (see below). One also uses the dual problem

(4.119)

whose solution u is an approximation to the Kantorovich potential.

An updating cycle in the genetic column generation (GenCol) method goes as follows:

1.

Given a set $$\Omega \subset X_{\mathrm {sym}}^N$$ of N-particle configurations, update the primal solution α and the dual solution u by solving (4.118) and (4.119).

2.

Given the updates αnew and unew, update Ω by the following genetic learning method:

(4.120)

Steps 1. and 2. are iterated until convergence, with the oldest configurations which do not contribute to the current optimal plan (i.e. satisfy $$\alpha ^{\mathrm {new}}_\nu = 0$$ ) being deleted from Ω whenever its size ℓ′ exceeds the maximum allowed size βℓ.

The simple but powerful genetic learning aspect of the search rule in (4.120) is that only “successful” N-electron configurations in Ω (i.e. ones that contribute to the current optimal plan (4.117) with a nonzero coefficient αν) are allowed to bear offspring. Numerical observations and theoretical considerations show that this is essential for overcoming the curse of dimension. An unbiased random search of new configurations, or the updating step in the classical column generation method of solving the so-called pricing problem,⁹ would merely turn the curse of dimension with respect to the size of the state space into a curse of dimension with respect to the number of search steps.

The rationale behind the acceptance criterion in (4.120) is that any new configuration r∗ satisfying it represents a constraint of the full dual problem (Eq. (4.119) with the r(ν) being replaced by all configurations in $$X_{\mathrm {sym}}^N$$ ) which the current dual solution unew violates. Adding this configuration to the set Ω “cuts off” unew from the optimization domain of the dual problem, yielding a new dual solution and an energy decrease. For a rigorous justification see [52].

Figure 4.10, taken from [52], shows the solution of the SIL problem (4.25) computed by the GenCol algorithm for 10 electrons in a 1D interval discretized by 100 gridpoints. In this example, the grid spacing is normalized to 1, the density is taken to be $$\rho (x)=\mathit {const} (0.2 + \sin ^2(\tfrac {x}{\ell +1}))$$ , and the interaction is the soft Coulomb potential (4.102) with a = 0.1. With the initial set of many-electron configurations chosen randomly, the algorithm always found the exact Seidl solution (see Sect. 4.3.11) of the discretized problem to machine precision using less than 7000 iterations and less than 5 samples per iteration. This means that only a tiny fraction of the configuration space was accessed. The energy decreased steadily at an exponential rate.

2 graphs of rho of r versus r plot 3 overlapping bell curves for unpolarized, polarized, and Q M C. In A, the curves begin from the y-axis, and N = 3 and omega = 0.005. In B, the curves begin from the origin and N = 3 and omega = 0.001.

Fig. 4.12

Self-consistent radial KS SCE densities ρ(r) for N = 3 electrons in 2D, with external potential $$v_{\mathrm {ne}}(r)= \frac {1}{2}\omega ^2\,r^2$$ [105], compared with accurate Quantum Monte Carlo (QMC) results [61, 73]. The KS SCE densities are shown for both the unpolarized (2 orbitals, of which only the lowest is doubly occupied) and spin polarized (3 different singly occupied KS orbitals) case

Tests reported in [52] on larger 1D systems with up to N = 30 electrons on 120 grid points (corresponding to a space of N-point densities of dimension ℓN ≈ 2.4 × 10⁶²) show only a slow polynomial growth in N of the number of iterations required to find the exact solution to machine precision, with the average number of samples needed per iteration to satisfy the acceptance criterion remaining approximately constant.

Apart from its simplicity and efficiency in high dimensions, attractive features of the genetic column generation method are that after discretization no further approximations are made (and the discrete SIL problem is solved accurately), and that the method also provides the Kantorovich potential for use within Kohn–Sham DFT.

Tests for accurately discretized three-dimensional densities are not yet available at the time of writing.

4.4.6 Approximations

As explained, there are presently no efficient algorithms to solve the SCE problem in an exact or very accurate way for the general three-dimensional case. In the usual spirit of DFT, several approximations for the functional $$\operatorname {V}_{\mathrm {ee}}^{\operatorname {SCE}}[\rho ]$$ have been proposed and used in combination with Kohn–Sham DFT. We review the approximations in this section, and their use within Kohn–Sham DFT in Sect. 4.5.

4.4.6.1 Gradient Expansion: Point-Charge-Plus-Continuum Model (PC)

The first gradient expansion approximation (GEA) for the indirect energy functional $$W_\infty [\rho ]=\operatorname {V}_{\mathrm {ee}}^{\operatorname {SCE}}[\rho ]-U[\rho ]$$ has been proposed by Seidl et al. [128], and it is called the point-charge-plus continuum (PC) model,

(4.121)

where $$A=-\frac 9{10}(\frac {4\pi }3)^{1/3}$$ and $$B=\frac 3{350}(\frac 3{4\pi })^{1/3}$$ . The model is built from the physical interpretation of W∞[ρ] as the electrostatic energy of a system of perfectly correlated electrons with density ρ inside a classical background with the same charge density ρ of opposite sign [128]. Notice that the electrons are not allowed to relax in this fictitious external potential, as they are kept in the SCE state with the prescribed density. Only when the density is uniform is the energy of the SCE state the same as the one we would obtain by letting the electrons relax in the positive background external potential [96]. The idea of the PC model is that when the density is slowly varying the energy should be well approximated by surrounding each electron by a PC cell (given by the combined effect of the background and the remaining electrons) that neutralizes its charge and it is such that the electron plus its cell have zero dipole moment [128].

The PC approximation works rather well: for example, for the atomic densities from He to Ne, the values $$W_\infty ^{\mathrm {PC}}[\rho ]$$ agree within 1% with the values obtained by using the radial co-motion functions (maps) described in Sect. 4.3.12, as shown in Table I of Ref. [127]. This is quite remarkable as, usually, gradient expansions for the exchange-correlation functionals fail in providing accurate quantitative results.

4.4.6.2 Generalized Gradient Approximations: The Modified PC Model

Although quantitatively accurate for the SIL energy, the main drawback of the PC model is that its functional derivative,

(4.122)

diverges to −∞ in the tail of atomic and molecular densities [45], making a self-consistent Kohn–Sham calculation impossible. Moreover, the PC model fails for quasi-2D and quasi-1D systems [31] .

To overcome these problems, Constantin [31] has proposed a generalized gradient approximation (GGA) for W∞[ρ], called the modified PC model (mPC), which reads

(4.123)

where A has the same value as in the original PC model, and a = 2. This approximation is less accurate for the SIL of atomic densities with respect to the original PC model (with errors around 9–10%), but has the advantage of a well-behaved functional derivative, and of achieving a physical description of the crossover from three to two dimensions.

4.4.6.3 Approximations with Some Non-Locality: The Non-local Radius (NLR) and the Shell Model

The PC and mPC are semilocal approximations, while, as we have seen, the exact SIL physics has an extreme non-local dependence on the density. Approximations that retain some (albeit limited) non-locality are the non-local radius (NLR) [140] and the shell models [6]. Both approximations use as key ingredient the spherically averaged density $$\widetilde {\rho }({\mathbf {r}},u)$$ around a given position r, obtained by integrating out the angular dependence of u,

(4.124)

and, in analogy with the SCE structure for spherical densities conjectured in Ref. [127] and illustrated in Sect. 4.3.12, its cumulant

(4.125)

In the NLR model [140] the functional W∞[ρ] is approximated as

(4.126)

where the radius R(r) is defined by the condition that the underlying exchange-correlation hole be normalized:

(4.127)

This simple approximation is less accurate than the PC and mPC models for the case of the uniform electron gas, giving a too high energy. For non-uniform densities, the NLR has the advantage, with respect to the PC and mPC models, of being exact for one-electron systems. For atomic densities, NLR makes errors, with respect to the SCE results of Ref. [127], of the order of 8–9% [140].

The shell model [6] substantially improves the NLR approximation, by making it exact for a uniform density, and reducing its error with respect to the SCE results for atomic densities by almost a factor of 10. While the NLR model approximates the exchange-correlation hole with a sphere depleting one electron from the spherically averaged density, the shell model adds a single positive oscillation, and reads

(4.128)

where for all r we have us = 0.849488 uc, which is the condition needed to make the model exact for a uniform density. The value of uc(r) is then obtained again by the normalization condition,

(4.129)

4.5 Kohn–Sham Combined with the Strong-Interaction Limit

4.5.1 Kohn–Sham with the SCE Functional (KS SCE)

The Kohn–Sham scheme with the SCE functional (KS SCE) was first proposed and implemented in [100], and corresponds to a crude, but well-defined approximation for the HK functional,

(4.130)

in which we replace the minimum of the sum of kinetic energy and electron-electron repulsion at fixed density, with the sum of the two minima. As such, the KS SCE will always provide a lower bound for the HK functional. When implemented self-consistently, the KS SCE scheme yields the usual KS equations with the Hartree-exchange-correlation potential given by the SCE or Kantorovich potential (written below for simplicity for a closed-shell system),

(4.131)

where the SCE potential is equal to

(4.132)

with u(r, [ρ]) the maximizer in Eq. (4.55), and the constant C[ρ] a shift that ensures limr→∞vSCE(r) = 0, see Eq. (4.64). This shift is the same, in the λ →∞ limit of the density-fixed adiabatic connection, as the one introduced by Levy and Zahariev [94, 139]. If we want to compute the ground-state density and the ground state energy only, one could better work with u instead of vSCE, as with the former the energy becomes simply [22, 94, 139] the sum of the occupied orbital energies, $$E_0=2\sum _{i=1}^{N/2}\epsilon _i$$ . The shift is needed if we want to estimate the ionisation potential $$I=E_0^{N-1}-E_0^N$$ from the highest occupied molecular orbital energy (HOMO), as I = −𝜖N∕2 holds only when the exchange-correlation potential goes to zero far from the barycentre of nuclear charge [3, 93]. For further discussion of this point see Chap. 1 by Toulouse in this volume.

4.5.1.1 1D Case

The self-consistent KS SCE equations have been solved for 1D systems with the interaction $$v_{\mathrm {ee}}^{\mathrm {wire}}(x)$$ of Eq. (4.101) when the external potential is harmonic, $$v_{\mathrm {ne}}(x)=\frac {1}{2}\omega ^2\,x^2$$ , [72, 100, 101], and with the soft Coulomb interaction $$v_{\mathrm {ee}}^{\mathrm {soft}}(x)$$ of Eq. (4.102) for model 1D atoms and molecules with ‘nuclei’ that attract the electrons with the same soft Coulomb potential [102]. At each KS iteration, the 1D co-motion functions [123] were computed numerically as explained in Sect. 4.4.1.1, and the potential vSCE(x, [ρ]) was obtained by simply integrating the force equation

(4.133)

with boundary condition vSCE(x →±∞, [ρ]) = 0, and where w(x) is the chosen 1D interaction (wire or soft Coulomb, see Sect. 4.4.1.1). In addition, at low density the highest occupied KS SCE eigenvalue gives a very accurate ionization energy of the system [101].

Harmonic External Potential

In Fig. 4.11 we show the self-consistent KS SCE densities for N = 4 electrons interacting with $$v_{\mathrm {ee}}^{\mathrm {wire}}(x)$$ of Eq. (4.101) when the external potential is harmonic, using scaled units in terms of L = 2 ω−1∕2, compared with accurate many-body results from configuration interaction (CI) and with KS within the local density approximation (LDA), provided for this interaction in Ref. [18]. We see that, as the system is driven to low density by reducing the strength of the harmonic confinement (large L), the exact many-body solution undergoes a so-called “2kF → 4kF” transition, in which the number of peaks in the density is doubled. At high density, in fact, the number of peaks is dictated by the number of occupied orbitals, N∕2 for a closed shell system. At low density, we have an incipient Wigner molecular structure, in which the electrons are well separated. Notice that with the Coulomb interaction this Wigner molecular phase exhibits different properties than the simpler case of very short-range interactions, in which the physics can be captured by making the system spin-polarized (i.e. by occupying N orbitals instead of N∕2). This is clearly illustrated in Ref. [142].

It is well-known that the local and semilocal approximations to the XC functional, as well as exact exchange, are not able to capture this “2kF → 4kF” transition [134, 135] without introducing artificial symmetry breaking. This is also clearly shown by the KS LDA results of Fig. 4.11, which become very close in this limit to the Thomas–Fermi result (minus the external potential in the classically allowed region) predicting a too delocalized density. The KS SCE self-consistent density, although not quantitatively very accurate, has the correct qualitative behavior, with two peaks at high density and four at low density, and with the correct extension. The KS SCE HOMO energy is also very close to the exact many-body ionisation potential [100, 101]. In the right panel of Fig. 4.11 we show the total KS potential at self consistency, vKS = vne + vSCE, for the most correlated case. The horizontal lines are the two occupied KS SCE eigenvalues. We see that the SCE functional is able to self-consistently build barriers that create classically forbidden regions inside the harmonic trap. Classically forbidden regions for the KS orbitals created by the Hartree-exchange-correlation potential seem to play a crucial role in describing strong correlation within KS DFT [12, 75, 144].

Model 1D Chemistry with Soft Coulomb Potential

In Ref. [102] the KS SCE method has been tested for model chemical systems in 1D, consisting of “nuclei” and electrons attracting each other with the soft-Coulomb potential (for the use of these 1D models to test DFT approximations, see also Refs. [74, 141]). While in the harmonic external potential we can drive the system to low density where the SCE becomes a very good approximation to the exact KS exchange-correlation functional, chemical systems (bound by the Coulomb external potential) are never in this regime. For this reason, KS SCE does not in general yield accurate results, with total energies that are way too low. An exception seems to be the good agreement between the eigenvalue of the highest occupied KS SCE orbital and the many-body chemical potential, as shown in Table 2 of Ref. [102].

4.5.1.2 2D Case

The circularly-symmetric 2D case of electrons interacting with the 1∕r repulsion in the harmonic external potential has been studied with KS SCE in Ref. [105], using the SGS radial co-motion functions and the reduced radial cost of Eq. (4.78) implemented as described in Sect. 4.4.1.2. As in 1D, the aim is to model electrons strongly confined in one direction, found, for example, at the interface of semiconductor etherostructures. For this reason, the interaction remains the same as the 3D Coulomb one.

As discussed in Sect. 4.3.12, the SGS state defined by (4.79)–(4.80) is not guaranteed to yield the absolute minimum for the electron-electron interaction in a given radial density ρ(r). Nonetheless, it can be proven [125] that, for a spherically-symmetric density, if we reduce the admissible class of maps $${\mathcal {T}}_\rho $$ in the SCE functional (4.37) to a class $${\mathcal {T}}^{{\mathrm {SGS}}}_\rho \subset {\mathcal {T}}_\rho $$ of maps given by the SGS ansatz defined in Eqs. (4.79)–(4.80) as an approximation for $$\operatorname {V}_{\mathrm {ee}}^{\operatorname {SCE}}[\rho ]$$ , even when the SGS maps are not optimal the functional derivative of this approximate $$\operatorname {V}_{\mathrm {ee}}^{\operatorname {SCE}}[\rho ]$$ with respect to ρ(r) still satisfies the force equation (written using the notation of Eq. (4.59)),

(4.134)

which we can integrate to obtain a potential vSGS(r). In other words, the SGS maps provide a well-defined approximation to the exact SCE functional, with an easy to evaluate functional derivative, which, in turn, can be used in the KS equations.

In Fig. 4.12 we show the resulting KS SCE self-consistent radial density for N = 3 electrons for two low-density cases, compared with accurate Quantum Monte Carlo (QMC) results from Refs. [61, 73]. The KS SCE calculations have been done for both the unpolarized case (2 KS orbitals, of which only the lowest is doubly occupied) and the spin-polarized case (3 different singly occupied KS orbitals). We see that the KS SCE densities are very close to the QMC ones, predicting the right shell structure with one peak. Total energies are in agreement with QMC within ∼ 4 − 6% [105]. Notice that at such low densities it is very hard to even obtain converged results using KS with the local-spin density (LSD) approximation. We thus see that even if the SGS maps are not optimal for these densities (see [125]), they yield very good results when used in the self-consistent KS equations at low density. However, we have to mention that QMC predicts that at such small ω’s the ground state is spin-polarized, while in KS SCE the unpolarized case always yields the lowest energy, due to the lack of any spin dependence in the SCE functional.

2 line graphs plot increasing lines for omega power negative 2 by 3, v subscript K S and decreasing lines for omega power negative 4 by 3 rho with respect to values of omega power 2 by 3 r. Both lines have fluctuations. The value of omega in A is 0.001 and in B is 1. A horizontal line at 6.35 and 7.4, approximately are plotted in A and B, respectively.

Fig. 4.13

Self-consistent KS SCE radial potential $$v_{\mathrm {KS}}(r)=v_{\mathrm {SCE}}(r)+ \frac {1}{2}\omega ^2 r^2$$ (blue solid line) and radial densities (red dashed line) for a strongly and weakly correlated case (top and bottom, respectively) of a 2D system composed of N = 10 electrons inside a circularly symmetric harmonic trap. The green dashed horizontal lines correspond to the energies of the highest occupied KS orbital. Notice the presence of classically forbidden regions inside the trap in the strongly correlated case (ω = 0.001)

Figure 4.13 shows the self-consistent KS SCE total potential and density for N = 10 electrons (spin unpolarized) [105]. The green dashed curve is the energy of the highest occupied KS orbital. We clearly see, as in the 1D case of Fig. 4.11, that when the system is driven to low-density (small ω case), KS SCE is able to self-consistently create classically forbidden regions inside the trap.

2 line graphs. A, the graph of electron density versus molecular axis plots 2 overlapping lines for S C E and L D A that form 2 peaks at negative 3 and 3. B, the graph of bond energy versus bond length plots 3 lines for exact, S C E, and L D A. The lines decrease below 0 initially and then increase gradually.

Fig. 4.14

Right: Dissociation curve of H₂ in KS SCE [22], that is, energy of H₂ minus twice the energy of the isolated H atom. For comparison, the KS LDA curve computed on the same mesh and the exact curve from Ref. [86] are also shown. Note that KS SCE, unlike the local density approximation, dissociates H₂ correctly. Left: Corresponding self-consistent KS SCE density and KS LDA density near dissociation

4.5.1.3 3D Case

KS SCE has been tested on the anions of the He isoelectronic series [106] and on the dissociation curve of the H₂ molecule [22].

Anions of the He Isoelectronic Series

In this case, i.e., N = 2 electrons with $$v_{\mathrm {ne}}(r)=-\frac {Z}{r}$$ , where Z is lowered until the system can no longer bind two particles, the co-motion function and the SCE potential are simply built following the original work of Seidl [123] (see Example 4.20), which is a special case of the SGS maps. While very accurate wavefunction results predict [47] that one electron is lost by the system at a critical nuclear charge $$Z_{\mathrm {crit}}^{\mathrm {exact}}\approx 0.911$$ , KS SCE binds two electrons down to $$Z_{\mathrm {crit}}^{\mathrm {KSSCE}}\approx 0.7307$$ [106]. This is because in the KS SCE case the two electrons can get much closer to the nucleus by perfectly avoiding each other, without raising the kinetic energy too much, which is only treated within KS.

The H₂ Molecule

The dissociation curve of the H₂ molecule has been computed within KS SCE in Ref. [22]. The result is shown in Fig. 4.14. To compute the self-consistent density and energy, an accurate adaptive three-dimensional finite element discretization was used and the SIL problem was solved using linear programming, as described in Sect. 4.4.2.1. The co-motion function for H₂ was then obtained from the SIL density via Eq. (4.105), and the SCE potential via the force Eqs. (4.59) and (4.64).

Not surprisingly, KS SCE predicts a binding energy that is way too low. A remarkable feature, though, is the ability of KS SCE to correctly dissociate the H₂ molecule, i.e., the molecular energy tends to twice the energy of the isolated H atom as the internuclear distance R becomes very large (see [22] for a rigorous proof). Local and semilocal approximations to the XC functionals are unable to do that, and exact exchange (or Hartree–Fock) perform even worse, unless we allow spin-symmetry breaking. Indeed, the extremely stretched H₂ molecule is often regarded as a severe test for XC functionals to check whether they are able to describe strong (or “static”) correlation [24].

Although the SCE functional yields the exact energy when R →∞, we see that at large but finite R the KS SCE curve immediately start to deviate from the exact one. We can understand this error by making the following simple analysis. With the internuclear vector R directed along the x-axis, we can expand the electron-electron interaction at large R, which, without considering one-body terms and neglecting higher orders in R⁻¹ yields

(4.135)

where the origins of r₁ and r₂ are placed on their respective nuclei. The SCE functional for large R then corresponds to the minimization of this interaction at fixed one-body density (hence, the neglect of one-body terms that will not affect the minimizer). The SCE problem in this limit reduces then to the attractive harmonic cost [56] in the bond (x) direction and to the repulsive harmonic cost [42] in the two directions perpendicular to the bond axis. For large R, the optimal map will then approach the solution

(4.136)

which corresponds to perfectly coupled dipoles (see Fig. 4.15). Such maps will give a finite (negative) expectation value for the r.h.s. of Eq. (4.135) even when the total density of the molecule is given by the sum of two spherical atomic densities, yielding an interaction energy that is too attractive, decaying as ∼ R⁻³ instead of the exact ∼ R⁻⁶. What is missing in the KS SCE approach is the raising in kinetic energy associated with the perfectly correlated dipoles of Fig. 4.15. A strategy to include the raise in kinetic energy in this asymptotic large-R regime is described in Ref. [88].

2 ellipses with a plus sign and a circumcircle at the center. The points on the ellipse in 4 directions are labeled A to D. In diagram A, the points are arranged anticlockwise direction, starting from the north. In B, the points are arranged clockwise direction, starting from the south.

Fig. 4.15

When the distance R between the two atoms of the H₂ molecule gets very large, the optimal map describes the physics of perfectly coupled dipoles. The figure shows four pairs of electronic positions {r, f(r)}, labeled with the same letter A,B,C,D, with respect to the two positive nuclei

4.5.2 Interaction Strength Interpolation (ISI) Functionals

Another way to use the SIL in KS DFT is the interaction strength interpolation (ISI) construction, originally proposed by Seidl, Perdew and Levy (SPL) [130]. ISI is essentially the extension to non-uniform densities of Wigner’s original idea [143] of approximating the energy of the uniform electron gas by interpolating between its high- and low-density asymptotics, which, by scaling, correspond to the weak- and strong-interaction limits, respectively.

The starting point is to use the Hellmann–Feynman theorem to write the exchange-correlation energy as an integral over the coupling-strength parameter λ of (4.17)¹⁰

(4.137)

where

(4.138)

with ψλ[ρ] the minimizer in (4.17). The idea is then to construct approximations for the adiabatic connection integrand Wλ[ρ] by interpolating between the λ → 0 asymptotic expansion,

(4.139)

with Ex the exchange energy and $$E_{\mathrm {c}}^{\mathrm {GL2}}$$ the second-order Görling-Levy perturbation theory correlation energy [69], and the large-λ limit provided by the SIL, and possibly by the conjectured next leading term of Eq. (4.47),

(4.140)

For example, SPL [130] proposed the following simple form to interpolate between the two limits, without using the term with FZPE[ρ]:

(4.141)

with

(4.142)

The SPL XC functional then reads

(4.143)

Several other interpolating functions that may or may not include FZPE[ρ] have been proposed in the literature, [43, 68, 98, 128, 129] and are reported, for example, in the appendix of Ref. [87].

4.5.2.1 Global Interpolations

Interpolations such as the one of Eq. (4.141) have been implemented and tested on several chemical systems by using for the λ →∞ limit the PC model of Sect. 4.4.6.1 (and its extension [128] to FZPE[ρ] when needed). In a practical calculation, KS orbitals with a given approximate semilocal or hybrid functional are used to compute the density ρ, the exchange energy Ex, and the second-order energy $$E_{\mathrm {c}}^{\mathrm {GL2}}$$ , which are then fed into formulas such as (4.143) to obtain improved energies. The result is thus dependent on the chosen starting approximate functional used to generate the KS orbitals.

A basic problem of these global (in the sense that they are done on quantities that have been already integrated over all space) interpolations is the violation of size-consistency, i.e., if we take two different systems A and B that do not interact with each other, it is easy to verify from (4.143) that, in general,

(4.144)

an issue shared by all the other interpolation formulas proposed in the literature [25]. Notice that size-consistency of approximate electronic-structure methods is a very delicate issue when A and/or B have a degenerate ground state [65, 120]. Here we stress that even when degeneracy is not present, the fact that the input ingredients (Ex, $$E_{\mathrm {c}}^{\mathrm {GL2}}$$ and W∞) enter in a non-linear way in the ISI formulas introduces anyway a size-consistency error. However, this error can be easily corrected [137]. In fact, the reason why size-consistency is crucial in chemistry is that we are interested in interaction energies rather than total energies. All we need to do is to set the limit of a molecular dissociation curve (when A and B are infinitely far apart) at the value given by the left-hand side of Eq. (4.144) rather than the one given by the right-hand side. Notice that both sides of this equation can be evaluated at exactly the same computational cost, as all that is needed is the input ingredients of the fragments A and B [137]. With this size-consistency correction it is possible to extract meaningful interaction energies from the ISI functionals [137].

The ISI functionals have been tested on several chemical data sets and systems [44, 62, 137]. They have been found to work reasonably well for interaction energies (especially of non-covalent systems) when Hartree–Fock orbitals (rather than KS ones) are used as input. This observation has triggered the study of the strong-interaction limit in Hartree–Fock theory [37, 126], which, in turn, has led to new interpolation schemes in this framework which are able to give very accurate results for a large variety of non-covalent interaction energies, ranging from small to medium-large systems [36].

If one wants to overcome the dependence on the input orbitals, one should evaluate the energy using the ISI functionals within a fully self-consistent KS scheme. For this, their functional derivative with respect to the density is needed, which is challenging due to the presence of second-order perturbation theory. Nonetheless, first attempts in the computation of the ISI functional derivatives have been reported in Refs. [45, 132], and self-consistent calculations are likely to appear soon.

4.5.2.2 Local Interpolations

Another possibility is to build the interpolations locally, in each point of space, by defining an energy density wλ(r;[ρ]) for the coupling-constant integrand Wλ[ρ] of Eq. (4.138), writing Exc[ρ] as

(4.145)

Energy densities are obviously not uniquely defined, and the only important requirement here is to use local quantities defined in the same way at weak and strong coupling. Some different choices for energy densities in the λ-interpolation context have been analyzed in [139], where it has been found that the electrostatic potential of the exchange-correlation hole¹¹ $$h^\lambda _{\mathrm {xc}}({\mathbf {r}}_1, {\mathbf {r}}_2)$$ seems to be the most suitable ,

(4.146)

where $$h^\lambda _{\mathrm {xc}}({\mathbf {r}}_1, {\mathbf {r}}_2)$$ is defined in terms of the pair-density $$P_2^\lambda ({\mathbf {r}}_1, {\mathbf {r}}_2)$$ and the density ρ,

(4.147)

with $$P_2^\lambda $$ obtained from Ψλ[ρ],

(4.148)

Local interpolations within this definition have been analysed and tested in Refs. [87, 138] on small systems, with mixed results.

4.6 Appendix: Kantorovich Duality

The dual construction of the SCE functional and potential (see Theorem 4.13) relies on Kantorovich duality. In this Appendix we give a precise mathematical statement of Kantorovich duality for multi-marginal optimal transport, and show how it implies Theorem 4.13 (1).

Recall the general Kantorovich optimal transport problem introduced in Sect. 4.3.6: for given marginal measures μ₁, …, .μN defined on closed subsets X₁, …, XN of $$\mathbb {R}^d$$ , minimize a cost functional

over probability measures Π on the product space X₁ ×… × XN subject to the marginal constraints

Here $$c \, : \, X_1\times \ldots \times X_N\to \mathbb {R}\cup \{+\infty \}$$ is a given measurable cost function.

This problem is related to a certain dual variational problem: maximize the functional

over potentials $$u_i \, : \, X_i\to \mathbb {R}$$ (i = 1, …, N) which must satisfy the pointwise constraint

(4.149)

The following nontrivial statement, taken from the recent textbook [50], summarizes what is known in $$\mathbb {R}^d$$ , and is general enough to cover the Coulomb cost.

Theorem 4.32 (Kantorovich Duality)

For given probability measures μ₁, …, μNdefined on closed subsets X₁, …, XNof $$\mathbb {R}^d$$ , provided the cost function $$c \, : \, X=X_1\times \cdots \times X_N\to \mathbb {R}\cup \{ +\infty \}$$ is bounded from below and lower semi-continuous and the optimal cost is finite,

(4.150)

where $${\mathcal {A}}(c)$$ is any of the following increasingly general sets of admissible potentials:

(1)

(2)

$${\mathcal {A}}(c)$$ as in (1) , with Cb(Xi) in place of C₀(Xi)

(3)

$${\mathcal {A}}(c)$$ as in (1) , with $$B(X_i)=\{u : X_i\to \mathbb {R}\, : \, u \mathit{\mbox{ bounded measurable}}\}$$ in place of C₀(Xi)

(4)

.

Here we have used the standard notation Cb(Xi) for the space of bounded continuous functions on Xi, and C₀(Xi) for the space of decaying continuous functions on Xi (i.e. those u which in addition satisfy u(ri) → 0 if |ri|→∞).

In the special case of two marginals defined on compact sets, cost functions c which are metrics (such as |r₁ −r₂|), and the choice (2) for the potentials, this fundamental result was discovered by Kantorovich [80]. A great many variants and modifications have subsequently appeared in the mathematics literature. Some of them replace the Xi by abstract spaces; many are worked out only for two marginals; almost all of them differ in the precise assumptions on the cost function and the class of admissible potentials. For instance, [117] (Theorems 2.1.4(b) and 2.1.1) and [81] cover bounded continuous cost functions and the class (3) for N marginals; [136] (Theorem 1.3) covers bounded-below lower semi-continuous cost functions and the class (4) for two marginals. Strictly speaking, none of the versions published prior to the discovery of the optimal transport/SCE theory connection applied directly to the multi-marginal Coulomb case, even though the underlying ideas essentially did. For the proof of Theorem 4.32 we refer the reader to [50].

Technical Remark

From a functional analysis point of view, the natural class of admissible potentials in (4.150) is the smallest one, (1). This choice reflects the duality between potentials ui and measures μi in the integral $$\int \! u_i \mathrm{d} \mu _i$$ ; note that the linear hull of the space $${\mathcal {P}}(\mathbb {R}^d)$$ of probability measures, that is, the space $${\mathcal {M}}(\mathbb {R}^d)$$ of signed measures, is the dual of $$C_0(\mathbb {R}^d)$$ . Enlarging this class from (1) to (2)–(4) has the virtue that the supremum of the dual problem is attained for increasingly general cost functions c.

Proof of Theorem 4.13 (1) using Theorem 4.32

Applying the Kantorovich duality theorem with Xi, μi, and c as in Example 4.11, and making the choice (3) for the class of admissible potentials, one obtains

The left-hand side is the enlarged-search definition (4.26) of the SCE functional $$V_{\mathrm {ee}}^{\mathrm {SCE}}[\rho ]$$ (which, by Theorem 4.1, is equivalent to the original definition (4.15)). The right-hand side can be simplified. For any collection of potentials (u₁, …, uN), the sum of the integrals on the right-hand side is preserved under the replacement $$(u_1,\ldots ,u_N)\mapsto (\bar {u},\ldots ,\bar {u})$$ , where $$\bar {u}$$ denotes the average (u₁ + ⋯ + uN)∕N; moreover the constraint in (4.6) is also preserved, thanks to the symmetry of Vee. Thus the right-hand side of (4.6) stays unaltered if the supremization is restricted to N equal potentials, u₁ = ⋯ = uN = u. But in this case the right-hand side reduces to that of (4.55), establishing Theorem 4.13 (1). □

Acknowledgements

AG acknowledges partial support of his research by the Canada Research Chairs Program and Natural Sciences and Engineering Research Council of Canada. This work started when AG was at the Vrije Universiteit Amsterdam and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. [795942].

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Footnotes

1

We follow the usual convention to use the same letter Vee both for the total interaction potential, a function on $$\mathbb {R}^{dN}$$ , and the associated quadratic form, a functional on the wavefunction space $$\mathcal {W}^{N}$$ .

2

This is not cured by dropping the requirement in (4.7) that Ψ must have square-integrable gradient and requiring mere square-integrability, i.e. replacing $$\bigwedge _{i=1}^N H^1(\mathbb {R}^d\times \mathbb {Z}_2;\mathbb {C})$$ by $$\bigwedge _{i=1}^N L^2(\mathbb {R}^d\times \mathbb {Z}_2;\mathbb {C})$$ .

3

By a limit point Π of a sequence Πλ of probability measures we mean a limit point in the sense of narrow convergence, that is, convergence of the integrals $$\int f \, \mathrm{d} \Pi _\lambda $$ to $$\int f \, \mathrm{d} \Pi $$ for any bounded continuous function f.

4

Note added in proof: this problem has recently been solved in https://​arxiv.​org/​abs/​2210.​07830.

5

The original conjecture concerned the physical case d = 3. Subsequently, two-dimensional models have also been considered in the literature [125, 127].

6

These are the points that cannot be written as convex combinations of any other points in the set.

7

The KL divergence between two nonnegative densities with possibly unequal mass is formally defined as $${\mathrm {KL}}(f|g) = \int f\log \frac {f}{g}$$ .

8

Strictly speaking, this inequality and related ones are proved in [77] under the tacit assumption that $$\sqrt {\varPi }$$ (or related reduced quantities) belong to H¹ and can be differentiated by the chain rule. For further discussion of this point see Chap. 5 by Kvaal in this volume.

9

which consists in our case in finding a configuration r∗ which maximizes the difference $$u^{\mathrm {new}}({\mathbf {r}}_1^*)+\cdots +u^{\mathrm {new}}({\mathbf {r}}_N^*)-V_{\mathrm {ee}}({\mathbf {r}}^*)$$ .

10

For further discussion see also Chap. 1 by Toulouse in this volume.

11

For further discussion of this point, see Chap. 1 by Toulouse in this volume.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_5

5. Moreau–Yosida Regularization in DFT

Simen Kvaal1  

(1)

Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Oslo, Norway

Simen Kvaal

Email: simen.kvaal@kjemi.uio.no

Abstract

Moreau–Yosida regularization is introduced into the framework of exact DFT. Moreau–Yosida regularization is a lossless operation on lower semicontinuous proper convex functions over separable Hilbert spaces, and when applied to the universal functional of exact DFT (appropriately restricted to a bounded domain), gives a reformulation of the ubiquitous v-representability problem and a rigorous and illuminating derivation of Kohn–Sham theory.

The chapter comprises a self-contained introduction to exact DFT, basic tools from convex analysis such as sub- and superdifferentiability and convex conjugation, as well as basic results on the Moreau–Yosida regularization. The regularization is then applied to exact DFT and Kohn–Sham theory, and a basic iteration scheme based in the Optimal Damping Algorithm is analyzed. In particular, its global convergence established. Some perspectives are offered near the end of the chapter.

5.1 Introduction

In this chapter, we introduce Moreau–Yosida regularization into the exact DFT framework. Such regularization of convex optimization problems is lossless: Given a small 𝜖 > 0, a non-smooth convex functional F[ρ] is regularized into a differentiable convex functional 𝜖F[ρ]. However, this smoothing is invertible, so that the information encoded is not lost. Thus, the regularization gives a reformulation of the convex optimization problem as a differentiable convex optimization problem. When applied to exact DFT, we gain insight into the ubiquitous v-representability problem and also a rigorous formulation of Kohn–Sham theory. This comes at a small price, as we need to describe densities and potentials using Banach spaces that are reflexive, which excludes, say, exact Coulomb potentials on infinite domains. This trade-off is acceptable, as no physical system is truly infinite in extent. Indeed, placing the system in a finite but large box is the only approximation being made in this chapter.

The motivation for introducing Moreau–Yosida regularization comes from two directions: First, DFT is committed to treating “all possible” potentials at once. For the exact interacting problem, this is surely overkill: After all we are mostly interested in the atomic Coulomb potentials, or at most well-behaved potentials modeling, say, harmonic traps. On the other hand, passage to the non-interacting Kohn–Sham system turns the problem upside down: The density is the known quantity, while the effective potential is the unknown function of the density. The standard approach is to differentiate the universal functional to obtain the exchange-correlation potential, but this is not permitted in exact DFT. Indeed, the large class of potentials implies that the universal functional will be highly non-smooth. We note in passing that in the strongly correlated limit of DFT, where one obtains the purely-interacting universal functional, the exchange-correlation potential can in fact be obtained rigorously by differentiation, as shown in Chap. 4 by Friesecke, Gerolin and Gori-Giorgi. The second motivation for Moreau–Yosida regularization is that formal Kohn–Sham theory relies on densities to be both interacting and non-interacting v-representable. This problem vanishes into thin air in the regularized Kohn–Sham formulation.

While Moreau–Yosida regularization resolves some problems, it still leaves open the question of how to approximate the regularized universal functional 𝜖F[ρ] in a systematic manner. On the other hand, the method of Moreau–Yosida regularization may also be used to model density functional approximations that are already convex.

This chapter has the following structure. In Sect. 5.2 we give a brief introduction to the convex formulation of exact DFT as given by Lieb [16]. Some results from the theory of quadratic forms over Hilbert spaces are needed, and we will refer the reader to the excellent literature that exists. The article [27] by Simon lays out the formulation of quantum mechanics using quadratic forms, and the monograph [25] by Schmüdgen gives further details, including the representation theorem of bounded-below closed quadratic forms. We include some definitions and theorems from the analysis of convex functions over Banach spaces, and also give proofs whenever deemed instructive. For an excellent and accessible introduction to convex analysis, see the short monograph [29] by van Tiel. After introducing exact DFT, we proceed to introduce the only approximation we need, that is, truncation of infinite space $$\mathbb {R}^3$$ to a bounded domain $$\Omega \subset \mathbb {R}^3$$ . This allows the formulation of exact DFT in a Hilbert space setting, or more generally in a reflexive Banach space setting.

In Sect. 5.3 we introduce the Moreau–Yosida regularization of convex problems over Hilbert spaces. For a more detailed exposition in the Hilbert space setting, see the monograph [1] by Bauschke and Combettes. Most results can be wholly or partially generalized to convex functions over reflexive Banach spaces, see the monograph [2] by Barbu and Precupanu. In our exposition, we include some important definitions and results, with complete proofs in many cases due to the central nature of the material. In the following Sect. 5.4, we apply Moreau–Yosida regularization to box truncated exact DFT. In particular, we detail the rigorous derivation of Kohn–Sham theory in the regularized setting, including a basic analysis of the Moreau–Yosida Kohn–Sham optimal damping algorithm (MYKSODA). Our treatment is adapted from Refs. [9, 10, 17, 20, 21].

Finally, in Sect. 5.5 we provide a conclusion and discuss opportunities for future research.

The author gratefully acknowledges helpful feedback from Andre Laestadius and Markus Penz, and also Gero Friesecke who additionally provided the complete proof of the fact that finite-kinetic energy N-electron wavefunctions have densities with square roots in the Sobolev space $$H^{1}(\mathbb {R}^3)$$ , see Theorem 5.2.

5.2 Exact DFT

In this section, we give a brief outline of elements of exact DFT, paying special attention to the regularity, or lack thereof, of the universal functional and ground-state energy. We then regularize exact DFT to a finite but large subdomain $$\Omega \subset \mathbb {R}^3$$ . Our focus is on molecular electronic systems.

5.2.1 The Variational Ground-State Problem

In this section, we use the definitions

Here, Hk(SN) denotes the set of elements of L²(SN) whose 2N spatial components are elements of the standard k’th order Sobolev space $$H^k(\mathbb {R}^{3N})$$ , i.e., each spatial component has square-integrable weak partial derivatives up to order k [25]. The starting point for DFT is the ground-state problem of N interacting electrons in an external potential v on variational form, i.e.,

(5.1)

with $$\mathcal {W}_N = \{\psi \in H^1_N \mid \|\psi \|{ }_2 = 1\}$$ consisting of the normalized finite kinetic-energy wavefunctions, and with

Here, V  is the many-electron operator corresponding to the potential v, and ρψ is the one-electron density of ψ. In order to make connection with more standard formulations of quantum mechanics, the variational minimization in terms of quadratic forms in Eq. (5.1) needs to be connected with the spectral theory of self-adjoint operators. Intuitively, the kinetic energy part of the variational formulation is obtained by integration by parts, enlarging the domain of the kinetic energy operator term $$T = -\tfrac {1}{2}\nabla ^2$$ in the full Hamiltonian. However, the transition back and forth between a self-adjoint operator and a quadratic form is subtle, and for some potentials the domain of the Hamiltonian operator is not $$H^2_N$$ due to singular behavior of v. For this exposition, we let the following be sufficient: Whenever $$v \in L^{3/2}(\mathbb {R}^3) + L^\infty (\mathbb {R}^3)$$ , $$\mathcal {V}$$ will be relatively bounded with respect to $$\mathcal {E}_0$$ , which means that as a perturbation it is sufficiently gentle to allow the sum of the forms to be well-defined with domain $$H^1_N$$ by the KLMN theorem [25, 27]. The fundamental representation theorem of closed semibounded quadratic forms [25] guarantees the existence of a unique self-adjoint operator $$\widehat {H}[v] : D[v] \to L^2$$ such that for every ψ in the operator domain D[v], we have $$\mathcal {E}_0[\psi ] + \mathcal {V}[\psi ] = \langle {\psi ,\widehat {H}[v]\psi }\rangle $$ . We point out that $$D[v]\subset H^1_N$$ is dense in $$L^2_N$$ , and that $$v\in L^2(\mathbb {R}^3) + L^\infty (\mathbb {R}^3)$$ is sufficient to guarantee $$D[v] = H^2_N$$ , a fundamental result of Kato [8]. However, for the stronger singularities present in $$L^{3/2}(\mathbb {R}^3)+ L^\infty (\mathbb {R}^3)$$ it may happen that D[v] becomes a proper subset of $$H^2_N$$ . On the other hand, the form domain is always $$H^1_N$$ for every $$v \in L^{3/2}(\mathbb {R}^3) + L^\infty (\mathbb {R}^3)$$ . In any case, the infimum in Eq. (5.1) will be the bottom, i.e., infimum, of the spectrum of $$\widehat {H}[v]$$ , and hence the connection is complete.

5.2.2 Densities

Central to DFT is the density $$\rho _\psi \in L^1(\mathbb {R}^3)$$ associated with a (normalized) $$\psi \in L^2_N$$ .

Definition 5.1

For any $$\psi \in L^2_N$$ , we define the density $$\rho _\psi : \mathbb {R}^3 \to [0,+\infty )$$ almost everywhere (a.e.) by the formula

For a given $$\rho \in L^1(\mathbb {R}^3)$$ , we write ψ↦ρ if ρψ = ρ.

It is immediate that $$\rho _\psi \geqslant 0$$ a.e., and that ∥ρψ∥₁ = N∥ψ∥². In fact, we have a complete characterization of densities that come from elements $$\psi \in \mathcal {W}_N$$ . In the proof of the following theorem, the proof that the density of $$\psi \in \mathcal {W}_N$$ satisfies $$\rho _\psi ^{1/2}\in H^1(\mathbb {R}^3)$$ is due to G. Friesecke.

Theorem 5.2 (N-Representable Densities)

Let $$\mathcal {D}^N$$ be the set of measurable functions $$\rho : \mathbb {R}^3 \to \mathbb {R}$$ that satisfy $$\rho \geqslant 0$$ a.e., $$\rho ^{1/2} \in H^1(\mathbb {R}^3)$$ with ∥ρ∥₁ = N. Then for every $$\rho \in \mathcal {D}^N$$ there is a $$\psi \in \mathcal {W}_N$$ such that ψ↦ρ. Conversely, for every $$\psi \in \mathcal {W}_N$$ , $$\rho _\psi \in \mathcal {D}^N$$ . Moreover, $$\mathcal {D}^N$$ is convex.

Proof

For the construction of a $$\psi \in \mathcal {W}_N$$ with ρψ = ρ for a given $$\rho \in \mathcal {D}^N$$ , see [16, Theorem 1.2]. For the converse, we must prove that $$\rho _\psi ^{1/2} \in H^{1}(\mathbb {R}^3)$$ .

Let $$\psi \in \mathcal {W}_N$$ , and let ρ = ρψ. Differentiating, we obtain

Application of Cauchy–Schwarz gives

(5.2)

where

is the kinetic-energy density. Consequently, $$\rho \in W^{1,1}(\mathbb {R}^3)$$ , the standard Sobolev space of $$L^1(\mathbb {R}^3)$$ functions with first-order weak derivatives in $$L^1(\mathbb {R}^3)$$ .

Let $$G(\mathbf {r}) = \exp (-|\mathbf {r}|{ }^2)$$ , and consider ρ𝜖(r) = ρ(r) + 𝜖G(r), with 𝜖 > 0 a small parameter. We first prove that $$\rho ^{1/2}_\epsilon \to \rho ^{1/2}$$ in $$L^2(\mathbb {R}^3)$$ as 𝜖 → 0. It is clear that $$|\rho ^{1/2}_\epsilon - \rho ^{1/2}| \to 0$$ pointwise. By the elementary inequality $$(a-b)^2 \leqslant 2(a^2 + b^2)$$ we have, for all $$\epsilon \leqslant 1$$ ,

By the dominated convergence theorem, the integral of the left-hand side converges to zero, and hence $$\rho ^{1/2}_\epsilon \to \rho ^{1/2}$$ in $$L^2(\mathbb {R}^3)$$ as claimed.

We next prove that $$\rho _\epsilon ^{1/2}$$ is bounded in $$H^1(\mathbb {R}^3)$$ for $$\epsilon \leqslant 1$$ . Let h(z)                                                                                        = (z + 𝜖G(r))1∕2, where z ∈ [0, +∞) and r is in some compact subset of $$\mathbb {R}^3$$ . The function h is continuously differentiable with bounded derivative. By the chain rule for Sobolev functions, which says that (1) the composition of a C¹ function with bounded derivative and a function in $$W^{1,1}_{\mathrm {loc}}(\mathbb {R}^3)$$ is again in $$W^{1,1}_{\mathrm {loc}}(\mathbb {R}^3)$$ , and (2) that its partial derivatives can be computed with the usual chain rule [6, Ch. 5, Exercise 17], we obtain $$(\rho + \epsilon G)^{1/2}\in W^{1,1}(\mathbb {R}^3)_{\mathrm {loc}}$$ with $$\nabla (\rho + \epsilon G)^{1/2} = \tfrac {1}{2}(\rho + \epsilon G)^{-1/2}(\nabla \rho + \epsilon \nabla G)$$ . Key now, is that this expression is uniformly bounded in $$L^2(\mathbb {R}^3)$$ . Indeed,

(5.3)

Thus, $$\rho _\epsilon ^{1/2}$$ is bounded in $$H^1(\mathbb {R}^3)$$ , as claimed. By weak sequential compactness of bounded sets in the Hilbert space $$H^1(\mathbb {R}^3)$$ there exists a sequence {𝜖k}⊂ [0, 1] such that $$\rho _{\epsilon _k}^{1/2} \to u$$ weakly in $$H^1(\mathbb {R}^3)$$ . Since $$\rho _{\epsilon _k}^{1/2} \to \rho ^{1/2}$$ strongly in $$L^2(\mathbb {R}^3)$$ , u = ρ1∕2, and thus $$\rho ^{1/2} \in H^1(\mathbb {R}^3)$$ .

For convexity of $$\mathcal {D}^N$$ , we first note that $$\rho \in W^{1,1}(\mathbb {R}^3)$$ , since $$\rho ^{1/2}\in \mathcal {D}^N$$ implies that ρ = φ² for a unique φ ∈ H¹, $$\varphi \geqslant 0$$ almost everywhere, and thus $$\|\nabla \rho \|{ }_1 = \|2\varphi \nabla \varphi \|{ }_1\leqslant 2 \|\varphi \|{ }_2\|\nabla \varphi \|{ }_2&lt;+\infty $$ . Let λ ∈ [0, 1], $$\rho _1,\rho _2\in \mathcal {D}^N$$ , and set ρ = λρ₁ + (1 − λ)ρ₂. We need to check that $$\rho ^{1/2} \in H^1(\mathbb {R}^3)$$ , which is obtained by repeating the regularization argument carried out for $$\rho =\rho _\psi \in W^{1,1}(\mathbb {R}^3)$$ . For that, we need a bound on |∇ρ(r)|² in lieu of Eq. (5.2). We begin with the inequality

and apply Young’s inequality, i.e., $$2\varphi _1\varphi _2 \nabla \varphi _2\cdot \nabla \varphi _2 \leqslant \varphi _1^2|\nabla \varphi _1|{ }^2 + \varphi _2^2|\nabla \varphi _2|{ }^2$$ , and then rearrange to get

which is precisely what is needed. □

Remark 5.3

The need for some regularization to an everywhere positive density comes from the fact that the commonly used formula $$\nabla \rho ^{1/2} = \tfrac {1}{2}\rho ^{-1/2}\nabla \rho $$ only makes sense if $$\{\mathbf {r}\in \mathbb {R}^3\mid \rho (\mathbf {r})=0\}$$ has zero measure. When this can be verified, such as for ρ = ρψ everywhere strictly positive, the proof of the above result is somewhat simpler [16, Theorem 1.1].

Throughout this section, we let $$X := L^1(\mathbb {R}^3)\cap L^3(\mathbb {R}^3)$$ , a Banach space with norm $$\|\rho \|{ }_X := \max (\|\rho \|{ }_1,\|\rho \|{ }_3)$$ . We note that the dual is $$X^* = L^{3/2}(\mathbb {R}^3) + L^\infty (\mathbb {R}^3)$$ with norm $$\|v\|{ }_{X^*} = \inf \{ \|f\|{ }_\infty + \|g\|{ }_{3/2} \mid f+g=v \}$$ , also a Banach space [18].

Proposition 5.1

$$\mathcal {D}^N \subset X$$ .

Proof

We need only show that $$\rho \in L^3(\mathbb {R}^3)$$ . But $$H^1(\mathbb {R}^3) \subset L^6(\mathbb {R}^3)$$ by the Sobolev embedding theorem, i.e. $$\rho \in \mathcal {D}^N$$ implies $$\|\rho ^{1/2}\|{ }_6 = \|\rho \|{ }_3^{1/2} &lt; +\infty $$ . □

5.2.3 Constrained Search and Skew Conjugate Pairs

Having a well-defined space of densities, we now introduce constrained search into Eq. (5.1), i.e.,

(5.4)

where the Levy–Lieb functional is defined for any measurable ρ by the expression

We use the convention that FLL[ρ] = +∞ whenever $$\rho \notin \mathcal {D}^N$$ .

We now have that $$E : X^* \to \mathbb {R}$$ is given as a pointwise infimum over a nonempty family of continuous affine functions. Such functions are automatically concave and upper semicontinuous.

For any pair (v, ρ) ∈ X∗× X we have the inequality $$E[v] \leqslant F_{\mathrm {LL}}[\rho ] + \int v \rho $$ . Rearranging, we can define a new function $$F : X \to \mathbb {R} \cup \{+\infty \}$$ by the expression

(5.5a)

which is then automatically convex and lower semicontinuous, satisfying $$F[\rho ] \leqslant F_{\mathrm {LL}}[\rho ]$$ . As we will show, one also has

(5.5b)

For an extended-valued map such as F, we define the effective domain $$\operatorname {dom}(F)$$ to be the points where F is real-valued.

Definition 5.4 (Skew Conjugation)

Let B be a Banach space with dual B∗, consisting of the continuous linear functionals on B. Denote by 〈⋅, ⋅〉 the dual pairing. For any $$f : B \to \mathbb {R}\cup \{+\infty \}$$ not identically + ∞ (f is then called proper) we define the (skew) concave conjugate functional $$f^\wedge : B^* \to \mathbb {R}\cup \{-\infty \}$$ by

For any $$g : B^* \to \mathbb {R}\cup \{-\infty \}$$ , we define the (skew) convex conjugate functional $$g^\vee : B \to \mathbb {R}\cup \{+\infty \}$$ by

A pair of functionals (f, g) satisfying f = g∨ and g = f∧ are said to be a skew-conjugate pair of functionals.

The above introduced concept of (skew) convex/concave conjugate functionals and its notation is slightly unconventional [23], but useful in DFT. The conventional convex conjugate (Legendre–Fenchel transform) of $$f : B \to \mathbb {R}\cup \{+\infty \}$$ is the function $$f^* : B^* \to \mathbb {R}\cup \{\pm \infty \}$$ given by

(5.6)

We introduce the class Γ₀(B) of proper convex lower semicontinuous functions $$f : B \to \mathbb {R}\cup \{+\infty \}$$ , meaningful for the Banach space B but also for general topological vector spaces such as $$B_w^*$$ , the dual space B∗ equipped with the weak-∗ topology. A central fact is that the Legendre–Fenchel transformation is a bijection between these two classes of functions. This result gives insight into the role of convex conjugation in DFT.

Theorem 5.5

The Legendre–Fenchel transform f↦f∗is a bijection between Γ₀(B) and $$\Gamma _0(B^*_w)$$ , and (f∗)∗ = f, as well as (g∗)∗ = g for any $$g \in \Gamma _0(B_w^*)$$ .

Proof

We mention two facts. First, f ∈ Γ₀(B) if and only if f ∈ Γ₀(Bw), where Bw is B equipped with the weak topology. This follows from f being lower semicontinuous if and only if f has closed convex sublevel sets. By Barbu and Precupanu [2, Proposition 1.73], norm-closed convex sets are weakly closed and vice versa. Second, a fact from functional analysis [24, Theorem IV.20] is that the dual of Bw is $$B_w^*$$ , and that $$(B_w^*)^* = B_w$$ .

From [2, Corrolary 2.21], if V  is a locally convex topological vector space, $$f : V \to \mathbb {R}\cup \{+\infty \}$$ is proper if and only if $$f^* : V^* \to \mathbb {R}\cup \{+\infty \}$$ is proper. Since f∗ is the supremum of a family of lower semicontinuous functions, f ∈ Γ₀(Bw) implies $$f^* \in \Gamma _0(B^*_w)$$ . By reflexivity the same argument gives that $$g \in \Gamma _0(B_w^*)$$ implies g∗∈ Γ₀(Bw). The biconjugation theorem (Fenchel–Moreau) for locally convex spaces [2, Theorem 2.22] can for reflexive locally convex topological spaces be phrased as f ∈ Γ₀(V ) implies (f∗)∗ = f. Hence, the conjugation is a bijection between Γ₀(Bw) and $$\Gamma _0(B_w^*)$$ and vice versa. Together with Γ₀(B) = Γ₀(Bw), the bijection of Γ₀(B) and $$\Gamma _0(B_w^*)$$ has been established. □

Theorem 5.5 can be reformulated in terms of skew conjugation. In the following, we write f ∈ − Γ₀(B) if − f ∈ Γ₀(B), and so on.

Proposition 5.2

Let B be a Banach space.

1.

Let f ∈ Γ₀(B). Then $$f^\wedge \in -\Gamma _0(B^*_w)$$ , and f∧[y] = −f∗[−y].

2.

Let $$g \in -\Gamma _0(B^*_w)$$ . Then g∨∈ Γ₀(B) and g∨[x] = (−g)∗[−x].

3.

The map f↦f∧is a bijection between Γ₀(B) and $$-\Gamma _0(B^*_w)$$ , and the map g↦g∨is a bijection between $$-\Gamma _0(B^*_w)$$ and Γ₀(B).

Proof

Proof of 1:

Proof of 2:

Proof of 3: Follows from 1, 2, and Theorem 5.5. □

It is readily seen that $$F = E^\vee = (F^\wedge )^\vee = (F_{\mathrm {LL}}^\wedge )^\vee $$ , and thus that $$F = (F_{\mathrm {LL}}^\wedge )^\vee $$ . F is distinct from FLL, as the latter function is not convex. We have the following characterization:

Theorem 5.6

For any $$f : B \to \mathbb {R}\cup \{+\infty \}$$ , f∗∗ = (f∧)∨∈ Γ₀(B) is the largest convex lower semicontinuous minorant of f, often called the convex envelope or closed convex hull of f.

Proof

See Theorem 6.15 in Ref. [29]. □

A central result in Lieb’s analysis (with a proof attributed to Simon), is the following:

Theorem 5.7

F = FDM, the density-matrix constrained-search functional defined by

where the infimum extends over all trace-class operators $$\mathcal {D}^N_{\mathit{\text{op}}}$$ on the form gamma equals sigma summation Underscript k Overscript normal infinity Endscripts lamda Subscript k Baseline vertical bar psi Subscript k Baseline right angle bracket left angle bracket psi Subscript k Baseline vertical bar ), with $$0\leqslant \lambda _k \leqslant 0$$ , ∑kλk = 1, and $$\{\psi _k\} \subset \mathcal {W}_N$$ an L²-orthonormal sequence. The notation γ↦ρ means that $$\sum _k \lambda _k \rho _{\psi _k} = \rho $$ almost everywhere. The effective domain of FDMis $$\mathcal {D}^N$$ .

Proof

See Theorem 4.4 in Ref. [16], in which it is proven that FDM is lower semicontinuous. Since it is also convex, we must have F = FDM by Theorem 5.6. □

The ground-state problem can be written as a minimization over the set $$\mathcal {D}^N_{\text{op}}$$ of density operators as

(5.7)

Proposition 5.3

Equations (5.7) and (5.1) define the same function, E[v] = EDM[v] for all v ∈ X∗.

Proof

Clearly, $$E_{\mathrm {DM}}[v] \leqslant E[v]$$ , since script upper E 0 left bracket psi right bracket plus script upper V left bracket psi right bracket equals trace left parenthesis ModifyingAbove upper H With caret left bracket v right bracket vertical bar psi right angle bracket left angle bracket psi vertical bar right parenthesis ), i.e., the search domain is larger in Eq. (5.7). On the other hand, $$E_{\mathrm {DM}}[v] \geqslant E[v]$$ , since for any gamma equals sigma summation Underscript k Endscripts lamda Subscript k Baseline vertical bar psi Subscript k Baseline right angle bracket left angle bracket psi Subscript k Baseline vertical bar ), $${\mathrm {Tr}}(\widehat {H}[v]\gamma ) = \sum _k \lambda _k [\mathcal {E}_0[\psi _k] + \mathcal {V}[\psi _k]] \geqslant \sum _k \lambda _k E[v] = E[v]$$ . □

Both FLL and FDM have the important property that they are expectation-valued, i.e., that the infimums in their definition are attained [10] as expectation values of unique states:

Theorem 5.8

For every $$\rho \in \mathcal {D}^N$$ , there exists a unique $$\psi _\rho \in \mathcal {W}_N$$ such that $$F_{\mathrm {LL}}[\rho ] = \langle {\psi _0|\widehat {H}[0]|\psi _0}\rangle $$ , and a unique gamma Subscript rho Baseline equals sigma summation Underscript k Endscripts lamda Subscript k Baseline vertical bar psi Subscript k Baseline right angle bracket left angle bracket psi Subscript k Baseline vertical bar ) such that $$F_{\mathrm {DM}}[\rho ] = {\mathrm {Tr}}(\gamma \widehat {H}[0]) = \sum _{k} \lambda _k \langle {\psi _0,\widehat {H}[0]\psi _0}\rangle $$ , with $$\psi _k \in \mathcal {W}_N$$ .

Proof

Theorem 3.3 and Corollary 4.5(ii) in Ref. [16]. □

5.2.4 Sub- and Superdifferentiability

Equations (5.5a) and (5.5b) cannot in general be differentiated to find a critical point condition, even if this is routinely done in the physics literature, see for example the classic monograph [22] by Parr and Yang. Indeed, neither F nor E are differentiable in general. However, for convex optimization problems, the weakest useful notion of differentiability is not the usual Gâteaux or Fréchet differentiability, but that of subdifferentiability.

Definition 5.9

Let B and C be topological vector spaces. Let $$f: B \to \mathbb {R}\cup \{+\infty \}$$ , and let x ∈ B. The subdifferential of f at x is the set

The elements are called subgradients. Similarly, the superdifferential of $$g : C \to \mathbb {R}\cup \{-\infty \}$$ at x is the set

The elements are called supergradients.

In other words, the subdifferential is the set of slopes of tangent functionals of f at x that are nowhere above the graph of f, i.e., below-supporting tangent functionals. Similarly, the superdifferential consists of the set of slopes of above-supporting tangent functionals. Note, that for $$f \in \Gamma _0(B^*_w)$$ , C = B∗ with the weak-∗ topology, and thus $${\overline {\partial }} f[x] \subset B$$ .

Important properties of the subdifferential are summarized in the following proposition, whose proof is so easy we skip it:

Proposition 5.4

Suppose f ∈ Γ₀(B), $$g = f^\wedge \in -\Gamma _0(B_w^*)$$ (such that f = g∨) form a skew-conjugate pair of functionals. Then the following hold:

1.

The subdifferential $${ \underline {\partial }} f : B \to 2^{B^*}$$ is a monotone operator, i.e., for all x₁, x₂ ∈ B, and for all $$y_i \in { \underline {\partial }} f[x_i] \subset B^*$$ ,

(5.8)

The superdifferential $${\overline {\partial }} g : C \to 2^{C^*}$$ is similarly monotone, i.e., for all x₁, x₂ ∈ C, and for all $$y_i \in {\overline {\partial }} g[x_i]\subset C^*$$ ,

(5.9)

If f (g) is additionally strictly convex (concave), then Eq.(5.8) [(5.9)] holds strictly if x₁ ≠ x₂.

2.

Let x ∈ B and y ∈ B∗be given. Then Fenchel’s inequality holds,

3.

The following are equivalent:

a.

g[y] − f[x] = 〈y, x〉.

b.

$$-y \in { \underline {\partial }} f[x]$$ .

c.

$$x \in {\overline {\partial }} g[y]$$ .

d.

$$f[x] + \langle {y,x}\rangle \leqslant f[x'] + \langle {y,x'}\rangle $$ for all x′∈ B.

e.

$$g[y] - \langle {y,x}\rangle \geqslant g[y'] - \langle {y',x}\rangle $$ for all y′∈ B∗.

Proof

Easy exercise. □

The equivalence of 3b and 3c is particularly important, showing that two optimization problems are equivalent. Applied to exact DFT, we obtain the equivalence

giving the critical point conditions of Eqs. (5.5a) and (5.5b). We also have the following result, which relates the optimality condition to ground states of density operator form [10]:

Proposition 5.5

Let $$\rho \in \mathcal {D}^N$$ and v ∈ X∗be given. Define $$G[v] \subset \mathcal {D}^N_{\mathit{\text{op}}}$$ as the set of minimizers for Eq.(5.7).

1.

upper G left bracket v right bracket equals c o left brace vertical bar psi right angle bracket left angle bracket psi vertical bar vertical bar psi element of script upper W Subscript upper N Baseline comma upper E left bracket v right bracket equals left angle bracket psi comma ModifyingAbove upper H With caret left bracket v right bracket psi right angle bracket right brace ), the convex hull of pure-state ground-state density operators.

2.

$$E[v] = F[\rho ] + \int v\rho $$ if and only if there is a γ ∈ G[v], with γ↦ρ.

Proof

Proof of 1: Clearly the stated convex hull is a subset of G[v], since for gamma equals sigma summation Underscript k Endscripts lamda Subscript k Baseline vertical bar psi Subscript k Baseline right angle bracket left angle bracket psi Subscript k Baseline vertical bar ) with all the ψk ground-states, $${\mathrm {Tr}}(\widehat {H}[v]\gamma ) = E[v]$$ . On the other hand, assume that γ is not in this convex hull. Then for at least one ψk in its decomposition, $$\langle {\psi _k,\widehat {H}[v]\psi _k}\rangle &gt; E[v]$$ , and $${\mathrm {Tr}}(\widehat {H}[v]\gamma ) &gt; E[v]$$ . Proof of 2: If: If G[v] is empty, then $$E[v] &lt; F[\rho _\gamma ] + \int \rho _\gamma v$$ for any γ. Let γ ∈ G[v], γ↦ρ. Then $$E[v] = {\mathrm {Tr}}(\widehat {H}[v]\gamma ) = {\mathrm {Tr}}(\widehat {H}[0]\gamma ) + \int \rho v$$ . Now, by definition $$F[\rho ] \leqslant {\mathrm {Tr}}(\widehat {H}[0]\gamma )$$ , but clearly equality must hold, otherwise we obtain a contradiction. Thus $$E[v] = F[\rho ] + \int \rho v$$ . Only if: Since F is expectation-valued (Theorem 5.8), there is a γ↦ρ such that $$F[\rho ] = {\mathrm {Tr}}(\widehat {H}[0]\gamma )$$ , and hence $$E[v] = {\mathrm {Tr}}(\widehat {H}[0]\gamma ) + \int \rho v = {\mathrm {Tr}}(\widehat {H}[v]\gamma )$$ . Thus, γ ∈ G[v]. □

5.2.5 Regularity of E and F

We next state some regularity results for convex functions in Γ₀(B), with B being a Banach space throughout this section.

Definition 5.10 (Gâteaux and Fréchet Differentiability)

A proper functional $$f : B \to \mathbb {R} \cup \{+\infty \}$$ is called Gâteaux differentiable at a point $$x\in \operatorname {dom}(f)$$ if, for all h in B, the directional derivative $$f'(x;h) = \lim _{t\to 0^+} t^{-1}[f(x+th)-f(x)]$$ exists, such that f′(x;h) = 〈∇f(x), h〉 for some ∇f(x) ∈ B∗ called the Gâteaux derivative. If additionally

as ∥h∥→ 0, then f is said to be Fréchet differentiable at x.

Note that we require the directional derivative to be a continuous linear functional. Some authors do not require the directional derivative to be continuous or even not linear in the definition of Gâteaux differentiability.

Theorem 5.11

Let f ∈ Γ₀(B), and x ∈ B. If f is Gâteaux differentiable at x then $${ \underline {\partial }} f(x) = \{ g\}$$ , a singleton, where g = ∇f(x₀), the Gâteaux derivative. Conversely, if f is continuous at x₀and $${ \underline {\partial }} f(x_0)$$ is a singleton, then f is Gâteaux differentiable at x₀, and $$\{\nabla f(x_0)\} = { \underline {\partial }} f(x_0)$$ .

Proof

See Proposition 2.40 in Ref. [2]. □

Remark 5.12

Note the continuity requirement in the converse statement. The condition that the subgradient must be a singleton is not sufficient to guarantee Gâteaux differentiability.

The following theorem demonstrates that convexity together with local boundedness above is quite a strong assumption on an f ∈ Γ₀(B).

Theorem 5.13

Suppose f ∈ Γ₀(B), with B a Banach space.

1.

If f is locally bounded above near x ∈ B, then f is locally Lipschitz continuous near x, and $${ \underline {\partial }} f(x)$$ is nonempty.

2.

If f is defined on a convex open set C ⊂ B, and is locally bounded above near some x ∈ C, then f is locally Lipschitz near all of x ∈ C.

Proof

Proof of 1: Let x ∈ B be given, and let B be the closed ball of radius δ > 0 around x. Suppose $$f(x)\leqslant M$$ in B. Let x + h ∈ B be arbitrary. Since x = (x + h)∕2 + (x − h)∕2, convexity of f gives $$f(x) \leqslant f(x+h)/2 + f(x - h)/2$$ . Thus

which implies

Let B′⊂ B be a slightly smaller concentric ball of radius δ − 𝜖. Let y₁, y₂ ∈ B′. Consider the point

Now $$\|z - x\| \leqslant \delta $$ so that x ∈ B, and y₂ lies in the open interval (y₁, z). Explicitly, y₂ is given by the convex combination

Convexity of f now gives

which after rearrangement gives

Repeating the argument with y₁ and y₂ interchanged gives the desired Lipschitz continuity.

The existence of subgradients near x is a consequence of the geometric form of the Hahn–Banach theorem, see, e.g., Theorem 1.36 in Ref. [2]. For the complete proof, see Proposition 2.36 in Ref [2].

Proof of 2: Let B be a ball of radius δ around x ∈ C on which f is locally bounded above by a constant M. Let y ∈ C. There exists a z ∈ C such that y = λz + (1 − λ)x for some λ ∈ [0, 1]. Now, it is straightforward to see, that for all x′∈ C such that $$\|x' - y\| \leqslant (1-\lambda )\delta $$ , $$f(x') \leqslant \max \{ M, f(z) \}$$ . Thus, f is locally bounded above near y, and by (1), locally Lipschitz near y. Since y was arbitrary, we are done. □

We now discuss the behavior of the functionals $$E \in -\Gamma _0(X^*_w)$$ and F ∈ Γ₀(X) in terms of classical differentiability and subdifferentials. Note that E is upper semicontinuous in the weak-∗ topology on X∗, and indeed by Barbu and Precupanu [2, Theorem 2.16] it is continuous in the stronger norm topology on X∗ since E is everywhere defined and clearly upper semicontinuous as the infimum of a nonempty family of affine functions. However, we can say even more:

Theorem 5.14

1.

The map $$E : X^* \to \mathbb {R}$$ is locally Lipschitz continuous.

2.

Suppose v ∈ X∗. Then E is Gâteaux differentiable at v if and only if $$\widehat {H}[v]$$ has a smallest eigenvalue, and all normalized eigenvectors belonging to this eigenvalue share the same density.

Proof

Proof of 1: E is everywhere finite. In order to apply Theorem 5.13, we need to show that E is locally bounded above at some point in X∗, and we choose v = 0. The proof is adapted from Ref. [16].

Let $$\|v\|{ }_{X^*} &lt; L/6$$ , where L is the constant in the Sobolev embedding of $$H^1(\mathbb {R}^3)$$ in $$L^6(\mathbb {R}^3)$$ ( $$\|\nabla f\|{ }_2^2 \geqslant L \|f\|{ }_6^2$$ ). Write v = u + w with u ∈ L3∕2 and w ∈ L∞, and note that ∥u∥3∕2 + ∥w∥∞ < L∕6. Using the fact that the two-electron repulsion operator is positive, we get

Using Eq. (5.3), we get

where $$g = \int (2G)^{-1} |\nabla G|{ }^2 &lt; +\infty $$ , $$G(\mathbf {r}) = \exp (-|\mathbf {r}|{ }^2)$$ . Hence, E is locally bounded above at 0 ∈ X∗.

Proof of 2: Since E is continuous, Theorem 5.11 tells us that E is Gâteaux differentiable at v ∈ X∗ if and only if the superdifferential is a singleton. It follows from Proposition 5.5 that $${\overline {\partial }} E[v]$$ is the convex hull of all densities ρψ of all ground-state wavefunctions of $$\widehat {H}[v]$$ . Thus, the superdifferential is a singleton if and only if all ground-state densities are the same. □

The universal functional F is quite badly behaved. Our discussion is mostly based on Ref. [13], in which many more details can be found. Since F is only defined on elements ρ ∈ X for which $$\int \rho = N$$ , it is clear that F cannot be Gâteaux differentiable, since any change in the particle number $$\int \rho $$ would give infinities. On the other hand, it could be differentiable in a more restricted sense, e.g., we could consider F as a function on the mentioned affine space and study directional derivatives, or even on the smaller space $$\operatorname {aff}\operatorname {dom} F$$ (the affine hull of the effective domain). The following discussion shows that F is singular in these cases, too.

Theorem 5.15

Let $$X_N = \{\rho \in X \mid \int \rho = N \}$$ , an affine closed space of codimension 1. Let $$X_N^+ \subset X_N$$ be the subset of those elements that satisfy $$\rho \geqslant 0$$ a.e.

1.

The map $$F : X \to \mathbb {R}\cup \{+\infty \}$$ has effective domain $$\mathcal {D}^N$$ , i.e., F[ρ] < +∞ if and only if $$\rho \in \mathcal {D}^N$$ .

2.

$$\mathcal {D}^N$$ is dense in $$X_N^+$$ .

3.

$$\operatorname {aff} \mathcal {D}^N$$ has empty algebraic interior in $$\operatorname {aff} X_N^+ = X_N$$ . The algebraic interior of a subset A ⊂ X₊consists of the points ρ ∈ A such that, for every line ℓ ⊂ XNthrough ρ, ℓ ∩ A contains a line segment with ρ in its interior.

Proof

Proof of 1: The domain of FDM is the convex hull of the domain of FLL, which is $$\mathcal {D}^N$$ , a convex set.

Proof of 2: For any $$\sigma \in X_N^+$$ , we must find $$\rho _\epsilon \in \mathcal {D}^N$$ such that ∥ρ𝜖 − σ∥→ 0 as 𝜖 → 0. This can be done using standard mollification arguments: Let $$g_\epsilon \in \mathcal {C}^\infty _c(\mathbb {R}^3)$$ be a mollifier, and define φ = ρ1∕2∕N1∕2, ψ𝜖 = g𝜖 ∗ φ, φ𝜖 = ψ𝜖∕∥ψ𝜖∥, $$\rho _\epsilon = N \varphi _\epsilon ^2$$ . Now, φ𝜖 ∈ H¹, $$\|\varphi _\epsilon \|{ }^2_2 = 1$$ , and moreover $$\varphi _\epsilon \geqslant 0$$ a.e., so $$\rho _\epsilon \geqslant 0$$ a.e., and hence $$\rho _\epsilon \in \mathcal {D}^N$$ . Furthermore, φ𝜖 → φ in H¹, and $$\|\rho _\epsilon -\rho \|{ }_3 \leqslant C \|\varphi _\epsilon - \varphi \|{ }^2_{H^1}$$ by the same Sobolev embedding used in Theorem 5.2. Similarly, ∥ρ𝜖 − ρ∥₁ → 0.

Proof of 3, adapted from Ref. [13]: Let $$\rho _0 \in \mathcal {D}^N$$ be given. We need to find a line segment $$\{\rho _0 + s\delta \rho \mid s \in [0,1]\} \in \operatorname {aff} \mathcal {D}^N$$ such that $$\rho _0 + s\delta \rho \notin \mathcal {D}^N$$ for any s > 0.

Let $$\sigma \in \mathcal {C}^\infty (\mathbb {R}^3)$$ be such that σ(r) ∈ [0, 1], σ(r) = 0 outside the ball around the origin with unit volume, $$\sigma (\mathbf {r}) \geqslant 1/2$$ inside the ball around the origin with volume 1∕2 (implying that $$\int \sigma \geqslant 1/4$$ ). Furthermore, we require that σ1∕2 ∈ H¹. Find a sequence Bn of balls of volume μ(Bn) = 2−n with centers rn satisfying $$|{\mathbf {r}}_n - {\mathbf {r}}_m| \geqslant 1$$ for all n, m. Additionally we require that the measure of $$\{\rho _0 \geqslant 2^{-3n}\}\cap B_n \leqslant \mu (B_n)/4 = 2^{-n-2}$$ (by taking a subsequence if necessary). Such a sequence exists, since otherwise ρ₀ cannot have a finite integral.

We now define

where τ is smooth, nonnegative and with support that does not intersect any of the Bn, such that $$\int \delta \rho = 0$$ . Moreover, we require that τ1∕2 ∈ H¹. Now, $$\delta \rho \in \operatorname {aff}\mathcal {D}^N$$ .

On any given Bn, there is a region (in the inner part) of positive measure where

Clearly, for any given s > 0 we can take n sufficiently large so that these values are negative. Thus, $$\rho \notin \mathcal {D}^N$$ for any s > 0. □

Remark 5.16

The construction of the direction δρ that immediately exits $$\mathcal {D}^N$$ exploits the requirement that $$\rho \geqslant 0$$ almost everywhere, and thus a pathology of the domain as opposed to the behavior of Fon the domain. It is a fact that the restriction of F to $$\mathcal {D}^N$$ (with the X-topology) is everywhere discontinuous in the sense that we can construct a sequence ρn in $$\mathcal {D}^N$$ which is X-convergent to some $$\rho \in X_N^+$$ , but for which F[ρn] → +∞. Such a construction is outlined in Ref. [13]. This also indicates that the topology of X is not really that well suited for DFT, since unlike F, the chosen topology on X is insensitive to density gradients.

5.2.6 The Conjecture of Hohenberg and Kohn

The last discussion in this section pertains to the structure of $${ \underline {\partial }} F[\rho ]$$ , and this is the domain of the Hohenberg–Kohn theorem [7, 16, Theorem 3.2]: That the external potential v is a unique function of the density ρ, up to a constant shift. This is a very appealing notion, because if ρ determines v, then it determines also $$\widehat {H}[v]$$ and hence the ground-state wavefunction ψ, and hence all physical observables as functions Ω[ρ]. The system density is elevated to a basic variable, replacing the wavefunction as the state parameter of the quantum mechanics of N-electron systems. Unfortunately, the Hohenberg–Kohn theorem as originally stated is a conjecture, since its proof has one step which is not rigorous: One divides the Schrödinger equation by ψ pointwise, but this requires the unique continuation property, see [16, Theorem 3.2]. At the time of writing it is still a partially open question if the Hohenberg–Kohn conjecture is true for potentials v ∈ X∗, although Garrigue has established the unique continuation property for potentials $$v \in L^p_{\mathrm {loc}}(\mathbb {R}^3)$$ with p > 2, which include the Coulomb potentials. See also the discussion by Lammert [15].

Conjecture 5.17 (Hohenberg–Kohn)

The density $$\rho \in \mathcal {D}^N$$ determines the potential v ∈ X∗up to an additive constant, i.e., we have either

or

whenever there are no potentials for which $$\rho \in \mathcal {D}^N$$ is a ground-state density.

The concept of v-representability has received a lot of attention in the DFT literature.

Definition 5.18 (Ensemble v-Representability)

We say that ρ ∈ X is (ensemble)v-representable if there is a v ∈ X∗ such that ρ = ργ for some γ ∈ G[v], i.e., that γ is a ground-state density operator of v. Equivalently,

or, $$-v\in { \underline {\partial }} F[\rho ]$$ , or $$\rho \in {\overline {\partial }} E[v]$$ . The set of (ensemble) v-representable densities is thus $$\mathcal {B}_N := \operatorname {dom}({ \underline {\partial }} F)$$ (defined as those ρ ∈ X for which $${ \underline {\partial }} F[\rho ]\neq \emptyset $$ ).

Remark 5.19

The concept of (ensemble) v-representability introduced here is slightly less general than the one used in the classical DFT literature. Here, ρ ∈ L¹ is (ensemble) v-representable if it is the ground-state density of “some potential” $$v : \mathbb {R}^3\to \mathbb {R}$$ , with the function space otherwise unspecified. For example, a gaussian density is the ground-state density of a harmonic potential (for N = 1), but this potential is not in X∗. Lammert [14] distinguishes between (ensemble) v-representability and (ensemble) X∗-representability, which is identical to our concept.

One of the classical problems of DFT is to characterize the set $$\mathcal {B}_N$$ , as this is the effective domain of the classical Hohenberg–Kohn functional FHK, defined by restricting the effective domain of F to $$\mathcal {B}_N$$ . The motivation for this study has been that since one wishes to differentiate FHK, the set $$\mathcal {B}_N$$ needs to be sufficiently well-behaved. In particular, it needs to have an algebraic interior for directional derivatives to make sense. Since we have $$F \leqslant F_{\text{HK}}$$ and in particular $$\operatorname {dom}(F_{\text{HK}})\subset \operatorname {dom}(F)$$ , the algebraic interior of $$\mathcal {B}_N$$ must be empty. However, we can say the following based on the Brøndsted–Rockafellar theorem [1, Theorem 16.45], which implies that for f ∈ Γ₀(B) (with B a Banach space), $$\operatorname {dom}( \underline {\partial } f)$$ is dense in $$\operatorname {dom}(f)$$ .

Proposition 5.6

$$\mathcal {B}_N$$ is dense in $$\mathcal {D}^N$$ .

Proof

Simply note that $$\mathcal {B}_N = \operatorname {dom}({ \underline {\partial }} F)$$ and $$\mathcal {D}^N = \operatorname {dom}(F)$$ . □

The Hohenberg–Kohn conjecture is often taken to be the foundation of DFT in the sense that it implies the existence of a universal Hohenberg–Kohn functional FHK, but the present author considers this a red herring: Our developments so far have not relied on it, and there is actually not much extra to gain from elevating the conjecture to a theorem, since the mapping ρ↦v would be very ill-behaved, see the discussion by Lammert in Ref. [13]. First, not all v ∈ X∗ have ground states. Second, not all $$\rho \in \mathcal {D}^N$$ can be ground-state densities (“v-representable”), and we have little knowledge of how the one-dimensional affine space $${ \underline {\partial }} F[\rho ]$$ changes with ρ.

5.2.7 Some Remarks on Exact DFT

The previous section sets up exact DFT as a convex optimization problem, demonstrating a symmetry between the ground-state energy map E[v] and the “universal” functional F[ρ]. The framework differs significantly from the “traditional” treatment that was initiated by the publications of Hohenberg, Kohn, and Sham [7, 11].

The framework has both strengths and weaknesses from the point of view of formulating the quantum physics of N-electron systems in external fields. First, it is a mathematically rigorous formulation of exact DFT that pinpoints some of the deficiencies of the “traditional” formulation of DFT, in terms of a well-established body of results.

Second, the potential space X∗ feels both unnecessarily large and too small at the same time, containing not only the physical Coulomb potentials, but also a host of “wild” potentials that would never appear in actual physical problems, while lacking some obvious interesting potentials, like the harmonic oscillator. At the same time, the constrained-search formula (5.4) is valid for the latter, which pose no particular difficulties for quantum theory. Actually, Eq. (5.4) is valid for much more general potentials. For example, for v = v₊ − v₋∈ X∗ with v±∈ X∗ almost everywhere positive, one may modify v₊ or v₀ (but not both) to arbitrary elements of $$L^1_{\mathrm {loc}}(\mathbb {R}^3)$$ (even if we may not have a link to a self-adjoint $$\widehat {H}[v]$$ in such cases). It seems fortuitous that X∗ happens to contain the most interesting potentials, namely the Coulomb potentials, as X is chosen for its capacity to hold $$\mathcal {D}^N$$ .

Third, the usual norm topology on X is not so suitable for DFT. It seems difficult to introduce a “well-behaved” Banach space of densities with a dual that contains correspondingly “mostly” interesting potentials. Thus, the standard convex analytic setting could be refined to allow for a better description of exact DFT.

Finally, it seems hard to connect the above treatment of exact DFT with the development of density-functional approximations. Indeed, it is not clear that the existing functional approximations are approximations to FDM (and one may suspect that they are not).

At this point, it is worthwhile to mention alternative formulations of DFT, in particular the coarse-grained formulation of Lammert [12, 14]. Coarse-graining can be motivated by the fact that (a) experimental resolution is not infinite, and (b) the nucleus is not a point particle, so that the singular Coulomb potential is not even exact. The space $$\mathbb {R}^3$$ is covered by a disjoint set of cells of uniformly bounded volume, such as a rectangular grid of uniform cells, and one considers equivalence classes ρ of those $$\rho \in \mathcal {D}^N$$ that have the same average in each cell. The potentials are taken to be constant over each cell. In this setting, every strictly positive ρ is shown to be ensemble v-representable. Moreover, the universal functional becomes much more well-behaved, in particular it is Gâteaux differentiable. Furthermore, Lammert demonstrates that certain limits conforming with Lieb’s theory are obtained as the resolution is increased. In summary, Lammert’s coarse-grained DFT represents a regularization alternative to Moreau–Yosida regularization which we study in Sect. 5.4

5.2.8 Box Truncated Exact DFT

One basic problem with the spaces X and X∗ is that they are nonreflexive, and Moreau–Yosida regularization has the most powerful effect in a Hilbert space setting. We therefore consider the N-electron problem in a open, connected and bounded domain $$\Omega \subset \mathbb {R}^3$$ , assuming Dirichlet boundary conditions on the Schrödinger equation. All the results from exact DFT carry over to this situation. On the other hand, the finite domain allows some simplifications and stronger statements. In particular, the long-range part of Coulomb potentials, being the source of the nonreflexiveness, disappear. In fact, we may now include harmonic potentials and other unbounded potentials in the class of external potentials. Moreover, every Hamiltonian $$\widehat {H}[v]$$ will have a ground state.

We begin with a simple lemma.

Lemma 5.20

Let X( Ω) := L¹( Ω) ∩ L³( Ω). Then X( Ω) ⊂ L²( Ω), for Ω bounded or unbounded, and also L²( Ω) ⊂ X( Ω)′. For Ω bounded, we have L²( Ω) ⊂ L¹( Ω), with continuous embedding. Furthermore, L²( Ω) ⊂ X( Ω)∗with continuous embedding.

Proof

Let u ∈ X( Ω), i.e., ∥u∥₁ and ∥u∥₃ are both finite.

Thus u ∈ L²( Ω). It is a standard fact that Lq( Ω) ⊂ Lp( Ω) for $$1 \leqslant p &lt; q &lt; +\infty $$ with continuous embedding for bounded Ω, proven by a simple application of Hölder’s inequality. In particular, L²( Ω) ⊂ L¹( Ω). Moreover, L²( Ω) ⊂ L3∕2( Ω) ⊂ X( Ω)∗, and since $$\|u\|{ }_{3/2} \leqslant \|u\|{ }_{X^*(\Omega )}$$ , the embedding is continuous. □

The significance of Lemma 5.20 is that it makes sense to consider FDM as a function of ρ ∈ L²( Ω), since $$\mathcal {D}^N(\Omega ) \subset L^2(\Omega )$$ (with an obvious definition of $$\mathcal {D}^N(\Omega )$$ ):

Proposition 5.7

The functional $$F_{\mathrm {DM}} : L^2(\Omega ) \to \mathbb {R}\cup \{+\infty \}$$ is lower semicontinuous.

Proof

Suppose ρn → ρ in L²( Ω). Then ρn → ρ in L¹( Ω) by Lemma 5.20, and $$F[\rho ]\leqslant \liminf _n F_{\mathrm {DM}}[\rho _n]$$ by lower semicontinuity of $$F_{\mathrm {DM}}: L^1(\Omega )\to \mathbb {R}\cup \{+\infty \}$$ . □

The second consequence of Lemma 5.20 is that it makes sense to consider the ground-state energy of the operator $$\widehat {H}[v]$$ for v ∈ L²( Ω), i.e., the map $$E : L^2(\Omega ) \to \mathbb {R}$$ is meaningful. It is also readily seen that Coulomb potentials are in L²( Ω). We summarize this as a theorem:

Theorem 5.21

For $$\Omega \subset \mathbb {R}^3$$ bounded, $$E : L^2(\Omega )\to \mathbb {R}$$ and $$F_{\mathrm {DM}} : L^2(\Omega )\to \mathbb {R}\cup \{+\infty \}$$ form a skew-conjugate pair of functionals. Moreover, for any v ∈ L²( Ω) there exists a ρ ∈ L²( Ω) such that E[v] = FDM[ρ] + 〈v, ρ〉₂(i.e., $${\overline {\partial }} E[v] \neq \emptyset $$ ).

Proof

We only need to prove the existence of a ground-state density for any v ∈ L²( Ω). For v ∈ L²( Ω), the self-adjoint N-electron Hamiltonian associated with the Hamiltonian quadratic form has domain $$H^2_{N,0}(\Omega )$$ [8, Theorem 1]. The Rellich–Kondrachov theorem implies that this Hamiltonian has a compact resolvent, and thus a purely discrete spectrum. In particular, the ground-state energy along with a ground-state eigenvector $$\psi \in H^1_{N,0}(\Omega )$$ exists for any external potential v ∈ L²( Ω), and thus ρψ is a minimizer for E[v] =infρFDM[ρ] + 〈v, ρ〉₂. □

One may ask, why not use the unbounded domain $$\Omega = \mathbb {R}^3$$ when considering L²( Ω) as the density and potential space? In this case convergence in L² does not imply L¹ convergence, and one opens up the possibility that FDM is not lower semicontinuous, so that the Lieb functional F is strictly different from FDM. This is, at the very least, a conceptual problem. Indeed, from Ref [10] we have the following:

Theorem 5.22

Let $$E : L^2(\mathbb {R}^3) \to \mathbb {R}$$ be defined by

and let F = E∨.

1.

$$E[v] \leqslant 0$$ . If $$v\geqslant 0$$ almost everywhere, then E[v] = 0.

2.

F[0] = 0, and hence FDMis not lower semicontinuous.

3.

$$0 \in \underline {\partial } F[0]$$ and $$0 \in \overline {\partial } E[0]$$ , but there are no ground-state wavefunctions for the Hamiltonian $$\widehat {H}[0]$$ .

Proof

Proof of 1: Writing v = v₊ − v₋, with $$v_{\pm } \in L^2(\mathbb {R}^3)$$ almost everywhere positive, we obtain

It is therefore sufficient to show that $$E[v_+] \leqslant 0$$ .

Let $$v \geqslant 0$$ almost everywhere. Let λ > 0 be arbitrary and let Ωk,λ with $$k\in \mathbb {N}$$ be disjoint cubes of side length λ such that $$\cup _k \Omega _{k,\lambda } = \mathbb {R}^3$$ . Each Ωk,λ can be obtained by translation of Ω1,λ. Since $$v\in L^2(\mathbb {R}^3)$$ ,

implying that $$\int _{\Omega _{k,\lambda }} \!\! v(\mathbf r)^2\,\mathrm d \mathbf r \to 0$$ as k →∞. Let $$\psi \in \mathcal {W}_N$$ have smooth components with support in $$\Omega _{1,1}^{N}$$ so that $$\rho _\psi \in \mathcal {C}^\infty _c(\mathbb {R}^3)$$ has support contained in Ω1,1. By translating ψ (denoting the result by ψk), we can ensure that the support of ψk is inside $$\Omega _{k,1}^N$$ and

independently of k. We obtain

where we have used the fact that

as k →∞. We now increase the size of the boxes Ωk,λ by varying λ > 1. By dilating ψ in the manner

the support is still inside $$\Omega _{1,\lambda }^N$$ and the density is scaled as ρψ(r) → λ⁻³ρψ(λ⁻¹r). We obtain the scaling

By repeating the above argument for λ = 1 and letting λ →∞, we obtain $$E[v]\leqslant 0$$ . On the other hand, $$E[v] \geqslant 0$$ since the Hamiltonian $$\widehat {H}[v]$$ is positive, yielding E[v] = 0. Proof of 2: FDM[0] = +∞, but F[0] =supvE[v] = 0. Thus $$F[0] = (F_{\mathrm {DM}}^\wedge )^\vee [0] &lt; F_{\mathrm {DM}}[0]$$ , and $$F_{\mathrm {DM}} \notin \Gamma _0(L^2(\mathbb {R}^3))$$ . Proof of 3: Easy. □

5.3 Moreau–Yosida Regularization

In this section we introduce Moreau–Yosida regularization of convex lower-semicontinuous functions over separable Hilbert spaces. The Moreau–Yosida regularization of a convex optimization problem is invertible. Hence, we trade a possibly non-smooth optimization problem for a smooth and indeed quite well-behaved one. The invertibility implies that the solution of the exact problem is connected to the solution of the regularized problem. In this exposition, we present elementary proofs of most statements due to their central nature in this chapter. Most proofs are adapted from the excellent monograph [1] by Bauschke and Combettes, to which the reader is pointed for further details.

5.3.1 The Moreau Envelope

In this section, $$(\mathcal {H},\langle {\cdot ,\cdot }\rangle )$$ is a generic separable real Hilbert space. By the usual identification of $$\mathcal {H}$$ and $$\mathcal {H}^*$$ and the identification of weak and weak-∗ topologies, we have $$\varGamma _0(\mathcal {H}) = \varGamma _0(\mathcal {H}_w^*)$$ . Our central objects of study is the Moreau envelope $${ }^\epsilon \! f \in \Gamma _0(\mathcal {H})$$ of a convex lower semicontinuous function $$f\in \Gamma _0(\mathcal {H})$$ and the associated proximal mapping $$\operatorname {Prox}_{\epsilon f}:\mathcal {H}\to \mathcal {H}$$ , introduced by Moreau in Ref [19]. The name Yosida is also attached to the formalism: In his proof of the Hille–Yosida theorem in Ref. [30] characterizing the generators of strongly continuous one-parameter semigroups, Yosida introduced a certain approximation of the resolvent of maximal monotone operators $$A : \mathcal {H}\to 2^{\mathcal {H}}$$ . Moreau’s Theorem [19, 26] connects the two concepts in the case where $$A = { \underline {\partial }} f$$ , see Remark 5.26.

Definition 5.23 (Moreau Envelope)

Let 𝜖 > 0 be given, and let $$f :\mathcal {H} \to \mathbb {R}\cup \{+\infty \}$$ with nonempty $$\operatorname {dom}(f)$$ . The Moreau envelope $${ }^\epsilon \! f : \mathcal {H} \to \mathbb {R}\cup \{\pm \infty \}$$ is defined by infimal convolution with the function x↦(1∕2𝜖)∥x∥²,

(5.10)

The unique minimizer Eq. (5.10) for 𝜖 = 1 is defined as $$\operatorname {Prox}_{f} x$$ , i.e.,

(5.11)

The map $$\operatorname {Prox}_f$$ is called the proximal mapping, and for general 𝜖 > 0 we have

(5.12)

that is, the unique minimizer is in general $$\operatorname {Prox}_{\epsilon f}x$$ .

Proposition 5.8

Definition5.23makes sense, i.e., a minimizer of Eq.(5.10) exists and is unique.

Proof

The case 𝜖 = 1 defines $$\operatorname {Prox}_f x$$ , and we leave it to the reader to show that the minimizer is $$\operatorname {Prox}_{\epsilon f}$$ in general. Let $$x\in \mathcal {H}$$ be given, and let hx[z] := f[x] + (1∕2𝜖)∥z − x∥². Let $$\{z_n\}\subset \mathcal {H}$$ be a minimizing sequence for the infimal convolution at x, i.e., hx[zn] →𝜖f[x]. Such a sequence exists since $$\operatorname {dom}(f)\neq \emptyset $$ . The sequence must be bounded, since hx[zn] → +∞ whenever ∥zn∥→ +∞. By taking a subsequence if necessary, we can assume that zn converges weakly to some $$z_*\in \mathcal {H}$$ . We note that hx is sequentially weakly lower semicontinuous (any lower semicontinuous convex function is sequentially weakly lower semicontinuous by [1, Theorem 9.1], and thus

Thus, the infimum in the infimal convolution is attained at z∗. To show uniqueness, we use that strictly convex functions have unique minima: Suppose $$z_{\ast }^{\prime }$$ is a different minimizer. Since hx is strictly convex, $$h_x[\lambda z_* + (1-\lambda )z_{\ast }^{\prime }] &lt; \lambda h_x[z_*] + (1-\lambda )h_x[z_{\ast }^{\prime }]$$ for λ ∈ (0, 1). But then $$h_x[\lambda z_* + (1-\lambda ) z_{\ast }^{\prime }] &lt; { }^\epsilon \! f[x]$$ , a contradiction. □

Remark 5.24

In the proof of Proposition 5.8, we used that $$\mathcal {H}$$ is a Hilbert space (more precisely, reflexivity is used) when picking a weakly convergent subsequence. Thus, in the general nonreflexive setting, the proximal mapping may not be well-defined.

5.3.2 The Proximal Mapping

Some important properties of the proximal mapping are the following:

Proposition 5.9

Let $$f \in \Gamma _0(\mathcal {H})$$ and let 𝜖 > 0 be given. The following holds:

1.

$$p = \operatorname {Prox}_{\epsilon f} x \iff $$ for all $$x' \in \mathcal {H}$$ we have

(5.13)

2.

$$p = \operatorname {Prox}_{\epsilon f} x \iff \epsilon ^{-1}(x-p) \in { \underline {\partial }} f[p]\,$$ .

3.

$$\operatorname {Prox}_{\epsilon f} : \mathcal {H} \to \mathcal {H}$$ is firmly nonexpansive [1, Section 4], i.e., for all $$x,x'\in \mathcal {H}$$ ,

In particular $$\operatorname {Prox}_{\epsilon f}$$ and $$\operatorname {Id}-\operatorname {Prox}_{\epsilon f}$$ are both Lipschitz continuous with constant 1.

4.

If $$x \in \operatorname {dom}(f)$$ and 𝜖 → 0⁺, then

Proof

Proof of 1: Suppose $$p = \operatorname {Prox}_{f}x$$ , and let x′ be arbitrary. Set z = λx′ + (1 − λ)p. We have

so that

Hence, for λ > 0,

Since this holds for all λ ∈ (0, 1), we obtain Eq. (5.13) as λ → 0, and substituting 𝜖f for f. The converse statement is easy: We rearrange Eq. (5.13), to get

Since x′ was arbitrary, we conclude that $$p=\operatorname {Prox}_{\epsilon f}x$$ .

Proof of 2: Follows directly from 1 and the definition of the subgradient.

Proof of 3: Assume $$p = \operatorname {Prox}_{\epsilon f} x$$ and $$p'=\operatorname {Prox}_{\epsilon f} x'$$ . From 1 we get

and

Adding these inequalities and rearranging, we get

(5.14)

Subtracting ∥x − x′∥²∕2𝜖 from each side and rearranging yields the equivalent condition of firm nonexpansiveness,

(5.15)

which also shows that $$\operatorname {Prox}_{\epsilon f}$$ and $$\operatorname {Id}-\operatorname {Prox}_{\epsilon f}$$ are both Lipschitz with constant 1.

Proof of 4: It is clear from the definition of the Moreau envelope that $${ }^\epsilon \! f[x] \leqslant f[x]$$ . Let $$\mu = \sup _{\epsilon &gt;0} { }^\epsilon \! f[x] \leqslant f[x] &lt; +\infty $$ , and thus,

Consider the map hx[z] := f[z] + ∥x − z∥²∕2. Then for all 𝜖 ∈ (0, 1), $$\operatorname {Prox}_{\epsilon f} x$$ is in the sublevel set $$\{z \in \mathcal {H} \mid | h_x[z] &lt; \mu \}$$ . This set is bounded. Hence $$\nu := \sup _{\epsilon \in (0,1)} \|\operatorname {Prox}_{\epsilon f} x\| &lt; +\infty $$ . We next use the fact that any $$f\in \Gamma _0(\mathcal {H})$$ possesses a continuous affine minorant [1, Theorem 9.19], i.e., there exist $$\omega \in \mathcal {H}$$ and $$a\in \mathbb {R}$$ such that $$\langle {\omega , \cdot }\rangle + a \leqslant f$$ . Therefore,

Rearranging gives $$\|x - \operatorname {Prox}_{\epsilon f} x\|{ }^2 \leqslant 2\epsilon (\mu + \nu \|\omega \|-a) = O(\epsilon )$$ as 𝜖 → 0⁺. □

The Moreau envelope of a function in $$\Gamma _0(\mathcal {H})$$ has several nice properties, summarized in the following proposition.

Proposition 5.10

Let $$f \in \Gamma _0(\mathcal {H})$$ and let 𝜖 > 0 be given. The following holds:

1.

$${ }^\epsilon \! f \in \Gamma _0(\mathcal {H})$$ , and $$\operatorname {dom}({ }^\epsilon \! f) = \mathcal {H}$$ .

2.

For every δ > 𝜖, and for all $$x\in \mathcal {H}$$ ,

3.

For every $$x \in \mathcal {H}$$ , 𝜖f[x] → f[x] from below as 𝜖 → 0⁺(even if $$x\neq \operatorname {dom}(f)$$ ).

4.

𝜖f is Fréchet differentiable, with derivative

The derivative is Lipschitz continuous with constant 𝜖⁻¹.

Proof

Proof of 1: The domain is already established. We show convexity. With differentiability shown in 3, it follows that $${ }^\epsilon \! f \in \Gamma _0(\mathcal {H})$$ . Let $$h_{x'}[x] = f[x'] + \|x-x'\|{ }^2/2\epsilon $$ , which is convex. Thus, $$h_{x'}[\lambda x_1 + (1-\lambda )x_2] \leqslant \lambda h_{x'}[x_1] + (1-\lambda )h_{x'}[x_2]$$ . Taking the infimum with respect to x′ on both sides yields that 𝜖f is convex.

Proof of 2: Easy.

Proof of 3: As in the proof of Proposition 5.9(4), set μ =sup𝜖>0𝜖f[x]. From 2, $${ }^\epsilon \! f[x] \to \mu \leqslant f[x]$$ from below as 𝜖 → 0⁺. Therefore, we can assume that μ < +∞ and demonstrate that $$\lim _{\epsilon \to 0^+} { }^\epsilon \! f[x] \geqslant f[x]$$ . From Proposition 5.9(4) we get

where we used that f is lower semicontinuous and that $$\operatorname {Prox}_{\epsilon f} x \to x$$ .

Proof of 4: Let $$x,x'\in \mathcal {H}$$ , $$p=\operatorname {Prox}_{\epsilon f}x$$ , $$p'=\operatorname {Prox}_{\epsilon f}x'$$ . Using the definition of $$\operatorname {Prox}_{\epsilon f}$$ and Eq. (5.13), we derive two inequalities:

A similar calculation gives the second bound $${ }^\epsilon \! f[x'] - { }^\epsilon \! f[x] \leqslant \epsilon ^{-1} \langle {x'-x,x'-p'}\rangle $$ . Combining these two inequalities gives

Using Cauchy–Schwarz and the firm nonexpansiveness condition (Proposition 5.9(3)), we obtain

It follows that

which proves that 𝜖f is Fréchet differentiable, with

Since $$\operatorname {Id} - \operatorname {Prox}_{\epsilon f}$$ has Lipschitz constant 1, ∇𝜖f has Lipschitz constant 𝜖⁻¹. □

We obtain a simple, but interesting fact from the Fréchet derivative from Proposition 5.10(4) and the variational characterization of the proximal mapping in Proposition 5.9(2): While $$f \in \Gamma _0(\mathcal {H})$$ may not be differentiable, the gradient of 𝜖f is always a subgradient at $$\operatorname {Prox}_{\epsilon f} x$$ .

Corollary 5.25

Let $$f\in \Gamma _0(\mathcal {H})$$ , and 𝜖 > 0. Then $$ \nabla { }^\epsilon \! f[x] \in { \underline {\partial }} f[\operatorname {Prox}_{\epsilon f} x] $$ .

Remark 5.26

Moreau’s Theorem [19, 26] states that, for $$f \in \Gamma _0(\mathcal {H})$$ ,

is convex and Fréchet differentiable with ∇𝜖f[x] =𝜖A. Here, $${ }^\epsilon \! A = \epsilon ^{-1}(\operatorname {Id} - { }^\epsilon \! J)$$ is the Yosida approximation to the resolvent $${ }^\epsilon \! J = (\operatorname {Id} + \epsilon A)^{-1}$$ of the maximal monotone operator $$A= \underline {\partial } f$$ . Thus, $${ }^\epsilon \! J = \operatorname {Prox}_{\epsilon f}$$ (which is the only statement in Moreau’s Theorem we have not proved), and we also have the compelling identity $${ \underline {\partial }} { }^\epsilon \! f = { }^\epsilon \! { \underline {\partial }} f$$ .

5.3.3 Conjugate of the Moreau Envelope

We next discuss the skew concave conjugate of the Moreau envelope and its properties.

Proposition 5.11

Let $$f \in \Gamma _0(\mathcal {H})$$ , and let 𝜖 > 0. Then

which is strictly (indeed strongly) concave, and

(5.16)

Proof

Let g = (𝜖f)∧, and let $$y \in \mathcal {H}$$ . We first note that

By the definition of the conjugate,

where we have used that if φ[x] = ∥x∥²∕2𝜖, then φ∧[y] = −𝜖∥y∥²∕2, which is left as an exercise.

To establish Eq. (5.16), we appeal to [29, Theorem 5.38], which, in the Hilbert space setting, states that if $$f_1,f_2\in \Gamma _0(\mathcal {H})$$ and there is a point in $$\operatorname {dom}(f_1)\cap \operatorname {dom}(f_2)$$ where f₁ is continuous, then $${ \underline {\partial }}(f_1+f_2)[x] = { \underline {\partial }} f_1[x] + { \underline {\partial }} f_2[x]$$ for every $$x\in \mathcal {H}$$ . (The inclusion $${ \underline {\partial }}f_1[x]+{ \underline {\partial }}f_2[x]\subset { \underline {\partial }}(f_1+f_2)[x]$$ is easy to prove, but the converse inclusion does not hold in general.) Our result is established by noting that $$\varphi ,-f^\wedge \in -\Gamma _0(\mathcal {H})$$ , and that φ is everywhere continuous. □

Remark 5.27

Proposition 5.11 implies that the Moreau–Yosida regularization is lossless, i.e., for any 𝜖 > 0

an explicit formula for the inverse of the regularization.

5.4 Moreau–Yosida Regularized Exact DFT

We apply Moreau–Yosida regularization to exact DFT in a box domain, as outlined in Sect. 5.2. The treatment in this section closely follows that of Ref. [9]. We describe Moreau–Yosida Kohn–Sham (MYKS) theory, and set up a basic self-consistent field (SCF) iteration, an abstract algorithm for the solution of the Kohn–Sham problem. We describe the Moreau–Yosida Kohn–Sham Optimal Damping Algorithm (MYKSODA), and prove a weak convergence result from Ref. [17].

In this section, we assume $$\Omega = [-\ell ,\ell ]^3 \subset \mathbb {R}$$ is a finite box, and we let $$\mathcal {H} = L^2(\Omega )$$ throughout. We omit all specifications of Ω in symbols like $$\mathcal {D}^N(\Omega )$$ for brevity. We let the Moreau–Yosida parameter 𝜖 > 0 be fixed unless otherwise stated.

5.4.1 Regularized Universal and Ground-State Energy Functionals

We consider the Moreau envelope $${ }^\epsilon \! F : \mathcal {H} \to \mathbb {R}$$ and its concave skew conjugate. Thus, the central functionals of this section are

These are related via skew conjugation,

The original infimum and supremum are a minimum and a maximum, since any $$v\in \mathcal {H}$$ has a ground state by Theorem 5.21, and since 𝜖E is strongly concave. The map 𝜖E is defined not as the Moreau envelope of E, but is instead related to the unregularized energy E by an explicit and easy-to-evaluate function of $$v \in \mathcal {H}$$ . By Proposition 5.10, 𝜖F is everywhere Fréchet differentiable with $$\nabla { }^\epsilon \! F[\rho ] = \epsilon ^{-1}(\rho - \operatorname {Prox}_{\epsilon F}\rho ) \in { \underline {\partial }} F[\operatorname {Prox}_{\epsilon F}\rho ]$$ . Thus $$\operatorname {Prox}_{\epsilon F}\rho \in \mathcal {B}_N$$ . The converse is also true, which gives the remarkable fact that $$\mathcal {B}_N$$ is the range of the proximal mapping, so that the v-representability problem, i.e., the problem of characterizing those $$\rho \in \mathcal {D}^N$$ that are ground-state densities of a given $$v\in \mathcal {H}$$ , is traded for the problem of evaluating 𝜖F, since differentiating this function is not an issue.

Proposition 5.12

The proximal mapping $$\operatorname {Prox}_{\epsilon F} : \mathcal {H} \to \mathcal {B}_N$$ is onto.

Proof

That $$\operatorname {Prox}_{\epsilon F}^{-1} = \operatorname {Id} + \epsilon { \underline {\partial }} F$$ comes directly from Proposition 5.9(2). By definition of range and domain, $$\operatorname {ran} A = \operatorname {dom} A^{-1}$$ for any set-valued operator $$A : \mathcal {H} \to 2^{\mathcal {H}}$$ [1, Definition 12.23], it follows immediately that $$\operatorname {ran}( \operatorname {Prox}_{\epsilon F}) = \operatorname {dom} (\operatorname {Id} + { \underline {\partial }}F) = \mathcal {B}_N$$ . □

Having established that $$\operatorname {ran}(\operatorname {Prox}_{\epsilon F}) = \mathcal {B}_N$$ , we ask what is the preimage of some $$\rho _\epsilon \in \mathcal {B}_N$$ ? The Hohenberg–Kohn Conjecture states that $${ \underline {\partial }} F[\rho _0]$$ is unique up to an additive constant. Assuming the conjecture to hold, we obtain an interesting property of the proximal mapping for a fixed 𝜖 > 0.

Proposition 5.13

Let $$\rho _0 \in \mathcal {B}_N$$ be the ground-state density of $$v\in \mathcal {H}$$ (which exists since any v has a ground-state in the box-truncated setting). Assume Conjecture5.17, i.e.,

Then, $$\rho \in \operatorname {Prox}_{\epsilon F}^{-1}\rho _0$$ is unique up to a constant shift, i.e.,

Proof

Any potential for which $$\operatorname {Prox}_{\epsilon F}\rho =\rho _0$$ is a ground state is of the form v = −𝜖⁻¹(ρ − ρ₀) + μ, with $$\mu \in \mathbb {R}$$ , and $$\rho \in \operatorname {Prox}_{\epsilon F}^{-1}[\rho _0]$$ . Rearranging, ρ = −𝜖v + ρ₀ − 𝜖μ, from which the result follows. □

We can view $$\operatorname {Prox}_{\epsilon F}$$ as a kind of nonlinear “projection” that takes a $$\rho \in \mathcal {H}$$ and maps it to an (ensemble) v-representable density $$\rho _\epsilon = \operatorname {Prox}_{\epsilon F}\rho $$ . Moreover, any (ensemble) v-representable density can be reached from some $$\rho \in \mathcal {H}$$ . However, it is not true in general that $$\operatorname {Prox}_{\epsilon F}^{\circ 2} = \operatorname {Prox}_{\epsilon F}$$ , so it is not a true projection.

The fact that $$\operatorname {dom}({ }^{\epsilon } F)=\mathcal {H}$$ may seem like a severe shortcoming of the formalism, as $$\mathcal {H}$$ contains mostly “unphysical densities”. On the other hand, for a given $$\rho \in \mathcal {H}$$ , it is $$\operatorname {Prox}_{\epsilon F}\rho $$ which is the physical density. We therefore call general elements of $$\mathcal {H}$$ “quasidensities”.

Definition 5.28

Let $$\rho \in \mathcal {H}$$ be given. We define the proximal density $$\rho _\epsilon [\rho ] := \operatorname {Prox}_{\epsilon F}\rho $$ , the resolvent of $${ \underline {\partial }} F$$ , and the proximal potential $$v_\epsilon [\rho ] := -{ }^\epsilon \! \underline {\partial } F[\rho ] = - \epsilon ^{-1}(\operatorname {Id} - \operatorname {Prox}_{\epsilon F})\rho $$ , with the Yosida approximation $${ }^\epsilon \! \underline {\partial } F = \nabla { }^\epsilon \! F$$ to the resolvent of $${ \underline {\partial }} F$$ , cf. Remark 5.26.

Remark 5.29

The proximal density and potential satisfies

demonstrating how the proximal mapping generates a v-representable density and a corresponding potential via the Yosida approximation.

The regularized functionals are approximations to the exact functionals in the sense of Proposition 5.10. For example, for every quasidensity $$\rho \in \mathcal {D}^N$$ , 𝜖F[ρ] → F[ρ] from below as 𝜖 → 0⁺. Moreover, from Proposition 5.9(4), we get ρ𝜖 → ρ. On the other hand, if $$\rho \notin \mathcal {D}^N$$ , then 𝜖F[ρ] → +∞.

5.4.2 Regularized Kohn–Sham Theory

Like in traditional Kohn–Sham theory, we set up a fictitious non-interacting system with an effective potential $$v_{\mathrm {eff}} \in \mathcal {H}$$ to be determined in a self-consistent manner. However, whereas in traditional Kohn–Sham the interacting and non-interacting systems are required to have the same physical densities, regularized Kohn–Sham theory requires the systems to have the same quasidensities. The Moreau–Yosida regularization resolves the two major issues with traditional Kohn–Sham theory: First, that ρ needs to be both interacting and non-interacting v-representable, resolved by $$\rho \in \mathcal {H}$$ being v-representable for any interaction strength in the Moreau–Yosida regularized formulation. Second, that F is not at all differentiable, which makes the set-up of the self-consistent field equations non-rigorous, resolved by 𝜖F being continuously Fréchet differentiable by Proposition 5.10(4).

5.4.2.1 Adiabatic Connection

To prepare for Kohn–Sham theory, we briefly discuss the adiabatic connection, which relates the interacting N-electron problem to a non-interacting one. We introduce a connection parameter λ ∈ [0, 1], and consider the modified energy functional on $$\mathcal {W}_N$$ given by

implying that the ground-state energy is a function of λ as well as the potential,

At λ = 0, the problem is “solvable” in the sense that the N-electron Schdrödinger equation becomes separable, a key motivation behind Kohn–Sham theory. Similarly, we obtain a λ-dependent universal functional,

The connection to spectral theory of self-adjoint operators in Sect. 5.2.1 does not depend on choosing λ = 1. Indeed, any $$\lambda \in \mathbb {R}$$ is acceptable. Moreover, the present discussion is valid for both the case $$\Omega = \mathbb {R}^3$$ and Ω a finite box.

Proposition 5.14

Fix v ∈ X∗and $$\rho \in \mathcal {D}^N$$ . The maps from $$\mathbb {R}$$ to $$\mathbb {R}$$ defined by λ↦Eλ[v] and λ↦Fλ[ρ] are everywhere finite, concave, left and right differentiable, and almost everywhere differentiable.

Proof

We need to show concavity of λ↦Eλ[v]. This follows from $$E^\lambda [v] = \inf _{\gamma \in \mathcal {D}^N_{\text{op}}} \{ A_\gamma + \lambda B_\gamma \}$$ with Aγ = Tr((T + V )γ) and Bγ = Tr(Wγ), i.e., a pointwise infimum of a family of affine functions $$\lambda \mapsto {\mathrm {Tr}}(\widehat {H}_0[v] + \lambda {\mathrm {Tr}}(\widehat {W}\gamma )$$ . Concavity of λ↦Fλ[ρ] follows immediately. It is clear that the both the maps are finite for all $$\lambda \in \mathbb {R}$$ . By [29, Theorem 1.6], concave functions are everywhere left and right differentiable on the interior of their domains, and by Theorem 1.8 in the same reference, almost everywhere differentiable. □

One immediate complication that arises in the adiabatic connection is that the set of ensemble v-representable densities may change with λ, i.e., the set-valued map $$\lambda \mapsto \mathcal {B}_N^\lambda \subset \mathcal {D}^N$$ is non-trivial. There are no known results that relate the different $$\mathcal {B}_N^\lambda $$ .

Turning to the Moreau–Yosida regularization in the domain Ω = [−ℓ, ℓ]³, we obtain λ-dependent functionals $${ }^\epsilon \! E^\lambda : \mathcal {H} \to \mathbb {R}$$ and $${ }^\epsilon \! F^\lambda : \mathcal {H} \to \mathbb {R}$$ . Fix $$\widehat {\rho } \in \mathcal {H}$$ , a target density. For any $$\lambda \in \mathbb {R}$$ , there exists a unique proximal potential,

The regularized ground-state energy of that potential is

related to the unregularized energy as

Associated with $$\widehat {\rho }\in \mathcal {H}$$ is also the proximal density $$\rho _\epsilon ^\lambda [\widehat {\rho }] = \operatorname {Prox}_{\epsilon F^\lambda }[\widehat {\rho }]$$ , being the true ground-state density of the proximal potential,

This is the adiabatic connection in the Moreau–Yosida formulation of DFT. However, we leave open the perhaps important question of the regularity properties of the maps $$(\widehat {\rho },\lambda ) \mapsto v^\lambda _\epsilon [\widehat {\rho }]$$ and $$(\widehat {\rho },\lambda ) \mapsto \rho ^\lambda _\epsilon [\widehat {\rho }]$$ .

5.4.2.2 Kohn–Sham Decomposition

Let $$v_{\mathrm {ext}} \in \mathcal {H}$$ be the external potential for which we wish to determine E¹[vext] and its corresponding ground-state density, known to exist by Theorem 5.21. (If the ground state of $$\widehat {H}[v_{\mathrm {ext}}]$$ is degenerate, there will be a convex set of solutions.) By Proposition 5.11, this problem is in a simple manner connected to the Moreau–Yosida regularized problem by Proposition 5.11,

and hence,

We connect this problem at λ = 1 to the non-interacting problem at λ = 0, where now the quasidensity is known, whereas the corresponding potential $$v_{\mathrm {eff}} \in \mathcal {H}$$ is unknown,

We know that the potential $$v_{\mathrm {eff}}\in \mathcal {H}$$ exists, since 𝜖F⁰ is differentiable at ρ. By Proposition 5.10, we have the stationary conditions

(5.17)

(5.18)

We introduce the Hartree-exchange correlation energy𝜖EHxc[ρ] :=𝜖F¹[ρ] −𝜖F⁰[ρ], which is continuously differentiable with Lipschitz continuous gradient 𝜖vHxc[ρ] := ∇𝜖EHxc[ρ], called the Hartree-exchange correlation potential. The condition (5.17) becomes

(5.17’)

The idea now is that it is EHxc, and not 𝜖F¹, which is available, at least as some kind of approximation. Equations (5.18) and (5.17’) form a self-consistent field problem where the density ρ and potential veff are the unknowns.

This motivates the following abstract algorithm:

Algorithm 1: Basic MYKS-SCF iteration scheme

An algorithm of basic M Y K S, S C F iteration scheme in 3 points. 1, choose an initial guess rho naught belongs to bold script H. 2, for i = 0, 1, and so on, 3 iterations are listed. 3, return the necessary density and energy equations as ground state density and ground state energy.

Remark 5.30

In Step 2.1 of Algorithm 1, the problem $$\widetilde {\rho }_{i+1} \in {\overline {\partial }} E^0[v_{\mathrm {eff},i+1}]$$ is equivalent to solving the eigenvalue equation of a one-electron operator $$\widehat {h} = -\nabla ^2/2 + v_{\mathrm {eff},i+1}$$ . By the Rellich–Kondrachov theorem, this operator has domain $$H^2_0(\Omega )$$ , a compact resolvent and hence a purely discrete spectrum. The first N eigenfunctions $$\varphi _k\in H^2_0(\Omega )$$ define a solution $$\widetilde {\rho }_{i+1} = \sum _{k=1}^N |\varphi _k|{ }^2$$ . Any degeneracies in the spectrum lead to a finite number of solutions, the convex hull of which is $${\overline {\partial }} E^0[v_{\mathrm {eff},i+1}]$$ . In particular, the superdifferential is always nonempty, so that the noninteracting problem can always be solved.

Remark 5.31

The formal limit 𝜖 → 0⁺ in Algorithm 1 gives the traditional Kohn–Sham self-consistent field iterations for exact DFT. Of course, the Hartree-exchange correlation potential is not rigorously defined in this case.

5.4.3 Optimal-Damping Algorithm

Simple SCF iterations often suffer from bad convergence properties, with phenomena like asymptotic oscillations. This particular behavior can be traced to the fact that each SCF iteration takes too large a step, so that one “jumps over” a minimum. The optimal-damping algorithm (ODA) for the Hartree–Fock SCF iterations were introduced by Cancès and Le Bris in Refs. [3, 5], in which the “bare” SCF density update is damped by a factor in (0, 1]. In this section, we apply the idea of optimal damping for extended Kohn–Sham iterations from Ref. [4] to modify Algorithm 1, leading to the Moreau–Yosida regularized Kohn–Sham optimal-damping algorithm (MYKSODA). This algorithm was introduced in Ref. [17], where the convergence of the energy to an upper bound, Theorem 5.32, was proven. A stronger convergence proof for a finite-dimensional setting, Theorem 5.33, was introduced in Refs. [20, 21]. We here present the full proof here with some additional details and corrections.

The ODA modification of Algorithm 1 consists of a relaxation of the density step, i.e., not setting $$\rho _{i+1}=\rho ^{\prime }_{i+1}$$ , but instead $$\rho _{i+1} = \rho _i + t_i\rho ^{\prime }_{i+1}$$ for some ti ∈ (0, 1]. In Theorem 5.33, the step lengths ti are actually not constrained, but convergence is still guaranteed.

Algorithm 2: MYKSODA

An algorithm of M Y K S O D A in 3 points. 1, set v subscript e f f, 0 = v subscript e x t and rho naught belongs to partial derivative bar belongs to E power 0 of v subscript e x t. 2, for i = 0, 1, and so on, 4 iterations are listed. 3, return the necessary density and energy equations as ground state density and ground state energy.

The MYKSODA algorithm is weakly convergent in the sense that the energy converges to an upper bound:

Theorem 5.32

In Algorithm 2, the sequence of energy estimates

is strictly descending and hence convergent. Denoting the limit by e[vext], we have

that is, an upper bound property of the computed energy limit.

Proof

It is clear that step 2.2 produces $$\rho ^{\prime }_{i+1}\in \underline {\partial }{ }^\epsilon \! E^0[v_{\mathrm {eff},i+1}]$$ . We need to check that step 2.3 is well-defined. To that end, consider the function G[ρ] :=𝜖F¹[ρ] + 〈vext, ρ〉, which is to be minimized by the iterations. We note that step 2.1 can be rewritten as veff,i+1 + ∇𝜖F⁰[ρi] = vext + ∇𝜖F¹[ρi]. The directional derivative of G at ρ in the direction $$\Delta \rho _i := \rho _{i+1}^{\prime }-\rho _i$$ is $$G'[\rho ;\Delta \rho _i] = \langle {\nabla { }^\epsilon \! F^1[\rho ] + v_{\mathrm {ext}},\rho _{i+1}^{\prime }-\rho _i}\rangle $$ , which gives

If the termination criterion was met at the previous iteration, $$G'[\rho _i;\rho _{i+1}^{\prime }-\rho _i]=0$$ , and step 2.3 will not take place. Otherwise, since $$\rho _i \in \overline {\partial } { }^\epsilon \! E^0[-\nabla { }^\epsilon \! F^0[\rho _i]]$$ (see Remark 5.29), write

This allows us to rewrite the directional derivative as

By monotonicity of the superdifferential (Propositions 5.4 and 1), the bracket is always nonpositive, which by step 2.1 in Algorithm 2 gives

(5.20)

Since now G[ρi] is strictly decreasing and convex in the direction Δρi (t↦G[ρi + t Δρi] is strongly convex at t = 0), it follows that there is a maximum step length ti > 0 such that ei+1 = G[ρi+1] < G[ρi] = ei, and such that Eq. (5.19) holds, i.e., $$G'[\rho _{i+1},\rho _{i+1}^{\prime }-\rho _i] \leqslant 0$$ . Since $$e_i \geqslant \inf _\rho G[\rho ] = { }^\epsilon \! E^1[v_{\mathrm {ext}}]$$ for all i, it follows that the monotonically decreasing sequence ei converges to an upper bound of 𝜖E¹[vext] = E¹[vext] − 𝜖∥vext∥²∕2. □

In the above, the step lengths ti ∈ (0, 1] were not specified. It is clear that any stronger convergence properties of the algorithm depend crucially on these. The final result of this section, paraphrased from Refs. [20, 21], exploits the definition of the Moreau–Yosida regularization at every step. The key idea is the following interpretation of Proposition 5.10(4): At every $$x \in \mathcal {H}$$ , 𝜖f is tangent to the regularization parabola $$h_x(z) := f(\operatorname {Prox}_{\epsilon f} x) + \|z-\operatorname {Prox}_{\epsilon f}x\|{ }^2/(2\epsilon )$$ , i.e., evaluated at x, both have gradient $$\epsilon ^{-1}(x -\operatorname {Prox}_{\epsilon f} x)$$ and their graphs are touching at x. Moreover, by definition $${ }^\epsilon \! f \leqslant h_z$$ . The step sizes ti are now chosen such that ρi+1 minimizes the regularization parabola associated with G[ρi] along the search direction Δρi. However, one must assume that $$\mathcal {H}$$ is finite-dimensional for this to guarantee convergence. A finite-dimensional $$\mathcal {H}$$ arises, for example, when the N-electron Hilbert space is discretized by a finite-dimensional basis generated by a finite set of single-particle functions, and does not change the properties of E or F in any way relevant for the current discussion.

Theorem 5.33

Assume that $$\mathcal {H}$$ is finite-dimensional. Then there exists a sequence of step lengths {ti}ifor Algorithm 2 such that {ρi} has energies ei =𝜖F[ρi] + 〈vext, ρi〉 converging to the exact result𝜖E[vext]. Moreover, {ρi} is the union of convergent subsequences $$\rho _{i_n} \to \sigma \in \mathcal {H}$$ , such that $$\sigma \in \overline {\partial } { }^\epsilon \! E[v_{\mathrm {ext}}]$$ is an exact ground-state quasidensity, and veff,i → vext +𝜖vHxc[σ].

Proof

Let $$p_i := \operatorname {Prox}_{\epsilon F}\rho _i$$ . Rewritten in terms of this proximal point, we have

(5.21)

The regularization parabola associated with G[ρ] is similarly obtained, noting that the linear term 〈vext, ρ〉 of G is easily absorbed,

(5.22)

We note that the global minimum of $$h_{\rho _i}$$ is at p∗ := pi − 𝜖vext, and that $$\nabla h_{\rho _i}[\rho _i] = \nabla G[\rho _i]$$ , by the remarks preceding the theorem. Let f(t) = G[ρi + t Δρi], which by Eq. (5.20) satisfies f′(0) < 0. At all t, $$f(t) \leqslant g(t) := h_{\rho _i}[\rho _i + t\Delta \rho ]$$ . Its derivative is

Define ti > 0 uniquely (and hence ρi+1 = ρi + ti Δρi) by the condition g′(t₀) = 0, which gives

and the equivalent orthogonality condition 〈ρi+1 − p∗, ρi+1 − ρi〉 = 0. Geometrically, ρi+1 − ρi is the orthogonal projection of ρ∗− ρi onto the search line. A brief calculation gives

(5.23)

Let $$m_i := h_{\rho _i}[\rho _{i+1}]$$ be the minimum of the parabola section. We have $$e_{i+1} \leqslant m_i$$ . Subtracting Eq. (5.22) with σ = ρi+1 from Eq. (5.21) and using orthogonality, we obtain

We now show that ρi converges. By assumption, $$\operatorname {dom}(F)$$ is bounded in $$\mathcal {H}$$ . It is easy to see that 𝜖F[ρ] = O(∥ρ∥²) as ∥ρ∥→ +∞. Since $$e_i = { }^\epsilon \! F[\rho _i] + \langle {v_{\mathrm {ext}},\rho _i}\rangle \leqslant e_1$$ for all i, {ρi} (which is not a subset of $$\operatorname {dom}(F)$$ !) is bounded in $$\mathcal {H}$$ . This again implies, by Proposition 5.104, that ∥∇𝜖Fλ(ρi)∥ is bounded, hence {veff,i} is bounded. Equation (5.20) combined with Eq. (5.23) gives

Since ∥ Δρi∥ti → 0, we now have ∥∇𝜖F⁰[ρi] + veff,i∥→ 0 and ∥∇𝜖F[ρi] + vext∥→ 0. The latter implies that vext = −limi→+∞∇𝜖F[ρi]. Let $$\sigma \in \mathcal {H}$$ be an accumulation point of {ρi}, which is guaranteed to exist by the Bolzano–Weierstrass theorem. There is a convergent subsequence $$\rho _{i_n}\to \sigma $$ for which we have $$v_{\mathrm {ext}} = -\lim _{n} \nabla { }^\epsilon \! F[\rho _{i_n}] = -\nabla { }^\epsilon \! F[\sigma ]$$ . But then $$\sigma \in \overline {\partial } { }^\epsilon \! E[v_{\mathrm {ext}}]$$ is a ground-state quasidensity of the exact regularized problem. We next obtain . Since the accumulation point was arbitrary, the sequence {ρi} splits into subsequences converging to different ground-state densities σ, as claimed. □

Remark 5.34

The step length choices in the proof of Theorem 5.33 are not practical (if one can consider the MYKS scheme practical at all), since they require the whereabouts of the proximal point $$\operatorname {Prox}_{\epsilon F}\rho _i$$ . But knowledge of that would make the Moreau–Yosida regularization approach redundant.

5.4.4 Density-Functional Approximations

So far our discussion has pertained to the exact ground-state energy functional $$E^\lambda \in \Gamma _0(\mathcal {H})$$ , which is an abstract setting far from actual numerical calculations. We now make some simple observations that open up the possibility of a rigorous treatment of Kohn–Sham theory for model density functionals [22], including the application and further analysis of the MYKSODA iteration scheme.

The Kohn–Sham decomposition relied on a family $$\mathcal {U} = \{ E^\lambda \in -\Gamma _0(\mathcal {H}) \mid \lambda \in [0,1]\}$$ . This family was such that $$\overline {\partial } E^{\lambda }[v] \neq \emptyset $$ for any $$v\in \mathcal {H}$$ , and generates the family $$\{ { }^\epsilon \! E^\lambda : \mathcal {H}\to \mathbb {R} \mid \lambda \in [0,1] \}$$ and the corresponding Moreau envelopes $$\{ { }^\epsilon \! F^\lambda : \mathcal {H}\to \mathbb {R} \mid \lambda \in [0,1] \}$$ . Moreover, the idea was present that solving for $$\rho \in \overline {\partial } { }^\epsilon \! E^0[v]$$ for arbitrary $$v\in \mathcal {H}$$ , i.e., the noninteracting problem, was in some sense easy compared to solving for $$\rho _\epsilon ^1 \in \overline {\partial } { }^\epsilon \! E^1[v_{\mathrm {ext}}]$$ . A property of the family $$\mathcal {U}$$ that was not used in the Kohn–Sham decomposition was the concavity in the parameter λ ∈ [0, 1].

In Algorithm 1, the basic MYKS-SCF iteration scheme, the existence of solutions $$\rho \in \overline {\partial } { }^\epsilon \! E^0[v]$$ for arbitrary $$v\in \mathcal {H}$$ was essential, but otherwise no additional properties of $$\mathcal {U}$$ were used. In Algorithm 2, MYKSODA, and Theorem 5.32, the same is true.

This motivates the introduction of an approximation, or model, $${ }^\epsilon \! \widetilde {E}_{\mathrm {Hxc}}$$ such that $${ }^\epsilon \! \widetilde {F}^1 = { }^\epsilon \! F^0 + { }^\epsilon \! \widetilde {E}_{\mathrm {Hxc}} \in \Gamma _0(\mathcal {H})$$ is continuously differentiable, replacing the exact Moreau–Yosida regularization in Algorithms 1 and 2. Of course, how to obtain such models is a different matter.

A second approach is to consider the exactλ-dependent Hartree-exchange correlation functional $$E^\lambda _{\mathrm {Hxc}} = F^\lambda -F^0$$ , a functional with domain $$\mathcal {D}^N$$ but of otherwise unknown composition, and introduce a model functional $$\widetilde {E}^\lambda _{\mathrm {Hxc}} : \mathcal {D}^N \to \mathbb {R}$$ and corresponding model universal functionals, viz.,

This family may or may not be convex, but suppose for simplicity that it is, and that $$\widetilde {E}_{\mathrm {Hxc}}^\lambda $$ is in some sense “cheap” but also sufficiently regular as to keep $$\widetilde {F}^\lambda \in \Gamma _0(\mathcal {H})$$ . The family induces a model energy $$\widetilde {E}^\lambda \in -\Gamma _0(\mathcal {H})$$ such that $$\widetilde {E}^0 = E^0$$ , retaining the ease of solution in the non-interacting limit.

The next step is to consider the Moreau–Yosida regularized model Hartree-exchange correlation energy,

The main problem is now to identify how and when $${ }^\epsilon \! \widetilde {E}^\lambda _{\mathrm {Hxc}}$$ can be easily calculated given that it involves a non-trivial convolution, which may negate the ease of evaluation of $$\widetilde {E}^\lambda _{\mathrm {Hxc}}[\rho ]$$ .

5.5 Conclusion

In this chapter, we have reformulated exact DFT as introduced by Lieb in terms of the Moreau–Yosida regularization. Our starting point was the N-electron problem. Not considered were grand-canonical ensembles at finite or zero temperature, which adds some interesting aspects to the formalism [22]. Resulting in a compelling reformulation of DFT, including a rigorous formulation of Kohn–Sham theory, Moreau–Yosida regularization still leaves open the question of how to find functional approximations to the Hartree-exchange correlation energy.

The mathematical understanding of Moreau–Yosida regularized Kohn–Sham theory is in its early stages. For example, Theorem 5.33 requires a finite-dimensional density space. Stronger and more general results are of course highly desired, and could turn Moreau–Yosida Kohn–Sham theory into a practical tool when combined with approximate Hartree-exchange correlation functionals.

In current-density functional theory (CDFT), the ground-state energy E[v, A] of an N-electron system in the external potential v and magnetic potential A is considered, leading to both the density ρ and the current density $$\mathbf {j} : \mathbb {R}^3\to \mathbb {R}^3$$ being variables in the constrained-search functional [28]. This requires a more general setting than that of a Hilbert space of densities. In Ref. [17], Moreau–Yosida regularized DFT was introduced in the abstract setting of a reflexive Banach space of densities, thereby accommodating CDFT and demonstrating that the regularization approach may also be important in other density-functional settings than the standard one considered in this chapter.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_6

6. Thermodynamic Limits of Electronic Systems

David Gontier1  , Jianfeng Lu2   and Christoph Ortner3  

(1)

CEREMADE, Université Paris-Dauphine, PSL University, Paris, France

(2)

Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, NC, USA

(3)

Department of Mathematics, University of British Columbia, Vancouver, Canada

David Gontier (Corresponding author)

Email: gontier@ceremade.dauphine.fr

Jianfeng Lu

Email: jianfeng@math.duke.edu

Christoph Ortner

Email: christoph.ortner@ubc.ca

Abstract

We review thermodynamic limits and scaling limits of electronic structure models for condensed matter. We discuss several mathematical ways to implement these limits in three models of increasing chemical complexity and mathematical difficulty: (1) Thomas–Fermi-like models; (2) Hartree–Fock-like models; and (3) Kohn–Sham density functional theory models.

JL was supported in part by National Science Foundation under grant NSF-DMS1454939 and NSF-CHE2037263

CO was supported by ERC Starting Grant 335120 and EPSRC Grant EP/R043612/1

6.1 Introduction

The goal of thermodynamic limits, as introduced in the 1960’s [23, 39], is to obtain mathematical models for infinite systems of particles. The overarching strategy is to study systems with a finite number of particles (which can be described efficiently by well-posed mathematical models), to let the number of particles go to infinity while filling the space, and to pass to the limit in the governing equations in order to obtain a limit model. The purpose of the present chapter is to review results of this kind in the context of electronic structure models in condensed matter.

Two prototypical applications of thermodynamic limits are (1) to justify models of the energy per unit cell of a homogeneous crystal (infinite periodic system); (2) to obtain models for the formation energy of a crystalline defects without artefacts due to the boundary conditions. In this chapter, we review different models and mathematical methods to treat both of these scenarios. Extensive references will be provided throughout the chapter.

In both cases, one would like to describe an infinite system of electrons in a potential generated by an infinite collection of nuclei at positions $$\mathscr {R} \subset \mathbb {R}^3$$ . In most studies, $$\mathscr {R}$$ is a periodic lattice (we write $$\mathscr {R} = \mathbb {L}$$ in this case), or a perturbation of it, describing for instance a crystal with a defect, or a deformed crystal (there are some studies for amorphous solids, in which case $$\mathscr {R}$$ is a random set [12]). In order to highlight the main ideas of the thermodynamic limit, we restrict ourselves to the simple case of a periodic crystal with one nucleus of charge 1 per unit cell. We denote by ma the charge density of a single nucleus, which we take to be smooth to avoid some technical details: $$m_{\mathrm {a}} \in C^\infty (\mathbb {R}^3)$$ with compact support and $$\int m_{\mathrm {a}} = 1$$ . The total nuclear density of the crystal is then given by

(6.1)

In order to approximate this infinite distribution of charges, we consider a sequence of finite systems that converges to the infinite one: we choose $$\mathscr {R}_N \subset \mathscr {R}$$ a finite subset of size $$| \mathscr {R}_N | = N$$ , and study the finite electronic problem with N electrons, in the external potential generated by N nuclei arranged along $$\mathscr {R}_N$$ . The total nuclear density is

(6.2)

Given this finite distribution of charges, $$m^{\mathrm {nuc}}_N$$ , one formulates a variational problem to equilibrate the electrons,

where the infimum is taken over all admissible states γN representing a system of N electrons, and E describes the energy of finite electronic systems. Usually, γN represents the electron density or density matrix. One then aims to make various statements about the limits of the energy, $$I(\mathscr {R}_N)$$ and the optimal electron variable $$\gamma _N^0$$ . Examples of important questions in this context include:

Does the sequence $$N^{-1} I(\mathscr {R}_N)$$ converge to some limit $$W(\mathscr {R})$$ asN →∞and $$\mathscr {R}_N \to \mathscr {R}$$ ? In this case, $$W(\mathscr {R})$$ would correspond to an average energy per electron or energy per unit volume.

Does the sequence of minimisers $$\gamma _N^0$$ have a limitγ⁰asN →∞? The limiting object would describe an infinite sea of electrons in a crystal. Which equations are satisfied by the limiting object γ⁰?

If $$\mathscr {R} \approx \mathscr {R}^{\prime }$$ , then does the energy difference $$\delta I_N := I(\mathscr {R}_N) - I(\mathscr {R}_N^{\prime })$$ have a limitδI? If $$\mathscr {R}$$ describes a crystal and $$\mathscr {R}^{\prime }$$ the same crystal with a local defect, then δI is the defect formation energy.

In the following sections, we focus on the case where the energy E is given by one of the following three models: the Thomas–Fermi–von Weizsäcker model in Sect. 6.2, the (reduced) Hartree–Fock model in Sect. 6.3, as well as Kohn–Sham density functional theory models, in Sect. 6.4.

In what follows, the energy of an N-electron state γN is denoted by E(γN). The infimum of this energy is $$I_N = \inf _{\gamma _N} E(\gamma _N)$$ , and represents the ground state energy of an N-electron system. The energy per unit electron is WN = N⁻¹IN. For the orbital-free TFW model the N-electron state is given by the electron density ρN.

Remark 6.1

Throughout this review we are technically misapplying the term “thermodynamic limit”, but we do so in a way that is consistent with its usage in the analysis literature. In a strict sense, the thermodynamic limit was introduced to describe many-particle systems in a limit where boundary effects can be neglected, and to employ the law of large numbers, large deviation theory and ergodic theory as a transition from microscopic states to macroscopic states (variational principles, PDEs etc). A key goal of this framework was to model the situation when the corresponding thermodynamic functions (pressure, free energy, susceptibility, magnetisation, etc) can have singularities which appear at the critical value of the intensive parameter (temperature, chemical potential, etc). We refer to [23, 39] for detailed treatments of the subject. The connection between the present review and the classical usage of the term “thermodynamic limit” is the study of the many-particle limit in which boundary and domain size effects can be ignored, however there is no (genuine) statistical mechanics aspect.

6.2 The Thomas–Fermi–von Weizsäcker Model

Thomas–Fermi models describe electronic structure purely in terms of the electron density and electrostatic potential, and can therefore be interpreted as a system of two nonlinear PDEs. In this setting there is a mature theory and general results on the structure of the model and in particular the thermodynamic limit. The original Thomas–Fermi model, while attractive due to its simplicity, does not allow for the existence of molecules [31]. We will therefore focus on the Thomas–Fermi–von Weizsäcker (TFW) model [42]. Our presentation is primarily based on the monograph [14], but also incorporates more recent results [3, 35].

6.2.1 TFW Model for a Cluster

We consider N nuclei at locations $$\mathscr {R}_N$$ and with total charge $$m^{\mathrm {nuc}}_N$$ , see Eq. (6.2). The non-dimensionalised TFW energy, parametrised by $$\mathscr {R}_N$$ as a functional of the electron density ρ, is given by

(6.3)

where the Coulomb quadratic form is defined by

(6.4)

The first two terms of (6.3) represent the kinetic energy, and the third term is the Coulomb energy. This term can further be split into

The first term is the direct term, or Hartree term, and describes the mean-field self-interaction of the electrons. The second is the electron-nuclei Coulomb interaction and the last term is the nuclei-nuclei one. Since we fixed the lattice $$\mathscr {R}$$ beforehand, the last term is constant, and does not play role in the minimisation problem. In addition, cW and cTF are positive physical constants that are irrelevant from a mathematical perspective; hence, for the sake of notational convenience, we set them to cW = cTF = 1.

The charge-neutral electronic ground state is obtained by solving

(6.5)

A direct computation shows that $$\rho \mapsto \int _{\mathbb {R}^3} |\nabla \sqrt {\rho }|{ }^2$$ is convex, which is a key ingredient to obtain the following result (see [2] for the proof).

Proposition 6.1

There exists a unique minimiser ρNof(6.​5). In addition, ρN > 0.

It can then be readily checked, at least formally, that the minimiser satisfies the Euler–Lagrange equation

for some Lagrange multiplier $$\theta _N \in \mathbb {R}$$ associated with the charge neutrality constraint $$\int _{\mathbb {R}^3} \rho = N$$ . It now becomes convenient to make the transformation $$\rho _N = u_N^2$$ , where we may again assume that uN > 0, and to introduce the total electrostatic potential

to obtain the Euler–Lagrange system

(6.6a)

(6.6b)

We have absorbed the Lagrange multiplier θN into the electrostatic potential $$V_N^{\mathrm {tot}}$$ , shifting it by a constant, which in particular implies that we need not have $$V_N^{\mathrm {tot}}({\mathbf {r}}) \to 0$$ as |r|→∞. See [2, 14, 30] for the details of this argument.

In the remainder of our treatment of the TFW model we review results establishing the convergence of the electron ground state $$(u_N, V_{N}^{\mathrm {tot}})$$ as N →∞, as the nuclei configuration $$\mathscr {R}_N$$ grows. To establish this limit, a convenient function space setting is provided by the spaces (we denote by $$B_R({\mathbf {r}}) := \{ {\mathbf {r}}^{\prime } \in \mathbb {R}^3, \ | {\mathbf {r}}^{\prime } - {\mathbf {r}} | &lt; R \}$$ )

6.2.2 Thermodynamic Limit Model

To pass to the thermodynamic limit N →∞ we begin with an infinite collection of (smeared) nuclei at positions $$\mathscr {R} \subset \mathbb {R}^3$$ . Here, $$\mathscr {R}$$ need not be a periodic lattice. Since the energy of an infinite system is not well-defined, the associated electronic ground state cannot be immediately characterised by an analogue of the variational problem (6.5). However, the nonlinear PDE representation (6.6) has a straightforward generalisation. Indeed, let $$m^{\mathrm {nuc}}({\mathbf {r}}) := \sum _{{\mathbf {R}} \in \mathscr {R}} m_{\mathrm {a}}({\mathbf {r}} - {\mathbf {R}})$$ , then it is natural to suppose that the electronic ground state for the nuclei arrangement $$\mathscr {R}$$ is given by ρ = u², where (u, Vtot) solves

(6.7a)

(6.7b)

To justify this model we will establish its well-posedness and show that it indeed arises as the thermodynamic limit of (6.5) (or, equivalently, (6.6)).

To that end, we need to impose restrictions on the configuration $$\mathscr {R}$$ . We assume that $$\mathscr {R}$$ describes roughly uniformly distributed matter, and in particular contains no clusters with arbitrary high densities, and no holes of arbitrary large volume. More precisely, we require that there exist constants c1,2, C1,2 > 0 such that

(6.8)

This condition is equivalent to (H1) and (H2) in [14]. One of the main results of [14] is the well-posedness of (6.7).

Theorem 6.2 (Well-Posedness [14, Thm 6.10])

Under the condition (6.8), there exists a unique pair $$(u, V^{\mathrm {tot}}) \in H^4_{\mathrm {unif}} \times H^2_{\mathrm {unif}}$$ , with $$u \geqslant 0$$ , solving (6.7). Moreover, $$\inf u &gt; 0$$ .

The majority of the monograph [14] is devoted to the proof of Theorem 6.2. Let us recall some key ideas: A crucial observation is that the linear operator

which is a kind of linearisation of (6.6a), is non-negative. This already hints at the existence of a strong stability property. Indeed, adapting this observation (see, for example, the proof of [14, Lemma 5.3]), the following result is shown in [35, Thm. 3.1], closely following variants of the same result in [14, Sec. 5.3] and [3].

Lemma 6.3 (Stability and Uniqueness [35, Thm. 3.1])

Let $$\mathscr {R}, \mathscr {R}_* \subset \mathbb {R}^3$$ satisfy (6.8), let $$m^{\mathrm {nuc}}_1, m^{\mathrm {nuc}}_2$$ be associated nuclear charge densities, and suppose that $$(u, V^{\mathrm {tot}}), (u_*, V^{\mathrm {tot}}_*) \in H^4_{\mathrm {unif}} \times H^2_{\mathrm {unif}}$$ are corresponding solutions to (6.7) with $$\inf u_i &gt; 0$$ . Then, there exist constants C ≥ 0 and α > 0, depending only on maand on the constants in (6.8), such that

(6.9)

Lemma 6.3 immediately implies uniqueness of solutions to (6.7), but it is much stronger in that it also provides a pointwise stability that quantifies the dependence of the local electronic structure on the far-field. We will return to this result in Sect. 6.2.3.

To establish the existence of solutions, we use a thermodynamic limit argument. At the same time, this also justifies the model (6.7). To that end, we specify a sequence of clusters approximating $$\mathscr {R}$$ : let $$\mathscr {R}_N \subset \mathscr {R}$$ and rN ↑∞, c > 0 such that

(6.10)

For each N, Proposition 6.1 yields the existence and uniqueness of an electronic ground state $$(u_N, V^{\mathrm {tot}}_N)$$ solving (6.6).

The stability result stated in Lemma 6.3 already hints at the possibility of uniform a priori estimates on the solutions (uN, φN), and indeed one can prove that

(6.11)

where C depends only on ma and on the constants in (6.8), see [14, Prop. 3.5] for the (involved and technical) details. A key technical step estimating the Lagrange multiplier, which we have hidden, is due to [40]. A summary of the proof, providing also quantitative estimates, can be found in [35, Prop. 6.1].

With the a priori estimate (6.11) in hand, we may extract a subsequence $$(u_{N_j}, V^{\mathrm {tot}}_{N_j}) \rightharpoonup (u, V^{\mathrm {tot}})$$ weakly in $$H^2_{\mathrm {loc}}$$ (we say $$f_j \rightharpoonup f$$ weakly in $$H^k_{\mathrm {loc}}$$ if $$f_j|{ }_D \rightharpoonup f|{ }_D$$ weakly in Hk(D) for all bounded domains D) and it is straightforward to deduce that the limit satisfies the PDE (6.7). Since the limit is unique, it follows that the entire sequence converges. We obtain the following result.

Theorem 6.4 (Convergence)

Let $$\mathscr {R}_N, \mathscr {R}$$ satisfy (6.10) and (6.8), and let $$(u_N, V^{\mathrm {tot}}_N) \in H^4_{\mathrm {loc}} \times H^4_{\mathrm {loc}}$$ and $$(u, V^{\mathrm {tot}}) \in H^4_{\mathrm {loc}} \times H^4_{\mathrm {loc}}$$ be the corresponding solutions of (6.6) and (6.7) respectively. Then

In particular, the convergence is locally uniform.

6.2.3 Discussion

We conclude this section with a series of further remarks about possible extensions and consequences of the results.

(1) Convergence rates:

In Lemma 6.3 the conditions on the second solution u∗ can be weakened. This allows us to prove that “well inside” the approximate domain $$\mathscr {R}_N$$ , the solutions $$(u_N, V^{\mathrm {tot}}_N)$$ and (u, Vtot) are exponentially close. Specifically, in [35, Proposition 4.1] it is shown that there are constants C > 0 and α > 0 independent of $${\mathbf {r}} \in B_{r_N}$$ and $$N \in \mathbb {N}$$ such that

Choosing $$r_N^{\prime } \ll r_N$$ this readily translates into the convergence rate

The same argument also shows that boundary effects decay exponentially into the bulk of the cluster and justifies the common usage of buffer regions in electronic structure calculations.

(2) Surfaces and sheets:

Our assumption (6.8) expressly disallows configurations with large sections of vacuum, in particular surfaces and 2D materials. Indeed, not only the mathematics but also the underlying physics changes in such situations. We refer to [3, 6, 32] for related results that go beyond this limitation.

(3) The Dirac correction:

The Thomas–Fermi–Dirac–von Weizsäcker model adds an additional correction term to the energy functional,

where the additional term can be interpreted as a model for the exchange of energy of the electrons. The additional challenge is that ETFDW is no longer convex. We are unaware of an in-depth treatment of this model, but refer to [14, Sec 3.6.3] for a discussion of possible avenues and [20] for results on the related Cauchy–Born scaling limit for this model.

(5) Further orbital-free DFT models:

Most orbital-free DFT models used in computations for practical materials have a more complicated functional form for kinetic energy and exchange-correlation energy, see e.g., [43, 44]. It is shown in [4] that the Wang–Teter kinetic energy [43] is not bounded from below, and thus the thermodynamic limit is ill-posed. The more complicated density-dependent orbital free kinetic-energy functionals, such as the Wang–Govind–Carter functional [44], are yet to be mathematically understood.

(6) Charge screening:

The stability result Lemma 6.3 clearly shows that interaction in the TFW model is exponentially localised, despite the presence of the long-range Coulomb interaction. This can be interpreted as a very general screening result. Consider two configurations $$\mathscr {R}, \mathscr {R}_*$$ satisfying (6.8), and which coincide outside a large ball of radius r > 0: $$\mathscr {R} \setminus B_r = \mathscr {R}_* \setminus B_r$$ . Let $$(u, V^{\mathrm {tot}}), (u_*, V^{\mathrm {tot}}_*)$$ be the corresponding solutions. Then Lemma 6.3 implies

(6.12)

For instance, if $$\mathscr {R}_*$$ contains more atoms than $$\mathscr {R}$$ , one might expect the potentials to satisfy $$V_*^{\mathrm {tot}}({\mathbf {r}}) \approx V^{\mathrm {tot}}({\mathbf {r}}) + Q/| {\mathbf {r}} |$$ as r →∞, where Q would be the extra effective charge. However, according to (6.12), this is not the case: the extra charge is completely screened. This is a very general fact in TFW theory, see [9, 35] for details.

One can also take into account the relaxation of the configuration $$\mathscr {R}_*$$ due to the presence of the defect. In this case, instead of an exponential decay, we obtain (see [17])

and, since |r|⁻² = o(|r|⁻¹), we deduce that charges are screened, see [9] and [35, Thm. 4.1] for the details.

6.2.4 Scaling Limit

A question related to but distinct from the thermodynamic limit arises when considering the derivation of a continuum model for elastic material response from an underlying electronic structure model. Such scaling limits for the TFW model were first studied in [5], but the following discussion builds on the results of [35]. Specifically, the stability and locality estimate (6.3) yields stronger and quantitative results. In addition, for the sake of consistency with the KS-DFT case in Sect. 6.4, we consider a periodic instead of an infinite-domain setting.

6.2.4.1 Spatial Decomposition of Energy

In preparation we first mention another useful consequence of the stability and locality estimate found in Lemma 6.3, which is also of independent interest: Let $$\mathscr {R} \subset \mathbb {R}^3$$ be a finite configuration of nuclei or an infinite configuration satisfying (6.8) and let (u, Vtot) be the associated solutions to (6.6), then we define the energy density

If $$\mathscr {R}$$ is finite then one may readily check [35, Eq. 4.18] that $$I^{\mathrm {TFW}}(\mathscr {R}) = \int _{\mathbb {R}^3} \mathcal {E}(\mathscr {R}; {\mathbf {r}}) \,\mathrm{d} {\mathbf {r}}.$$ We may therefore think of

as the energy stored in a compact sub-domain $$\Omega \subset \mathbb {R}^3$$ . This intuition is further supported by the following result, proven in [35, Proof of Thm. 4.2], which is closely related to (6.3): there exist constants C, γ such that

(6.13)

In [35, Sec. 4.4] this observation is used to demonstrate exponential locality of interatomic forces in the TFW model. In the following section we use it for an easy derivation of the Cauchy–Born scaling limit.

6.2.4.2 The Cauchy–Born Scaling Limit

Consider a periodic arrangement $$\mathscr {R} = B \mathbb {Z}^3$$ of nuclei, where $$B \in \mathbb {R}^{3 \times 3}$$ is a non-singular matrix, and let (u, Vtot) describe the corresponding TFW ground-state. Then, uniqueness of solutions to (6.6) implies that they must observe the same periodicity. In particular, we can define the Cauchy–Born energy function, which represents the energy stored in ΩB := B[0, 1)³, that is

A deformed configuration of the crystal is described by a continuum deformation field Y (x) = x + U(x) where U is smooth and $$\mathscr {R}$$ -periodic. We assume that U is chosen such that Y  is bijective, i.e. a proper deformation.

Given a parameter 𝜖 > 0 describing the inverse length-scale over which the deformation varies we define a deformed crystalline configuration by

This definition encodes the Cauchy–Born hypothesis that nuclei in a crystal follow the continuum deformation field. An example of such an atomistic configuration shadowing a continuum field is given in Fig. 6.1(left). It must be emphasised that this is a simplifying assumption that is only approximately valid in specific deformation regimes and for simple crystals; see [22, 36] for in-depth discussions.

3 illustrations of the scaling limit. Left, a square consists of several small dots and a parallelogram at the center. Center, a square consists of several dots and a parallelogram at the center. Right, a square with a dot inside a parallelogram.

Fig. 6.1

Illustration of the scaling limit. Left: a deformation of a homogeneous crystal varying slowly relative to the scale of atoms; Center: the blow-up of a small section is nearly homogeneous; Right: The near-homogeneous section can be approximately represented by a single unit cell

In the following we concern ourselves with the scaling limit𝜖 → 0 of the stored elastic energy per unit undeformed volume. To that end, let Ω𝜖 := 𝜖⁻¹ Ω then $$\mathscr {R}^\epsilon $$ has periodic cells Ω𝜖 or alternatively also Y𝜖( Ω𝜖). That is, we may write the elastic energy per unit volume as

We remark that the electronic coordinates r belong to the deformed space, i.e. it is natural to write r = Y𝜖(x).

The key observation now is that the locality estimate (6.9) on the electronic structure and its extension to the energy density (6.13) suggest that to predict the value of $$\mathcal {E}(\mathscr {R}^\epsilon , {\mathbf {r}})$$ for a non-uniform deformation varying at the macroscopic scale, it is not required to be aware of the global configuration $$\mathscr {R}^\epsilon $$ but it is sufficient to know the local deformation near r. This is illustrated by Fig. 6.1.

To make this precise, let $${\mathbf {n}} \in \mathbb {L}$$ , tn = Y𝜖(n) and Fn = ∇Y𝜖(n), then

for x in any bounded neighbourhood of n. A Taylor expansion of $$\mathcal {E}$$ with respect to $$\mathscr {R}^\epsilon $$ , employing (6.13), implies that for r ∈ Y𝜖(n +  Ω) we have

After integrating over one cell in the undeformed crystal, which in deformed coordinates becomes Y𝜖(n +  Ω), and making a further elementary approximation, we obtain

Finally, after summing over all such unit cells, using the scale-invariance of the deformation gradient one obtains the following convergence result. The second order error O(𝜖²) is obtained by a more careful exploitation of the point symmetry in simple crystal lattices.

Theorem 6.5

Let $$Y \in C^4_{\mathrm {per}}(\Omega )$$ , then

The presentation of this section follows unpublished notes. Related results using different techniques were first presented in [5, Theorem 5, case (i)]. The paper [5] also considers several generalisations, including domains with boundaries and alternative scaling regimes.

6.3 The Reduced Hartree–Fock Model

We now focus on the reduced Hartree–Fock (rHF) model. In this model, a fermionic system with N electrons is described by a one-body density matrix γ, which is a self-adjoint operator on $$L^2(\mathbb {R}^3)$$ satisfying the Pauli principle 0 ≤ γ ≤ 1, and with trace N.

Together with the spectral theorem, this implies that γ is of the form

Here, the functions (φi)i form an orthonormal basis of eigenfunctions in $$L^2(\mathbb {R}^3)$$ , and are called the orbitals, and the numbers 0 ≤ ni ≤ 1 are the occupation numbers. To such a one-body density matrix, we can associate its density ργ(r) := γ(r, r) =∑ini|φi|²(r).

In the potential generated by the nuclei at $$\mathscr {R}_N \subset \mathbb {R}^3$$ , the rHF energy of a state γ is

(6.14)

where $$m^{\mathrm {nuc}}_N$$ is the total nuclear density defined in (6.1). Compared with (6.3), we see that the kinetic energy part $$\int | \nabla \sqrt { \rho } |{ }^2 + \rho ^{5/3}$$ has been replaced by

In particular, this model is no longer a function of the density ρ, but of the one-body density matrix γ. The energy of the configuration $$\mathscr {R}_N$$ is given by the minimisation problem

(6.15)

Closely related to the rHF model is the Hartree–Fock (HF) model, where the exchange term is considered. This term is a correction to the direct Hartree energy, and is due to the fermionic nature of the particles. The HF model reads

(6.16)

Since the HF model is not convex in γ, we only have partial results for it, and most of the following facts only hold for the rHF model.

In the thermodynamic limit, we consider a regular periodic lattice $$\mathbb {L} \subset \mathbb {R}^3$$ , and a sequence of arrangements $$\mathscr {R}_N \subset \mathbb {L}$$ with $$| \mathscr {R}_N | = N$$ , and satisfying (6.10). We want to study the energy per unit cell $$N^{-1} I^{\mathrm {rHF}}(\mathscr {R}_N)$$ as N goes to infinity.

The finite electron model (6.15) was introduced and studied by Solovej [41]. The existence of an optimiser $$\gamma _N^0$$ is provided here. The thermodynamic limit was latter studied by Catto, Le Bris and Lions in a series of papers. In [13], the authors announced their results, later proved in [15] (for the models presented here) and [16] (for the pure-state version of these problems, i.e. when γ is further constrained to be a rank-N projector). They prove the thermodynamic limit for the rHF model:

(6.17)

where $$I^{\mathrm {rHF}}_ {\mathrm {per}}$$ can be characterised by a minimisation periodic problem, that we describe in the next section, and $$\mathfrak {m}$$ is the Madelung constant, see (6.18) below. They conjectured that a similar result should hold for the HF model. Finally, they proved that the limiting problem $$I^{\mathrm {rHF}}_ {\mathrm {per}}$$ (and its HF counterpart $$I^{\mathrm {HF}}_ {\mathrm {per}}$$ ) is indeed well-posed. We discuss this point in the next section.

6.3.1 The Periodic Model

In order to write the limiting periodic model, as introduced by Catto, Le Bris and Lions, we define the set of periodic one-body density matrices (recall that in our simple setting, we expect one electron per unit cell)

where τℓ is the translation operator τℓf(x) := f(x − ℓ). Such a periodic density matrix has an $$\mathbb {L}$$ -periodic density ργ(r) := γ(r, r). Its trace per unit cell $$ \underline {\mathrm {Tr}} \, \gamma $$ is defined by

where Ω is a unit cell associated to the lattice $$\mathscr {R} = \mathbb {L}$$ . A periodic density matrix $$\gamma \in \mathcal {P}_ {\mathrm {per}}$$ has an $$\mathbb {L}$$ -periodic density $$\rho _\gamma ({\mathbf {x}}) := \gamma ({\mathbf {x}}, {\mathbf {x}}) \in L^1_ {\mathrm {per}}(\Omega )$$ . We let G be the $$\mathbb {L}$$ -periodic Coulomb kernel, solution to

and we introduce the periodic Coulomb quadratic form Dper(⋅, ⋅) defined for periodic functions by (compare with (6.4))

The Madelung constant appearing in (6.17) is defined to be

(6.18)

Note that since F(r) := G(r) −|r|⁻¹ satisfies ΔF = 0 on Ω, the function F is indeed smooth on Ω, hence has a well-defined value at r = 0. The Madelung constant somehow describes the mismatch between the full space Coulomb kernel |r|⁻¹ and the periodic kernel G(r) (which can a priori be defined up to a constant).

With these notations, the limit $$W_ {\mathrm {per}}^{\mathrm {rHF}}$$ for the perfect crystal is defined as the minimisation problem [15]

(6.19)

where the energy per unit cell $$E^{\mathrm {rHF}}_ {\mathrm {per}}$$ is

(6.20)

and where $$m^{\mathrm {nuc}}_ {\mathrm {per}}({\mathbf {x}}) := \sum _{\ell \in \mathbb {L}} m_{\mathrm {a}}({\mathbf {x}} - \ell )$$ is the periodic nuclear density. Comparing this expression with (6.14), we see that all terms have been “normalised” to take into account the periodicity of the infinite system.

The fact that $$I^{\mathrm {rHF}}_ {\mathrm {per}}$$ is a well-posed problem was proved by Catto et al. in [15]. Later in [7], Cancès, Deleurence and Lewin proved that the minimiser γ satisfies the Euler–Lagrange equations

Here, $$\varepsilon _F \in \mathbb {R}$$ is the Fermi level. The operator Hγ is the mean-field one-body Hamiltonian of the crystal, which is a self-adjoint operator that commutes with $$\mathbb {L}$$ -translations. Its spectral properties are well understood thanks to the Bloch transform [37, Chapter XIII], and its spectrum is composed of bands and gaps. When εF is in a gap, the crystal is an insulator, while when εF is in a band, it is a metal.

6.3.2 Supercell Methods, and Periodic Thermodynamic Limits

Once the periodic problem (6.19) has been written and justified, it is possible to understand its properties from other approaches. In [7] (see also [18]), Cancès, Deleurence and Lewin proved that this problem was also the limit of another thermodynamic limit. Their idea was to start directly with a periodic problem on the large supercell ΩL := L Ω with N = L³ electrons, and take the limit L →∞. In other words, instead of working with one-body density matrices γ acting on the whole space $$L^2(\mathbb {R}^3)$$ , they looked at one-body density matrices acting on the supercell $$L^2_{\mathrm {per}}(\Omega _L)$$ . We therefore define

where we set for simplicity $${\mathrm {Tr}}_L \, := {\mathrm {Tr}}_{{\mathcal S}(L^2_{\mathrm {per}}(\Omega _L))}$$ . A one-body operator $$\gamma \in \mathcal {P}_ {\mathrm {per}}^L$$ has an $$L\mathbb {L}$$ -periodic density $$\rho _\gamma \in L^1_ {\mathrm {loc}}$$ . We also define the $$L \mathbb {L}$$ -periodic Coulomb kernel as GL(x) := L⁻¹G(L⁻¹x), and the L-periodic Coulomb quadratic form defined for $$L\mathbb {L}$$ -periodic functions by

The supercell model is given by a periodic minimisation problem of the form

(6.21)

with the supercell energy

(6.22)

Here, $$\mathscr {R}^L$$ is an $$L \mathbb {L}$$ -periodic lattice (for instance $$\mathscr {R}^L = \mathbb {L}$$ , or a deformation of it, see below), and $$m^{\mathrm {nuc}}_L$$ is the nuclear density $$m^{\mathrm {nuc}}_L := \sum _{{\mathbf {R}} \in \mathscr {R}_L} m_{\mathrm {a}}({\mathbf {x}} - {\mathbf {R}})$$ . In the case $$\mathscr {R}^L = \mathbb {L}$$ , there are L³ nuclei and electrons per supercell.

Even in the perfect crystal case, that is when $$\mathscr {R}^L = \mathbb {L}$$ , the problems (6.19)–(6.20) and (6.21)–(6.22) differ. In (6.19), the minimisation is performed for γ acting on the whole space $$L^2(\mathbb {R}^3)$$ , while in (6.21), it is performed for γ acting on the supercell L²( ΩL). These two types of operators cannot be compared, and it is not obvious a priori that there is a link between the two problems. Still, both operators give $$\mathbb {L}$$ -periodic densities, which can be compared. This important fact allows us to prove the convergence [7]

(6.23)

The result was later refined in [25], where the authors proved that, in the insulating case (see [10] for the metallic case), the convergence is exponential, in the sense that there exist constants $$C \in \mathbb {R}$$ and α > 0 such that

(6.24)

where $$\rho ^0_ {\mathrm {per}}$$ and $$\rho ^0_{ {\mathrm {per}},L}$$ are the electronic densities of the periodic and supercell minimisers respectively, seen here as $$\mathbb {L}$$ -periodic functions. This exponential convergence comes from the analyticity of the Bloch representation.

This means that the full space problem $$W^{\mathrm {rHF}}_ {\mathrm {per}}$$ can be well-approximated by the supercell model $$I^{\mathrm {rHF}}_{ {\mathrm {per}}, L}$$ . This latter problem can be studied efficiently from a numerical point of view, thanks to the Bloch transform. As noticed in [25], the supercell model corresponds exactly to a uniform discretisation of the Brillouin zone, as described in a famous paper by Monkhorst [34]. For metallic systems, the exact rate of convergence is unknown in the general case (see [10] for details).

Remark 6.6

When studying supercell methods for non-convex problems, symmetry breaking may happen (see e.g. [27, 38]). In this case, the density of the $$L \mathbb {L}$$ -periodic problem may not be $$\mathbb {L}$$ -periodic, and the periodic problem may not be the limit of supercell models.

To sum up, the energy per unit cell $$W_ {\mathrm {per}}^{\mathrm {rHF}}$$ is the limit of two different sequences, namely

In the first limit, the crystal is seen as the limit of finite systems. This is the correct physical limit, as a real crystal is indeed always finite. However, we expect the convergence to be slow, due to boundary effects. On the other hand, the second limit has no real physical meaning, but gives an exponential rate of convergence in the insulating case.

6.3.3 Local Defects in Crystals, in the Reduced Hartree–Fock Model

We now discuss how to define the energy of a defect inside a crystal. We would like to define this energy as the difference between the energy of a crystal with a defect, and the energy of the crystal without the defect. Unfortunately, these two quantities are infinite. Also, the model with defect does not have an underlying periodicity, hence there is no notion of energy per unit cell in this case. One way to define the energy of a defect is through a thermodynamic limit procedure.

Let $$\mathscr {R} := \mathbb {L}$$ be the arrangement of nuclei for the perfect crystal, and let $$\mathscr {R}_*$$ be the one for the crystal with (local) defect, that is such that $$\mathscr {R}$$ and $$\mathscr {R}_*$$ coincide outside a ball of radius r > 0. The nuclear charge of the defect is therefore

For $$L \in \mathbb {N}^*$$ , we can consider $$\mathscr {R}_*^L$$ , the $$L \mathbb {L}$$ -periodic arrangement which is equal to $$\mathscr {R}_*$$ on ΩL (note that $$\mathscr {R}^L = \mathscr {R} = \mathbb {L}$$ ). In [7], the authors consider the supercell energy of the defect ν, defined by

Here there is a slight complication: since we do not know a priori how many electrons should be in the system, we should not fix the number of electrons, but rather the Fermi level (grand canonical ensemble). To keep this presentation simple, we do not comment on this point, but refer to [7] for details.

Although the two quantities $$I^{\mathrm {rHF}}_{ {\mathrm {per}}, L}(\mathscr {R}_*^L)$$ and $$I^{\mathrm {rHF}}_{ {\mathrm {per}},L}(\mathbb {L})$$ diverge to infinity with rate O(L³), the difference of the two quantities stays finite in the limit, and we can define the energy of the defect as

In [26], the authors prove that the corresponding rate of convergence is O(L⁻¹). This slow rate of convergence is due to the spurious interaction between the defect and its periodic images, a fact predicted in [29, 33]. This makes the supercell method quite a poor numerical method in this case.

It turns out that the limit $$\mathcal {J}_\infty (\nu )$$ can be characterised as a minimisation problem on a set of “defect” operators. It is unclear whether this last problem can be tackled directly with efficient numerical methods (see also [8]).

6.4 Scaling Limit for Kohn–Sham DFT

In this section, we discuss Kohn–Sham models. For a finite system with N electrons described by a one-body density matrix γ, the Kohn–Sham density functional takes the form

(6.25)

where $$m^{\mathrm {nuc}}_N$$ is the total nuclear density defined in (6.1). Compared with the reduced Hartree–Fock model (6.14), the Kohn–Sham model includes the exchange-correlation energy Exc[ργ], where we have adopted the notation for an LDA or GGA type functional, and thus it can be explicitly written in terms of ργ as

Even the simplest exchange-correlation functionals used in practice have complicated expressions, and hence will not be given explicitly here. Most of them are non-convex in ρ, such as, for instance the Dirac exchange term $$E_{\mathrm {x}}^{\text{Dirac}}[\rho ] = - C_D\int \rho ({\mathbf {x}})^{4/3} \mathrm{d} {\mathbf {x}}$$ . Thus, even the existence of a minimiser to the Kohn–Sham DFT problem becomes a difficult question. The existence of minimisers for LDA functionals has been proved in [1, 24, 28], while for GGA-type functionals, it remains open with only preliminary results available (see the case of N = 1 in [1]).

We will henceforth assume LDA-type exchange-correlation functionals. The variation of the functional (6.25) gives rise to the Kohn–Sham equations for $$\gamma = \sum _{i=1}^N | \psi _i \rangle \langle \psi _i |$$ , where ψi are the Kohn–Sham orbitals, solutions to

(6.26)

with ργ the density associated with γ and the Hartree and exchange-correlation potentials respectively given by

The Kohn–Sham equations (6.26) form a set of nonlinear eigenvalue problems, as the effective Hamiltonian operator HKS[ργ] depends on the solution γ. We remark that in general there is no guarantee that the Kohn–Sham orbitals of the minimisers of (6.25) correspond to the lowest N eigenvalues of the self-consistent Hamiltonian, though in practice this is often assumed and known as the Aufbau principle.

Due to the non-convexity and hence possible symmetry breaking, see Remark 6.6, the thermodynamic limit of Kohn–Sham DFT with exchange-correlation functionals is very challenging and not much progress has been made.

To understand the behaviour of electronic structure in materials, we take a typical starting point of modelling in materials science—the periodic Kohn–Sham model with supercell Ω. This can be formulated using the density matrix similar to the periodic Hartree–Fock model discussed in Sect. 6.3.1. A periodic Kohn–Sham energy is of the form

where the rHF energy $$E^{\mathrm {rHF}}_{\mathrm {per}}$$ was defined in (6.20). This is the rHF model with the addition of a periodic exchange-correlation energy. One could follow the lines of Sect. 6.3.1 to study the thermodynamic limit. We can also consider the following alternative formulation, presented in [21], and that we present now.

The self-consistent Kohn–Sham eigenvalue problem (6.26) can be reformulated as a fixed point equation for the density

(6.27)

where $$\mathscr {C}$$ is a contour in the resolvent set separating the first N eigenvalues of HKS from the rest of the spectrum (assuming a spectral gap). The right-hand side of (6.27) denotes the diagonal of the kernel of the density matrix viewed as an integral operator.

6.4.1 The Periodic Kohn–Sham DFT Model

For a periodic system with Bravais lattice $$\mathbb {L}$$ , we can write a similar equation. We introduce the periodic Kohn–Sham Hamiltonian associated with some $$\mathbb {L}$$ -periodic density ρ, given by

where the periodic Hartree potential solves

with periodic boundary condition, where mnuc is understood as a background charge density given by the nuclei (to be specified below). As the potential is periodic, the Bloch–Floquet theory applies to the Hamiltonian. In particular, the spectrum of $$H^{{\mathrm {KS}}}_{ {\mathrm {per}}}[\rho ]$$ has a band structure. For each k ∈ Ω∗, the first Brillouin zone, the Bloch waves solve the eigenvalue problem

with periodic boundary condition on Ω. The spectrum is given by

This is known as the band structure, see Fig. 6.2 for an illustration.

A schematic of band structure of e v versus band structure of silicon. The bands are in fluctuating trend and band gap approximately at e v of 11.

Fig. 6.2

Schematic band structure of crystalline silicon, along various lines connecting high-symmetry points in the first Brillouin zone Ω∗. The first 4 bands are occupied and separated by a band gap from the higher bands

If the first N bands are occupied and there exists a gap between the occupied and unoccupied spectrum (in physical terms, the system is an insulator), the Kohn–Sham map can be generalised to the periodic setting as

(6.28)

where the contour $$\mathscr {C}$$ lies in the resolvent set and separates the occupied and unoccupied spectra. For the periodic Kohn–Sham model, we thus recast the problem as a fixed point equation

(6.29)

Using the electron density ρ as the basic variable is more convenient in studying the scaling limit than the Kohn–Sham orbitals (Bloch waves), which will be discussed next.

6.4.2 Scaling Limit for the Periodic Model

Starting from the fixed point equation (6.29), valid for instance for a periodic configuration $$\mathbb {L}$$ , we would like to find other solutions when the crystal is deformed.

We consider the electronic structure of an elastically deformed system in the scaling limit where the lattice parameter goes to 0. To set up the atomic configuration, we assume a lattice structure for the undeformed system. The atoms are located at $$\varepsilon \mathbb {L}$$ , where $$\mathbb {L}$$ is a Bravais lattice, and the lattice parameter ε will serve as the scaling parameter in the limit, which can be understood as the ratio of the lattice parameter and the characteristic length scale of the system.

For simplicity, we assume that Ω coincides with the unit cell of $$\mathbb {L}$$ . Thus, in Ω, atoms are located at $$\Omega \cap \varepsilon \mathbb {L}$$ . The system consists of ε⁻³ atoms and correspondingly Nε⁻³ electrons, where N is the number of valence electrons per atom.

Fix a smooth function $$u: \mathbb {R}^3 \to \mathbb {R}^3$$ of the form u(x) = Bx + uper(x), where B is a 3 × 3 matrix and uper is periodic with respect to Ω. The deformed atom locations are

Correspondingly the background charge distribution is given by

where $$m_{\mathrm {a}}^{\varepsilon }$$ is the rescaled version of the charge contribution from each individual atom (recall that the lattice parameter is scaled to ε):

As the lattice parameter is scaled to be ε, the Kohn–Sham Hamiltonian needs to be rescaled correspondingly as

where the Hartree potential $$V_{\text{H}}^{\varepsilon }$$ solves

Thus the electron density of the deformed system is determined by the fixed point of the Kohn–Sham map

(6.30)

In order to make sense of the Kohn–Sham map defined in (6.27) and (6.30), we require a gap between the occupied and unoccupied spectrum, and thus we make the following assumption for the undeformed system. The gap of the effective Hamiltonian of the perturbed system follows from a perturbation argument.

Assumption 6.7 (Insulating Undeformed System)

There exists a Ω-periodic $$\rho _0 \in C^{\infty }(\mathbb {R}^3)$$ , that is positive and uniformly bounded away from zero, such that

The spectrum of the Hamiltonian $$H^{{\mathrm {KS}}}_{\varepsilon = 1, 0}[\rho _0]$$ has a positive gap between the occupied and unoccupied spectra.

ρ₀ is a fixed point of the Kohn–Sham map:

where $$\mathscr {C}$$ is a contour in the resolvent set enclosing the occupied spectrum.

6.4.3 Cauchy–Born Rule for Electronic Structure

The question of the scaling limit is to characterise the electron density, as a solution to the Kohn–Sham equation (6.30), when the deformation is elastic in the sense that the deformation gradient is not too large. This is motivated by the Cauchy–Born rule for passing from atomistic models to elastic models, where the analogous question for DFT is to pass from electronic structure models to continuum elastic models. The scaling limit for Thomas–Fermi–von Weizsäcker model was studied by Blanc, Le Bris and Lions in [5], see Sect. 6.2.4.

For the Kohn–Sham type models, the scaling limit was proved by E and Lu in [19, 21] under the stability conditions on the level of linear response of the undeformed system. In order to state the stability assumptions, let us introduce the linearised Kohn–Sham map for the undeformed system

It has been established [21] that $$\mathcal {L}_0$$ is a bounded linear operator on the space $$\mathcal {X}_n := \dot {H}^{-1}_{ {\mathrm {per}}}(n \Omega ) \cap H^2_{ {\mathrm {per}}}(n \Omega )$$ for every $$n \in \mathbb {N}$$ , where $$H_{ {\mathrm {per}}}^2(n\Omega )$$ stands for the periodic Sobolev space with square integrable second derivatives and $$\dot {H}^{-1}_{ {\mathrm {per}}}(n\Omega )$$ is the homogeneous Sobolev space with index − 1 on the domain n Ω.

Assumption 6.8 (Stability of Charge Density Wave Response)

For every $$n \in \mathbb {N}$$ , the operator $$I - \mathcal {L}_0$$ is uniformly invertible as an operator on $$\mathcal {X}_n$$ .

Physically the stability assumption states that the undeformed crystal is stable with respect to spontaneous charge density wave perturbation at every scale. In particular, this prevents the possibility of symmetry breaking as ε → 0.

We also define the macroscopic permittivity tensor for the undeformed crystal as

where the 3 × 3 matrix A₀ is given by

where

and δρVeff is the linearisation of the effective potential operator: Veff[ρ] = VH[ρ] + Vxc[ρ] at ρ₀ for the undeformed crystal. The dielectric permittivity for the reduced Hartree–Fock theory has been studied in [11].

Assumption 6.9 (Stability of Dielectric Response)

The macroscopic permittivity tensor for the undeformed crystal E₀ is positive definite.

The main result of [21] establishes the Cauchy–Born rule for the electronic structure.

Theorem 6.10 ([21, Thm. 5.1])

Under Assumptions6.7, 6.8and6.9, if the deformation gradient is sufficiently small, then there exists a $$\rho ^{\varepsilon }_u$$ satisfying the Kohn–Sham fixed point equation (6.30), and furthermore, $$\rho ^{\varepsilon }_u$$ can be locally approximated by the Cauchy–Born rule:

where ρCB(⋅;A) is the electron density of a homogeneous deformed system with u(x) = Ax(which is well-defined provided $$\lvert A \rvert $$ is not too large).

The main technical ingredients of the proof of the theorem is a two-scale analysis of the linearised Kohn–Sham map. As a part of the analysis, the effective potential and the macroscopic dielectric response of the deformed crystal can also be characterised, we refer the readers to [21] for details.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_7

7. Numerical Methods for Kohn–Sham Models: Discretization, Algorithms, and Error Analysis

Eric Cancès1  , Antoine Levitt1  , Yvon Maday2, 3   and Chao Yang4  

(1)

CERMICS, Ecole des Ponts and Inria Paris, Marne-la-Vallée, France

(2)

Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), Paris, France

(3)

Institut Universitaire de France, Paris, France

(4)

Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Eric Cancès (Corresponding author)

Email: eric.cances@enpc.fr

Antoine Levitt

Email: antoine.levitt@inria.fr

Yvon Maday

Email: yvon.maday@sorbonne-universite.fr

Chao Yang

Email: cyang@lbl.gov

Abstract

This chapter aims at presenting the main algorithms used to solve the Kohn–Sham models, as well as the current state-of-the-art of the numerical analysis of these models and algorithms.

7.1 Introduction

There are several avatars of the Kohn–Sham model, each characterized by

the type of systems it describes: molecules or clusters, perfect crystals, disordered materials,

the choice of the exchange-correlation functional: local, semilocal, nonlocal, orbital-dependent (see Chap. 1 by Toulouse in this volume),

the electrons explicitly taken into account: all electrons vs pseudo-electron models (see Sect. 7.4),

the underlying variational model: the standard Kohn–Sham models derived from the Levy-Lieb formulation of DFT, the extended Kohn–Sham models derived from the Valone–Lieb construction of DFT, the positive-temperature Kohn–Sham models [50].

The most appropriate discretization method and the algorithm best suited to solve the resulting discretized problem depend on these four criteria.

The molecular orbital formulation of the standard Kohn–Sham model for finite systems, and the most common types of exchange-correlation functionals are presented in Sect. 7.2.1. The standard Kohn–Sham model can be reformulated in terms of the (one-body) density matrix. The density matrix formalism is also the right one to set the extended and finite-temperature Kohn–Sham models, introduced in Sect. 7.2.2. All Kohn–Sham type models are variational models: some energy (or free energy) functional is minimized over some “nice” set, which can be a manifold of N-tuples of orthonormal vectors (molecular orbital formulation of the standard Kohn–Sham model), a manifold of orthogonal projectors (density matrix formulation of the standard Kohn–Sham model), or the convex hull of a manifold of orthogonal projectors (extended and finite-temperature Kohn–Sham models). The geometries of these objects are discussed in Sect. 7.2.3.

Kohn–Sham models are used in both quantum chemistry and condensed matter physics (we will use this terminology here to refer to solid-state physics, the physics of liquids, and materials science all together). The models and numerical techniques used in these two fields are closely related, but however significantly different.

Quantum chemistry aims at modeling molecules, and understanding their properties and reactions. Most chemical reactions of practical interest take place in the liquid phase, and involve solvated molecules. For some systems, ignoring the solvent effects and considering the molecule of interest in vacuum is a sufficient approximation. For others, the solvent has to be taken into account. Explicit solvent models, in which each and every solvent molecule in a large simulation box are dealt with at the same level of theory as the solute molecule of interest, are commonly used in classical molecular dynamics, but are much too computationally expensive for DFT. A much cheaper alternative is to use implicit solvent models, in which the solute molecule is placed in a molecular-shape cavity and the solvent is modeled by a continuous dielectric medium of permittivity equal to the zero-frequency dielectric permittivity of the solvent. We refer to [138] for reviews of implicit solvent models. The rate-limiting computation is then the electronic state of a molecule in the environment of the solvent, which is mathematically and computationally similar to isolated molecules in vacuum. We therefore limit ourselves to this case in this chapter. The resulting equations are most often discretized in atomic basis sets; this approach is presented in Sect. 7.3.1.

In contrast, models used in condensed matter physics deal with extended systems containing virtually an infinite number of nuclei and electrons. The latter models can be seen as thermodynamic limits of the former models when the number of particles tends to infinity. Of course, it is not possible to simulate an infinite number of individual particles on a computer. There are two usual ways to circumvent this difficulty:

1.

if the system is periodic (perfect crystals), a Bloch–Floquet transform can be used to reduce the problem with infinitely many particles set on the whole space to a family of problems with finitely many particles set on the periodic cell with periodic boundary conditions (see e.g. [121]);

2.

infinite aperiodic systems (“real” crystals, involving defects, amorphous solids, liquids) are usually dealt with using the supercell method: a chunk of the system is placed in a large rectangular box and appropriate boundary conditions are imposed on the Kohn–Sham orbitals to get rid of surface effects. This approach is also used in the field of homogenization, where the supercell is called a representative volume element (RVE).

It follows that in practice Kohn–Sham problems encountered in condensed matter physics are set on periodic cells, which either have a physical meaning (as for perfect crystals) or are chosen for numerical convenience (as in the supercell method). For this reason, it is natural to discretize them in plane-wave basis sets, in which Kohn–Sham orbitals and electronic densities are approximated by a finite linear combination of Fourier modes. This discretization method is presented in Sect. 7.3.2. Notably, plane-wave discretization methods perform extremely poorly on genuine all-electron Kohn–Sham models because of the singularities of the Coulomb potential generated by the nuclei. Most Kohn–Sham simulations in the condensed phase make use of pseudopotentials: only valence electrons are explicitly dealt with, valence Kohn–Sham orbitals (which have cusps at nuclear positions) are replaced by smoother valence pseudo-orbitals, and the local singular Kohn–Sham potential, by a smoother but nonlocal pseudopotential. This approximation is discussed in Sect. 7.4.

Whatever the system, the exchange-correlation functional and the discretization method that is used, the resulting discretized Kohn–Sham models can be formulated as constrained optimization problems. The generic methodology of constrained optimization proves too general, and the specific simple form of the constraints (which form Riemannian manifolds) allow for efficient custom algorithms, termed direct minimization algorithms. The Euler–Lagrange equations of the minimization problem can be recast as a nonlinear eigenvector problem, which has an appealing physical interpretation as a mean-field equation. Algorithms based on a fixed-point solution of these equations are known as self-consistent field (SCF) algorithms. In both cases, two challenges of different natures have to be faced: (i) ensure convergence toward a (hopefully) global minimum, and (ii) accelerate local convergence, which can be very slow due to ill-conditioning.

Algorithms to solve Kohn–Sham problems in atomic basis sets on the one hand, and in plane-wave basis sets on the other hand, have been developed almost independently by chemists and physicists respectively. This can be explained by the fact that the corresponding problems are quite different from a numerical point of view. Indeed, the performance of a given algorithm heavily depends on the physical properties of the system as well as on the discretization methods. Problems of global convergence are usually more severe in quantum chemistry, where bond breaking, open shells and limited basis sets contribute to the existence of local minima. These problems are relatively less common in the more rigid solid-state context. Similarly, ill-conditioning comes from different sources: overcompleteness of the basis set and degeneracies in quantum chemistry, low and high-frequency modes in condensed-matter. Finally, the problems caused by large systems are also different, and treated using specific methodologies. We will present the quantum chemistry perspective (atomic basis sets) in Sect. 7.5.3, and the materials science perspective (planewave basis sets) in Sect. 7.5.4.

Let us emphasize that very few algorithms for solving the Kohn–Sham model are well understood mathematically. Sometimes, convergence results have only been proved for the special cases of Hartree and Hartree–Fock models, for which the energy functional is quadratic in the density matrix. The mathematical understanding of most Kohn–Sham models and algorithms is only partial.

Likewise, error analysis for Kohn–Sham models still is in its infancy. The a priori error analysis of the Kohn–Sham LDA model was carried out in [24] for the plane-wave discretization of the supercell model. The proof was adapted in [41] to cover the cases of any variational discretization of the Kohn–Sham LDA model in a bounded domain with Dirichlet boundary conditions. The construction of a posteriori error estimators for Kohn–Sham, and more generally for electronic structure models, is a major challenge in molecular simulation. The a priori error analysis and the few results available to date on a posteriori error analysis are discussed in Sect. 7.6.

7.2 Mathematical Structures of Discretized Kohn–Sham Problems

For simplicity, we limit ourselves to non-magnetic spin-unpolarized systems, in which Kohn–Sham orbitals can be chosen real-valued and are occupied by two electrons (one with spin-up and one with spin-down).

7.2.1 Molecular Orbital Formulation of the Standard Kohn–Sham Model

The standard Kohn–Sham model for a molecule or a cluster (finite number of electrons and nuclei) is

(7.1)

where N is the total number of electron pairs, and EKS the Kohn–Sham energy functional given by

(7.2)

In the above expression, Vext is the external potential generated by the M nuclei of atomic charges $$z_k \in \mathbb {N}^\ast $$ and positions $${\mathbf {R}}_k \in \mathbb {R}^3$$ , ρΦ the electronic density associated with the orbitals Φ, and Exc the exchange-correlation energy functional:

(7.3)

The factor 2 in (7.3) is due to the fact that each φi hosts two electrons. The expression of Exc depends on the chosen approximate exchange-correlation functional. We will limit ourselves here to some simple examples: the Hartree (also called reduced Hartree–Fock, rHF) model for which $$E_{\mathrm {xc}}^{\mathrm {rHF}}=0$$ , the local density approximation (LDA), and the Xα model. We refer to the contributions of Lewin, Lieb, and Seiringer (Chap. 3) for a mathematical analysis of the LDA, and to the contribution by Toulouse (Chap. 1) for a detailed description of the many approximate exchange-correlation functionals that have been proposed by physicists and chemists. The LDA exchange-correlation functional is given by

(7.4)

where $$e_{\mathrm {xc}}^{\mathrm {HEG}}:\mathbb {R}_+ \to \mathbb {R}$$ is the exchange-correlation energy density of the homogeneous electron gas. The function $$e_{\mathrm {xc}}^{\mathrm {HEG}}$$ is not known explicitly. Several approximations of this function have been proposed, based on theoretically obtained asymptotic expansions in the high-density regime, and quantum Monte Carlo (QMC) simulations at lower densities [38]. The current reference approximation (PW92) has been proposed by Perdew and Wang [111]. In mathematical and numerical studies, it is convenient to replace the LDA functional by the Xα functional, which is explicit and shares the most important features of the LDA exchange-correlation functional:

where CD is the Dirac constant.

In GGA (generalized gradient approximations), the exchange-correlation functional is of the form

(7.5)

where $$e_{\mathrm {xc}}^{\mathrm {GGA}}: \mathbb {R}_+ \times \mathbb {R}_+ \to \mathbb {R}$$ is a function such that $$e_{\mathrm {xc}}^{\mathrm {GGA}}(\overline \rho ,0)=e_{\mathrm {xc}}^{\mathrm {HEG}}(\overline \rho )$$ and whose dependence in |∇ρ(r)| is constructed by picking up a suitable functional form depending on a few parameters, and fitting these parameters using exact conditions and/or empirical data. The most popular GGA functionals are the PBE [110] and PBE0 [2] functionals.

Meta-GGA and hybrid functionals involve respectively the kinetic energy density

and the one-body density matrix

and are therefore not explicit in the density ρΦ. In order to include all the above exchange-correlation functionals in a simple unified framework, we redefine the Kohn–Sham energy functional as

(7.6)

where we allow $$\mathcal {E}_{\mathrm {xc}}$$ to depend explicitly on the density matrix (note that τΦ(r) can be easily obtained from γΦ). Another advantage of this framework is that it contains Hartree–Fock as a special case:

(7.7)

where $$\mathcal {E}_{\mathrm {xHF}}(\gamma )$$ is the exact-exchange functional defined as

Once the model is set, variational discretization methods consist in choosing a set of Nb linearly-independent functions $$\Xi :=(\xi _1,\ldots ,\xi _{N_b}) \in (H^1(\mathbb {R}^3))^{N_b}$$ and solving the finite-dimensional minimization problem

(7.8)

where $$V_\Xi =\mbox{span}(\xi _1,\ldots ,\xi _{N_b}) \subset H^1(\mathbb {R}^3)$$ . Denoting by $$X \in \mathbb {R}^{N_b \times N}$$ the matrix collecting the coefficients of the orbitals φi in the basis ξμ i.e.

problem (7.8) can be rewritten as

(7.9)

where

(7.10)

and

(7.11)

where IN is the rank-N identity matrix. The matrices $$S^\Xi \in \mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}$$ and $$h^\Xi \in \mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}$$ (here and in the sequel, $$\mathbb {R}^{n \times n}_{\mathrm {sym}}$$ stands for the vector space of n × n real symmetric matrices) are the overlap and one-electron Hamiltonian matrix

(7.12)

The functions $$J^\Xi : \mathbb {R}^{N_b \times N_b}_{\mathrm {sym}} \to \mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}$$ and $${\mathcal E}^\Xi _{\mathrm {xc}}:\mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}\to \mathbb {R}$$ are defined by

where AΞ is the 4th-order tensor of two-electron integrals:

(7.13)

Recall that the notation AΞ : P stands for the double contraction

Let us finally make a few remarks on the mathematical properties of the minimization problem (7.9). First, the function $$E^\Xi :\mathbb {R}^{N_b \times N}\to \mathbb {R}$$ is continuous and the minimization set $${\mathcal M}_{\mathrm {MO}}^{\Xi }$$ is a non-empty compact set of $$\mathbb {R}^{N_b \times N}$$ , which guarantees the existence of a minimizer to (7.9). Second, problem (7.9) is gauge invariant: denoting by O(N) the group of N × N real orthogonal matrices, we have

(7.14)

In particular, if X₀ is a minimizer of (7.10), then X₀U also is a minimizer of (7.10). Uniqueness of the minimizer up this gauge invariance is an open problem.

7.2.2 Density Matrix Formulations of the Various Kohn–Sham Models

We see from (7.10) that the discrete Kohn–Sham energy functional is in fact a function of the symmetric matrix P = XXT:

where $${\mathcal E}^\Xi :\mathbb {R}^{N_b\times N_b}_{\mathrm {sym}} \to \mathbb {R}$$ is defined by

(7.15)

This is not surprising since the matrix P = XXT is in fact the matrix in the tensor basis ξμ ⊗ ξν of the one-body reduced density matrix γΦ:

The gauge invariance (7.14) simply originates from the fact that X and XU for U ∈ O(N) generate the same density matrix. Note that in the Hartree–Fock setting, the energy functional can be written as

(7.16)

with

(7.17)

It is also easy to check that when X spans the minimization set $$ {\mathcal M}_{\mathrm {MO}}^{\Xi }$$ arising in (7.9) (and defined in (7.11)), the matrix P = XXT spans the set

This set $$\mathcal {M}_{\mathrm {DM}}$$ is the set of rank-NS-orthogonal projectors in the Nb-dimensional real space.

We therefore have

(7.18)

which corresponds to the variational approximation in the basis Ξ of the density matrix formulation of the continuous standard Kohn–Sham model:

(7.19)

where $${\mathcal S}(L^2(\mathbb {R}^3))$$ is the vector space of bounded self-adjoint operators on $$L^2(\mathbb {R}^3)$$ and

ργ denoting twice the density of the trace-class operator γ, i.e. ργ(r) = 2γ(r, r), where γ(r, r′) is the kernel of the operator γ. We refer to [122] for an introduction to the theory of trace-class operators.

Let us now turn to the discretized extended and finite-temperature Kohn–Sham models. These models are more conveniently formulated in the density matrix formalism. For both of them the minimization set is the convex hull of $${\mathcal M}_{\mathrm {DM}}^\Xi $$ , that is

The inequality PSΞP ≤ P is to be understood in the sense of square symmetric matrices:

Note that any matrix $$P \in \mathrm {Conv}({\mathcal M}_{\mathrm {DM}}^\Xi )$$ can be decomposed as

where ni is called the occupation number of the orbital Φi. The orbital Φi is called fully occupied if ni = 1, partially occupied if 0 < ni < 1 and unoccupied, or virtual if ni = 0. The discretized extended Kohn–Sham model is defined as

(7.20)

and corresponds to the discretization in the basis set Ξ of the continuous problem

(7.21)

obtained from Valone–Lieb formulation of DFT [94, 141]. The minima E₀ in (7.19) and (7.21) are equal for the exact Levy-Lieb and Valone–Lieb DFT functionals, but differ in general if approximate exchange-correlation functionals and finite basis sets are used.

A discretized finite-temperature Kohn–Sham model can be defined as

(7.22)

where T is the temperature and kB the Boltzmann constant. The functional

is the fermionic entropy for spin 1∕2-unpolarized systems, favoring configurations with fractional occupation numbers. The functional $${\mathcal E}^\Xi (P) - T S(P)$$ can therefore be interpreted as a free energy. It should be mentioned that while the above problem always has a minimizer, its continuous counterpart diverges: the infimum of the free energy $${\mathcal E}(\gamma )+ 2 k_{\mathrm {B}}T \, \mathrm {Tr}(\gamma \ln \gamma + (1-\gamma ) \ln (1-\gamma ))$$ over the set of admissible density matrices is equal to −∞, due to the presence of a continuous spectrum in the underlying Kohn–Sham Hamiltonian. On the other hand, the continuous finite-temperature Kohn–Sham model is well-defined for confined systems as well as, more interestingly, for periodic crystals, in the sense that the free-energy per unit volume has a minimizer. In both settings, the extended Kohn–Sham model is the limit of the finite-temperature Kohn–Sham model when the temperature goes to zero.

7.2.3 Geometrical Structure of the Kohn–Sham Problems

The minimization sets $${\mathcal M}_{\mathrm {MO}}^\Xi $$ and $${\mathcal M}_{\mathrm {DM}}^\Xi $$ in (7.9) and (7.18) have simple geometrical structures. They are respectively diffeomorphic to the smooth manifolds

through the linear transforms

(7.23)

Note that $$\mathcal {M}_{\mathrm {MO}}$$ and $$\mathcal {M}_{\mathrm {DM}}$$ are themselves representations in $$\mathbb {R}^{N_b \times N}$$ and $$\mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}$$ respectively of the Stiefel and Grassmann manifolds $$\mathrm {V}_N(\mathbb {R}^{N_b})$$ and G(N, Nb) of orthogonal bases and subspaces widely studied in differential geometry (see e.g. [54] and references therein).

In order to simplify the presentation, we will assume in the rest of the chapter that the basis set Ξ is orthonormal. The general case of a non-orthonormal basis can be recovered by using the change of variable (7.23) (we will however make some comments on the additional difficulties arising when the overlap matrix SΞ is ill-conditioned). The minimization sets then are the manifold $$\mathcal {M}_{\mathrm {MO}}$$ (orbital formulation) or $$\mathcal {M}_{\mathrm {DM}}$$ (density matrix formulation). Still in order to simplify the notation, we will set

the functions $$E: \mathbb {R}^{N_b \times N} \to \mathbb {R}$$ and $$\mathcal {E}: \mathbb {R}^{N_b \times N_b}_{\mathrm {sym}} \to \mathbb {R}$$ being related by the equality

Let us recall some of the basic properties of the sets $$\mathcal {M}_{\mathrm {MO}}$$ and $$\mathcal {M}_{\mathrm {DM}}$$ that will be useful in our analysis. Let $$g:\mathbb {R}^{N_b \times N}\to \mathbb {R}^{N \times N}_{\mathrm {sym}}$$ be the polynomial map defined by g(X) = XTX − IN, so that $$\mathcal {M}_{\mathrm {MO}}=g^{-1}(\{0\})$$ . It is easily checked that for each $$X \in \mathcal {M}_{\mathrm {MO}}$$ ,

(7.24)

is surjective. This proves that g is a submersion [88], and therefore that $$\mathcal {M}_{\mathrm {MO}}$$ is a smooth submanifold of $$\mathbb {R}^{N_b \times N}$$ of dimension NbN − N(N + 1)∕2. The tangent space to $$\mathcal {M}_{\mathrm {MO}}$$ at some $$X \in \mathcal {M}_{\mathrm {MO}}$$ is

(7.25)

Similar computations show that $$\mathcal {M}_{\mathrm {DM}}$$ is a smooth submanifold of $$\mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}$$ of dimension N(Nb − N), and that its tangent space at $$P \in \mathcal {M}_{\mathrm {DM}}$$ is the vector space of matrices

If $$P \in \mathcal {M}_{\mathrm {DM}}$$ , the range and kernel of P are respectively of dimension N and Nb − N, and are orthogonal. They are respectively called the occupied and virtual subspaces. Introducing an orthogonal matrix $$U_0 \in \mathbb {R}^{N_b \times N_b}$$ whose first N columns form an orthonormal basis of Ran(P) and the last Nb − N an orthonormal basis of Ker(P), we have

(7.26)

and—dropping the subscripts indicating the sizes of the null matrices to simplify the notation—

(7.27)

where the subscript ov stands for occupied-virtual. The map $$\Phi _P : \mathbb {R}^{N \times (N_b - N)} \rightarrow \mathcal {M}_{\mathrm {DM}}$$ defined by

(7.28)

(7.29)

is surjective and its restriction to a neighborhood of 0 defines a local map of $$\mathcal {M}_{\mathrm {DM}}$$ in the neighborhood of P.

7.2.4 First-Order Optimality Conditions

As we have seen in the previous sections, the standard Kohn–Sham model boils down after discretization to an equality constrained optimization problem, which admits two equivalent natural formulations

the orbital formulation (7.9) (i.e. an optimization problem on a Stiefel manifold);

the density matrix formulation (7.18) (i.e. an optimization problem on a Grassmann manifold).

The orbital formulation (7.9) of the standard Kohn–Sham model is an equality constrained optimization problem of the form

(7.30)

where $$g:\mathbb {R}^{N_b \times N} \to \mathbb {R}^{N \times N}_{\mathrm {sym}}$$ is defined in (7.24) (recall that we assume an orthonormal basis and have set E(X) := EΞ(X)). As seen in the previous section, $${\mathcal M}_{\mathrm {MO}}$$ is a smooth submanifold of $$\mathbb {R}^{N_b \times N}$$ and $$ T_X{{\mathcal M}_{\mathrm {MO}}}$$ is defined in (7.25). Assuming that the function $$P \mapsto {\mathcal E}_{\mathrm {xc}}^\Xi (P)$$ is differentiable at each point of $${\mathcal M}_{\mathrm {DM}}$$ , the energy functional E is differentiable at each point of $${\mathcal M}_{\mathrm {MO}}$$ . Endowing $$\mathbb {R}^{N_b \times N}$$ with the Frobenius inner product defined by $$(X_1,X_2)_{\mathrm {F}}:=\mathrm {Tr}(X_1^TX_2)$$ , its gradient is given by

(7.31)

where

is the Kohn–Sham matrix (also called mean-field Hamiltonian) associated with the density matrix P = XXT. The first-order optimality condition associated with (7.30) is then

(7.32)

where $$g'(X)^\ast : \mathbb {R}^{N \times N}_{\mathrm {sym}} \to \mathbb {R}^{N_b \times N}$$ is the adjoint of g′(X). A simple calculation shows that

(7.33)

Putting together (7.31)–(7.33) we finally obtain the Euler–Lagrange equations associated with (7.9):

(7.34)

Interestingly, the gauge invariance (7.14) can be used to simplify the search for critical points of (7.9). Consider a solution (X,  Λ) to the Euler–Lagrange equations (7.34). Since the Lagrange multiplier Λ of the constraint g(X) = 0 is a symmetric matrix, it can be diagonalized in an orthonormal basis:

where U ∈ O(N) is an orthogonal matrix and ε₁ ≤⋯ ≤ εN are the N eigenvalues of Λ, counted with their multiplicities. Observing that (XU)T(XU) = IN, (XU)(XU)T = XXT = P, and $$H(P)(XU)=XU \mbox{diag}\left ( \varepsilon _1,\ldots ,\varepsilon _N \right )$$ , we obtain that, up to replacing X by XU, all the solutions to the Euler–Lagrange equations (7.34) can be obtained by orthogonal transforms from the solutions to the so-called Kohn–Sham equations

(7.35)

where $$\widetilde X_1,\ldots ,\widetilde X_N$$ are the N columns of the matrix XU. In the following, we drop the tildes for simplicity of notation.

Remark

In the general case of a possibly non-orthonormal basis, the Kohn–Sham equations read

(7.36)

The Kohn–Sham equations therefore have the structure of a nonlinear generalized eigenvalue (or rather eigenvector) problem since the Kohn–Sham matrix H(P) depends on the vector space spanned by its low-energy generalized eigenvectors Xi’s, 1 ≤ i ≤ N, through the density matrix P.

Remark

The equations above are the discrete versions of the continuous Kohn–Sham equations, which are derived from the Euler–Lagrange equations of the variational problem (7.1) in the same way as above, and which read, in the LDA case, as

(7.37)

where $$V_{\mathrm {xc}}[\rho ](\mathbf {r}) = \frac {\mathrm{d} e_{\mathrm {xc}}^{\mathrm {HEG}}}{\mathrm{d} \rho }(\rho (\mathbf {r}))$$ .

It can be checked that any critical point P of the density matrix formulation of the standard Kohn–Sham model (7.18) is such that

In addition, the above conditions are equivalent to

(7.38)

Conditions (7.38) also read

(7.39)

or equivalently

(7.40)

Indeed, considering a generic element P in $$\mathcal {M}_{\mathrm {DM}}$$ , and using the representation (7.26) of P, we have

The matrix $$4H_{\mathrm {ov}} \in \mathbb {R}^{N \times (N_b-N)}$$ can be identified with the gradient at Aov = 0 of the function $$A_{\mathrm {ov}} \mapsto {\mathcal E}(\Phi _P(A_{\mathrm {ov}}))$$ , that is, of the discretized Kohn–Sham functional in the local map ΦP defined by (7.28)–(7.29). Likewise, the matrix 2[[H(P), P], P] is the orthogonal projection of the gradient $$\nabla {\mathcal E}(P)=2H(P)$$ on $$T_P\mathcal {M}_{\mathrm {DM}}$$ , the tangent space to the manifold $$\mathcal {M}_{\mathrm {DM}}$$ at P, for the Frobenius inner product. The geometrical interpretation of the equivalent conditions (7.39) and (7.40) is that the gradient at Aov = 0 of the Kohn–Sham energy functional in the local map ΦP vanishes, and that the gradient of $$ {\mathcal E}$$ at P is orthogonal to the manifold $$\mathcal {M}_{\mathrm {DM}}$$ at this point.

The reader familiar with the Hartree–Fock model may have noticed that the discretized Kohn–Sham equations (7.35) have a similar structure as the discretized Hartree–Fock equations, which is not surprising since the minimizing sets and the gauge invariance properties are the same for the two models. There is however an important difference between the two models. For the (spin-unrestricted) Hartree–Fock model, it can be proved that if Φ = (φ₁, …, φN) is a minimizer of the Hartree–Fock model, then ε₁ ≤⋯ ≤ εN are the lowest N eigenvalues of the Hartree–Fock Hamiltonian (Aufbau principle), and that there is a positive gap between εN and εN+1 (no unfilled-shell property [9]). In addition, any minimizer of the Hartree–Fock energy functional on the convex hull $$\mbox{Conv}(\mathcal {M}_{\mathrm {DM}})$$ is on $$\mathcal {M}_{\mathrm {DM}}$$ . For Kohn–Sham models, these properties do not hold true for all molecular systems.

Let us now turn to the extended and finite-temperature Kohn–Sham models. We assume here that the functional $$P \mapsto {\mathcal E}^\Xi (P)$$ is differentiable at each point of $$\mathrm {Conv}(\mathcal {M}_{\mathrm {DM}})$$ . Since $$\mathrm {Conv}(\mathcal {M}_{\mathrm {DM}})$$ is a non-empty closed convex subset of $$\mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}$$ , the first-order optimality conditions for (7.20) are given by the Euler inequality

which also reads

(7.41)

It is well-known (see e.g. [25]) that any $$P \in \mathrm {Conv}(\mathcal {M}_{\mathrm {DM}})$$ for which (7.41) holds is of the form

(7.42)

Otherwise stated, any critical point P of (7.20) is of the form

(7.43)

where the occupation numbers of the Kohn–Sham orbitals Xi satisfy

(7.44)

where μ is the Lagrange multiplier of the constraint Tr(P) = N, called the Fermi level. In words, the Kohn–Sham energy levels below the Fermi level μ are fully occupied, the ones above the Fermi level are unoccupied, while the ones just at the Fermi level can be partially occupied. Two cases can be encountered:

1.

either there is a gap between the Nth and (N + 1)st energy levels (i.e. if εN < εN+1), in which case $$P \in \mathcal {M}_{\mathrm {DM}}$$ is a critical point of the standard Kohn–Sham problem (7.18) satisfying the Aufbau principle, and μ can be any number in the range (εN, εN+1);

2.

or μ is a degenerate energy level of H(P), in which case the occupation numbers of the Kohn–Sham orbitals at the Fermi level are not determined by the first-order optimality condition and are not known a priori; if P is a minimizer of (7.41), they can be obtained numerically by considering the second-order optimality condition (or higher-order conditions in case of degeneracies).

Lastly, the first-order optimality conditions associated with the finite-temperature Kohn–Sham model (7.22) inferred from Euler’s inequality are

(7.45)

where $$\beta =\frac {1}{k_{\mathrm {B}}T}$$ is the inverse temperature, fβ the Fermi-Dirac function (fβ(ε) = (1 + eβε)⁻¹) and μ the Lagrange multiplier of the constraint Tr(P) = N. In terms of Kohn–Sham orbitals, (7.45) can be reformulated as (7.43), together with

(7.46)

Formally, the condition (7.44) on the occupation numbers for the extended Kohn–Sham model is the limit when T goes to zero of the condition (7.46) on the occupation numbers for the finite-temperature Kohn–Sham model.

7.2.5 Quantities of Interest

We have above explained how to define the ground-state energy of a molecular system in the standard Kohn–Sham approximation in a given atomic configuration $$\mathbf {R}=({\mathbf {R}}_1,\ldots ,{\mathbf {R}}_M) \in \mathbb {R}^{3M}$$ and for a given basis set Ξ as a problem of the form

(7.47)

where $$\mathbf {R} \in \mathbb {R}^{3M}$$ are the atomic positions, and dropping the explicit dependence on Ξ for simplicity.

That computation is by itself of limited interest. It however opens the door to a variety of useful analyses, grouped under the umbrella term of “properties”. For instance, the force exerted on nucleus k by the electrons is given by

Computing forces allows for the efficient optimization of geometries, and for molecular dynamics (integrating the classical equations of motion for the nuclei).

We now detail how to compute these forces. For simplicity, we will assume that the overlap matrix SΞ is set to the identity for all R. When this is not the case (as for instance with atom-centered basis functions), then additional terms, called Pulay forces, appear, which can easily be computed through Lagrange multipliers [25].

Assume that the variational problem (7.47) has a unique global minimizer P(R) that depends smoothly on R. Then we can compute

This formula can be greatly simplified by noticing that, since P(R) minimizes $${\mathcal E}$$ , by the first-order optimality conditions, the linear functional $$\frac {\partial \mathcal E}{\partial P}(\mathbf {R},P(\mathbf {R})) = H(\mathbf {R}, P(\mathbf {R}))$$ vanishes on the tangent space to $${\mathcal M}_{\mathrm {DM}}$$ at P(R), to which $$\frac {\partial P}{\partial {\mathbf {R}}_{k}}(\mathbf {R})$$ belongs. It follows then that

(7.48)

which can be computed easily (it is simply the discrete version of the classical electrostatic force exerted on nucleus k by the electron cloud). The above statement, that the derivatives of the energy with respect to external parameters do not involve the derivatives of variational quantities, is known as the (generalized) Hellmann–Feynman theorem.

In this fashion, one can compute first derivatives of the energy with respect to other external parameters, such as an external field, or cell parameters in the case of solids (giving access to stresses). Other properties of interest, such as force constants (the second derivatives of the energy with respect to atomic positions at equilibrium geometry), polarizabilities (the derivatives of the polarization with respect to an external electric field) and even more complex quantities (piezoelectric properties, anharmonic effects …) can be accessed through perturbation theory, as we outline now. Consider the problem of computing $$\delta P = \frac {\partial P}{\partial {\mathbf {R}}_{k}}\cdot \delta {\mathbf {R}}_{k}$$ , the variation δP in the density matrix caused by a displacement δRk of nucleus k. Starting from the equation

we get

(7.49)

We now decompose this equation in an orthonormal basis $$(X_{n})_{n=1,\dots ,N_{b}}$$ of eigenvectors of H: HXn = εnXn with $$\varepsilon _{1} \le \dots \le \varepsilon _{N_{b}}$$ . We assume a gap εN < εN+1. In this basis,

Differentiating the equation P² = P, we obtain that δPmn = 0 whenever 1 ≤ m ≤ N < n or 1 ≤ n ≤ N < m, and using (7.49), we obtain

(7.50)

Since the right-hand side depends on δP, this is a self-consistent linear equation, called the Dyson equation. It is the linearized version of the nonlinear self-consistent equation $$P = {\mathbbm 1}_{(-\infty ,\mu )}(H(P))$$ , and can be solved using the same tools.

This methodology can be extended in several directions:

other types of perturbations (such as the derivative of the density with respect to an external field, or second derivatives of the energy) can be computed in the same way;

equation (7.50) can be written in terms of orbitals (the resulting equation sometimes being called the Sternheimer equation [136]), avoiding a full diagonalization of the Hamiltonian;

higher-order properties can be computed. A notable guiding principle is Wigner’s 2n + 1 rule, which generalizes the Hellmann–Feynman theorem: to compute the (2n)th and (2n + 1)st derivatives of the energy, one only needs to compute the first nth derivatives of the density matrix;

the methodology can be applied to time-dependent models, resulting in frequency-dependent response properties.

We refer to [13] for the condensed-matter perspective, and to [109] for the quantum chemistry one.

7.3 Discretization Methods

The current most popular software is based on one of the following two discretization methods: Gaussian-type orbitals (GTOs), and planewaves (PW). The former is the method of choice in quantum chemistry for molecules and clusters. The latter is widely used in solid-state and condensed-matter physics.

GTOs were first introduced by Boys [19] in 1950 to discretize the Hartree–Fock model and have been at the origin of the rapid development of quantum chemistry in the second half of the 20th century. PW basis sets are natural discretization methods for solving Kohn–Sham-type equations on rectangular boxes with periodic boundary conditions [44, 56, 71, 86], as is the case for periodic solids (perfect crystals), and for supercell methods used to simulate disordered crystals, amorphous solids, and liquids.

Other approaches include general-purpose methods for discretizing partial differential equations such as (extended) finite element [80, 147], finite difference [40, 58, 64], spline [105], and wavelet methods [8, 63], as well as specific methods: other kinds of atomic orbitals (Slater-type orbitals [133], numerical atomic orbitals [134]), Augmented plane waves (APW)-type orbitals [4, 42, 129, 132], periodic sinc function basis [131], adaptive local-basis sets [95, 148]. For the sake of brevity, we focus in this chapter on GTO and PW methods. Other types of discretization methods are dealt with in subsequent chapters.

7.3.1 Atomic Basis Sets for Molecules

In atomic basis sets methods, each chemical element A is equipped with a finite number of rapidly decaying L²-normalized basis functions $$(\xi _\mu ^{\mathrm {A}})_{1 \le \mu \le N_{\mathrm {A}}} \in H^1(\mathbb {R}^3)$$ centered at the origin such that $$\mbox{Span}(\xi _\mu ^{\mathrm {A}}, \, 1 \le \mu \le N_{\mathrm {A}})$$ is rotationally invariant. An atomic orbital basis set is a collection $$((\xi _\mu ^{z})_{1 \le \mu \le N_{z}})_{z}$$ , where z spans the periodic table of the elements, or usually only a part of it (e.g. the first four rows, from hydrogen to xenon). Consider now a molecular system consisting of M nuclei of atomic numbers z₁, …zM in a configuration $$({\mathbf {R}}_1,\ldots , {\mathbf {R}}_M) \in (\mathbb {R}^3)^M$$ . The basis set used to carry out a Kohn–Sham calculation in the atomic basis set $$((\xi _\mu ^{z})_{1 \le \mu \le N_{z}})_{z}$$ is then

Note that in atomic basis sets, the basis functions are attached to the nuclei and therefore move with them: for a given molecular system and a given atomic orbital basis set $$((\xi _\mu ^{z})_{1 \le \mu \le N_{z}})_{z}$$ , the discretization basis depends on the configuration (i.e. of the positions of the nuclei). As already mentioned in Sect. 7.2.5, this gives rise to the so-called Pulay forces [118] in molecular dynamics.

For consistency with atomic symmetries, the functions $$\xi _\mu ^{z}$$ usually are of the form

(7.51)

where $$(Y_l^m)_{l \ge 0, \, -l \le m \le l}$$ are the real spherical harmonics (which forms an orthonormal basis of $$L^2({\mathbb S}^2)$$ ), and $$f:\mathbb {R}_+ \to \mathbb {R}$$ a smooth, fast decaying function. Note that if $$(\xi _\mu ^{z})_{1 \le \mu \le N_z} \in H^1(\mathbb {R}^3)$$ contains a function of the form (7.51), then $$Y_l^{m'}\left ( \frac {\mathbf {r}}{|\mathbf {r}|} \right ) |\mathbf {r}|{ }^l f(|\mathbf {r}|) \in \mbox{Span}((\xi _\mu ^{z})_{1 \le \mu \le N_z})$$ for all − l ≤ m′≤ l in order to fulfill rotation invariance. Recall that $$|\mathbf {r}|{ }^l Y_l^m\left ( \frac {\mathbf {r}}{|\mathbf {r}|} \right )$$ is a homogenous harmonic polynomial of total degree l.

In most quantum chemistry codes, the functions $$\xi _\mu ^{z}$$ are Gaussian-type orbitals (GTOs), namely finite linear combinations of polynomials times Gaussian functions, that is

with αn > 0 and $$c_n \in \mathbb {R}$$ . A few quantum chemistry codes use Slater-types orbitals (STOs) for which

Compared to GTOs, STOs better account for the cusps of the Kohn–Sham orbitals at the nuclear positions as well as their decay rates at infinity. On the other hand, GTOs have a tremendous advantage over STOs: as pointed out by Boys [19], all the integrals in (7.12)–(7.13) can be computed explicitly for GTOs using three key properties of Gaussian functions:

1.

the product of two Gaussian functions is a Gaussian function;

2.

the Fourier transform of a Gaussian function is a Gaussian function;

3.

the product of a Gaussian function by a homogeneous polynomial of order l is a linear combination of partial derivatives of order l of this Gaussian function, and vice-versa.

As a consequence, the kinetic, nuclei-electrons interaction, and Coulomb terms can be computed explicitly for GTOs. Only the exchange-correlation term must be computed on a grid. In most quantum chemistry packages, the quadrature points are obtained by unions of atomic quadrature grids. More precisely,

each chemical element A is associated with a spherical quadrature grid $$({\mathbf {r}}_g^A)_{1 \le g \le N_A}$$ which is itself the tensor product of a radial grid by a set of Lebedev points on the unit sphere $${\mathbb S}^2$$ , and associated weights $$(w_g^A)_{1 \le g \le N_A}$$ ;

the quadrature grid for a molecular system with M nuclei of type Ak located at Rk, 1 ≤ k ≤ M, is the union of all the points $${\mathbf {r}}_{k,g}:={\mathbf {r}}_g^{A_k}+ {\mathbf {R}}_k$$ for 1 ≤ k ≤ M and $$1 \le g \le N_{A_k}$$ . The weight of the quadrature point rk,g is given by the formula

where

and $$f:[-1,1]\to \mathbb {R}_+$$ is a smooth function satisfying f(−1) = 0, f(0) = 1∕2 and f(1) = 0. Usual choices include Becke and Stratmann–Scuseria–Frisch (SSF) partitioning functions. As a matter of example, the former is defined as

For very large molecular systems, Hamiltonian and overlap matrices are sparse due to the extremely fast decay of Gaussian atomic orbitals. In addition, so are density matrices, at least for insulators, and metals at finite temperature. This phenomenon, sometimes called the nearsightedness principle in the physics and chemistry literature [115], originates from the decay properties of the Green kernel of the resolvent of Schrödinger operators. This allows the use of sparse linear algebra methods (see [16] and references therein). The main bottleneck is then the building of the Kohn–Sham matrix

for the current density matrix P. The first term hΞ is simple to deal with since its O(Nb) non-zero entries (at numerical precision) are explicit and easy to compute for Gaussian orbitals. The third term is local and can be computed in O(Nb) operations also. The truly nonlocal second term involves the Coulomb integrals

The evaluation of the Coulomb term AΞ : P can be accelerated using Fast Multipole Methods (FMMs). The original FMM for point charge distributions, due to Greengard and Rokhlin [72, 73], must be adapted to handle Gaussian-polynomial charge distributions. This has been done by several authors from the 90s on. We will not detail these technicalities here for the sake of brevity and refer the interested reader to [39] and references therein.

7.3.2 Plane-Wave Discretization

We examine now the plane-wave method. Although it was originally formulated for computing properties of crystals and is still mostly used for that purpose, for pedagogical purposes we introduce it here in the case of an isolated neutral molecule.

Note that the solutions φi of the self-consistent Eq. (7.37) for negative εi are exponentially localized (with characteristic length $$1/\sqrt {-\varepsilon _{i}}$$ ). It is therefore justified, with exponentially small error, to limit ourselves to a large box, which we will take for simplicity to be $$\Omega = \left [-\frac L 2,\frac L 2\right ]^{3}$$ . We will replace the space $$\mathbb {R}^{3}$$ by Ω equipped with the topology of a torus (therefore imposing periodic boundary conditions). We can then expand orbitals φi in the orthonormal Fourier basis $$\{e_{\mathbf {K}}\}_{\mathbf {K} \in \mathcal R^*_{L}}$$ , with

and $$\mathcal R^*_{L} = \frac {2 \pi }{L} \mathbb {Z}^{3}$$ . We then have the expansion

The kinetic, potential and exchange-correlation terms in (7.2) and (7.4)–(7.5) adapt easily by simply truncating the integrals to Γ. To approximate the Hartree term in a way that leads to simple computations, we replace the Coulomb kernel $$\frac 1 {|\mathbf {r}|}$$ by the periodic Coulomb kernel

(7.52)

Then, we seek to minimize the total energy

StartLayout 1st Row 1st Column upper E Subscript upper L Baseline left parenthesis upper Phi right parenthesis 2nd Column equals integral Underscript upper Omega Endscripts left parenthesis sigma summation Underscript i equals 1 Overscript upper N Endscripts StartAbsoluteValue nabla phi Subscript i Baseline EndAbsoluteValue squared plus upper V rho Subscript upper Phi Baseline right parenthesis 2nd Row 1st Column Blank 2nd Column plus one half integral Underscript upper Omega times upper Omega Endscripts rho Subscript upper Phi Baseline left parenthesis bold r right parenthesis rho Subscript phi Baseline left parenthesis bold r prime right parenthesis upper G Subscript upper L Baseline left parenthesis bold r minus bold r prime right parenthesis normal d bold r normal d bold r prime plus upper E Subscript x c Baseline left parenthesis rho Subscript upper Phi Baseline right parenthesis EndLayout

variationally by limiting the discretization space $$\{e_{\mathbf {K}}\}_{\mathbf {K} \in \mathcal R^*_{L}}$$ to the subspace

where Ecut > 0 is a truncation parameter that controls the maximum kinetic energy allowed in the system.

The resulting energy as a function of the coefficients ciK is

where

are the Fourier coefficients of the periodic extension of V .

Naively evaluated, these terms scale quadratically with both N and the number of plane waves $$|\mathcal R^*_{E_{\mathrm {cut}}}|$$ . However, many of these computations are convolutions, arising from pointwise multiplication in real space. These convolutions can be evaluated efficiently in real space, using the discrete convolution theorem: the cyclic convolution of two arrays can be computed by a discrete Fourier transform, evaluated efficiently using fast Fourier transforms (FFTs). To avoid aliasing effects arising from the cyclic convolution, zero-padding is used: the discrete Fourier transforms are performed on a Cartesian grid that contains all the K + K′, for $$\mathbf {K},\mathbf {K}' \in \mathcal R^*_{E_{\mathrm {cut}}}$$ .

It is important to note that this procedure allows us to treat the kinetic, potential and Hartree terms exactly: for any given set of coefficients ciK, the energy terms computed in this way are the exact energy terms of the orbitals $$\varphi _{i}(\mathbf {r}) = \sum _{\mathbf {K}\in \mathcal R^*_{E_{\mathrm {cut}}}} c_{i \mathbf {K}} e_{\mathbf {K}}(\mathbf {r})$$ . Therefore, for the rHF model, the plane-wave method (for a given L) is variational: the ground state energy decreases with Ecut. However, the exchange-correlation term is a non-polynomial function of the density, and cannot be evaluated exactly in this fashion. In practice, this term is approximated by an integration on the same real-space grid as for the other terms.

The numerical analysis of this method and of the related Gross–Pitaevskii equation is discussed in Sect. 7.6. Note that this method is based on an expansion of the orbitals φi in a Fourier basis. In order for this to be effective, the orbitals need to be smooth, because of the equivalence between smoothness in real space and decay in reciprocal space. However, the singularity of the Coulomb potential imposes cusps on the φi: for instance, the first eigenfunction of the Hydrogen atom is proportional to e−|r|. Further, even if we remove these cusps (for instance, by mollifying the Coulomb potential), the φi still need to oscillate to satisfy the orthogonality conditions φi ⊥ φj, and these oscillations need a large number of plane waves to represent properly. The plane-wave discretization is therefore not applicable directly to atomic systems; in practice, these problems are remedied through the use of the pseudopotential approximation.

7.4 Pseudopotentials

The pseudopotential method is a technique developed in solid-state physics to modify Vext so that electrons close to a nucleus are treated as part of an ionic core. Only wavefunctions associated with valence electrons are to be computed. We refer to [65] and references therein for an overview of the pseudopotential method in practice. A very similar method is used in quantum chemistry, where it goes by the name of “effective core potential”; we refer to [49] for details. To simplify the discussion, we assume an isolated system of non-interacting electrons (we ignore the Hartree and exchange-correlation terms). With appropriate modifications, pseudopotentials can be extended to treat the Kohn–Sham equations.

Reintroducing spin temporarily, recall that the states of an atomic radial Hamiltonian $$H_{\mathrm {at}} = -\frac 1 2 \Delta + V_{\mathrm {at}}$$ can be labelled as φnℓmσ:

where n ≥ 1, $$\ell \geqslant 0$$ , m = −ℓ, …, ℓ and σ ∈{α, β} are the principal, azimuthal, magnetic and spin quantum numbers. The functions Yℓm are the real spherical harmonics. The functions Rnℓ are solutions of the radial Schrödinger equation

for r > 0, with Rnℓ(0) = 0 and $$\int _{0}^{\infty } |R_{n\ell }|{ }^{2} = 1$$ .

There are 2(2ℓ + 1) available states (φnℓmσ)m=−ℓ,…,ℓ,σ={α,β} for a given energy level εnℓ. The electronic state of an atom is conventionally given as a sequence of terms of the form (n + ℓ)ℓk, meaning that k of the 4ℓ + 2 available states with a given (n, ℓ) are occupied. The angular momentum quantum number ℓ is labelled using letters: s, p, d, f, etc. For instance, silicon has 14 electrons, and its electronic structure is 1s² 2s²2p⁶ 3s²3p². This means: two electrons in the φn=1,ℓ=0,m=0,σ={α,β} orbitals, two electrons in the φn=2,ℓ=0,m=0,σ={α,β} orbitals, six electrons in the φn=1,ℓ=1,m={−1,0,1},σ={α,β} orbitals, etc. The spatial extension $$\langle r \rangle := {\int _{\mathbb {R}^{3}}|\mathbf {r}| |\varphi |{ }^{2}(\mathbf {r}) \mathrm{d} \mathbf {r}}$$ and energies ε of these states, computed using the PBE density functional [110], are given Table 7.1, and the orbitals are plotted Fig. 7.1 (left panel).

2-line graphs. Both graphs plot the phi function of r versus r. The first graph has declining and then stable curves for 1 s, 2 s, 2 p, 3 s, and 3 p. They reach 0 functional value. Second graph has curves for 3 p all electron, 3 s pseudo, 3 s all electron and 3 p pseudo.

Fig. 7.1

Orbitals of the Si atom at the PBE level (left), and pseudized 3s and 3p orbitals (obtained with use of the pseudo-potential) (right). The pseudized orbitals are nodeless, and free of cusps

Table 7.1

Energy ε and spatial extension 〈r〉 for the orbitals of the silicon atom at the PBE level. Data from the atomic code included in the Quantum Espresso distribution. For comparison, the energy of a typical covalent bond is about 0.1 Ha, and the interatomic distance in bulk silicon at ambient conditions is about 4.5 Bohr

State

(n, ℓ)

εnℓ (Ha)

〈r〉 (Bohr)

1s²

(1, 0)

−64.4

0.11

2s²

(2, 0)

−5.10

0.57

2p⁶

(1, 1)

−3.51

0.54

3s²

(3, 0)

−0.40

2.17

3p²

(2, 1)

−0.15

2.79

From the data in Table 7.1, it can be seen that electrons can be clearly separated into two classes. The 10 electrons in the 1s, 2s and 2p configurations are core electrons. They are mainly localized close to the nucleus, with a low energy. This means that they react very little to their environment, as can be shown using the following heuristic argument. Assume that atom A has only one orbital φA, with energy EA, and that atom B has only one orbital φB, with energy EB > EA. When in the diatomic configuration, the total Hamiltonian, expanded in the basis of the φA and φB, has the approximate form

where the coupling term α is due to the overlap between φA and φB. The eigenvalues of this matrix are

In order to have E± significantly different from the isolated energies EA and EB (which leads to chemical bonding), we need α to be at least of comparable magnitude to EB − EA. In other words, to get a bond between two atoms, there needs to be orbitals with a significant overlap which are close in energy.

To a very good approximation, we can therefore consider the 1s, 2s and 2p orbitals of silicon as core orbitals, that are frozen: they are the same in any chemical environment. By contrast, the 3s and 3p orbitals are valence orbitals, and bind to the orbitals of other atoms.

It follows from this splitting that representing all the φi is wasteful, since the core electrons do not contribute significantly to chemical bonding. On the other hand, the valence electrons still feel the influence of the core electrons, and have strong oscillations near the nuclei. Combined with the cusp at the nucleus, this makes them very hard to represent in a plane wave basis. A further complication is that the core electrons of heavy atoms, having a high velocity, are subject to non-negligible relativistic effects. These impact the valence orbitals indirectly through the orthogonalization requirements and the mean-field potential. All these difficulties are remedied through pseudopotentials, at the cost of an approximation.

To introduce pseudopotentials, we first split the energy states εnℓ of the atom into core and valence orbitals. We ignore spin in the following discussion, and label the orbitals only by φnℓm. We take ℓ = 0, …, ℓmax to be set the of angular momenta for which there is a valence orbital, $$\mathcal I_{\ell } = \{n \in \mathbb {N}, \varepsilon _{n\ell } \text{ is a valence state}\}$$ and $$n_{\mathrm {core},\ell } = \min \mathcal I_{\ell } - 1$$ the number of core orbitals with angular momentum ℓ. For instance, for the Silicon atom a natural choice is to set the 1s, 2s and 2p orbitals as core, and the 3s and 3p orbitals as valence: $$\ell _{\mathrm {max}} = 1, \mathcal I_{0} = \{3\}, \mathcal I_{1} = \{2\}$$ , ncore,0 = 2, ncore,1 = 1.

The pseudopotential method then replaces the atomic potential Vat by a pseudo-potential $$\widetilde V_{\mathrm {at}}$$ (usually a nonlocal operator) such that the lowest energy eigenfunctions of the pseudo-Hamiltonian $$\widetilde H_{\mathrm {at}} = -\frac 1 2 \Delta + \widetilde {V}_{\mathrm {at}}$$ , the pseudo-orbitals $$\widetilde \varphi _{n\ell m }$$ for $$\ell = 0,\dots ,\ell _{\mathrm {max}}, n = 1,\dots ,|\mathcal I_{\ell }|$$ , are smooth and match the valence orbitals $$\varphi _{n+n_{\mathrm {core},\ell },\ell m }$$ of the real Hamiltonian $$H_{\mathrm {at}}=-\frac 1 2 \Delta + {V_{\mathrm {at}}}$$ outside the core of the atom. In silicon, this means for instance that the new 1s state of $$\widetilde H$$ matches the 3s state of H outside a chosen cutoff radius rc (see Fig. 7.1 (right panel)). Since the decay rate of an eigenstate with a negative eigenvalue ε is $$\sqrt {-\varepsilon }$$ , it follows that the eigenvalues of $$\widetilde H_{\mathrm {at}}$$ should match those of Hat:

for $$\ell = 0,\dots ,\ell _{\mathrm {max}}, n = 1,\dots ,|\mathcal I_{\ell }|, m = -\ell ,\dots ,\ell $$ .

Once this procedure is established for an atom, all the individual potentials of the atoms in a molecule can be replaced (“pseudized”) in this fashion, and the pseudo-Hamiltonian of the molecule is then solved.

Setting aside the construction of these pseudopotentials for a moment, two main issues have to be considered. First, transferability: to what extent do the results of a pseudopotential computation on a molecule match the results of the corresponding reference (all-electron) computation? Second, softness: how many Fourier modes are needed to represent the pseudo-orbitals accurately? Pseudopotentials seek an optimal compromise between these two competing objectives. Transferability is usually assessed empirically, by comparing the properties of a molecular system to those obtained using an all-electron computation. It is improved by choosing a small cutoff radius rc and ensuring that all relevant electrons are included as valence. Softness is ensured by choosing a smooth pseudopotential $$\widetilde V_{\mathrm {at}}$$ and by the fact that the lowest energy eigenfunction of a radial local Hamiltonian is nodeless, which reduces oscillations (see Fig. 7.1).

Building pseudopotentials is often done in two parts: given a local potential $$V_{\mathrm {at}}(\mathbf {r}) = -\frac Z{|\mathbf {r}|}$$ , first construct smooth pseudo-orbitals $$\widetilde \varphi _{n\ell m }$$ , and second construct a pseudopotential $$\widetilde V_{\mathrm {at}}$$ such that the eigenstates of $$\widetilde H$$ are the $$\widetilde \varphi _{n\ell m }$$ .

The radial part $$\widetilde R_{n\ell }$$ of the pseudo-orbitals $$\widetilde \varphi _{n\ell m }$$ should match the original orbitals outside of a cutoff radius:

As eigenstates of $$\widetilde H$$ , they should also be orthonormal. Orbitals with different angular momenta are automatically orthogonal; the orbitals with the same angular momenta must satisfy

which implies the “generalized norm-conservation condition”

(7.53)

for all $$m, n \in \mathcal I_{\ell }$$ . Finally, they should be smooth, and, as low-energy eigenfunctions, have the correct amount of nodes.

Historically, most pseudopotentials have been constructed assuming the pseudization of only one orbital per angular momentum: $$|\mathcal I_{\ell }| = 1$$ for ℓ = 0, …, ℓmax. This is for instance reasonable in the silicon example above. For simplicity, we denote these orbitals and pseudo-orbitals by φℓm and φℓm, with energies εℓ. The Troullier–Martins procedure [139] then postulates a simple functional form for $$\widetilde R_{\ell }$$ , and adjusts its parameters to match the values and the derivatives up to fourth order of $$\widetilde R_{\ell }$$ and Rℓ at rc. The RRKJ procedure expands $$\widetilde R_{\ell }$$ in a basis of Bessel functions, and minimizes their kinetic energy above a certain cutoff energy to ensure smoothness [120].

Once the $$\widetilde \varphi _{\ell m }$$ are constructed, a potential $$\widetilde V_{\mathrm {at}}$$ is constructed. A solution is to first choose the local radial potential $$\widetilde V_{\ell }$$ so that $$\widetilde \varphi _{\ell m }$$ is a solution of

This can be done by simply solving the equation above for $$\widetilde V_{\ell }$$ . Then, we can choose the semilocal form

(7.54)

where Pℓ =∑m=−ℓ,…,ℓ|Yℓm〉〈Yℓm| is the projector on angular momentum ℓ. This operator automatically has the $$\widetilde \varphi _{\ell m }$$ as eigenstates with correct eigenvalues, completing the pseudo-potential.

This procedure however has a major defect: the operator $$\widetilde V_{\mathrm {at}}$$ is a nonlocal operator whose evaluation in plane-wave codes requires the computation of a dense matrix. This problem was solved by Kleinmann and Bylander [82], who realized that the above procedure was wasteful; the most economical operator that will produce the $$\widetilde \varphi _{\ell m }$$ as eigenstates is the low-rank form

(7.55)

where $$\widetilde V_{\mathrm {loc}}$$ is an arbitrary radial local potential and

By construction, $$(-\frac 1 2 \Delta + \widetilde V_{\mathrm {at}}^{\mathrm {KB}})\widetilde \varphi _{\ell m } = \varepsilon _{\ell }\widetilde \varphi _{\ell m }$$ , as desired. The potential $$\widetilde V_{\mathrm {loc}}$$ is usually chosen to remove one of the ℓ components from the sum.

Note that we have only insisted on the fact that the $$\widetilde \varphi _{\ell m }$$ are eigenfunctions of $$\widetilde H_{\mathrm {at}}$$ at energies εℓ, but have said nothing of the other states; it may very well happen that $$\widetilde H_{\mathrm {at}}$$ has states with lower energies (“ghosts”), invalidating the pseudopotential. This is usually treated heuristically, by varying parameters in the pseudopotential construction to avoid the appearance of ghosts.

The procedure outlined above is the basis for the construction of the historically successful Troullier–Martins and RRKJ pseudopotentials. More modern pseudopotentials try to reproduce more than one state per angular momentum. This is especially useful for highly-localized valence orbitals (like the 2p orbitals of oxygen), or semicore orbitals, which are not well isolated from the valence orbitals. In this case, one cannot use the semilocal form (7.54), because no single potential Vℓ can reproduce all the pseudo-orbitals at the required energies. However, the form (7.55) naturally generalizes to this case: we seek a low-rank potential such that $$\widetilde H_{\mathrm {at}} \widetilde \varphi _{n\ell m } = \varepsilon _{n\ell } \widetilde \varphi _{n\ell m }$$ for i = 1, …, nℓ, and obtain

(7.56)

where the βnℓm are defined as above, and

From the generalized norm-conservation property (7.53), $$\langle \widetilde \varphi _{n\ell m}, \widetilde \varphi _{n' \ell m} \rangle = \delta _{nn'}$$ , and it follows that B is Hermitian.

The scheme above was used by Hamann [74], who constructed pseudo-orbitals satisfying the generalized norm-conservation property (7.53). That work built on an earlier method of Vanderbilt, the “ultra-soft pseudopotentials”, which relaxed the constraint (7.53) by introducing a modified inner product in which (7.53) holds, resulting in a generalized eigenvalue problem.

An alternative to norm-conserving or ultra-soft pseudopotentials is the projector-augmented wave (PAW) method of Blöchl [18], which uses a transformation that allows the recovery of the original all-electron orbitals. This is especially useful for probing properties that depend on fine details of the electrons close to the nuclei, such as nuclear magnetic resonance effects [112], but comes at the price of a more complex formalism, the response computations in particular being much more cumbersome.

Also notable is the “black-box” approach used in [70, 76], where a very simple functional form for the local and nonlocal part of the pseudopotential is postulated, and its parameters adjusted by least-squares fitting to a number of desirable properties, such as the eigenvalues εi and norm conservation.

Despite being extremely successful in practice, the various pseudopotential methods remain uncontrolled approximations which are not systematically improvable. For a given density functional, they are often the major source of variation between computational codes [89]. Their error analysis is still underdeveloped; see [30] for the existence of norm-conserving pseudopotentials which optimally reproduce the first-order Stark effect, and [51] for an analysis of the PAW method for a particular one-dimensional system.

7.5 Algorithms

The development of algorithms for solving Kohn–Sham problems in quantum chemistry, on the one hand, and condensed matter physics, on the other hand, has followed two different paths. Until the mid 90s and the introduction of hybrid functionals, DFT was barely used in quantum chemistry, while the Hartree–Fock model was very popular, on its own—in the early days—or as a first step toward correlated methods (Møller–Plesset perturbation theory, coupled cluster, quantum Monte Carlo…). From a mathematical viewpoint, the Hartree–Fock and zero-temperature standard Kohn–Sham model are very similar: they are minimization problems on the same sets (orbitals or density matrix formulations), and the energy functionals only differ by the exchange-correlation term, which is in some sense a “small” perturbation of the other three terms (kinetic energy, electrons-nuclei interaction, and Hartree term). It was therefore natural for quantum chemists to try and adapt to the Kohn–Sham setting the many algorithms for Hartree–Fock problems discretized in Gaussian basis sets developed since the 50s.

In parallel, in the mid 60s physicists started to develop algorithms for solving Kohn–Sham LDA and GGA models for solids discretized in plane-wave basis sets, and came up with specific techniques such as Anderson’s density mixing [5], Thomas–Fermi charge mixing [119], Kerker’s preconditioning in Fourier [81] or real [130] spaces, etc.

From a numerical point of view, solving the Kohn–Sham problem for molecules in a Gaussian basis set or for solids in a plane-wave basis set are quite different games. In particular,

the self-consistent Hamiltonians in quantum chemistry tend to have a gap between occupied and virtual states, whereas that is not the case for metallic solids and numerical devices such as an artificial finite temperature are needed;

the Kohn–Sham Hamiltonian of a large molecule is sparse and can be stored in memory for a Gaussian basis set, while it is dense and too large to be stored in memory for a plane-wave basis set (only matrix-vector products can be computed in practice);

the number of discretization functions per electron pair is of the order of 2 to 10 for Gaussian basis sets, and typically 100 or more for plane-wave basis sets;

Gaussian basis sets tend to become overcomplete when the size of the basis set increases (i.e. the overlap matrix becomes ill-conditioned), while plane-wave bases remain orthonormal when the energy cut-off increases.

This can explain why the two communities barely interacted until the beginning of the 21st century. The situation is rapidly changing since quantum chemists and materials scientists now share many common interests: hybrid functionals originally developed in quantum chemistry are now used in materials science, quantum chemists simulate extended systems to study, for example, heterogeneous catalysis…

In the following, we will consider the zero-temperature standard Kohn–Sham problem, which can be formulated either as

(7.57)

or

(7.58)

where we recall that $$E(X)=\mathcal {E}(XX^T)$$ . Both formulations are useful in different contexts. The first stores N orbitals of size Nb, and is therefore more economical. However, it breaks a natural symmetry of the problem. Indeed, the energy does not depend on the orbitals themselves, but rather on the subspace they span:

Density matrices quotient this gauge invariance automatically, and are therefore more convenient to work with theoretically. They are also useful in formulating linear scaling methods. However, they require storing and operating with an Nb × Nb matrix, which is prohibitively expensive when large basis sets are used. It is often most convenient to formulate and analyze the algorithms using density matrices, but implement them using orbitals and related quantities (such as the density ρ and self-consistent potential V ). We follow this viewpoint in this presentation.

We will first describe in Sects. 7.5.1 and 7.5.2 the different algorithms used to solve the above problem in its density matrix formulation, without consideration of their community of origin, implementation or computational cost. Then, in Sects. 7.5.3 and 7.5.4, we will consider their implementation in the quantum chemistry and solid-state contexts respectively.

In order to lighten the notation, we assume throughout this section that the discretization basis Ξ is orthonormal, and we will omit the superscript Ξ in the expressions of the core Hamiltonian hΞ, the linear map GΞ defined in (7.17), the discretized Hartree–Fock energy functional $${\mathcal E}^{\mathrm {HF},\Xi }$$ , etc.

7.5.1 Self-Consistent Field Algorithms

7.5.1.1 Simple SCF and Damping

As mentioned above, the main algorithms for solving the Kohn–Sham problem in quantum chemistry were originally developed for the Hartree–Fock (HF) model. In the HF setting, the energy functional $${\mathcal E}^{\mathrm {HF}}$$ is given by (7.16). When the no-unfilled shell property holds true (which is always the case for spin-unrestricted models, see Sect. 7.2.4), the ground state density matrix satisfies

(7.59)

where

where $$H^{\mathrm {HF}}(P):=\nabla {\mathcal E}^{\mathrm {HF}}(P) = h+G(P)$$ is the Fock matrix (the HF Hamiltonian matrix) and μ a Fermi level, which depends on P and is chosen so that $$\mathrm {Tr}({\mathcal F}_{\mathrm {HF-DM}}(P))=N$$ . One way to compute $${\mathcal F}_{\mathrm {HF-DM}}(P)$$ for a given $$P \in {\mathcal M}_{\mathrm {DM}}$$ , is to assemble the matrix HHF(P), diagonalize it, and compute the orthogonal projector on the space spanned by N orthonormal eigenvectors associated with the lowest N eigenvalues (counting multiplicities). It is efficient for small or moderate size basis sets, but other methods must be used for large systems and basis sets. The function $${\mathcal F}_{\mathrm {HF-DM}}(P)$$ is not uniquely defined if the Nth and (N + 1)st eigenvalues of HHF(P) are equal, but it is in the neighborhood of any ground state density matrix by virtue of the no-unfilled shell property.

Self-consistent field algorithms for HF are iterative algorithms aiming at solving (7.59). The simplest of them is the Roothaan algorithm [125] defined by

(7.60)

This algorithm, as well as the level-shifting algorithm [127], has been analyzed in [28] from a mathematical point of view. Introducing the symmetric functional $$E^{\mathrm {HF}}: \mathbb {R}^{N_b \times N_b}_{\mathrm {sym}} \times \mathbb {R}^{N_b \times N_b}_{\mathrm {sym}} \to \mathbb {R}$$ defined by

it can be easily checked, using the symmetry property $$\mathrm {Tr} \left ( G(P) P' \right )=\mathrm {Tr} \left ( G(P') P \right )$$ , that, in the case when the sequence of density matrices generated by the Roothaan algorithm is well defined, it satisfies for all $$k \in \mathbb {N}$$ ,

In other words, the simple SCF scheme amounts to an alternate minimization of the functional EHF on $${\mathcal M}_{\mathrm {DM}} \times {\mathcal M}_{\mathrm {DM}}$$ . It behaves as follows:

either the whole sequence $$(P_k)_{k \in \mathbb {N}}$$ converges to the global minimizer of the Hartree–Fock model;

or the sequences of even and odd iterates $$(P_{2k})_{k \in \mathbb {N}}$$ and $$(P_{2k+1})_{k \in \mathbb {N}}$$ both converge, but to different values Peven and Podd, none of them being a minimizer of the Hartree–Fock energy functional.

The latter behavior is called charge sloshing in the physics and chemistry literature: when k is large (close to convergence), the electronic density $$\rho _{P_k}$$ moves from one part of the molecule for k even to another one for k odd (e.g. from a functional group to another one) and back at the next iteration. This phenomenon is also encountered for isolated atoms where charge sloshing between e.g. 3d and 4s shells can be observed.

Let us now turn to the adaptation of this algorithm to the Kohn–Sham model. We assume that at each step of the algorithm under consideration there is a gap between the Nth and (N + 1)st eigenvalues of the Kohn–Sham Hamiltonian $$H(P_k):=\nabla \mathcal {E}(P_k)$$ , uniformly bounded from below by a positive constant independent of k (uniform well-posedness—UWP—assumption). The Roothaan algorithm for Kohn–Sham, also called the simple SCF algorithm in the physics literature, reads

(7.61)

where, under the UWP assumption,

where μ is chosen as above. In the special case of the Hartree model, for which the exchange-correlation energy functional is set to zero, the energy is quadratic in the density matrix and this algorithm behaves as in the HF setting (convergence or oscillation between two states which are not solutions to the problem). For a non-quadratic exchange-correlation functional, the iterates can oscillate between more than two states, or even display chaotic behavior for some systems [37].

A way to prevent charge sloshing is to use simple mixing, for which the iterates are defined as

(7.62)

where $$\widetilde P_0=P_0$$ and 0 < α < 1 is a fixed mixing parameter (α = 1 corresponds to the simple SCF scheme, in which case $$\widetilde P_k=P_k$$ for each k). Note that unlike the Pk’s, the iterates $$\widetilde P_{k+1}$$ in (7.62) do not live on $${\mathcal M}_{\mathrm {DM}}$$ but in its convex hull $$\mathrm {Conv}({\mathcal M}_{\mathrm {DM}})$$ . From a geometrical viewpoint, $${\mathcal F}_{\mathrm {DM}}(\widetilde P_k)$$ is the point P on $${\mathcal M}_{\mathrm {DM}}$$ , and also actually in $$\mathrm {Conv}({\mathcal M}_{\mathrm {DM}})$$ , for which the slope at t = 0 of the function

is minimal, linking the mixing scheme with the Frank–Wolfe algorithm popular in convex optimization [108]. In particular, this implies that the energy $$\mathcal {E}(\widetilde P_k)$$ decreases along the iterations, for 0 < α small enough. Under the UWP assumption that there is a uniform positive gap between the Nth and (N + 1)st eigenvalues of the Kohn–Sham Hamiltonians $$H(\widetilde P_k):=\nabla \mathcal {E}(\widetilde P_k)$$ and regularity assumptions on $${\mathcal E}$$ , it can be shown that this algorithm “numerically converges” (in the sense that Pk+1 − Pk → 0 and $$\widetilde P_{k} - P_{k} \to 0$$ ) and that the limit (if the iteration actually converges) is a critical point—a local minimizer in practice—of $${\mathcal E}$$ on $${\mathcal M}_{\mathrm {DM}}$$ as well as a critical point—a local minimizer in practice—of $${\mathcal E}$$ in $$\mathrm {Conv}({\mathcal M}_{\mathrm {DM}})$$ [25].

Remark 7.1

For a majority of chemical systems, it is observed that the UWP assumption is satisfied for k large enough, and that the iterates $$\widetilde P_k$$ numerically converge toward a local, hopefully global, minimizer of $$\mathcal {E}$$ on $${\mathcal M}_{\mathrm {DM}}$$ . However, for some systems, the iterates $$\widetilde P_k$$ numerically converge toward a local minimizer of $$\mathrm {Conv}({\mathcal M}_{\mathrm {DM}})$$ which does not belong to $$\mathcal {M}_{\mathrm {DM}}$$ , in other words to a local minimizer of the extended Kohn–Sham model with fractional occupation numbers at the Fermi level, while the iterates Pk do not converge in general. It can be argued that for such systems, the standard Kohn–Sham problem is not appropriate since the Kohn–Sham energy can be lowered by allowing fractional occupancies.

The optimal damping algorithm [22] (ODA), also originally introduced in the HF setting [27] based on ideas in [93], is a variant of the simple mixing algorithm using an optimal step αk:

(7.63)

with

(7.64)

A proof of numerical convergence of the ODA—still under a UWP assumption—can be read in [25]. Stronger convergence results require more information, such as second-order conditions [37] or analyticity [90]. Note that Remark 7.1 is also valid for ODA.

Remark 7.2

Implementing the ODA algorithm for LDA or GGA functionals does not require density matrices to be stored: it suffices to store in memory electronic densities and Kohn–Sham orbitals. This is of course very important in practice since, for example, density matrices cannot be stored for plane-wave calculations on large systems.

Even when it converges, the solutions to the self-consistent Kohn–Sham equations obtained by the simple SCF scheme sometimes correspond to critical points which are local, non-global, minima, or to critical points which do not satisfy the Aufbau principle. This situation is sometimes encountered with GTO basis sets. Minimization based methods such as the ODA seem to be free from such problems. The main limitation of the ODA is that it may converge very slowly in the vicinity of the solution. Acceleration methods are then needed.

7.5.1.2 Anderson–Pulay Acceleration Schemes

Consider a function $$g:\mathcal {X} \to \mathcal {X}$$ where $$\mathcal {X}$$ is a Euclidean (or Hermitian) space and the fixed-point problem: find x∗ in $$\mathcal {X}$$ such that

(7.65)

Even when the convergence of the Picard iterations xk+1 = g(xk) locally converge, convergence may be very slow. The original formulation of the Anderson acceleration scheme [5], introduced in 1965, is:

(7.66)

where the coefficients θk,i,∗ satisfy $$\sum _{i=0}^{m_k} \theta _{k,i,*}=1$$ and are obtained by solving

(7.67)

which has the structure of a linear least-square problem. When mk = k for all k, the scheme is called Anderson acceleration without truncation. An additional mixing step can be added, in which case the iterates are defined as

(7.68)

where βk > 0 is allowed to vary with k (the original Anderson acceleration scheme corresponds to βk = 1 for all k).

When applied to the affine function g(x) : (A + I)x − b where $$x,b \in \mathbb {R}^n$$ and $$A \in \mathbb {R}^{n \times n}$$ , the Anderson method is equivalent to the GMRES algorithm [126] for solving the linear system Ax = b, provided Anderson’s iterates do not stagnate [124, 142]. It can thus be interpreted as a nonlinear version of GMRES (although less robust than GMRES in the linear case). It can also be interpreted as an extrapolation method [20].

Anderson acceleration has been rediscovered a number of times under various names, corresponding to different interpretations: “multisecant Broyden” and “inversion in the subspace”. Indeed, it has recently been understood that the Anderson acceleration scheme can be seen as a special instance of a family of algorithms called Anderson–Pulay schemes [43]. This family also includes Pulay’s Direct Inversion in the Iterative Space (DIIS) schemes [116, 117] originally introduced to solve the Hartree–Fock equations (7.59), and widely used as well to solve the Kohn–Sham equations. This approach is often referred to as the Pulay mixing in the condensed matter physics literature. Note that Pulay’s first DIIS scheme [116] is in fact identical to Anderson acceleration scheme, while Pulay’s second DIIS [117] (also called C-DIIS, for commutator-DIIS) does not fit into Anderson’s framework. In their most general formulation, Anderson–Pulay schemes are designed to accelerate local convergence toward a point x∗ of a function g, which may, as for Hartree–Fock and Kohn–Sham models, take its values in a differentiable submanifold $$\mathcal {M}$$ of the Euclidean space $$\mathcal {X}$$ . We therefore assume, following [43], that

A1.

$$\mathcal {X}$$ is a Euclidean space, $$\mathcal {M}$$ is a differentiable submanifold of $$\mathcal {X}$$ , $$x_* \in \mathcal {M}$$ , U is an open neighborhood of x∗ in $$\mathcal {X}$$ , and $$g:U \to \mathcal {M}$$ is a function of class C² satisfying g(x∗) = x∗;

A2.

There exist a Euclidean space $$\mathcal {Y}$$ , and a C² function $$f:U \to \mathcal {Y}$$ satisfying the following properties: f(x∗) = 0 and there exists a constant σ > 0 such that

(7.69)

From Assumption A2, we have that for all $$x \in U \cap \mathcal {M}$$ , f(x) = 0 if and only if x = x∗. The function f can in fact be interpreted as a residual function: $$\sigma ^{-1}\|f(x)\|{ }_{\mathcal {Y}}$$ provides an upper bound of the distance from x to the solution x∗.

In Pulay’s schemes, $$\mathcal {X}=\mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}$$ , $$\mathcal {M}=\mathcal {M}_{\mathrm {DM}}$$ , g is the Hartree–Fock or Kohn–Sham map $$g=\mathcal {F}_{\mathrm {HF-DM}}$$ or $$g=\mathcal {F}_{\mathrm {DM}}$$ , and the residual function f is defined as

the natural residual map f(P) = g(P) − P in the first scheme [116];

the commutator f(P) = [H(P), P] in the (more efficient) second scheme [117]. This is indeed a legitimate residual map in view of (7.39).

There exist several formulations of the Anderson–Pulay scheme differing by the way the previous iterates are taken into account. In the fixed-depth Anderson–Pulay scheme implemented in most electronic structure codes, the depth mk is chosen equal to $$m_k=\min (k,m_{\mathrm {max}})$$ , where mmax a fixed number, typically mmax = 20, chosen empirically to optimize the performance of the algorithm in both computational time and memory requirement. Restarted versions [12, 114] have also been proposed. We detail here the recently introduced adaptive-depth formulation of the Anderson–Pulay scheme [43], which seems to outperform the previous versions:

1.

Initialization and first iteration:x₀, $$m_{\mathrm {max}} \in \mathbb {N}^*$$ , δ ≥ 0 and η > 0 being given, set

2.

Subsequent iterations: while ∥rk∥₂ > η do

a.

solve

(7.70)

and set

b.

set

c.

set mk+1 the largest integer $$m \le \min (m_k+1,m_{\mathrm {max}})$$ such that for k + 1 − m ≤ i ≤ k, δ∥ri∥₂ < ∥rk+1∥₂

d.

set k = k + 1.

Taking δ = 0 gives back the usual fixed-depth scheme. For δ > 0, the depth mk+1 is adjusted so that the residuals at the previous mk+1 residuals are not too large compared to the current residual rk+1. In view of (7.69), this ensures that the previous mk+1 iterates are not too far away from the current iterate xk+1. This scheme can be proved to converge superlinearly under some assumptions on f and g [43].

Numerically, this procedure is often very efficient, and can converge quickly even if the underlying fixed-point iteration $$P_{k+1} = {\mathcal F}_{\mathrm {DM}}(P_{k})$$ is divergent. It however lacks in robustness when started far away from a solution.

7.5.1.3 Newton and Quasi-Newton Methods for the SCF Equations

Problem (7.59) can also be seen as a nonlinear equation

to which the standard Newton method can be applied:

where Dk solves the correction equation

(7.71)

and Jk is the Jacobian of $$\mathcal {F}_{\mathrm {DM}}$$ at Pk.

However, a practical limitation of Newton’s method is the high cost of evaluating the Jacobian and solving the Newton correction equation (7.71). One workaround is to use a Jacobian-free Krylov–Newton technique [83] and solve (7.71) iteratively using, for example, the GMRES algorithm [126]. The matrix vector multiplication of the form JkD, which is required in each GMRES iteration, can be computed using methods similar to those used to compute response properties (see Sect. 7.2.5), or simply approximated by the finite difference formula

for an appropriately chosen small scalar t. The finite difference calculation requires one additional function evaluation of $$\mathcal {F}_{\mathrm {DM}}$$ per GMRES step. Therefore, even though Newton’s method may exhibit quadratic convergence, each Newton step may be expensive if the number of GMRES steps required to solve the correction equation is large. Numerical experiments presented in [62] show that without a good preconditioner, too many GMRES iterations are needed to ensure convergence.

Quasi-Newton methods can also be used, where the inverse Jacobian (1 − Jk)⁻¹ is approximated by a simpler matrix Bk. For instance, the simple mixing scheme above corresponds to Bk = αI. More sophisticated schemes can be obtained by using variants of Broyden’s method [21, 79, 135], which builds the Bk from previous iterates. In such an approach, Bk is obtained by performing a sequence of low-rank modifications to some initial approximation B₀ of the Jacobian inverse using a recursive formula. One particular choice among several is [57, 103]

(7.72)

where Bk−1 is the approximation to the Jacobian inverse constructed in the (k − 1)st Broyden step. The matrices Υk and Yk above are defined as

(7.73)

where sj and yj are defined by

respectively. Here the residual matrices are given by $$R(P) = \mathcal {F}_{\mathrm {DM}}(P) - P,$$ and the $$\mathrm {vec}\left [\cdot \right ]$$ notation is used to turn a matrix into a vector by stacking its columns on top of each other. The equation Υk = BYk is the multisecant condition and imposes the consistency of the inverse Jacobian with the observed relationships between the Pj and R(Pj).

It is easy to show that the solution to (7.72) is

(7.74)

where $$Y_k^{\dagger }$$ denotes the pseudo-inverse of Yk. If Bk−1 is reset to αI, we obtain Anderson’s method

(7.75)

commonly used in KSDFT solvers [5].

7.5.2 Direct Minimization

Instead of solving the HF equation (7.59) by an SCF algorithm, the HF ground state can be obtained by direct minimization of the HF energy functional. A quadratically-convergent algorithm, consisting in obtaining Pk+1 by performing one Newton step in the local map $$\Phi _{P_k}$$ defined by (7.28)–(7.29), was proposed by Bacskay in [10]. Trust region methods were introduced to the Hartree–Fock setting in [60, 137].

Because the Kohn–Sham problem is a constrained minimization problem as indicated in Sect. 7.2.3, we can try to solve the optimization problem directly. The simplest type of optimization procedure is the projected gradient algorithm

(7.76)

where $$H(P_k)=\nabla \mathcal {E}(P_k)$$ and where [[H(Pk), Pk], Pk] is the projected gradient on the tangent space to $$\mathcal {M}_{\mathrm {DM}}$$ at Pk, and α is a line search parameter chosen to reduce $$\mathcal {E}(P)$$ along the projected gradient. A simple and often effective linear search strategy is the Barzilai–Borwein (BB) strategy [15]. As soon as P moves away from Pk along the projected gradient, it violates the idempotency constraint. To bring $$\widetilde P_{k+1}$$ back to the manifold $$\mathcal {M}_{\mathrm {DM}}$$ and obtain Pk+1, a retraction (also called purification) step is performed. The algorithms proposed in [31, 128, 149] are examples of this approach.

As usual in optimization methods, several strategies can be employed to enhance convergence. First, the quality of the search step can be improved using a preconditioner [3, 107], although the design of good preconditioners is less explored than for SCF algorithms [7]. Second, information from past iterations can be used to obtain curvature information, using for instance the conjugate gradient or the L-BFGS algorithm [1]. Finally, Newton’s method can be used, yielding the QC-SCF algorithm [10].

Lastly, minimization algorithms can be combined with the SCF idea. The basic representative of this class is the ODA algorithm described above, where the SCF iteration is used to generate a search direction in which an energy minimization is performed. An extension of that idea is the EDIIS algorithm, in which the minimization is performed on a subspace using all past iterates rather than just a segment [87].

7.5.3 Implementation for Molecular Systems

The Gaussian basis sets used in most quantum chemistry packages are in some sense reduced bases: a very small number of basis functions per atom are optimized on a few reference simple systems (mostly isolated atoms) and are then used to simulate a whole variety of molecular systems. As a consequence, it is possible to simulate systems containing a few hundred electrons in basis sets with a few thousand elements. Density matrices can then be stored in memory and optimized dense linear algebra packages can be used to efficiently solve a generalized eigenvalue problem at each iteration of the SCF cycle.

For most molecular systems and small to medium size basis sets, an efficient iterative strategy to solve the Kohn–Sham equations is [87]:

to make a few iterations of the EDIIS algorithm to ensure a decrease of the energy;

to switch to DIIS to obtain fast convergence when some coarse convergence threshold is reached.

On the other hand, existing algorithms usually perform poorly for very large basis sets, giving rise to ill-conditioned overlap matrices (“overcomplete” Gaussian basis sets). This observation deserves a detailed mathematical analysis that has not yet been made to our knowledge.

When the Fermi level is a degenerate eigenvalue of the Kohn–Sham Hamiltonian, all the above described algorithms break down. An extension of Bacskay’s quadratically convergent algorithm [10] was introduced in [26]. It works well for small and medium size-systems but, as for all Newton-type algorithms, cannot be applied to large systems.

7.5.4 Application to Condensed Matter

We first note that condensed-matter models are often formulated using a finite temperature T = 1∕β (setting to 1 the value of the Boltzmann constant kB). This is usually not to capture physical effects due to the influence of the temperature on electronic properties (usually negligible at room temperature since the energy scale of the electrons is much larger than that of thermal fluctuations), but a numerical device to accelerate the convergence of properties of metallic infinite systems with respect to the supercell size. Indeed, metals present strong long-range correlations, and the convergence of properties with respect to the supercell size L is slow (1∕L for naive methods). Adding an artificial temperature introduces an exponential locality, and improves this to exponential convergence [36], at the price of slightly distorted properties.

As mentioned in Sects. 7.2.2 and 7.2.4, the fixed-point equation at finite temperature takes the form

where the Fermi level μ is chosen such that $$\mathrm {Tr} {\mathcal F}_{\mathrm {DM}}(P) = N$$ . The fixed-point mapping $${\mathcal F}_{\mathrm {DM}}$$ is well-defined even in the absence of a gap, as there is no notion of “occupied” or “virtual” states (but only partially occupied states).

Planewave expansion, finite difference and finite element methods are often used to discretize the Kohn–Sham problem for condensed matter systems. In these approaches, the number of degrees of freedom per electron pair is often over 100. The dimension of the Hamiltonian Nb is therefore much larger than the number of occupied states N. For large systems that contain many atoms (in a unit cell), it is generally not practical to form and work with the density matrix P directly, and only orbitals, densities, and potentials are manipulated directly. This feature makes the implementation of SCF and minimization algorithms somewhat different from those discussed in the previous section. We will highlight some of these differences in this section.

7.5.4.1 Charge Density Mixing and Preconditioning

When the exchange-correlation functional is an explicit function of the charge density ρ (i.e. for LDA or GGA), the ground state density ρGS satisfies a Kohn–Sham equation of the form

(7.77)

Consequently, SCF iteration and acceleration techniques can be designed to find the fixed point of $${\mathcal F}_{\mathrm {KS}}$$ .

At a finite temperature T = 1∕β, and in a real-space grid representation, $${\mathcal F}_{\mathrm {KS}}$$ has the form

(7.78)

Here diag(⋅) denotes the diagonal elements of a matrix, and I is the identity matrix. The map $${\mathcal F}_{\mathrm {KS}}$$ is well defined even if H[ρ] has degenerate eigenvalues near the Fermi level μ, which is chosen to ensure that

(7.79)

The Hamiltonian H[ρ] is the discrete counterpart of the continuous Kohn–Sham Hamiltonian

where the Kohn–Sham potential vKS[ρ] is the sum of the fixed nuclear (pseudo-) potential and the density-dependent Hartree and exchange-correlation potentials:

GL being the periodic Couloumb kernel defined in (7.52).

In practice, the solution (ρ, μ) to the coupled Eqs. (7.78) and (7.79) is obtained in an alternating fashion. For a fixed μ, we may update ρ by a simple SCF iteration

(7.80)

starting from an initial guess ρ₀. For each fixed ρk, a bisection method can be used to find the solution to (7.79) since the right-hand side of (7.79) is monotonic with respect to μ.

The SCF iteration can be viewed as a fixed point iteration applied to the Kohn–Sham map. To analyze the convergence of the SCF iteration and develop acceleration strategies, it is useful to examine the Jacobian associated with $$\mathcal F_{\mathrm {KS}}$$ , which we denote by $$\mathcal J_{\mathrm {KS}}[\rho ] := \partial \mathcal F_{\mathrm {KS}}[\rho ]/\partial \rho $$ .

Introducing the potential-to-density map

where T is the kinetic-energy matrix, we have $$\mathcal F_{\mathrm {KS}}(\rho )=\mathcal {F}_{P \to D}(v_{\mathrm {KS}}[\rho ])$$ and it therefore follows from the chain rule that

(7.81)

where the first term on the right-hand side of (7.81) is the functional derivative of $$\mathcal {F}_{P\to D}$$ . Its value at the ground-state Kohn–Sham potential is the irreducible (or independent-particle) polarizability

of the system. At the continuous level, its kernel is given by

(7.82)

where the εm are the eigenvalues of the Kohn–Sham Hamiltonian $$-\frac 12 \Delta + v_{\mathrm {KS}}[\rho ^{\mathrm {GS}}]$$ , φm an orthonormal basis of associated real-valued eigenfunctions, and

(7.83)

is the occupation number associated with εm (by convention, (fm − fn)∕(εm − εn) = f′(εm) if εm = εn).

The second term in (7.81) is the functional derivative of vKS[ρ], and its kernel is given by

(7.84)

where GL is again the periodic Coulomb kernel defined in (7.52). The second term on the right-hand side, which is called the exchange-correlation kernel, is the functional derivative of the exchange-correction potential with respect to ρ, which (at least formally) defines a self-adjoint operator. We denote it by Kxc(r, r′).

Subtracting the exact solution ρ to (7.77) from both sides of (7.80), and performing a Taylor expansion of $$\mathcal F_{\mathrm {KS}}$$ at ρk up to the first order yields

(7.85)

where $$\mathcal J_{\mathrm {KS}}(\rho _k)$$ is the Jacobian of $$\mathcal F_{\mathrm {KS}}$$ evaluated at ρk. Therefore, when ρk is sufficiently close to the exact solution, the fixed point iteration (7.80) converges if the spectral radius of $$\mathcal J_{\mathrm {KS}}(\rho _k)$$ , denoted by $$\sigma (\mathcal J_{\mathrm {KS}}(\rho _k))$$ , is less than 1 for all k’s. However, this condition does not usually hold. In this case, the simple mixing scheme introduced in (7.62) can be used to update ρ by

(7.86)

where 0 < α < 1 is a mixing parameter. From linear stability analysis, when ρk is sufficiently close to the exact solution ρGS, simple mixing will lead to convergence if

(7.87)

where $$\epsilon = I - \mathcal J_{\mathrm {KS}}(\rho ^{\mathrm {GS}})$$ is known as the dielectric operator (function) in the physics literature. It can be shown that, when ρGS is an energy minimum, the eigenvalues of $$\mathcal J_{\mathrm {KS}}(\rho ^{\mathrm {GS}})$$ are positive, and therefore it is always possible to choose an α so that (7.87) holds [37]. However, such a choice may lead to slow convergence, depending on the conditioning of 𝜖.

To analyze this it is convenient to neglect the Kxc term, an often reasonable assumption known as the random phase approximation, 𝜖 ≃ 𝜖RPA := 1 − χ₀vH. The eigenvalues of this operator are easily seen to be greater than one, so that ill-conditioning can only appear through large eigenvalues (for more complicated situations where the random phase approximation is too crude, such as systems close to magnetic phase transitions, small eigenvalues of 𝜖 can also be problematic). These large eigenvalues can be caused either by large eigenvalues of χ₀, which are by (7.82) related to small gaps, or by large eigenvalues of vH. From the expression (7.52) of GL in Fourier space, it can easily be seen that the divergences of vH are related to long-wavelength modes. Both these types of divergences can lead to the phenomenon of charge sloshing, wherein the density undergoes large oscillations during the iterations.

The behavior of 𝜖 for large systems in the random phase approximation is markedly different depending on its physical nature. In insulators, the long-wavelength divergence of the Coulomb potential is compensated by χ₀, so that 𝜖RPA is bounded independently of the system size [29]. It follows that charge sloshing is absent, and such systems are usually found to converge quickly. On the other hand, for metals, χ₀ does not compensate for the divergence of vH, and charge sloshing hampers convergence. A simple model for this divergence is the case of the homogeneous electron gas, where vKS is constant. In this case, χ₀ is a multiplication operator in Fourier space, characterized by its multiplier $$\chi _{0}^{\mathrm {HEG}}(q)$$ , which can be computed explicitly [66]. It can in particular be seen that $$\chi ^{\mathrm {HEG}}_{0}(q)$$ approaches a negative constant − ζ as q → 0, where ζ is proportional to the inverse square of the Thomas–Fermi screening length. 𝜖HEG is therefore a multiplication operator in Fourier space, such that $$\epsilon ^{\mathrm {HEG}}(q) \simeq 1 + \frac {4\pi \zeta }{|q|{ }^{2}}$$ for low q.

In the case of the homogeneous electron gas, the appropriate iteration scheme is then the preconditioned iteration

(7.88)

with K a fixed preconditioner, given as a multiplication in Fourier space by

a choice known as Kerker preconditioning [81]. When combined with Anderson–Pulay acceleration, this has been found to be very efficient in practice for a wide range of metals, and is the default iteration scheme in most plane-wave codes. A mathematical proof of the independence of the convergence rate of (7.88) on the system size in the reduced Hartree–Fock model can be found in [91]. For general systems, and in particular those containing hybrid insulating/conducting character, finding an efficient preconditioner is still a challenge; see [96, 151] and [144] for a review. Very recently, an inexpensive and parameter-free preconditioner based on the local density of states has been introduced [77]. It is based on a physically motivated approximation to the independent-particle susceptibility operator, and its efficiency has been demonstrated on several heterogeneous systems such as clusters and surfaces.

7.5.4.2 Iterative Diagonalization

An accelerated fixed point iteration for seeking the solution of (7.77) requires evaluating the Kohn–Sham map $$\mathcal F_{\mathrm {KS}}$$ at each ρk. This evaluation is often performed by making use of spectral decomposition H(ρk) = U ΛU∗, where U contains the eigenvectors of H(ρk) and $$\Lambda = \mathrm {diag}(\varepsilon _1,\varepsilon _2, \ldots , \varepsilon _{N_b})$$ contains the corresponding eigenvalues. Using such a decomposition, we can write $$\mathcal F_{\mathrm {KS}}(\rho _k)$$ as

(7.89)

where fm is the occupation number defined in (7.83), and φm is the mth column of U.

Because the dimension of H (Nb) can be extremely large, using full diagonalization algorithms for dense matrices implemented in LAPACK [6], ScaLAPACK [17] or ELPA [102] can be prohibitively expensive. For large basis sets such as planewaves and discretization methods based on finite elements and finite differences, Nb is typically much larger than the number of required eigenpairs, which is N (at zero temperature), or slightly larger (in the usual range of positive temperatures). Iterative eigensolvers such as the Davidson–Liu algorithm [46, 99] or the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm [84] are then preferred. These iterative eigensolvers can take advantage of an efficient representation of H that allows the multiplication of H by one or multiple vectors to be performed efficiently, through fast Fourier transforms (plane waves) or sparse matrix techniques (finite elements, finite differences).

One additional advantage of using iterative diagonalization methods is that approximate eigenvectors from the kth SCF iteration can be used as the starting guess of eigenvectors for the (k + 1)st SCF iteration so that only a few iterative diagonalization steps may be needed per SCF iteration. This is certainly true when ρn has nearly converged. But even when ρn is far from the solution at the beginning of the SCF procedure, only a few iterative diagonalization steps are needed to yield approximate eigenpairs. This is due to the fact that in the first few SCF iterations, the error in nonlinear Eq. (7.77) is still relatively large. Hence, as long as the error in $$\mathcal F_{\mathrm {KS}}$$ evaluation is less than the difference between ρ and $$\mathcal F_{\mathrm {KS}}(\rho )$$ even when the latter is evaluated exactly, the convergence of the SCF procedure can still be achieved in subsequent iterations.

Another way to evaluate $$\mathcal F_{\mathrm {KS}}(\rho )$$ is to apply a subspace iteration to Td(H[ρ]) where Td(t) is a d-th degree scaled and shifted Chebyshev polynomial of the second kind that satisfies the property

This technique was pioneered in [152, 153], and is now widely used in a number of software tools. At zero temperature, the subspace iteration converges to an approximate invariant subspace spanned by eigenvectors associated with the lowest N eigenvalues. The vectors ui, i = 1, 2…, N, used in (7.89) do not need to be eigenvectors. Any basis of the invariant subspace suffices. At finite temperature, only a small fraction of the eigenpairs of H corresponding to eigenvalues slightly below and above μ are computed using the method presented in [11, 106].

For large problems consisting of many atoms, the cost of an iterative diagonalization procedure is dominated by the computation required to obtain orthonormalized basis vectors of the desired subspace and extract eigenvalue and eigenvector approximations from such a subspace. The complexity of this computation is typically $$\mathcal {O}(N^3)$$ . Furthermore, some of these computations do not scale to a large number of processors on a massively parallel computer. The recent effort in developing spectrum slicing algorithms has the potential to address this issue, but requires solving difficult interior eigenvalue problems.

Methods for reducing the $$\mathcal {O}(N^3)$$ have also developed since the 90s. These methods often try to approximate the matrix function $$\left [ 1+\mathrm{e} ^{\beta (H[\rho ]-\mu )} \right ]^{-1}$$ directly through either polynomial or rational expansions [14, 67–69, 85, 92, 98]. Some of these methods can scale linearly with respect to N, although with a large prefactor. Details of these methods can be found in the contribution by Lin et al. (Chap. 8).

7.5.4.3 Computing the Hartree and Fock (Exact) Exchange Potential

When the Kohn–Sham equations are discretized by GTOs for molecular systems, both the Hartree and the Fock (exact) exchange potentials are computed by using the two-electron integrals (7.13) evaluated in advance or on the fly.

When the problem is discretized by a large basis set such as planewaves, the Hartree potential vH[ρ] is obtained by solving the periodic Poisson problem − ΔvH[ρ] = 4πρ in a unit cell by using FFT. In a continuous formulation, the non-local Fock exchange potential is defined (at zero temperature for simplicity) as

(7.90)

where φi(r), i = 1, 2, …, N are the occupied Kohn–Sham orbitals, and GL(r, r′) is the periodic Coulomb kernel defined in (7.52). When an iterative diagonalization method is used to compute the desired eigenpairs, we need to find a vX which can be efficiently applied to a set of N wavefunctions, i.e.,

(7.91)

For each j, N integrals on the right-hand side of the above equation can be evaluated by performing N cyclic convolutions via FFTs. Since j = 1, 2, …, N, the total number of FFTs that need to be performed is $$\mathcal {O}(N^2)$$ . An efficient low rank approximation technique called interpolative separable density fitting (ISDF) [78] has recently been developed to reduce the number of FFTs to a small multiple of N by constructing a set of numerical auxiliary basis functions that nearly span the same subspace as φi(r)φj(r), for i, j = 1, 2, …, N. Furthermore, by recognizing that the application of vX to {ψj}, j = 1, 2, …, N, only needs to be correct in the space spanned by {ψj}, further reduction in complexity can be achieved in the ISDF construction of the vX operator through a technique called the adaptively compressed exchange operator [97]. The technique of using a small set of auxiliary basis functions to compactly represent pair product basis functions (to reduce the cost of the Fock exact exchange calculation) is also known as resolution of identity or density fitting in general. (See the review article [123] and the references therein.)

7.5.4.4 Direct Minimization

The algorithms in Sect. 7.5.2 can be adapted to the condensed matter setting by using orbitals. The basic gradient algorithm then takes the form

where R(Xk) is the residual that takes the form of

(7.92)

followed by an orthogonalization step; choices include standard orthogonalization methods (based on the Gram–Schmidt, Cholesky or singular value decomposition algorithms), as well as the so-called WY update [143].

However, the separation of line search and the orthonormalization procedure used to keep X feasible is suboptimal. A better strategy is to carry out these two tasks simultaneously in a constrained optimization framework.

In order to do that, we express Xk+1 as

(7.93)

where G₁, G₂ and G₃ are N × N real matrices to be determined, and where $$K \in \mathbb {R}^{N_b \times N_b}_{\mathrm {sym}}$$ is a preconditioner. We choose these matrices by solving the following constrained minimization problem

(7.94)

where Y = (X(k), K⁻¹R(X(k)), X(k) − X(k−1)) and $$G = (G_1^T \: G_2^T \: G_3^T)^T$$ . This approach is referred to as a direct constrained minimization (DCM) method in [146]. It is a nonlinear extension of the locally optimal block preconditioned conjugate gradient (LOBPCG) method [84]. When Pk−1 term is not included in (7.93), the DCM algorithm can be viewed as a nonlinear extension of the Davidson–Liu algorithm [99] for solving a linear eigenvalue problem. The problem (7.94) is of reduced size, that can be solved using stabilized SCF algorithms for instance; we refer to [146] for details.

Faster convergence can be achieved by a Riemannian Newton’s method [1, 150] that preserves the orthonormality constraint while minimizing the total energy objective function. The key step in the Newton method is to compute the Newton search direction by solving the following equation

(7.95)

where D²E(X(k)) is the Hessian of E at X(k) restricted to the manifold $$\mathcal {M}_{\mathrm {MO}}$$ associated with the orthonormality constraint in (7.57), and ∇E(X(k)) is the projected gradient along the tangent of this constraint. Although it is not practical to write down the Hessian and solve the Newton equation to high accuracy, the Hessian does have a special structure that allows one to multiply it by a vector efficiently. This structure is described in [62, 100]. An iterative method can be used to solve the Newton equation to obtain a search direction ΔX(k) that satisfies

(7.96)

where σ is a dynamically chosen parameter. Even though Newton’s method can yield quadratic convergence, each step of the method requires a large linear system of equations to be solved. Without a good preconditioner, the cost for obtaining a good solution can be high. Constructing a good preconditioner is not trivial.

The approach discussed above is applicable to Kohn–Sham models at zero temperature. At finite temperature, we minimize an objective function similar to the one given in (7.22). In addition to optimizing with respect to the wavefunction X, we also need to optimize the occupation numbers. An efficient algorithm for solving this optimization problem is given in [140]. Similar algorithms which optimize (7.22) with respect X and the occupation numbers simultaneously can be found in [61, 104].

7.5.4.5 Stopping Criteria

When an iterative algorithm is designed to solve (7.77), a natural stopping criterion at step k is

(7.97)

where τ is some user specified error tolerance. A similar criterion can be used to check the convergence of the potential that depends on ρ. Although such a criterion does not guarantee the convergence of ρ to the ground state density, it does allow us to check the self-consistency of the approximate solution.

In a direct minimization approach, we can terminate an iterative procedure if the Frobenius norm of the residual R(X) defined in (7.92) is sufficiently small. Again, such a stopping criterion does not guarantee X to be a global minimizer. But it ensures that X is a self-consistent approximation.

In the physics community, the difference between the Kohn–Sham total energy evaluated at an approximate X and an estimation known as the Harris–Foulke energy functional [59, 75] defined in the real space as

where ρin is the input density, and ρout and Pout are the output density and density matrix, is often used in a termination criterion. When $$\rho _{\mathrm {{in}}}=\rho _{\mathrm {{ }_{out}}}$$ is the solution to (7.77), EHarris–Foulke(ρ) is identical to the Kohn–Sham total energy. However, in general, these two energy functions are different. When ρin converges to the ground state density, the difference between the two becomes smaller.

7.6 Error Analysis

The mathematical analysis of the error between the exact solution to the Kohn–Sham model under consideration and the numerical result obtained by discretization and an approximately converged iterative algorithm encompasses two frameworks: the a priori analysis, and the a posteriori analysis.

The a priori analysis aims at answering two questions:

1.

algorithmic error: does the solution algorithm converge and how fast does it converge? The problems we are interested in this chapter are indeed highly nonlinear and thus iterative methods are needed; the question is then to understand if the solution algorithm converges when the number of iterations tends to infinity and—if so—to estimate the number of iterations necessary to reach a given accuracy. A related side question is to understand the fundamental reasons why different algorithms converge, and to identify key quantities such as the condition number of a specific matrix or the Lipschitz constant of some contraction map which control the convergence rates of these algorithms. Such an analysis then allows us to improve the existing algorithms by modifying them by acting on these key quantities, for example through the design of preconditioners or accelerated algorithms. The knowledge accumulated during the a priori analysis may also lead to reformulations of the problem which make it fall into a class of problems for which efficient and robust algorithms already exist (e.g. write it as a smooth convex minimization problem);

2.

discretization error: the goal is to prove that, eventually, when the number of degrees of freedom uniformly¹ increases to infinity, then the discrete solution converges to the exact solution in some sense. In this case, the discretization method is called systematically improvable. Such an analysis generally provides a theoretical convergence rate, that is, a lower bound of the asymptotic speed of convergence of the error when the number of degrees of freedom uniformly increases to infinity. The a priori analysis helps us to choose the more appropriate numerical discretization schemes for a given problem by highlighting the link between the convergence rate of a given discretization scheme and properties of the exact solution such as its regularity. For instance, fixed (low) order discretization methods (e.g. finite elements) are generally indicated if the exact solution is globally not very regular, whereas high (or infinite) order methods (e.g. spectral or planewave methods) are generally preferred when the exact solution and its derivatives are uniformly bounded by well-controlled constants.

The a priori analysis allows us to build a first, mathematically consistent, numerical strategy (discretization and iterative algorithm) for solving the problem under consideration, based on some qualitative analysis: characterization of the class of problems this specific problem belongs to, regularity of the exact solution…

Let us mention the existence of other contributions (not analyzed here) to the error between the exact solution and the result of the numerical simulation, in particular (i) finite arithmetics errors, which can become significant if part of the calculation is done in single precision (which is of upmost importance for hybrid architectures involving GPUs), and (ii) implementation errors (bugs, which are almost inevitable in software consisting of hundreds of thousands lines of code). The model error, due to the fact that the exchange-correlation functional used in the calculation is an approximation of the true one, is of course important as well, but is currently out of reach of the mathematical and numerical analysis. The situation is different for wavefunction methods (e.g. configuration interaction or coupled cluster) for which estimating the model error should be possible since the approximate ground state wavefunction generated by these methods can be used to define a residual for the exact N-body Schrödinger equation. Some preliminary works in this direction are in progress.

In contrast with the a priori analysis, the a posteriori analysis aims at answering more practical questions: what is the size of the error between the exact (unknown) solution and the discrete solution that comes out of the numerical simulation and what can be done to improve the accuracy of the numerical result? By “size” we mean here an upper (ε⁺) bound and a lower (ε⁻) bound of some appropriate and relevant norm of the error such that ε⁺ − ε⁻ is much smaller than ε⁺. Improved versions of this analysis further provide “indicators” that help us to decide what to do if these upper and lower bounds reveal that the accuracy is not sufficient to answer the question motivating the simulation. Examples are local a posteriori error indicators pointing out where to refine the mesh in finite element methods. This leads to adaptive schemes and to non-uniform solution-dependent quasi-optimal meshes. Other examples are iterative procedures in which the discretization and algorithmic components of the error are separately estimated at each iteration. This allows one to adaptively refine the discretization when the SCF or minimization algorithm gets closer to convergence, thus avoiding costly calculations in large discretization basis when the iterates are far from convergence. More generally, error balancing consists in designing numerical schemes which minimize the computational cost necessary to reach a given accuracy, or, conversely, minimize the error reachable at a given computational cost. A posteriori error estimators of the various components of the error are a key tool for the design of black-box error balancing strategies.

Note that most of the time, what we are looking for is not the knowledge of the global state (e.g. the Kohn–Sham ground state density matrix) but the computation of some properties or quantities of interest (e.g. the forces acting on the nuclei) that are evaluated from the numerical approximation of the global state. These quantities of interest are generally expressed as functionals that are continuous with respect to some norms. These are the norms for which the a posteriori error analysis has to be performed. Thus, estimates of the error on the global state for these norms automatically give rise to estimates of the error on the quantities of interest. Note that obtaining such bounds on these quantities of interest generally requires the introduction of auxiliary dual problems to obtain convergence rates consistent with the ones observed in numerical simulations, similar to the classical Aubin–Nitsche methodology in finite element methods [55].

7.6.1 A Priori Error Analysis of a Simple Model

For the sake of pedagogy, let us explain the strategy on the simple example of a one-dimensional periodic Gross–Pitaevskii model. The equations have the same form as those arising in Kohn–Sham DFT, with several important simplifications: the state of the system is described by a single orbital, the model is strictly convex in the density, and the Aufbau principle always holds.

Let $$\Omega :=(-\frac {1}{2},\frac {1}{2})$$ be the unit cell of the periodic lattice $$\mathcal {R}=\mathbb {Z}$$ of $$\mathbb {R}$$ . The Sobolev spaces of 1-periodic functions (or distributions) are defined for all $$s \in \mathbb {R}$$ by

For each $$s \in \mathbb {R}$$ , $$H^s_{\mathrm {per}}(\Omega )$$ is a Hilbert space, whose norm $$\|\cdot \|{ }_{H^s}$$ can be easily expressed in terms of Fourier coefficients. Denoting by eK(x) = eiKx, $$K \in 2\pi \mathbb {Z}$$ , the 1-periodic Fourier modes, any $$v \in H^s_{\mathrm {per}}(\Omega )$$ can be expanded as

and we can choose the following equivalent norm for $$H^s_{\mathrm {per}}(\Omega )$$ :

(7.98)

We also have

The one-dimensional periodic Gross–Pitaevskii problem reads

(7.99)

where the energy functional $$E^{\mathrm {GP}}:H^1_{\mathrm {per}}(\Omega )\to \mathbb {R}$$ is given by

(7.100)

where $$V : \mathbb {R} \to \mathbb {R}$$ is a 1-periodic bounded function. Problem (7.99) can be reformulated as

(7.101)

where we have introduced the function $${\mathcal E}^{\mathrm {GP}}(\rho ) := E^{\mathrm {GP}}(\sqrt {\rho })$$ .

The energy $${\mathcal E}^{\mathrm {GP}}$$ is strictly convex and smooth, and it is easy to prove that problem (7.101) has a unique solution ρ₀, which is continuous and positive on $$\mathbb {R}$$ —hence bounded away from zero since it is 1-periodic–, so that problem (7.99) has exactly two solutions: $$u_0=\sqrt {\rho }_0$$ and − u₀; we refer e.g. to [23] for a detailed proof of these results. The Euler–Lagrange equation associated to this problem involves an eigenvalue λ₀ (the Lagrange multiplier of the normalization constraint) and reads:

(7.102)

The variational formulation of the above problem is

(7.103)

Note that a solution u₀ to the nonlinear eigenvalue problem (7.102) may not be a minimizer of (7.99), but, as in the linear case, it will be if one of the following conditions is fulfilled:

(u₀, λ₀) is a solution to (7.102) and λ₀ is the smallest eigenvalue of the nonlinear problem

(u₀, λ₀) is a solution to (7.102) and λ₀ is the smallest eigenvalue to the bounded-below self-adjoint operator with compact resolvent $$-\Delta +V + u_0^2$$ .

The planewave approximation of this problem in the discretization space

(see Sect. 7.3.2 for the 3D setting) is given by

(7.104)

which, obviously, has—at least—one solution. This states the existence of a solution u0,N satisfying the constraint $$\|u_{0, N}\|{ }_{L^2_{\mathrm {per}}}=1$$ , ∫Ωu₀u0,N ≥ 0 (up to replacing u0,N by − u0,N), and the variational equation

(7.105)

From convexity arguments, there exists a positive constant α such that

(7.106)

Then let Π0,N be the orthogonal projector from $$H^s_{\mathrm {per}}(\Omega )$$ to its finite-dimensional subspace $$\mathcal X_{N}$$ :

We deduce from the smoothness of the functional EGP that

for some constant $$C \in \mathbb {R}_+$$ independent of N. Together with (7.106), this implies that $$\|u_{0, N} -u_0\|{ }_{H^1_{\mathrm {per}}}$$ converges to 0 as N tends to infinity. This also implies that, in any small enough neighborhood of u₀, there exists a unique solution u0,N to (7.104), provided N is large enough.

Our goal is now to compare the error between u₀ and u0,N (measured in the $$H^1_{\mathrm {per}}(\Omega )$$ -norm) on the one hand, and the error between λ₀ and λ0,N on the other hand, with the best fit error in $$\mathcal X_{N}$$ realized by Π0,Nu₀. To do this, we reformulate the problem (7.105) solved by (u0,N, λ0,N) as a nonlinear equation FN(u0,N, λ0,N) = 0, with $$F_{N} : H^1_{\mathrm {per}}(\Omega ) \times \mathbb {R} \to H^{-1}_{\mathrm {per}}(\Omega ) \times \mathbb {R}$$ defined by

(7.107)

This has the property that FN(u0,N, λ0,N) = 0, and furthermore

which is small. Now we can link the residual FN(u₀, λ₀) to the error (u₀, λ₀) − (u0,N, λ0,N) by the expansion

(7.108)

By showing that [DFN(u0,N, λ0,N)]⁻¹ is uniformly bounded with respect to N (which follows from quite tedious arguments, see [52] for details), and controlling higher order terms, one obtains that there exists a constant $$C \in \mathbb {R}_+$$ such that for N large enough

(7.109)

In the end, the convergence rate is thus only related to the Sobolev regularity of the eigenfunction u₀, which itself depends of the regularity of the external potential V : if $$V \in H^\sigma _{\mathrm {per}}(\Omega )$$ for some σ ≥ 0, then $$u_0 \in H^{\sigma +2}_{\mathrm {per}}(\Omega )$$ by elliptic regularity arguments. Since for all s > 1, we have

we obtain that if $$V \in H^\sigma _{\mathrm {per}}(\Omega )$$ for some σ ≥ 0, then

(7.110)

for some constant $$C \in \mathbb {R}_+$$ independent of N. The above estimate derived from (7.109) is optimal.

In contrast, the estimate on the eigenvalue error obtained from (7.109) does not scale optimally. Indeed, in the linear case

(7.111)

a quadratic convergence rate of the eigenvalue can be established following the Hellmann–Feynman type arguments in Sect. 7.2.5.

Using estimates à la Aubin–Nitsche in negative Sobolev norms, this approach can be extended to the nonlinear case of the Gross–Pitaevskii equation, yielding error estimates of the form

(7.112)

It can also be established that

(7.113)

Similarly, other quantities of interest computed from u0,N and λ0,N are at hand and (7.110) and (7.112) allow us to derive a priori error estimates for these outputs.

Note that it is impossible to get accurate values of the errors from the above estimates since the constants C, C′ and C″ in (7.110), (7.112) and (7.113) are not explicit, depending on quantities such as the gap that are not known a priori. This motivates the next section.

7.6.2 A Posteriori Error Analysis of a Simple Model

We follow up with the same problem (7.102), but now, as explained at the beginning of this section, we aim at an accurate numerical value for the size of the error between the exact solution (or quantity of interest) and the computed approximate solution (or computed QOI), and not a rate of convergence as the number of degrees of freedom of the discrete problem and the number of iterations of the SCF or minimization algorithm increase.

The general strategy to derive a posteriori error bounds is somewhat similar to the one for a priori error bounds described in the previous section. The difference is that it involves a nonlinear function F whose root is the exact solution: F(u₀, λ₀) = 0. Now the distance between some approximate solution (u, λ) and the exact solution (u₀, λ₀) is related to the size of F(u, λ). Indeed, similarly as in (7.108), we have

(7.114)

and (u − u₀, λ − λ₀) behaves as DF(u₀, λ₀)⁻¹F(u, λ), using this time the fact that F(u₀, λ₀) = 0.

The definition of $$F:H^1_{\mathrm {per}}(\Omega ) \times \mathbb {R} \to H^{-1}_{\mathrm {per}}(\Omega ) \times \mathbb {R}$$ is

(7.115)

and we have, assuming that (u, λ) is close enough to (u₀, λ₀) and neglecting higher-order terms,

Here, the approximate solution (u, λ) can be (u0,N, λ0,N) (converged numerical solution in the discretization space $$\mathcal {X}_{N}$$ ) for N large enough, but the argument also holds for the kth iterate of a SCF or minimization algorithm in some (possibly adaptively refined) discretization space.

Note that $$\|\lambda u + \Delta u -V u- u^3\|{ }_{H^{-1}_{\mathrm {per}}}$$ can be easily evaluated in the Fourier representation using (7.98). To turn the above inequalities into a practical tool to evaluate $$\|u-u_0\|{ }_{H^1_{\mathrm {per}}}$$ , we need an accurate bound for the operator norm of DF(u₀, λ₀)⁻¹ or an estimate for $$\left \| [DF(u_0,\lambda _0)]^{-1} F(u, \lambda ) \right \|{ }_{H^1_{\mathrm {per}} \times \mathbb {R}}$$ . For the sake of brevity, we will not detail this quite technical point here and refer the interested reader to [52]. The conclusion is that

with a constant $$C \in \mathbb {R}_+$$ which can be accurately estimated; this is a fundamental difference between a priori and a posteriori error bounds. As previously mentioned, the evaluation of this H⁻¹-norm of the residual λu +  Δu − V u − u³ is rather simple in the Fourier representation. Actually, the analysis can be improved to decompose the error into two terms: a discretization error due to the fact that u lives in a finite-dimensional discretization space $$\mathcal {X}_{N}$$ ; an algorithmic error corresponding to the fact that the SCF or minimization algorithm has not yet converged to (u0,N, λ0,N). The former component of the error vanishes in the limit N →∞, the latter when the iterative algorithm has converged in the current discretization space. This decomposition leads to adaptive algorithms in which the discretization space is increased as the SCF iterations proceed. We refer to [52] for more details.

7.6.3 Extensions to the Kohn–Sham Problem

Let us continue now to the a posteriori analysis of the Hartree–Fock and Kohn–Sham problems. The point is to write these problems again under the formal form F = 0, and an approximation like Fcut = 0 on the manifold

(7.116)

Due to the gauge invariance (7.14) of the Hartree–Fock or Kohn–Sham energy, the ground state problems do not have a unique solution and thus DF, at the ground state, is not invertible: it has a Kernel within the tangent space to $${\mathcal M}$$ . More precisely, for any $$\Phi = (\varphi _1,\ldots ,\varphi _N)^T \in \mathcal {M}$$ we denote by $$T_{\Phi } \mathcal {M}$$ the tangent space at Φ to $$\mathcal {M}$$ . It reads

Then, if we denote by

the set of all orbitals that are strongly orthogonal to Φ, we have (see e.g. Lemma 4 in [101] and [53]) that

where $$\mathcal {A} = \left \{A \in \mathbb {R}^{N \times N} \; | \; A^T=-A\right \}$$ is the space of the N × N antisymmetric real matrices. The gauge invariance implies that $$\mathcal {A} \Phi ^0$$ is a subset of the Kernel to DF. It is reasonable to assume that this Kernel is restricted to $$\mathcal {A} \Phi ^0$$ . Now, if this is indeed the case, due to the fact that the problem we are considering is a minimization problem, the second order condition further states

where for all Ψ = (ψ₁, …, ψN)T and Υ = (υ₁, …, υN)T in $$(H^1_{\mathrm {per}}(\Gamma ))^N$$ ,

Under the previous assumption that the Kernel to DF is restricted to $$\mathcal {A} \Phi ^0$$ , this implies that $$a_{\Phi ^0}$$ is positive definite on Φ0, ⊥ ⊥ , so that, using compactness arguments (see e.g. Proposition 1 in [101]), one can deduce a stronger statement: $$a_{\Phi ^0}$$ is coercive on Φ0, ⊥ ⊥ (for the $$H^1_{\mathrm {per}}$$ norm), i.e. the previous assumption is equivalent to assuming that there exists a positive constant $$c_{\Phi ^0}$$ such that²

(7.117)

This coercivity constant appears in the upper bound of [DF( Φ⁰, 𝜖⁰)]⁻¹ so that, similarly as above, there exists an orthogonal matrix U in O(N) such that

This orthogonal matrix U can be identified: let us denote by $$M_{\Phi ^0_{N}, \Phi ^0}$$ the N × N matrix with entries

If $$M_{\Phi ^0_{N}, \Phi ^0}$$ is invertible, then $$U = M_{\Phi ^0_{N},\Phi ^0}^T(M_{\Phi ^0_{N},\Phi ^0} M_{\Phi ^0_{N},\Phi ^0}^T)^{-1/2}$$ .

Pushing the analysis further, it is proven in [101] that, by using the coercivity property (7.117) we can actually solve a simple linear problem, the solution of which is named the reconstructed error, that can be used to improve the accuracy of the initial solution $$\Phi ^0_{N}$$ . This approach is of a similar nature as the two grids method in [32] or [145] or the perturbative methodology in [33].

A last remark stands on the relative simplicity of the periodic framework since the use of a Fourier basis allows us to refer to the Fourier coefficients definition of the norms that makes the H⁻¹-norm above rather easily to evaluate. In the non-periodic case, where such an easy evaluation of the dual norm of the residual is not feasible, for instance with a local basis of finite element approximations, people classically refer to local reconstruction of the residual by solving local problems over patches of mesh elements that allows bounds to be provided on the dual norm of the residual in a computable way. We refer to [113] and [47, 48] for an original approach to this theory of equilibrated fluxes, and to [34, 35] and [45] for applications to eigenvalue problems.

After these elements on “a posteriori analysis”, we can proceed to “a priori analysis”.

We follow here the same lines as in the first subsection above; we start by showing the existence of a discrete solution in a neighborhood of a solution Φ⁰.

The analysis needs to be slightly more subtle since Φ⁰ is not unique due to the already mentioned gauge invariance. We have thus to work in the quotient space with respect to this gauge condition.

First, we remark that any element $$\Psi _{N} in \mathcal X_{N}^N\cap \mathcal {M}^{\Phi _{N_c}}$$ can be written as

(7.118)

where $$W_{N} \in \mathcal X_{N}^N \cap [\Pi _{N} (\Phi ^0)]^{\perp \!\!\!\perp }$$ satisfies $$0 \le M_{W_{N_c},W_{N_c}} \le 1_N$$ and where, for any W, $${\mathcal S}(W) = (1_N-M_{W,W})^{1/2}-1_N$$ is a N × N symmetric matrix. This provides an interesting parametrization $${\mathcal X}_{N}^N \cap \mathcal {M}^{\Phi _{N_c}}$$ which allows us to translate the original discrete minimization Kohn–Sham problem into a minimization problem set over $$\mathcal X_{N}^N \cap [\Pi _{N} (\Phi ^0)]^{\perp \!\!\!\perp }$$ , which is easier to work with, and show that there exists a unique solution close to ΠN( Φ⁰) thanks to the coercivity of $$a_{\Phi ^0}$$ . We refer to [24] for the details on this property.

This allows even more because, following the same arguments involving the ground state energy as in the first subsection above, an estimate that states

(7.119)

already shows some convergence that can further be used in the next step.

Once a solution to the discrete problem is set, one can actually perform the a priori analysis, i.e. the analysis of the error between Φ⁰ and $$\Phi ^0_{N}$$ .

The basic tool is again related to the precise estimate of the norm of $$[DF_{N}((\Phi ^0_{N},\epsilon ^0_{N})]^{-1}$$ which is derived from the above estimate (7.117) on the norm of [DF( Φ⁰, 𝜖⁰)]⁻¹ and subtle inequalities to evaluate the non linear terms thanks to (7.119).

We thus obtain

(7.120)

for some constant $$C \in \mathbb {R}_+$$ independent of N and σ related to the regularity of Φ⁰ (see [24] for more results in this situation).

Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement EMC2 No 810367), and the Center for Applied Mathematics for Energy Research Applications (CAMERA) funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research under Contract No. DE-AC02-05CH11231.

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Footnotes

1

In this context, uniformly means that the discretization scheme is such that there exists a non-increasing function $$f:\mathbb {N} \to \mathbb {R}_+$$ going to zero at infinity such that for any function v in the natural function space $$\mathcal {Y}$$ on which the problem is set, for instance the space $$H^1_{\mathrm {per}}(\Omega )$$ for the simple model considered in Sect. 7.6.1, the distance from v to its best approximation in a discretization space with Ndof degrees of freedom is bounded by $$f(N_{\mathrm {dof}}) \|v\|{ }_{\mathcal {Y}}$$ .

2

Note that, as is proven in [CCM], in the simplified linear framework: $$h=-\frac 12\Delta + V_{\mathrm {local}} + V_{\mathrm {nl}}$$ , the coercivity condition (7.117) is satisfied if and only if there is a gap $$c_{\Phi ^0} &gt; 0$$ between the lowest Nth and (N + 1)st eigenvalues of h.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_8

8. Recent Progress in Evaluating the Kohn–Sham Map

Lin Lin1  , Jianfeng Lu2   and Lexing Ying3  

(1)

Department of Mathematics, University of California, Berkeley and Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

(2)

Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, NC, USA

(3)

Department of Mathematics and Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA

Lin Lin (Corresponding author)

Email: linlin@math.berkeley.edu

Jianfeng Lu

Email: jianfeng@math.duke.edu

Lexing Ying

Email: lexing@stanford.edu

Abstract

In the framework of self-consistent field iteration for solving Kohn–Sham density functional theory (DFT), the evaluation of the Kohn–Sham map is typically the most time-consuming step. We review some recent progress in numerical methods to evaluate the Kohn–Sham map, which are suitable for large scale electronic structure calculations.

8.1 Introduction

The evaluation of the Kohn–Sham map is typically the computational bottleneck when solving Kohn–Sham equations for large and complex quantum systems (i.e. systems with thousands to tens of thousands of atoms). The first two rungs on the ladder of exchange-correlation functionals [47] comprise the local and semi-local exchange-correlation functionals. These functionals include local density approximation (LDA) and generalized gradient approximation (GGA) functionals. The effective Kohn–Sham potential Veff(ρ) and hence the Hamiltonian depends only on the electron density ρ. Formally, the density-to-density Kohn–Sham map can be written as

(8.1)

Here fβ(ε) = (1 + eβε)⁻¹ is the Fermi–Dirac function, β is the inverse temperature, and the chemical potential μ should be chosen to satisfy the normalization condition (N is the number of electrons)

(8.2)

The potential to density mapping is defined accordingly as $$V_{\mathrm {eff}}\mapsto \rho _{f_{\beta }\left (-\frac {1}{2} \Delta +V_{\mathrm {eff}}-\mu \right )}$$ satisfying the constraint (8.2).

$$\mathcal {F}_{\text{KS}}$$ maps the effective potential (external potential due to the nuclei, and the Hartree-exchange-correlation potential) to the electron density. The third rung on the ladder of exchange-correlation functionals are the meta-GGA functionals, of which the treatment is similar to that of the GGA functionals. However, meta-GGA functionals require some modification to the Laplacian operator and will not be discussed here.

The main challenge is the cost of orthogonalizing the Kohn–Sham orbitals. While such cost is often insignificant for small systems, it scales cubically with respect to the system size, and can quickly dominate the computational cost for large systems. The orthogonalization procedure is essential in all diagonalization-based methods, and leads to the “cubic scaling wall”. As a result, any algorithm with improved computational complexity must not perform an eigen-decomposition of the Kohn–Sham Hamiltonian.

Kohn–Sham DFT calculations with nonlocal functionals, such as rung-4 functionals (hybrid functionals) and rung-5 functionals, can be considerably more costly than calculations with exchange-correlation functionals from the first three rungs of the ladder. More specifically, Kohn–Sham equations with local and semi-local functionals can be viewed as eigenvalue problems corresponding to differential operators. When a rung-4 functional is used, the Kohn–Sham Hamiltonian operator becomes an integro-differential operator due to the Fock exchange term. For rung-5 functionals, the self-consistency is computationally rather challenging, and most calculations are done as a post-processing step to obtain the correlation energy as a perturbation to the self-consistent solution for a semi-local functional. For rung-4 nonlocal exchange-correlation functionals, since the effective potential depends on the density matrix P, one needs to evaluate the density matrix-to-density matrix (generalized) Kohn–Sham map

(8.3)

where Veff might be a non-local effective potential, and $$\mathcal {F}_{\text{DM}}$$ maps the effective potential to the density matrix P. Hence the additional computational challenge of large scale Kohn–Sham DFT calculations with nonlocal functionals is the efficient evaluation of nonlocal exchange-correlation functionals.

There has been much progress in the past two decades towards addressing the challenges of evaluating the Kohn–Sham maps [2–4, 10, 16, 17, 21, 23, 29, 30, 32, 38, 39, 45, 48, 49, 55, 56]. The most works involve “linear scaling” methods. The linear scaling property relies on the near-sightedness of the electron matter, as formulated by Kohn [27]: Vaguely speaking, the dependence of electron density at r on the effective potential at r′ is negligible when $$\lvert {\mathbf {r}} - {\mathbf {r}}^{\prime }\rvert $$ is large. More precisely, the Jacobian matrix of the Kohn–Sham map (also called the irreducible polarizability operator in the physics literature) can be well-approximated by a sparse matrix. Therefore one may evaluate the Kohn–Sham map using divide-and-conquer and/or sparse matrix techniques. We refer the readers to [6, 16] for extensive reviews of linear scaling methods. In order to apply linear scaling methods, a necessary condition is that the system should be insulating, i.e. have a positive energy gap. On the other hand, there have been much fewer options for reducing the computational complexity of evaluating the Kohn–Sham map for both insulating and metallic systems, which is the focus of this chapter. Our discussion is heavily biased towards our own work in the past few years, and some omissions are inevitable. We also refer readers to [36] for a more detailed review of numerical methods for solving Kohn–Sham equations.

The rest of the chapter is organized as follows. Section 8.2 first introduces the filtering methods, which aim at “postponing” the cubic scaling barrier for large systems. In particular, we introduce Chebyshev filtering type techniques to reduce the preconstant of the cubic scaling component. Then Sect. 8.3 introduces the pole expansion and selected inversion (PEXSI) method, of which the cost for evaluating the Kohn–Sham map with semi-local functionals is at most quadratic with respect to the system size. In Section 8.4 we introduce a method called the interpolative separable density fitting (ISDF), which provides a compact representation of the pair product of Kohn–Sham orbitals, and is a useful tool for reducing the cost of calculations with nonlocal functionals. Section 8.5 discusses techniques for reducing the cost of hybrid functional calculations, where we also illustrate how to use ISDF to reduce the cost of evaluating hybrid functionals and a technique called the adaptive compression to mitigate the impact due to the cost of the hybrid functional in the context of self-consistent field iterations. For simplicity and without loss of generality, we consider isolated, charge neutral, spinless systems throughout this chapter.

8.2 Filtering Methods

When the matrix-vector multiplication Hψ can be performed efficiently, it is common to apply iterative methods, such as conjugate gradient type methods [26, 46], to obtain low-energy eigenvectors of H. In each iteration, these methods apply H to the occupied orbitals once, followed by a Rayleigh–Ritz procedure to update the occupied orbitals. The cost of the latter step scales cubically with respect to the system size, which can dominate the computational cost for large systems.

The main idea of filtering methods is to apply H to the occupied orbitals multiple times to obtain a better approximation to the subspace spanned by the desired low-energy eigenvectors. Once such a subspace is identified, we still need to perform a Rayleigh–Ritz operation to diagonalize the projected Hamiltonian (i.e. the Hamiltonian operator restricted to the subspace). However, the frequency of applying cubic scaling steps is much reduced, which leads to improved algorithmic efficiency for large systems.

In order to filter out the unwanted information, we may choose a polynomial p(z) which maximizes the magnitudes at the eigenvalues ε₁, …, εN and at the same time minimizes at the eigenvalues $$\varepsilon _{N+1},\ldots , \varepsilon _{N_b}$$ . This can be optimally done by using a properly shifted and scaled Chebyshev polynomial Pk(z), where the subscript k denotes the degree of the polynomial [56]. The Chebyshev polynomials are bounded by 1 within the interval [−1, 1] and grow rapidly outside this interval. Hence we can map the standard interval [−1, 1] to $$[\varepsilon _{N+1},\varepsilon _{N_b}]$$ , and the shifted-and-scaled Chebyshev polynomial Pk(z) only amplifies the occupied states of the H. At the beginning of the SCF iteration, one does not know a priori the values of εN+1, $$\varepsilon _{N_b}$$ , or the order of the Chebyshev polynomial. One way to estimate these values is to perform a few steps of the Lanczos iteration [55], which also allows us to update these parameters on the fly. When applying the Chebyshev filtering algorithm, we often choose the number of states Ns to be slightly larger than N to accelerate the convergence, and the convergence is monitored with respect to the orbitals corresponding to the lowest N eigenvalues (see Algorithm 1 for a pseudo-code of the Chebyshev filtering algorithm) .

Algorithm 1: Chebyshev filtering algorithm.

An 8-line computer program for the Chebyshev filtering algorithm. The input is Hamiltonian matrix H and an orthogonal matrix X belongs to C superscript N subscript b times N subscript s. The output is a set of eigenvalues, epsilon subscript i, where i = 1 to N, and a set of eigenfunctions, psi subscript i, where i = 1 to N. The while do command is used.

We remark that although the Chebyshev filtering algorithm can be used to accurately obtain the lowest N orbitals by repeatedly applying the Chebyshev polynomials, in many implementations, we often perform a small number of iterations and do not evaluate the Kohn–Sham map accurately. In other words, we may combine the outer SCF loop with the inner diagonalization loop, and sometimes even a single inner iteration per SCF step can be sufficient. For many problems, such a combination does not increase the number of outer SCF iterations and hence may significantly reduce the overall running time.

The idea of filtering the Hamiltonian matrix also appears in the spectral slicing approach [1, 48, 49, 52, 54], where a number of matrix functions fi(H) are applied to the occupied orbitals. Here the scalar functions fi(⋅) can be polynomials or rational functions approximately supported only on a small energy interval. One can then perform the Rayleigh–Ritz step to compute the eigenvalues restricted to each interval. At the end, the eigenvalues obtained from different intervals are merged together. The spectral splicing approach is naturally suited for massive parallelization to reduce the wall clock time.

The preconstants of the cubic scaling steps (more specifically, the Rayleigh–Ritz step in step 5 of Algorithm 1) can also be significantly reduced using the complementary subspace strategy with two levels of Chebyshev filtering (CS2CF) [3]. In the Chebyshev filtering approach, the density matrix is computed as

where $$\widetilde {P}$$ is the projected density matrix of size Ns, with its eigenvalues given by the occupation numbers $$\{f_i\}_{i=1}^{N_s}$$ . Here we have used the finite temperature formulation of Kohn–Sham DFT. Since Ns is only slightly larger than N, most of the occupation numbers are equal to 1. States 1 through N₁ are as those with occupation numbers equal to 1. The remaining states, from N₁ + 1 through Ns, have occupations numbers less than 1. Let Nt be the number of these fractionally occupied states, i.e. Nt = Ns − N₁. The eigenvectors of the projected density matrix are $$\{\widetilde {\psi }_i\}_{i=1}^{N_s}$$ . Denoting the identity matrix of size Ns by $$I_{N_s}$$ , from the resolution of the identity we may rewrite the expression for the projected density matrix as

(8.4)

Hence if the Nt top eigenvectors $$\widetilde {\psi _i}$$ and corresponding occupation numbers fi are known, the projected density matrix $$\widetilde {P}$$ may be computed. Thus, instead of determining the full Ns × Ns set of vectors, we need to determine only an extremal block of vectors (of dimension Ns × Nt), corresponding to the states i = N₁ + 1 to Ns.

Moreover, physical quantities such as the electron density, energy, entropy and atomic force can all be computed by knowing only the top eigenstates. These eigenstates can be efficiently evaluated via another Chebyshev polynomial filtering steps applied to the projected Hamiltonian matrix $$\widetilde {H}$$ . Using this technique, the Rayleigh–Ritz step may be avoided altogether for insulating systems. For metallic systems, the cost for the Rayleigh–Ritz step can also be significantly reduced. Table 8.1 shows that the CS2CF strategy can be applied to insulating and metallic systems with $$\mathcal {O}(10^4)$$ atoms and efficiently parallelized over $$\mathcal {O}(10^4)$$ computational cores. The wall clock time to solution can be of an order of magnitude faster than parallel dense diagonalization methods such as ScaLAPACK or ELPA [43] for large systems.

Table 8.1

Wall clock times for one SCF iteration large systems using the CS2CF strategy in DGDFT (Credit: [3])

System

# atoms

Computational

CS2CF

Diagonalization

(# electrons)

cores

(subspace time) [s]

via ELPA [s]

Electrolyte3D3×3×3

8586 (29, 808)

3456

34) (19)

647)

SiDiamond3D10×10×10

8000 (32, 000)

3456

40) (24)

648)

Graphene2D8×8

11, 520 (23, 040)

4608

35) (27)

262)

CuFCC3D10×10×10

4000 (44, 000)

3000

75) (46)

199)

LiBCC3D12×12×12

27, 648 (82, 944)

12, 960

180 (165)

5844

8.3 Pole Expansion and Selected Inversion Method

To reduce the computational cost of Kohn–Sham DFT calculations with semi-local exchange-correlation functionals, it is important to realize that we require only the electron density in the Kohn–Sham map, but not the eigenfunctions or eigenvalues. This offers the opportunity to develop more efficient algorithms for the Kohn–Sham map. Since one does not need each individual eigenvector, it is perfectly fine to use an alternative representation for the occupied subspace, such as a density matrix or localized orbitals.

One such example is the Fermi operator expansion (FOE) method, which was originally designed to be used as a linear scaling method. Consider the density matrix at the finite temperature

The right-hand side is a matrix function with respect to the Hamiltonian matrix H. Instead of diagonalizing H and evaluating the matrix function using the eigen-decomposition, the basic idea of FOE is to expand the Fermi–Dirac function fβ(⋅) into an m-term expansion as

(8.5)

The corresponding matrix function approximation is

(8.6)

The above formulation is quite general: we only require each term gn(H − μ) to be a simple function so that the corresponding matrix function can be evaluated directly without diagonalizing the matrix.

One example of FOE is to expand the Fermi–Dirac function into polynomials [17]

The corresponding matrix function version is

(8.7)

Note that each term of (8.7) is simply a matrix power (H − μ)n, which can be evaluated using only matrix-matrix multiplication recursively, without diagonalizing H. When H is a sparse matrix, this means that the polynomial approximation to fβ(H − μ) can be a sparse matrix as well.

Besides the polynomial expansion, another possibility is to approximate the Fermi–Dirac function using rational functions. A rational function can be decomposed into a linear combination of terms of the form (ε − z)−p, where $$z\in \mathbb {C}$$ and p ≥ 1. In particular, if all terms use p = 1 the resulting expansion is called a simple pole expansion, or just a pole expansion. Compared to the polynomial expansion, there are two main advantages of using the rational expansion: First, the number of terms needed for the rational expansion can be much smaller than that required for the polynomial expansion to achieve the same accuracy. This is particularly the case for small gapped systems. Second, the use of the pole expansion can yield fast algorithms with reduced complexity even for metallic systems. This is called the pole expansion and selected inversion algorithm (PEXSI). To our knowledge, PEXSI is so far the only algorithm allowing such reduction of complexity.

While linear scaling algorithms in principle yield fast algorithms for the evaluation of the Kohn–Sham map, their accuracy often crucially depends on the decay of orbitals or density matrices, making them usually only suitable for insulating systems with a large gap. Another drawback from the point of view of practical use is that they often require user input on the support of truncation and other tuning parameters to achieve a balance between efficiency and accuracy. The pole expansion and selected inversion method (PEXSI) [30, 32] is a reduced scaling algorithm with computational scaling at most $$\mathcal {O}(N^2)$$ and smaller for lower dimensional systems. While it has a worse computational scaling than linear, the PEXSI algorithm can be applied to general systems and gives accurate results.

As an example of the FOE method, the PEXSI algorithm uses the following pole expansion to approximate the Fermi–Dirac distribution

(8.8)

Here we focus on the scaling with respect to the dimensionless quantity β ΔE, where ΔE is the spectral radius of the shifted operator H − μ. To reach a fixed target accuracy, although the number of terms of a straightforward construction of the pole expansion also scales as $$\mathcal {O}(\beta \Delta E)$$ [4], it has subsequently been improved to $$\mathcal {O}\left ( (\beta \Delta E)^{\frac 12}\right )$$ [45], $$\mathcal {O}\left ( (\beta \Delta E)^{\frac 13}\right )$$ [8], and finally to $$\mathcal {O}\left ( \log (\beta \Delta E)\right )$$ [33]. The work [44] introduces a near-optimal min-max approximation to the Fermi–Dirac function, which further reduces the number of poles needed in practice.

In order to obtain such a pole expansion, one possibility is to use the Cauchy contour integral formulation. Note that the Fermi–Dirac function fβ(ε) is a meromorphic function in $$\mathbb {C}$$ , and the only poles are at $$\varepsilon =(2n+1)\mathrm {i}\pi /\beta , n\in \mathbb {Z}$$ . Furthermore, fβ(ε) can be expanded using the following Matsubara expansion [42]

(8.9)

Note that (8.9) only converges conditionally, and the infinite summation must be performed symmetrically with respect to the position and negative choice of n. The number of terms needed in the direct truncation of the Matsubara series naturally scales as $$\mathcal {O}(\beta \Delta E)$$ .

The efficiency of the pole expansion can be improved by using a contour integral formulation:

(8.10)

Here the contour $$\mathscr {C}$$ should be chosen so that it encloses all the (real) eigenvalues of H − μ, but without any poles of fβ(z), i.e., $$(2n+1)\mathrm {i}\pi /\beta , n\in \mathbb {Z}$$ . This leads to the “dumbbell shaped” contour used in [33] (see Fig. 8.1 for an illustration). The contour is symmetric with respect to the chemical potential μ. The discretization points are chosen to be denser around μ in order to resolve the sharp transition of the Fermi–Dirac function at μ. At finite temperature, the contour integral formulation remains well defined for gapless systems, i.e. εN = εN+1.

A 4-quadrant graph of imaginary versus the real axes. It plots a Cassini oval-like diagram with its pinched area at the y-axis. The points epsilon subscript 1 to N is on the negative side of the real axis and epsilon subscript N+1, N+2, and so on, are on the positive side of the real axis. The units on the y-axis are i times pi by beta, 3 i times pi by beta, and so on.

Fig. 8.1

Illustration of the contour integral representation and pole expansion for the density matrix at finite temperature β⁻¹

Each term in the pole expansion corresponds to a matrix inverse, or Green’s function (zl − H)⁻¹, which can be evaluated directly without diagonalizing the matrix H. Equation (8.8) converts the problem of computing P to the problem of computing m Green’s functions. In order to find the Kohn–Sham map, instead of the entire density matrix P we only need the electron density, which corresponds to the diagonal matrix of P (again for simplicity we assume that a real space representation is used, e.g. the Kohn–Sham model is discretized on a grid). This amounts to the question of finding the diagonal of a Green’s function. Note that even if H is a sparse matrix, the matrix inverse (zl − H)⁻¹ can be a fully dense matrix. One direct method is to first evaluate each Green’s function and extracts its diagonal elements. However, when H is a sparse matrix, the computation of diagonal entries, and more generally the entries of Gl corresponding to the sparsity pattern of H, can be evaluated much more efficiently by means of the selected inversion method [11, 22, 32, 35].

Although H has been a general Hermitian matrix throughout this chapter, for simplicity we assume H to be a real symmetric matrix. This makes $$A=z_{l}-H\in \mathbb {C}^{N_{b}\times N_{b}}$$ a complex symmetric, non-singular matrix. For such a matrix, the standard approach for computing A⁻¹ is to first decompose A as

(8.11)

where L is a unit lower triangular matrix and D is a diagonal or a block-diagonal matrix. Equation (8.11) is often known as the LDL⊤ factorization of A. Given such a factorization, we can obtain $$A^{-1} = (x_1, x_2, \ldots , x_{N_{b}})$$ by solving a number of triangular systems

(8.12)

for j = 1, 2, …, Nb, where ej is the j-th column of the identity matrix I. The computational cost of such algorithm is generally $$\mathcal {O}(N_{b}^3)$$ . However, when A is sparse, we can exploit the sparsity structure of L and ej to reduce the complexity of computing selected components of A⁻¹.

The selected inversion algorithm can be heuristically understood as follows [35]. Let A be partitioned into a 2 × 2 matrix block form as

(8.13)

The first step of an LDL⊤ factorization produces a decomposition of A that can be expressed by

(8.14)

where α is often referred to as a pivot, ℓ = bα⁻¹ and $$S = \widetilde {A} - b\alpha ^{-1} b^{\top }$$ is known as the Schur complement. The same type of decomposition can be applied recursively to the Schur complement S until its dimension becomes 1. The product of lower triangular matrices produced from the recursive procedure, which all have the form

where ℓ⁽¹⁾ = ℓ = bα⁻¹, yields the final L factor. At this last step the matrix in the middle becomes diagonal, which is the D matrix.

From (8.14), A⁻¹ can be expressed by

(8.15)

This expression suggests that, once α and ℓ are known, the task of computing A⁻¹ can be reduced to that of computing S⁻¹. Because a sequence of Schur complements is produced recursively in the LDL⊤ factorization of A, the computation of A⁻¹ can be organized in a recursive fashion too. Clearly, the reciprocal of the last entry of D is the (Nb, Nb)-th entry of A⁻¹. Starting from this entry, which is also the 1 × 1 Schur complement produced in the (Nb − 1)-th step of the LDL⊤ factorization procedure, we can construct the inverse of the 2 × 2 Schur complement produced at the (Nb − 2)-th step of the factorization procedure, using the recipe given by (8.15). This 2 × 2 matrix is the trailing 2 × 2 block of A⁻¹. As we proceed from the lower right corner of L and D towards their upper left corner, more and more elements of A⁻¹ are recovered. At the end, we recover all the diagonal entries of A⁻¹exactly. In fact, given the factorization A = LDL⊤, the selected inversion algorithm can be used to efficiently compute all entries $$\{A^{-1}_{i,j}\mid (L+L^{\top })_{i,j} \ne 0\}$$ .

In order to understand the asymptotic complexity of the selected inversion algorithm, we assume without loss of generality that the sparsity pattern of H is similar to that obtained by the second-order central difference discretization of a Laplacian operator. The computational cost associated with the LDL⊤ factorization, as well as the selected inversion algorithm, scales as $$\mathcal {O}(N_b)$$ , $$\mathcal {O}(N_{b}^{1.5})$$ and $$\mathcal {O}(N_{b}^{2})$$ for one, two and three-dimensional systems, respectively [33]. We remark that this complexity count is robust under changes of the discretization scheme as long as local basis sets are used. Hence for quasi-1D systems (such as nanotubes) and quasi-2D systems (such as monolayer systems and surfaces), the computational cost also scales as $$\mathcal {O}(N_{b})$$ and $$\mathcal {O}(N_{b}^{1.5})$$ , respectively. A pseudo-code for the selected inversion algorithm is given in Algorithm 2.

Algorithm 2: Selected inversion algorithm based on LDL⊤ factorization.

A 7-line computer program for the selected inversion algorithm based on L D L transpose factorization. The input is L D L transpose factorization of a symmetric matrix A belongs to C superscript N subscript b times N subscript b. The output is selected elements of A inverse. The for do command is used.

The pole expansion and selected inversion (PEXSI) method [22, 33, 35] therefore combines the pole expansion and the selected inversion, and evaluates the Kohn–Sham map without solving any eigenvalues or eigenfunctions. The selected inversion method is an exact method if exact arithmetic is used, i.e. the only error in the selected inversion method is due to round off errors. Hence the accuracy of the PEXSI method is determined by the pole expansion, which can be systematically improved by increasing the number of poles. The PEXSI method is ideally suited for massively parallel computers. The treatment of the poles can be parallelized in a straightforward fashion with communication needed only at the end to construct the density matrix. The selected inversion method itself can also be massively parallelizable to thousands of processors [22], and the total number of processors that can be efficiently used by PEXSI can be over 100, 000. The PEXSI software package (available at http://​www.​pexsi.​org, distributed under the BSD license) has now been integrated into electronic structure software packages such as BigDFT, CP2K, DFTB+, DGDFT, FHI-aims, QuantumWise ATK, SIESTA, and is part of the “Electronic Structure Infrastructure” (ELSI) package [53].

In addition to computing the charge density at a reduced computational complexity in each SCF iteration, PEXSI can be leveraged to compute energy, free energy and the atomic forces efficiently without diagonalizing the Kohn–Sham Hamiltonian, using the same set of poles as those used for computing the charge density [30]. Another numerical issue associated with the PEXSI technique, as well as the Fermi operator expansion techniques in general, is to determine the chemical potential, so that the condition

(8.16)

is satisfied. Note that Nβ(⋅) is a non-decreasing function of μ. Hence the chemical potential can be determined via a bisection strategy, or Newton’s method. When Newton’s method is used, the chemical potential converges rapidly near its correct value. However, the standard Newton’s method may not be robust enough when the initial guess is far away from the correct chemical potential. It may give, for example, too large a correction when $$N_{\beta }^{\prime }(\mu )$$ is close to zero, as when μ is near the edge or in the middle of a band gap.

One way to overcome the above difficulty is to use an approximation to the function Nβ(ε) to narrow down the region in which the correct μ must lie. This function can be seen effectively as a (temperature smeared) cumulative density of states, counting the number of eigenvalues in the interval (−∞, ε). We can evaluate such zero temperature limit, denoted by N∞(ε), again without computing any eigenvalues of H, by using Sylvester’s law of inertia [50]. It requires fewer floating point operations than the complex arithmetic direct sparse factorization used in PEXSI. During the self-consistency field iteration, the zero temperature limit N∞(ε) evaluated from the inertia counting procedure may be used to construct upper and lower bounds of the chemical potential. This makes it possible to perform PEXSI calculations over multiple energy points only once per SCF iteration, without sacrificing the accuracy at convergence [24].

As an example, we apply the parallel PEXSI method to two systems DG_Graphene_2048 and DG_Graphene_8192, which are disordered graphene systems with 2048 and 8192 atoms, respectively, and compare its performance with a standard approach that requires a partial diagonalization of (H, S). The ScaLAPACK subroutine pdsyevr [51] based on the multiple relatively robust representations (MRRR) algorithm is used to perform such diagonalization. Though both H and S are sparse matrices, the MRRR algorithm treats them as dense matrices. For $$H,S\in \mathbb {R}^{N\times N}$$ , the MRRR algorithm first performs a tridiagonalization procedure with $$\mathcal {O}(N^3)$$ cost, then solves the eigenvalues and eigenvectors of the tridiagonal system with $$\mathcal {O}(N^2)$$ cost, and finally constructs the eigenvectors with $$\mathcal {O}(N^3)$$ cost. Figure 8.2 shows that for graphene problem with 2048 and 8192 processors, the PEXSI technique is nearly two orders of magnitude faster than the ScaLAPACK routine pdsyevr, and can be scalable to a much larger number of processors. The advantage of PEXSI becomes even clearer for a disordered graphene system with 32, 768 atoms. For this case, the diagonalization routine is no longer feasible, while the time to solution for the PEXSI technique can be as small as 241 s (Fig. 8.3).

2 connected line graphs of time versus proc number for disordered graphene 2048 and 8192 plot 2 decreasing lines for P E X S I and P S e l l n v, and a fluctuating line for diagonalize that decreases and increases. Only 3 vertices are present for the line diagonalize and the line is above the other 2 lines.

Fig. 8.2

The wall clock time versus the number of cores for a graphene system (Credit: [22])

A connected line graph of time versus proc number for disordered graphene 32768 plots 2 decreasing lines for P E X S I and P S e l l n v, and a decreasing dashed slant line for ideal diagonalize which is above the other 2 lines.

Fig. 8.3

The wall clock time versus the number of cores for a graphene system with 32, 768 atoms (Credit: [22])

8.4 Interpolative Separable Density Fitting Method

Kohn–Sham DFT calculations with rung-4 functionals require the evaluation of the generalized Kohn–Sham map. They often involve the pair product of orbitals, in the form $$\{\varphi ^*_{i}({\mathbf {r}})\psi _{j}({\mathbf {r}})\}_{i,j=1}^{N}$$ , and the number of the pair products is $$\mathcal {O}(N^2)$$ . However, since these pair products are not independent from each other, it is possible to approximate pair products using only $$\mathcal {O}(N)$$ functions, by a technique called interpolative separable density fitting (ISDF) [39, 40]. This can be used to reduce the cost of rung-4 functional calculations with a large basis set [9, 21], large scale phonon calculations [34], and post-Hartree–Fock calculations such as RPA correlation energy [38] and second/third order Møller–Plesset perturbation theory calculations [28].

As an intuitive explanation that a compressed representation of the pair products is possible, let us take a real space representation and note that the number of grid points Ng to represent the orbital pairs scales linearly with respect to N, but not quadratically as the number of orbital pairs. This suggests that the numerical rank of $$\{\varphi ^{\ast }_{i}\psi _{j}\}$$ , viewed as a matrix of size Ng × N², must scale asymptotically as $$\mathcal {O}(N)$$ . It has been proved in [37] that for any 𝜖 and N, there exists a subspace BN of dimension on the order of 𝜖−δN1+δ such that

where {ψi}, i = 1, …, N are the first N eigenfunctions of an effective Hamiltonian on a compact domain or closed manifold, and δ is an arbitrary positive constant. This shows that the N² pair products of eigenfunctions can be approximated with nearly $$\mathcal {O}(N)$$ auxiliary basis functions, regardless of the discretization level.

The ISDF method uses the following compression format

f Subscript beta Baseline left parenthesis upper H minus mu right parenthesis almost equals sigma summation Underscript n equals 1 Overscript m Endscripts c Subscript n Baseline left parenthesis upper H minus mu right parenthesis Superscript n minus 1 Baseline period

(8.17)

Here is a subset of real space grid points $$\{{\mathbf {r}}_{i}\}_{i=1}^{N_{g}}$$ on which the orbitals are evaluated. We will refer to as the interpolation points, and $$\{\zeta _{\mu }({\mathbf {r}})\}_{\mu =1}^{N_{\mu }}$$ sampled on $$\{{\mathbf {r}}_{i}\}_{i=1}^{N_{g}}$$ the interpolation vectors. These terms come from the understanding of ISDF from the perspective of interpolation. Let denote a set of grid points in the real space, and let ζμ(r) be the Lagrange interpolation function on these grid points satisfying

f Subscript beta Baseline left parenthesis upper H minus mu right parenthesis almost equals sigma summation Underscript n equals 1 Overscript m Endscripts c Subscript n Baseline left parenthesis upper H minus mu right parenthesis Superscript n minus 1 Baseline period

The ISDF decomposition becomes sufficiently accurate as one systematically refines the set . In the extreme case, all grid points need to be selected and Nμ = Ng. In practice, the interpolation often becomes sufficiently accurate when N < Nμ ≪ Ng, especially when both {φi(r)} and {ψi(r)} consist of functions that are sufficiently smooth. In general we may set Nμ = cN where c is a relatively small constant (4 ∼ 10).

Let us first discuss how to find the interpolation vectors, assuming that the interpolation points are given. Define

(8.18)

Equation (8.17) can be written as

(8.19)

where each column of Z is defined by Eq. (8.18) sampled on real space grids $$\{{\mathbf {r}}_{i}\}_{i=1}^{N_g}$$ . $$\Theta = [\zeta _1, \zeta _2, \ldots , \zeta _{N_{\mu }}]$$ contains the interpolating vectors, and each column of C with a multi-index (i, j) is given by

f Subscript beta Baseline left parenthesis upper H minus mu right parenthesis almost equals sigma summation Underscript n equals 1 Overscript m Endscripts c Subscript n Baseline left parenthesis upper H minus mu right parenthesis Superscript n minus 1 Baseline period

Equation (8.19) is an over-determined linear system with respect to the interpolation vectors Θ. One possible way to solve the over-determined system is to impose the Galerkin condition

(8.20)

The solution to the above is given by

(8.21)

which can also be understood as a least squares approximation to the solution of Eq. (8.17).

It may appear that the cost for matrix-matrix multiplications ZC∗ and CC∗ scales as $$\mathcal {O}(N_g N^{3})$$ and $$\mathcal {O}(N^4)$$ , respectively, because the size of Z is Ng × N² and the size of C is Nμ × N² (here for simplicity we use that Nμ ∼ N). However, both multiplications can be carried out with fewer operations due to the separable structure of Z and C, and the computational complexity for computing the interpolation vectors can be reduced to $$\mathcal {O}(N_gN^{2}+N^3)$$ [39].

In order to optimize the set of interpolation points, a general strategy is to use the QR factorization with column pivoting (QRCP). The pseudocode of the ISDF algorithm is given in Algorithm 3.

Algorithm 3: Interpolative separable density fitting (ISDF) method

An algorithm for the interpolative separable density fitting method. The inputs are functions of phi subscript i of r and psi subscript j or r. The output is interpolation points of r subscript mu cap, C, and theta. It includes 4 points.

The above procedure can be further accelerated. Using Eq. (8.19) directly is sub-optimal as the storage requirement for the matrix Z is $$\mathcal {O}(N_gN^{2})$$ and the computational cost associated with a standard QRCP procedure is $$\mathcal {O}(N_g^2N^{2})$$ . One possibility is to lower the cost of QRCP by using a random matrix to subsample columns of the matrix Z to form a smaller matrix $$\widetilde {Z}$$ of size $$N_{g}\times \widetilde {N}_{\mu }$$ , where $$\widetilde {N}_{\mu }$$ is only slightly larger than Nμ [39, 40]. The reduced matrix size allows the computational cost of the QRCP procedure to be reduced to $$\mathcal {O}(N_gN^{2})$$ . The implementation and parallelization of ISDF is then relatively straightforward, as QRCP algorithms are available in standard linear algebra software packages such as LAPACK and ScaLAPACK,

Another possibility for choosing interpolation points is to use a heuristic strategy. Note that an effective choice of the set of interpolation points should satisfy the following two conditions. (1) The distribution of the interpolation points should roughly follow the electron density. In particular, there should be more points in regions with higher electron density. (2) The interpolation points should be separated from each other, as otherwise the matrix formed by the interpolation vectors will be highly ill-conditioned. The QRCP procedure satisfies both (1) and (2) simultaneously. On the other hand, the conditions above can also be satisfied through a much simpler centroidal Voronoi tessellation (CVT) procedure applied to a weight vector, such as the electron density [9]. More specifically, in the centroidal Voronoi tessellation (CVT) approach, we may partition the grid points in the global domain into Nμ Voronoi cells. The interpolation points can then be simply chosen to be the centroids corresponding to each cell. The CVT procedure can be effectively implemented through a k-Means algorithm [41]. Besides reduction of the computational cost, the use of a k-Means algorithm also produces a smoother potential energy surface particularly in the context of ab initio molecular dynamics. We refer to [9] for more details of this approach.

8.5 Hybrid Functionals

For rung-4 exchange-correlation functionals, the Kohn–Sham equations take the form

(8.22)

Here VHxc is the Hartree and exchange-correlation contribution from the electron density ρ only, and $$V_{\text{x}}^{\text{EX}}[P]$$ is derived from the exact exchange functional $$E_{\text{x}}^{\text{EX}}[P]$$ , with kernel

(8.23)

Here K(r, r′) is the kernel for the electron-electron interaction. When the (bare) Hartree–Fock exchange is used, $$K({\mathbf {r}},{\mathbf {r}}^{\prime })=1/\lvert {\mathbf {r}}-{\mathbf {r}}^{\prime }\rvert $$ is the Coulomb kernel. When screened Fock exchange interactions [19] are used, K takes the form of a screened Coulomb kernel: $$K({\mathbf {r}},{\mathbf {r}}^{\prime })=\text{erfc}(\alpha _s\lvert {\mathbf {r}}-{\mathbf {r}}^{\prime }\rvert )/\lvert {\mathbf {r}}-{\mathbf {r}}^{\prime }\rvert $$ , where αs is called the inverse screening length parameter. Hence the Kohn–Sham Hamiltonian is a nonlocal operator. $$V_{\text{x}}^{\text{EX}}[P]$$ is also called the Fock exchange operator.

The kernel of $$V_{\text{x}}^{\text{EX}}[P]$$ is a low rank operator, due to the Hadamard product (i.e., element-wise product) between the kernels of P and K. $$V_{\text{x}}^{\text{EX}}[P]$$ is also negative semi-definite. In quantum chemistry calculations with a small basis set, denoted by $$\Phi =[\varphi _{1},\ldots ,\varphi _{N_{b}}]$$ , the matrix elements of the Fock exchange operator projected to the span of the basis set is often computed:

(8.24)

Here we have expanded the occupied orbital as

The computation of the last term in Eq. (8.24) involves the so-called two-electron integrals, and the computational cost for constructing the Fock exchange matrix $$(V_{\text{x}}^{\text{EX}}[P])_{pq}$$ scales as $$\mathcal {O}(N^{4})$$ .

The formation of the Fock exchange matrix allows us to solve the Kohn–Sham equations by directly diagonalizing the Kohn–Sham Hamiltonian. When iterative methods (conjugate gradient, Chebyshev filtering etc) are used, the associated asymptotic complexity can in fact be reduced to $$\mathcal {O}(N^{3})$$ without any approximation. We only have access to the Fock exchange operator via its application to an occupied orbital φj as

(8.25)

for $$P = \sum _{i=1}^N |\varphi _i\rangle \langle \varphi _i|$$ . Here we deliberately distinguish the orbitals in the density matrix ({φi}) and the orbitals $$V_{\text{x}}^{\text{EX}}$$ acts on ({φj}) to emphasize that they may correspond to different density matrices before self-consistency is achieved. The integral in Eq. (8.25) can be implemented by solving N Poisson-like equations, with the effective charge given by the pair product $$\varphi _{i}^{\ast }({\mathbf {r}})\varphi _{j}({\mathbf {r}})$$ . Let Ng denote the grid size, and the computational cost of solving each Poisson-like equation using FFT scales as $$\mathcal {O}(N_{g} \log N_{g})$$ . Therefore the cost for applying the Fock exchange operator to all occupied orbitals scales as $$\mathcal {O}(N_{g} \log N_{g} N^2)\sim \mathcal {O}(N^{3})$$ . However, the preconstant of this cubic scaling component can be very large. In practical Hartree–Fock calculations, the application of the Fock exchange operator can often take more than 95% of the overall computational time.

8.5.1 Adaptive Compression Method

The adaptively compressed exchange operator (ACE) method [29] accelerates hybrid functional calculations by reducing the frequency of applying the Fock exchange operator, without compromising the accuracy of the self-consistent solution. This is similar in spirit to the filtering method introduced in Sect. 8.2, which improves the efficiency by reducing the frequency of performing the expensive Rayleigh–Ritz procedure. There is a related approach called the projector based compression of the exchange operator [5, 10], which can also reduce the frequency of applying the Fock exchange operator.

Although the Fock exchange operator $$V_{\text{x}}^{\text{EX}}$$ is a full rank operator, ACE constructs a low rank surrogate operator, denoted by $$\widetilde {V}_{\text{x}}^{\text{EX}}$$ , to approximate $$V_{\text{x}}^{\text{EX}}$$ . Note that $$V_{\text{x}}^{\text{EX}}$$ is generally a dense, full rank operator. Hence the low rank surrogate cannot be expected to be accurate when applied to an arbitrary vector. Instead we only require $$\widetilde {V}_{\text{x}}^{\text{EX}}$$ to be accurate when applied to all occupied orbitals.

Consider a set of orbitals $$\{\varphi _{i}\}_{i=1}^{N}$$ which defines implicitly the density matrix P, which further gives $$V_{\text{x}}^{\text{EX}}[P]$$ . We first apply $$V_{\text{x}}^{\text{EX}}[P]$$ to $$\{\varphi _{i}\}_{i=1}^{N}$$ as

(8.26)

The adaptively compressed exchange operator should satisfy the conditions

(8.27)

The choice of this surrogate operator is not unique. One possible choice satisfying the conditions (8.27) is given by

(8.28)

where B = M⁻¹ is a negative definite matrix, and

(8.29)

Let us perform Cholesky factorization for − M, i.e., M = −LL∗, where L is a lower triangular matrix. Then we get B = −L−∗L⁻¹. Define the projection vector in the ACE formulation as

(8.30)

The adaptively compressed exchange operator is then given by

(8.31)

Once ACE is constructed, the cost of applying $$\widetilde {V}_{\text{x}}^{\text{EX}}$$ to any orbital φ is similar to the application of a nonlocal pseudopotential operator, thanks to its low-rank structure. ACE only needs to be constructed once per outer iteration, and we can repeatedly use the operator for all the subsequent SCF iterations, until the exchange operator is updated. The ACE formulation has been integrated in electronic structure software packages such as Quantum ESPRESSO [7, 13] and ABINIT [18].

Table 8.2 demonstrates the accuracy of the ACE formulation for Kohn–Sham DFT calculations with the HSE06 [19] hybrid functional, for silicon systems ranging from 64 to 1000 atoms. The ACE formulation can evaluate the HF energy and energy gap accurately (total energy difference is under 10⁻⁴ Hartree) even for large systems. Furthermore, ACE can perform hybrid functional calculations at a fraction of the cost of conventional methods [20].

Table 8.2

Comparison between the conventional hybrid DFT calculations and ACE enabled hybrid DFT calculations in terms of the HF energy $$E_{\text{x}}^{\text{EX}}$$ (Hartree) and the energy gap Egap (Hartree) for the Si₆₄, Si₂₁₆, Si₅₁₂ and Si₁₀₀₀ systems. The corresponding relative errors of the HF energy are shown in parentheses (Credit: [20])

Methods

ACE HSE06

Conventional HSE06

systems

Egap

Egap

Si₆₄

− 13.541616 (10⁻⁶)

1.488335

−13.541629

1.488352

Si₂₁₆

− 45.471192 (10⁻⁷)

1.449790

−45.471190

1.449790

Si₅₁₂

− 107.698011 (10⁻⁷)

1.324901

−107.698016

1.324902

Si₁₀₀₀

− 210.300628 (10⁻⁶)

1.289162

−210.300524

1.289128

The efficiency of the ACE formulation also rests on the assumption that the magnitude of the Fock exchange operator is relatively small compared to other components of the Hamiltonian. In particular, in linearized Hartree–Fock-like equations, the convergence properties of the adaptive compression formulation can be rigorously analyzed for Hamiltonians of the form H = A + B, where the operator norm $$ \left \| A \right \|{ }_2$$ is much larger than that of $$ \left \| B \right \|{ }_2$$ . We refer readers to [31] for details.

8.5.2 Interpolative Separable Density Fitting Method

Since Eq. (8.25) involves solving Poisson-like equations for all pair products of occupied orbitals, we may reduce the cost by using the interpolative separable density fitting (ISDF) technique in Sect. 8.4. Using the decomposition (8.17), we can rewrite (8.25) as

f Subscript beta Baseline left parenthesis upper H minus mu right parenthesis almost equals sigma summation Underscript n equals 1 Overscript m Endscripts c Subscript n Baseline left parenthesis upper H minus mu right parenthesis Superscript n minus 1 Baseline period

(8.32)

Here $$V^{\zeta }_{\mu }({\mathbf {r}}) = \int K({\mathbf {r}},{\mathbf {r}}^{\prime }) \zeta _{\mu }({\mathbf {r}}^{\prime }) \,\mathrm {d} {\mathbf {r}}^{\prime }$$ is the Coulomb-like potential with the interpolation vector ζμ being the charge-like quantity.

One may also combine ISDF and ACE together to further improve the accuracy. Note that the matrix M in Eq. (8.29) is negative semidefinite. Since the ISDF compression introduces numerical error, if we directly use Eq. (8.29) to evaluate M, the resulting matrix may be neither Hermitian nor negative semidefinite, which leads to numerical instability. One way to overcome this problem is to apply the ISDF decomposition to both sides of the kernel K as

(8.33)

Then M is Hermitian negative semidefinite, which directly follows from the fact that $$\widetilde {M}$$ is constructed to be Hermitian negative semidefinite. We remark that although the number of Poisson-like equations is reduced from $$\mathcal {O}(N^2)$$ to $$\mathcal {O}(N)$$ , the cost of ISDF-based hybrid functional calculation still scales cubically due to the other matrix-matrix multiplication and inversion operations. However, the preconstant associated with such operations can be smaller than that associated with solving Poisson-like equations. Hence ISDF is effective when it is relatively costly to solve Poisson-like equations, such as in the context of planewave methods, but even more so for finite difference and finite element methods.

As an example, the accuracy of ACE-ISDF in hybrid functional calculation for the semiconducting Si₂₁₆ system (with 216 atoms) and metallic Al₁₇₆Si₂₄ system (with 200 atoms) respectively are shown in Fig. 8.4. The accuracy is controlled by a single rank parameter c, which sets the number of interpolation points to be Nμ = cNe. The accuracy of ACE-ISDF improves systematically as the rank parameter c increases. When the rank parameter is large enough (≥ 20.0), the results are fully comparable to those obtained from the benchmark calculations. Furthermore, for a moderate choice of the rank parameter c = 6.0, the error of the energy per atom reaches below the chemical accuracy of 1 kcal/mol. This demonstrates that ACE-ISDF is applicable to both insulating and metallic systems.

2 connected line graphs of delta E subscript H F and delta E versus rank parameter plot 4 decreasing lines for S i 216 C V T, S i 216 Q R C P, A l 176 S i 24 C V T, and A l 176 S i 24 Q R C P. The lines are very close to each other. In A, the line for A l 176 S i 24 Q R C P is above all other lines.

Fig. 8.4

The accuracy of ACE-ISDF-based hybrid functional calculations (HSE06) obtained by using the centroidal Voronoi tessellation (CVT) and QR decomposition with column pivoting (QRCP) procedures to select the interpolation points, with varying rank parameter c from 4 to 20 for Si₂₁₆ and Al₁₇₆Si₂₄, including the error of (a) Hartree–Fock exchange energy ΔEHF (Ha/atom) and (b) total energy ΔE (Ha/atom) (Credit: [9])

8.6 Conclusion and Future Directions

The past decade has witnessed significant progress in accelerating the evaluation of the Kohn–Sham map and the generalized Kohn–Sham map in large scale electronic structure calculations. The methods reviewed in this paper are heavily biased towards our own work in this direction. The accuracy and the applicability range of Kohn–Sham DFT are ultimately determined by the choice of exchange-correlation functionals. In recent years, an increasing number of numerical calculations are now performed using the more accurate rung-4 and rung-5 functionals. In particular, the family of rung-5 functionals is still under development and the formulation and numerical treatment of self-consistency for such functionals is much more complicated, since these functionals involve virtual orbitals and orbital energies. Most current strategies for rung-5 functional treat them as post-DFT methods: First obtain the effective Hamiltonian and corresponding Kohn–Sham orbitals based on a semi-local or hybrid functional, and then calculate the energy of the rung-5 functional as a post-processing step (using perturbation theory). Self-consistency calculations have been performed using the optimized effective potential (OEP) framework [12, 14, 15] or more recently the generalized optimized effective potential approach [25], but they require considerable numerical effort. An active research direction is to find an efficient approach for a self-consistent treatment of rung-5 functionals for large scale systems.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_9

9. Augmented Plane Wave Methods for Full-Potential Calculations

Huajie Chen1   and Reinhold Schneider2  

(1)

School of Mathematical Sciences, Beijing Normal University, Beijing, China

(2)

Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

Huajie Chen

Email: chen.huajie@bnu.edu.cn

Reinhold Schneider (Corresponding author)

Email: schneidr@math.tu-berlin.de

Abstract

In this chapter, we review the family of augmented plane wave methods, which is among the most accurate methods for full-potential/all-electron DFT calculations. Further, we view this type of approach as a nonconforming mortar method and provide some rigorous numerical analysis results. This provides an understanding of the efficiency of the methods from a mathematical point of view.

9.1 Introduction

First principles simulations using density functional theory (DFT) [33, 35] have become one of the most frequently used and computationally tractable tools in condensed matter physics. Along with the advances in computing technology that have occurred during the last decade, there have been important algorithmic improvements, particularly for plane wave-based methods. For certain classes of materials it is now feasible to simulate systems containing thousands of atoms in a unit cell. This opens the door for the direct application of these techniques in studying a substantial set of “real materials” problems. Further, it is possible to use first principles calculations to create sophisticated data sets that can parameterize model Hamiltonians, which then can be used to model even more complex materials problems. At this time, DFT practitioners are divided into two nearly disjoint communities: one employing pseudopotentials [44] and relatively simple basis sets, particularly plane waves; and the other using methods with complex but efficient basis sets, such as the augmented plane wave, the muffin-tin orbital and related methods [44]. The latter community has traditionally dominated research on transition metals and their compounds.

Plane waves with pseudopotentials are natural methods for periodic systems, which are simple to implement and give relatively accurate simulations. The pseudopotential approximations replace singular nuclear attraction potentials and complicated effects of the motion of core electrons by smooth potentials. They give satisfactory results in many cases, but sometimes fail, especially for transition metals. The mathematical analysis of the pseudopotential approximations is very rare, and we refer to [12, 13, 16, 21] for some recent works. Moreover, the core states are fixed in an atomic reference configuration. The frozen core approximation is generally reliable, but breaks down for some elements with extended core states. The core electrons also have to be considered sometimes and are responsible for some properties. Therefore, the full-potential/all-electron calculations are necessary.

For eigenvalue problems with singular potentials in full-potential calculations, plane waves are inefficient basis functions for describing the cusps and rapidly varying wavefunctions near the nuclei [28, 29, 32]. In contrast, it is observed that a significant part of the rapid oscillations can be captured by atomic orbitals such as Gaussians and Slater-type orbitals [44], which have been widely used in quantum chemistry and studied through rigorous mathematical analysis [8, 17, 38, 39]. Therefore, it would be practically efficient to approximate the wavefunction in a crystal by using combinations of plane waves and appropriate atomic orbitals, which comes to the idea of augmented plane waves (APW) [53], linearized augmented plane waves (LAPW) [51], and their extensions by including local atomic orbitals (APW+lo) [43, 49, 52]. In this chapter, we will mainly discuss this type of augmented plane wave-based approach.

A more realistic scenario is that these traditionally distinct approaches will eventually converge, and there are strong relationships between plane wave pseudopotential methods and the APW-based methods [51]. Both approaches have a common starting point, i.e. a plane wave basis set. Further, both approaches are motivated by the observation that plane waves themselves are inefficient for direct simulations of the wavefunctions in a crystal. In both cases, the basis functions are labeled as plane waves, by wave vector G, and the Hamiltonian matrix elements are modified in such a way that rapid convergence with the maximum |G| (energy cutoff) can be obtained. In the plane wave pseudopotential approach, this problem is avoided by replacing the Hamiltonian near the nuclei with a smoother pseudo-Hamiltonian in such a way that the valence energy spectrum is reproduced, but the core states are removed. Although this approach may have offered significant computational advantages, much of the complexity of the method is transferred from the calculation itself to the generation of the pseudopotentials. In the APW-based methods, the plane waves are modified near the atoms rather than the Hamiltonian. This modification/augmentation is such that augmented plane waves with small |G| can reproduce the rapid variations in the wavefunctions. Thus, as in the pseudopotential method, the valence energy spectrum can be reproduced with a low energy cutoff for plane waves. The appearance and success of APW-based methods have already triggered the development of improved pseudopotentials, the so-called projector augmented wave (PAW) method [14, 37], which has been used, for example, in the VASP code [55], see also the discussions in [12, 13, 21].

The goal of this chapter is twofold. First, it provides a detailed and self-contained exposition of the APW-based methods. Secondly, it gives an understanding of this approach from a mathematical point of view. The organization of this chapter is as follows. In Sect. 9.2, we give a self-contained exposition of the APW-based methods. In Sect. 9.3, we discuss the numerical discretizations based on domain decomposition techniques, which are based on a similar idea as the APW-based methods. In Sect. 9.4, we provide some numerical analysis of the APW-based methods. In Sect. 9.5, we give some conclusions and future perspectives.

9.2 The Augmented Plane Wave Methods

The augmented plane wave (APW) method was originally proposed by Slater [53] in 1937. In spite of its demanding computational cost due to the energy dependency, it has been successfully used (see e.g., [23]). Several improvements of the basis set were tried to get rid of the energy dependency, the first really successful one was the linearization scheme introduced by Andersen [1, 2], leading to the linearized augmented plane wave (LAPW) method [34]. It was further developed by including local atomic orbitals (APW+lo) to have enough variational flexibility in the radial basis functions [43, 49, 52]. Several widely used quantum chemistry and solid-state physics software programs are based on these methods, such as Exciting [25], FLEUR [27], and WIEN2k [58].

The philosophy behind the family of APW methods is to simulate many-electron systems by introducing a basis set that is in some ways the “best of both worlds”. The smoothly varying parts of the wave functions between the atoms are represented by plane waves, and the rapidly varying parts near the nuclei are represented by radial atomic functions time spherical harmonics. We will review relevant aspects of these methods in the following.

For simplicity, we will consider a linear Schrödinger type equation on $$\mathbb {R}^3$$ . Let $$A\in \mathbb {R}^{3\times 3}$$ be a non-singular matrix and $$\mathbb {L}=A\mathbb {Z}^3$$ be the Bravais lattice. Let $$\mathbb {L}^{\prime }$$ be the dual/reciprocal lattice of $$\mathbb {L}$$ and Ω the unit cell of $$\mathbb {L}$$ . Then we consider the eigenvalue problem: Find $$(\varepsilon _i,u_i)\in \mathbb {R}\times H^1_{\mathrm {per}}(\Omega )$$ such that

(9.1)

where Vper(r) is a periodic potential with respect to $$\mathbb {L}$$ and has singularities at nuclear positions. This linear eigenvalue problem can be viewed as a linearization of the Kohn–Sham equations in DFT [44], and has to be solved to obtain the low-lying eigenvalues in each step of the self-consistent field (SCF) iterations. We mention that the algorithms and analysis discussed in this chapter can be extended to nonlinear Kohn–Sham DFT problems [18, 51].

Throughout this chapter, we shall denote by r = |r| and $$\hat {r} = \mathbf {r}/r$$ for a vector $$\mathbf{r}\in \mathbb {R}^d$$ . Moreover, we shall denote $$\sum _{l=0}^{\infty }\sum _{m=-l}^l$$ by ∑lm, and $$\sum _{l=0}^{L}\sum _{m=-l}^l$$ by $$\sum _{lm}^L$$ for simplicity. We will denote by (⋅, ⋅) the inner-product $$L^2_{\mathrm {per}}(\Omega ) \times L^2_{\mathrm {per}}(\Omega )\rightarrow \mathbb {R}$$ :

Since we only consider $$\mathbb {L}$$ -periodic functions in this chapter, the subscript $$L^2_{\mathrm {per}}$$ of the inner-product will be suppressed for simplicity.

9.2.1 The APW Method

In the APW method, the unit cell Ω is partitioned into two types of regions (the so-called “muffin-tin” division [44, 51], see Fig. 9.1): (1) spheres $$\mathscr {C}_i$$ centered at atomic sites Ri with radius Ri, in which the effective potential is assumed to be spherically symmetric, i.e.

(2) the remaining interstitial region $$\mathscr {D}$$ . In fact, the muffin-tin approximation is widely used and very good for close packed (fcc and ideal hcp) materials. It is not as good but still reasonable for bcc and related materials [24], and becomes increasingly less reliable as the site symmetry and coordination decrease.

A diagram of the muffin-tin division. A square of interstitial region D with two circles labeled script c 1 and script c 2. Script c 1 is bigger than script c 2.

Fig. 9.1

The “muffin-tin” division of the unit cell Ω into spheres $$\mathscr {C}_i$$ centered at atoms and an interstitial region $$\mathscr {D}$$

The APW method constructs a set of basis functions that are “augmented” from plane waves. Each individual APW basis function, also labeled by the wave vector G, consists of a single plane wave (with the same wave vector G) in the interstitial region matched to radial functions in the atomic spheres. More precisely, the augmentation from plane waves to APW basis functions $$\psi _{{\mathbf {G}}}^{\varepsilon L}({\mathbf {r}})$$ can be written as follows

(9.2)

where L is the truncation of the angular momentum (i.e. $$|l|\leqslant L$$ ), ri = r −Ri, ri = |ri|, $$\widehat {\mathbf {r}}_i={\mathbf {r}}_i/r_i$$ , $$Y_{lm}(\widehat {\mathbf {r}})$$ denotes the spherical harmonic functions on S², χl is the solution of the radial Schrödinger equation at energy parameter ε

(9.3)

and the coefficients $$\alpha ^{\mathbf {G}}_{lm}$$ are determined such that each angular component of the basis function is matched through the spherical surface. There are two remaining issues to implement the APW method: (a) how to determine the coefficients $$\alpha ^{\mathbf {G}}_{lm}$$ , and (b) the choice of energy parameter ε for the basis functions. For simplicity of presentation, we will assume in the following that there is only one atom located at the origin and denoted $$\mathscr {C}_i$$ by $$\mathscr {C}$$ . All results can be generalized to many-atom systems without any difficulty.

In the APW method, the continuity of the basis functions through the spherical surface $$\Gamma (=\partial \mathscr {C})$$ is guaranteed by imposing the constraint

(9.4)

and v± are the traces of v taken from outside and inside the sphere, respectively. This is essentially a “weak” continuity that guarantees each $$Y_{lm}~(|l|\leqslant L)$$ component is matched at the spherical surface, but the basis functions cannot be exactly continuous due to the truncation $$|l|\leqslant L$$ . Then by using the scattering expansion [45]

(9.5)

one can derive the formula for the coefficients so that the constraint in (9.4) can be satisfied

(9.6)

For the choice of energy parameter ε, if it was taken as a fixed parameter, then the APW method would result in a standard linear eigenvalue problem. The solution/eigenpairs would then yield the band energies and wavefunctions, which is unfortunately not a workable scheme. The APW basis functions are solutions of the Schrödinger equation inside the spheres, but only at the energy parameter ε. They lack variational freedom to allow for changes in the wavefunction as the band energy deviates from this reference. Therefore, ε must be set equal to the band energy εi (eigenvalues of (9.1)) to be computed. This means that the energy bands cannot be obtained from a single diagonalization. Rather, it is necessary to solve the determinant as a function of energy parameter ε and determine its roots (or equivalent nonlinear problems [18]), which is a much more computationally demanding procedure.

Another, less serious, difficulty of the APW method is the so-called asymptote problem [51]. The eigenvalue dependent APW basis functions must be evaluated for a large number of parameters, and sometimes one might hit a parameter ε for which χl(R, ε) equals zero at the spherical surface. Inserting this into (9.6) will yield infinite coefficients $$\alpha ^{\mathbf {G}}_{lm}$$ and cannot match the basis functions at the spherical surface. Therefore, we shall always assume that $$\chi _l(R,\varepsilon )\geqslant \epsilon $$ with some constant 𝜖 > 0 for any $$|l|\leqslant L$$ , which can be achieved by varying the radii of the atomic spheres.

9.2.2 The LAPW Method

Several modifications of the APW method were proposed with the aim of circumventing the difficulties mentioned above, in particular, to get rid of the energy dependency. The first really successful one was the linearization scheme introduced by Andersen [2]. The LAPW method uses the combination of χl(r, ε) and its energy derivative at fixed parameter ε as the radial basis function. The energy derivative is defined by

where χl is kept normalized to the same value in the atomic sphere. The properties of $$\dot {\chi }_l(r,\varepsilon )$$ can be read in [2, 44, 51]); this function satisfies the following equation

Then the LAPW basis functions $$\zeta ^{\varepsilon L}_{{\mathbf {G}}}({\mathbf {r}})$$ can be written as

(9.7)

The coefficients $$\alpha ^{\mathbf {G}}_{lm}$$ and $$\beta ^{\mathbf {G} }_{lm}$$ are determined by requiring that the basis functions match both the value and slope, in the weak sense of (9.4).

The LAPW method provides flexible basis functions with fixed parameter ε to properly describe eigenfunctions that correspond to eigenvalues near ε. In contrast to the APW method, this scheme allows us to obtain eigenvalues of (9.1) by solving a linear eigenvalue problem. The LAPW method introduces errors of order |ε − εi|² for the eigenfunction in the atomic sphere (which can be easily observed by a Taylor expansion with respect to the energy parameter [44, 51]), and it yields errors of order |ε − εi|⁴ for the eigenvalues. Although the LAPW method cannot achieve a systematically convergence due to this error, the LAPW basis functions form a good basis set over a relatively large energy region. In the few instances for which this is not possible, the energy region of interest may be divided into a few (very rarely more than 2) windows and separate calculations need to be carried out for each. This is an enormous simplification over the standard APW method, since the energy independent LAPW basis functions result in a linear eigenvalue problem and the accurate energy bands can be obtained with a single diagonalization.

9.2.3 The APW+lo Method

Singh [50] modified the LAPW approach by adding specially constructed local orbitals $$\varphi ^{\varepsilon ^{\prime }}_{lm}$$ , with energy parameters ε′, to the basis set

(9.8)

where the coefficients $$\alpha ^{\varepsilon ^{\prime }}_{lm}$$ and $$\beta ^{\varepsilon ^{\prime }}_{lm}$$ do not depend on the plane wave vector G, but are determined by the requirement that $$\varphi _{lm}^{\varepsilon ^{\prime }}$$ is zero at the sphere boundary and normalized. Note that different parameters ε′ can be chosen so that different states can be described simultaneously.

This method, known as APW+lo [51, 52], was shown to be highly effective in simulations of many systems, especially for materials with large interstitial spaces or mixtures of atoms that require high plane wave cutoffs with those requiring lower cutoffs. The added local atomic orbitals not only permit relaxation of the original basis set without an increase in the plane wave cutoff, but also suggest the possibility of avoiding the non-vanishing error of the LAPW method. Therefore by using a similar construction, the APW+lo method can enhance the accuracy with nearly no extra cost. Moreover, the APW+lo method can be used for general full-potentials, and is not restricted to muffin-tin potentials or other symmetric approximations of the charge density and potential.

9.3 Discretizations Based on Domain Decompositions

The essence and motivation of the APW-based methods have been clearly and concisely stated before: near nuclear positions, the potential and wavefunctions are strongly varying; while in the interstitial space between the nuclei, both the potential and wavefunctions are smoother; accordingly, space is divided into regions and different basis expansions are used in different regions. Similar ideas have long been explored in the community of numerical mathematics: the discretizations based on domain decompositions, which have been widely used to design efficient and parallel algorithms for solving partial differential equations. They use separate finite-dimensional discretization on non-overlapping subdomains, the basis functions on different subdomains do not match on the interface, and the “continuity” of the solution is enforced by additional techniques, for example, Lagrange multipliers, penalties and so on. The domain decomposition technique allows us to benefit from the presence of the subdomains in order to choose the discretization method best adapted to the local behavior of the solution. A further advantage is that more flexible and economical adaptive procedures such as refinements can be made on the subdomains, where needed.

In this section, we will review several widely used discretization methods based on domain decomposition techniques. Since only the mortar method is closely related to the family of APW-based methods (which essentially belong to the mortar method), we will discuss the other two techniques, discontinuous Galerkin (DG) and weak Galerkin (WG) methods, only very briefly.

9.3.1 The Mortar Method

The idea of the mortar method is to use different basis functions to approximate the solutions on different subdomains, and then match incompatible discretizations with a suitable variational operator, which ensure an optimal transmission of information between adjacent subdomains (see, e.g., [9–11]).

The discrete problem is constructed via the Galerkin process applied to the variational formulation of the partial differential equations. Even if the local discrete spaces, i.e., the discrete spaces on each subdomain, are included in the local variational spaces, this is usually no longer the case for the global space, since the matching conditions that are enforced on the interfaces between the subdomains are too weak to ensure the conformity. The mortar approximations involve constraints on the space, which is the so-called “weak continuity”. The continuity of the solution across the subdomain interfaces is imposed in a weak sense by using the multiplier space. These constraints could be treated as Lagrange multipliers leading to a saddle point problem.

Let $$\mathcal {S}(\Omega )$$ be the finite-dimensional space spanned by the basis functions on all subdomains, and $$\mathcal {T}(\Gamma )$$ be the multiplier space on the interface. The discretization of (9.1) by mortar method reads: Find $$\varepsilon _i^{\mathrm {M}} \in \mathbb {R}$$ and $$\varphi _i^{\mathrm {M}}\in \mathcal {S}(\Omega )$$ , such that $$\|\varphi _i^{\mathrm {M}}\|{ }_{L^2(\Omega )} =1$$ and

(9.9)

where the bilinear forms a and b are defined by $$a(\cdot ,\cdot ):\mathcal {S}(\Omega ) \times \mathcal {S}(\Omega ) \rightarrow \mathbb {C}$$ :

and $$b(\cdot ,\cdot ):\mathcal {S}(\Omega ) \times \mathcal {T}(\Gamma ) \rightarrow \mathbb {C}$$ :

A key argument is that the choice of integral type matching conditions on the interfaces leads to an optimal evaluation of the consistency error issued from the nonconformity of the discretization [10]. The mortar method has been successfully applied to quantum eigenvalue problems (see, e.g. [20, 31]).

We mention that the family of APW approaches actually belongs to the category of mortar methods. The convergence of APW-based methods will be derived in the framework of the mortar nonconforming methods, see Sect. 9.4.

9.3.2 The Discontinuous Galerkin Method

The DG framework has been widely used in numerical solutions of partial differential equations and investigated theoretically in a lot of works (see, e.g., [6, 7, 15, 54] and references cited therein). For second-order elliptic problems, the development of DG methods is based on the idea of introducing an interior penalty [5, 47, 57]. DG discretization methods have also recently been developed for DFT calculations [40–42, 60].

We shall briefly discuss the DG discretization for (9.1), which is related to the framework in [4]. For respectively vector-valued and scalar-valued functions w and u which are not continuous on the spherical surface Γ, we define the jumps by

and the averages by

where w± and u± are traces of w and u on Γ taken from inside and outside the sphere, and n± are the normal unit vectors. Let $$\mathcal {S}(\Omega )$$ be the finite-dimensional space spanned by the basis functions on all subdomains. We then construct the DG methods for eigenvalue problem (9.1): Find $$\varepsilon _i^{\mathrm {DG}} \in \mathbb {R}$$ and $$\varphi _i^{\mathrm {DG}}\in \mathcal {S}(\Omega )$$ , such that $$\|\varphi _i^{\mathrm {DG}}\|{ }_{L^2(\Omega )} =1$$ and

(9.10)

where the bilinear form $$a^{\mathrm {DG}}(\cdot ,\cdot ):\mathcal {S}(\Omega ) \times \mathcal {S}(\Omega ) \rightarrow \mathbb {C}$$ is defined by

with $$\sigma \in \mathbb {R}^+$$ the discontinuity-penalization parameter. Note that there are many other types of DG formulations, and (9.10) is the classical symmetric interior penalty method [4, 5, 41]. We refer to [48] for an overview of DG methods.

9.3.3 The Weak Galerkin Method

The WG method was first proposed in [56], and further developed in [46, 59, 61]. In the WG method, differential operators are approximated by weak forms as distributions over a set of generalized functions. It has been demonstrated that the WG method is highly flexible and robust as a numerical technique employing discontinuous piecewise polynomials on polygonal or polyhedral finite element partitions. For simplicity of presentations, we will skip the WG formulations of the eigenvalue problems (9.1) but refer to [59] for details.

9.4 Convergence Analysis of the APW-Based Methods

In this chapter, we will provide a mathematical understanding of the APW-based methods. The theory relies on two aspects: (a) the asymptotic regularity of the wavefunctions, and (b) viewing the APW family as mortar methods. Our analysis shows that, in principle, the APW and APW+lo approaches can provide super algebraic and nearly exponential convergence for full-potential calculations. This explains why the APW-based codes are frequently used for sufficiently accurate electronic structure calculations from a numerical point of view.

We shall first review the regularity of the eigenvalue problems (9.1), then construct a non-conforming mortar method based on polynomial radial basis functions and a similar construction of the APW methods, and finally discuss the convergence and error estimates of the APW methods in the framework of mortar methods.

9.4.1 Regularity

In full potential calculations, the potential Vper has singularities at nuclear positions, which gives rise to cusps at the singular points of the eigenfunctions. Then a first and natural question to ask is: What is the regularity of the solution of the eigenvalue problem (9.1)? By a careful analysis [28, 29, 32, 52] it has been shown that the solution to such systems is analytic away from the nuclei but satisfies some cusp condition at the nuclear positions, if the exact singular Coulomb potential is used. Due to the cusps at the nuclear positions, the plane wave approximations of (9.1) with singular Coulomb potential cannot achieve a good spectral convergence rate.

In our analysis of APW-based methods, we rely heavily on the regularity result in weighted Sobolev space for Schrödinger type eigenvalue problems developed in [26] (see also [42]). This type of analysis was introduced to investigate singularities of boundary value problems in conical domains with corners and edges (we refer to [19, 22, 30, 36] for more details). In our case the geometry is fairly simple, while the singular electrostatic potential generated by the nuclei fits this treatment perfectly. We will skip the definitions of weighted Sobolev spaces but refer to the above references for details. Instead, we state the following corollary, which can be used directly in the analysis.

Proposition 9.1

Let φ be an eigenfunction of (9.1). If $$V_{\mathrm {per}} \in L_{\mathrm {per}}^2(\Omega )$$ is equal to $$-Z/r~(Z\in \mathbb {R}^+)$$ in the neighborhood of a nuclear position and is sufficiently smooth away from it, then φ(r) ∈ Hs([0, R] × S²) in $$\mathscr {C}$$ and $$\varphi \in H_{\mathrm {per}}^s(\mathscr {D})$$ for any $$s\in \mathbb {R}^+$$ .

This regularity result indicates that locally, when expressed in spherical coordinates around the nuclei, the solution is infinitely differentiable. Therefore it is actually possible to propose combined approximations which have exponential convergence with respect to the number of degrees of freedom.

9.4.2 A Nonconforming Method

We construct a nonconforming mortar method analogous to the APW methods, but using polynomials as the radial basis functions in the atomic spheres. This method has the same philosophy as the APW methods. Meanwhile, it is easier to see that it has a systematic spectral convergence rate.

We first give the basis functions on the subdomains. Define the space of functions on $$\mathscr {D}$$ expanded by plane waves

and the space of functions on $$\mathscr {C}$$ expanded by polynomial times spherical harmonics

where $$\{\chi _n\}_{n=0}^N$$ forms a basis set of the space of polynomials on [0, R] with degree no more than N. Let

(9.11)

and $$\varrho =\min \{K,N,L\}$$ . We may assume in the sequel that $$\varrho \ge C\max \{K,N,L\}$$ with some constant C > 0. With the approximation space $$\mathcal {S}^K_{NL}(\Omega )$$ , we can follow (9.9) to write down the nonconforming mortar approximation of (9.1): Find $$\varepsilon _{i,\varrho }(=\varepsilon _{i,NL}^K)\in \mathbb {R}$$ and $$\varphi _{i,\varrho }(=\varphi _{i,NL}^K)\in \mathcal {S}^K_{NL}(\Omega )$$ such that $$\|\varphi _{i,\varrho }\|{ }_{L^2(\Omega )}=1$$ and

(9.12)

We then design an equivalent discretization form that uses the same construction as the APW methods. Let $$\{\chi _n\}_{n=0}^N$$ be a basis set that spans the space of polynomials on [0, R] with degree at most N and satisfies (see, Fig. 9.2)

(9.13)

Note that the subspace $$\widetilde {\mathcal {B}}_{NL}\subset \mathcal {B}_{NL}$$ defined by

is a finite-dimensional subspace of $$H_0^1(\mathscr {C})$$ .

4 graphs of radial basis functions. Graphs 1 and 2 plot a decreasing line of chi 0 and concave curve of chi 1 from 0 to R. Graphs 3 and 4 plot an oscillation curve of chi N - 1 and increasing line from 0 to R.

Fig. 9.2

Schematic plots of radial basis functions χn(r) n = 0, …, N satisfying (9.13)

We further augment the plane waves with the above radial basis functions as follows

(9.14)

where the coefficients

are determined by the continuity constraint (9.4) and the scattering expansion (9.5).

Set $$\widetilde {\mathcal {S}}_{NL}^K = \widetilde {\mathcal {B}}_{NL}\oplus {\mathrm {span}}\{\omega _{\mathbf {G}}({\mathbf {r}}):~|{\mathbf {G}}|\leqslant K\}$$ and define a direct Galerkin approximation of (9.1): Find $$\varepsilon _{i,\varrho }\in \mathbb {R}$$ and $$\varphi _{i,\varrho }\in \widetilde {\mathcal {S}}^K_{NL}(\Omega )$$ such that $$\|\varphi _{i,\varrho }\|{ }_{L^2(\Omega )}=1$$ and

(9.15)

It is obvious from the definition of ωG and comparison of the dimensions (see [18]) that the discretizations (9.12) and (9.15) are equivalent.

With the above constructions, we can then derive the following convergence and a priori error estimate of the approximations, by using a careful numerical analysis and the regularity of the eigenfunctions. We refer to [18] for the proof of this theorem. We mention that the result is not restricted to muffin-tin potentials but is valid for general potentials.

Theorem 9.1

Let εibe an eigenvalue of (9.1) with $$\dim \big (\mathscr {R}(\varepsilon _i)\big )=m$$ , where $$\mathscr {R}(\varepsilon _i)$$ denotes the eigenspace of εi. If the assumption of Proposition9.1for Vperis satisfied and ϱ is sufficiently large, then there exist m eigenvalues εi1,ϱ, …, εim,ϱof (9.12), or (9.15), such that

(9.16)

where Csis a constant depending only on s, Ω and Vper.

9.4.3 Error Estimates of the APW Method

The APW method can be viewed as a modified scheme of the nonconforming method introduced in Sect. 9.4.2, under the assumption that the potential is spherically symmetric inside the atomic sphere $$\mathscr {C}$$ , say Vper(r) = V (r) for $$r\leqslant R$$ .

Compared with the discretization introduced in Sect. 9.4.2, the APW method ignores the subspace $$\widetilde {\mathcal {B}}_{NL}$$ and replaces the basis function ωG in (9.14) by $$\psi _{\mathbf {G}}^{\varepsilon L}$$ in (9.2), where ε is the eigenvalue to be computed. This leads to a discrete nonlinear eigenvalue problem: Find $$\varepsilon _{KL}\in \mathbb {R}$$ and $$\varphi _{KL}\in \mathcal {S}_{KL}^{\varepsilon _{KL}}(\Omega ):={\mathrm {span}}\{\psi _{\mathbf {G}}^{\varepsilon _{KL} L},~|{\boldsymbol {G}}|\leqslant K\}$$ , such that

(9.17)

This is a nonlinear eigenvalue problem since the basis functions and the variational subspace depend on the eigenvalue to be computed. We have discussed its disadvantages in Sect. 9.2.

We can show that the solutions of (9.17) have limiting points as K and L go to infinity (see [18]). Let ε∞ be any accumulation point, i.e., there exists a subsequence of eigenpairs, which we still denote by $$\{(\varepsilon _{KL},\varphi _{KL})\}_{K,L\in \mathbb {N}}$$ , such that

We show in the following result that the limiting point is an eigenpair of (9.1) and the convergence rate for the APW approximations is optimal. We refer to [18] for the proof of this theorem.

Theorem 9.2

If Vperis a spherically symmetric potential in $$\mathscr {C}$$ and the assumption of Vperin Proposition9.1is satisfied, then the limiting pair (ε∞, φ∞) is an eigenpair of (9.1), that is

Moreover, for the sequence of eigenpairs $$\big \{(\varepsilon _{K_jL_j},\varphi _{K_jL_j})\big \}_{j\in \mathbb {N}}$$ that converges to (ε∞, φ∞), there exists a constant Csdepending only on s, Ω and Vper, such that

When the radius R is well chosen, the assumption of a spherically symmetric potential is reasonable for many systems, as discussed in Sect. 9.2, so this result explains why the APW method has been successfully used in many computations.

9.4.4 Error Estimates of the LAPW Method

Since the accuracy of the LAPW method depends heavily on the choices of parameters, we cannot provide a systematic convergence analysis but only some error estimates.

Compared with the mortar nonconforming method introduced in Sect. 9.4.2, the LAPW method ignores the subspace $$\widetilde {\mathcal {B}}_{NL}$$ and replaces the basis function ωG in (9.14) by $$\zeta _{\mathbf {G}}^{\varepsilon L}$$ in (9.7), where ε is a fixed parameter. Combined with the error estimate in Theorem 9.2, we can derive for the LAPW approximations that

where $$\bar {\varepsilon }$$ is the eigenvalue to be computed.

Although the errors of the eigenvalue approximations cannot systematically converge to zero and rely on the choice of the parameter ε, the high order of $$|\bar {\varepsilon }-\varepsilon |$$ means that the LAPW method forms a good basis set over a relatively large eigenvalue region. In most materials, it is quite adequate to choose ε near the center of the region of eigenvalues to be computed. However, in a few instances, there is no single choice of ε that is adequate for all the eigenvalues that must be considered. Then the eigenvalue region may be divided into a few windows and separate computations with different parameters should be carried out for each.

9.4.5 Error Estimates of the APW+lo Method

We shall finally discuss the convergence of APW+lo method. This method adds local basis functions to the LAPW method in order to have enough variational flexibility for the radial basis functions in the atomic spheres.

Compared with the mortar nonconforming method introduced in Sect. 9.4.2, the APW+lo method replaces the basis function ωG in (9.14) by $$\zeta _{\mathbf {G}}^{\varepsilon L}$$ in (9.7) with ε a fixed parameter, and uses the local atomic orbitals $$\varphi _{lm}^{\varepsilon ^{\prime }}$$ in (9.8) to span the subspace $$\widetilde {\mathcal {B}}_{NL}$$ . We observe that the APW+lo method is very similar to the method in Sect. 9.4.2, the only difference being that the polynomial radial basis functions are replaced by atomic orbitals with compact support. This method not only gets rid of the nonlinearity from energy dependence of the APW basis functions, but also systematically achieves convergence as long as the local atomic orbital sets are well chosen. By using the same arguments as for Theorem 9.1, we can easily justify the similar convergence and error estimate.

As shown by the numerical experiments in [43], the APW+lo method converges practically to the same result as the LAPW method, but allows a significantly smaller basis set (up to 50%) and thus reduces the computational cost drastically.

We finally use the package Exciting to perform some full-potential computations for aluminum (Al) and lithium-fluorine (LiF) crystals. Exciting is a full-potential DFT package based on the APW+lo method which uses an SCF iteration to solve the nonlinear Kohn–Sham equations. The numerical errors of the ground state energy approximations with respect to the plane wave and angular momentum cutoffs are presented in Fig. 9.3, from which we clearly observe exponential convergence rates with respect to both discretization parameters K and L.

2 line graphs plot energy error versus radius R times planewave cutoff K and angular momentum cut off L. 1. A I, y = exp of negative x, LiF, and y = 50 exp of negative 1.5 x in decreasing trend. 2. A I, y = exp of negative 1.3 x by 50, LiF, and y = 10 exp of negative 2 x in decreasing trend.

Fig. 9.3

Errors of the ground state energy approximations for Al and LiF, by using the package Exciting. Left: decay with respect to the wave vector cutoff. Right: decay with respect to the angular momentum cutoff

9.5 Conclusions and Perspectives

In this chapter, we have reviewed the family of APW methods that are widely used in full-potential electronic structure calculations. These methods enable highly accurate and reliable simulations for crystals, since they avoid the use of pseudopotentials that might be a source of modeling errors, and which are difficult to control.

We further introduce a nonconforming method based on the idea of the APW methods, which uses polynomial radial basis functions inside the atomic spheres. We obtain an a priori error estimate of the nonconforming approximations and generalize the results to APW-based methods. Due to the fast spectral convergence rate, the APW methods provide high precision using only approximately the same number of degrees of freedom as the plane wave methods.

As an extension, we mention at the end the muffin-tin orbital (MTO) approach [2, 3, 44], which is another type of atomic sphere method similar to the APW methods. It exploits the same idea of dividing the simulation domain and provides efficient representations for atomic-like features that are rapidly varying near each nucleus and smoothly varying between the atoms. The MTO method reformulates the multiple-scattering (MST, also called KKR since it was invented independently by Korringa, Kohn and Rostoker [44]) method, and leads to physically meaningful descriptions of the electronic states in terms of a small basis set of localized, augmented functions. Since the solutions obtained by the MTO method satisfy the equation both inside and outside the spheres, the convergence of the MTO approximations can be proven using similar arguments as those in this chapter.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_10

10. Finite Element Methods for Density Functional Theory

Xiaoying Dai1   and Aihui Zhou1  

(1)

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

Xiaoying Dai

Email: daixy@lsec.cc.ac.cn

Aihui Zhou (Corresponding author)

Email: azhou@lsec.cc.ac.cn

Abstract

In this chapter, we introduce the finite element method, an efficient and widely used discretization tool for the solution of partial differential equations, and apply it to density functional theory-based electronic structure calculations. We focus on the mathematical theory of the finite element method, including the a priori and the a posteriori error estimations of finite element approximations, the convergent rate and complexity of adaptive finite element approximations. In addition, we also address high performance computing for large scale electronic structure calculations based on finite element discretizations.

10.1 Introduction

The physical properties of a system of M nuclei and N interacting electrons can, in principle, be obtained by solving the time-independent Schrödinger equation if the relativistic effect is not taken into account:

(10.1)

with the external potential being the Coulomb potential

due to the M nuclei. Here, the spin state has been fixed, Zm and Rm are the charges and the positions of the nuclei (m = 1, 2, …, M), ri are the coordinates of the electrons (i = 1, 2, …, N), Eel is the total electronic energy of the ground state and Ψ is the associated electronic wave function. Atomic units are used throughout this chapter $$(\hbar =m_e=\frac {4 \pi \mathrm{e} ^2}{\epsilon _0}= 1)$$ .

The Schrödinger equation (10.1) is a linear eigenvalue problem defined in $$\mathbb {R}^{3N}$$ , which is of high dimension. In principle, it is an intractable model. Therefore, people have to resort to equivalent or reduced models that are computable. Among them, the density functional theory (DFT) Kohn–Sham model is the most successful one.

Instead of using the many-body wave function as the basic quantity as in the Schrödinger equation, DFT employs the electron density

as the basic physical variable, and any physical quantities can be obtained from the electron density ρ [51, 60, 64].

We may date DFT back to the early works of Thomas [84] and Fermi [42] in 1927, where the kinetic energy of the system is approximated as an explicit functional of density ρ, which is originally from free electron (or homogeneous non-interacting electron) gas systems. Although their approximations are too rough for modern electronic structure calculations, the approach of using the electron density ρ as the basic physical variable inspired later researchers. In 1964, Hohenberg and Kohn [51] presented their famous work, mainly consisting of the two Hohenberg–Kohn theorems: the Hohenberg–Kohn theorem and the Hohenberg–Kohn variational theorem. The two theorems formulate DFT as an exact theory of many-body systems [60]. We mention that by using the Fundamental Theorem of Algebra, Zhou [102] gave a simple, self-contained and mathematically rigorous proof of the Hohenberg–Kohn theorem in the case of the Coulomb type potential. We refer to [57, 59, 103] and references cited therein, and Chap. 3 by Lewin, Lieb and Seiringer in this volume for more discussions on Hohenberg–Kohn theorems.

The Kohn–Sham models are basic models of DFT and were introduced in [55]. Although the Kohn–Sham equation is much easier to deal with than the original Schrödinger equation, the solution of the Kohn–Sham equation is still full of difficulties and challenges. For instance, the Kohn–Sham equation is a nonlinear eigenvalue problem, the Coulomb potential vext is singular, the number of eigenpairs required is large, many eigenvalues are degenerate, and the gaps between two eigenvalues are usually of multiscale. We see that its solution requires the design of highly accurate and efficient algorithms for simulating large scale systems [32, 73]. In addition, it is important to provide a mathematical theory for the reliability and efficiency of the numerical approximations in electronic structure calculations. We also note that the error control of the computations is very much a concern of chemists, physicists, materials scientists as well as biologists.

Based on the choice of the basis functions, we may divide the existing discretization methods for solving the Kohn–Sham models into three classes: the plane wave method, the atomic orbital basis function method (local basis set method), and the real space method. Each method has its own advantages and disadvantages. Currently, most of the mature software for first principles electronic structure calculations are based on the plane wave discretization and/or the atomic orbital basis function discretization. The real space methods include the finite difference method, the wavelet method, the finite element method, and the finite volume method. Because of the locality of the real space method, the resulting matrix is sparse, which may make it more competitive than the other discretizations in parallel computing. Indeed, the real space method becomes more and more attractive to DFT calculations. Among these real space methods, due to its simple construction and completeness of the basis functions, and the easy implementation of adaptive computation, the finite element method should have more potential than the others. In addition, the ability of adaptive computation of the finite element discretization makes it convenient to deal with both the full potential calculation and the pseudopotential calculation.

The finite element method [9, 23] is widely used in the solution of partial differential and integral equations and has achieved great success. It has been used in electronic structure calculations since the 1980s. In 1985, Levin and Shertzer [58] used the finite element method on non-uniform meshes to solve the Schrödinger equation of the He atom. In 1993, Hackel et al. [48] applied higher order finite elements to calculate the total energy of the H, $$\mathrm {H}_2^{+}$$ and $$\mathrm {H}_3^{2+}$$ molecules based on non-uniform meshes. Ackermann and Roitzsch [1, 2] utilized an adaptive finite element method together with higher order elements to compute the electronic structure of the $$\mathrm {H}_3^{2+}$$ molecule to obtain highly accurate approximations. In 1995, Tsuchida and Tsukada et al. [87] calculated the H₂ molecule and Si atom by using non-uniform meshes. In 1999, Pask et al. [65] used the finite element method to compute the energy band of some solids together with a numerical analysis. Sterne [80] applied a finite element method to compute positron systems with hundreds of atoms. In 2004, Zheng et al. [100, 101] used the polar coordinate-based finite element approach to compute the Schrödinger equation of the H and Li atoms. Tsuchida [86] carried out the ab initio molecular-dynamics simulation for systems with hundreds of atoms based on the finite element discretization. Shen [75] and Zhang [95] introduced an adaptive finite element method together with a two-grid approach to solve systems with tens of atoms. Fang [36] used a hexahedral mesh-based finite element scheme to compute the electronic structure of systems with thousands of atoms using the symmetry of the systems, and has also carried out a relaxation simulation for systems with tens of atoms. In 2011, Pask et al. [67] introduced the PUFEM (Partition-of-unity finite-element method) to the solution of the Kohn–Sham equation. We refer to [14, 16, 17, 25, 37–40, 43, 66, 75, 83, 92, 95] and references therein for more applications of the finite element method in electronic structure calculations.

The numerical analysis of the Kohn–Sham models plays a central role in our understanding of the reliability and efficiency of the numerical methods of electronic structure calculations. Due to the difficulties and challenges, there is very little work on these topics and only in recent decades has some progress been made. The main difficulties involved in the numerical analysis of Kohn–Sham models is that we have to either handle global and nonconvex minimization problems with orthogonality constraints or deal with nonlinear eigenvalue problems with a large number of (possibly degenerate) eigenvalues. Under a coercivity assumption of the so-called second-order optimality condition, Cancès et al. [12] provided a numerical analysis of plane wave approximations and showed that some ground state solutions can be approximated by plane wave solutions with a certain convergence rate. Suryanarayana et al. [83] showed the convergence of ground state energy approximations based on finite element discretizations. In [18], Chen et al. presented a systematic analysis for a general finite-dimensional discretization and proved that all the limit points of finite-dimensional approximations are ground state solutions of the system and some ground state solutions can be well approximated by finite-dimensional solutions if the associated local isomorphism condition is satisfied. They provided not only convergence of the ground state energy approximations but also convergence rates of both eigenvalue and eigenfunction approximations in [18]. It should be pointed out that the local isomorphism condition used in [18] should be very mild and is indeed satisfied if the second-order optimality condition is provided. In further work, Chen et al. [15] carried out an a posteriori analysis for the finite element approximations of the Kohn–Sham equation. For instance, they proposed the residual type a posteriori error estimators and showed that the a posteriori error estimators can be used as the upper and lower bounds of the approximate errors.

We see that the eigenfunctions of (10.8) still vary rapidly around nuclei or chemical bonds even in the pseudopotential setting [7, 26, 47, 99]. It is natural and reasonable to apply adaptive computational approaches to improve the accuracy of the finite element approximations and reduce the computational cost. Indeed, Tsuchida and Tsukada combined the finite element method with the adaptive curvilinear coordinate approach for electronic structure calculations of some molecules [88, 89]. Shen and Zhang introduced some adaptive tetrahedral finite element discretizations in their theses [75, 95] and calculated several typical molecular systems efficiently [47, 76, 96, 97]. Bylaska et al. used an adaptive piecewise linear finite element method on completely unstructured simplex meshes to resolve the rapid variation electronic wave functions around atomic nuclei [10]. Dai et al. designed some parallel adaptive and localization based finite element algorithms for typical quantum chemistry and nanomaterials computations containing more than one thousand atoms using tens of hundreds of processors on computer cluster [25, 26, 29, 31]. Gavini et al. constructed a finite element mesh using an unstructured coarse-graining technique and computed materials systems [62, 83]. Yang successfully scaled their adaptive finite element simulations to over 6000 CPU cores on the Tianhe-1A supercomputer, see his thesis [92]. Bao et al. [6] proposed an adaptive method to solve the Kohn–Sham equation by optimizing the distribution of mesh points without increasing their number. Chen et al. [15] proved not only by numerical experiments but also by theory the robustness and efficiency of the adaptive finite element computations in electronic structure calculations. We refer to [15, 19, 32, 41, 43, 52, 61, 85] and references cited therein for other interesting discussions and investigations on the adaptive finite element method (AFEM).

Although there are many works on applying adaptive finite element approaches to electronic structure calculations, there was no work on carrying out the numerical analysis for adaptive finite element approximations to the Kohn–Sham models until [15]. Under some reasonable assumptions, Chen et al. [15] and Yang and Zhou [91] proved that all limit points of the adaptive finite element approximations are Kohn–Sham ground state solutions. In addition, they also proved that the Kohn–Sham ground state solutions can be well approximated by adaptive finite element approximations with a certain convergence rate, and the computational complexity is quasi-optimal. We refer to [19, 20] for more analyses of AFEM for DFT models.

Due to the needs of science and engineering and the availability of high performance computers, there is a great demand for highly efficient and scalable algorithms for electronic structure calculations. There have been many contributions toward this effort to design high performance algorithms (see, e.g., [27, 28, 36, 49, 61, 63] and references cited therein). We mention here, for instance, the symmetry-based decomposition approach [36, 40], the parallel orbital-updating approach [27, 28, 63], the local and high order finite element discretizations [25, 29], the three-scale schemes [31, 43, 44], and the hexahedron-grid-based two-scale higher order finite element scheme [21]. The first two approaches can be carried out in the two-level parallelization setting.

In this chapter, we expose some existing mathematical theory for the finite element method in DFT calculations, mainly focusing on its numerical analysis, including the a priori error estimate and the a posteriori error estimate for the finite element approximations, the convergence rate and complexity for adaptive finite element approximations. In addition, we also address the symmetry-based decomposition approach and the parallel orbital-updating approaches for high performance computing for large scale electronic structure calculations in modern supercomputers.

The rest of this chapter is organized as follows. In Sect. 10.2, we give a brief introduction to the Kohn–Sham DFT models as well as some necessary preliminaries. In Sect. 10.3, we introduce the standard finite element discretization of the Kohn–Sham equation. Then, we present the existing analysis for the a priori error estimate and the a posteriori error estimate for finite element discretizations of the Kohn–Sham equation in Sect. 10.4. In Sect. 10.5, we provide a mathematical theory of the adaptive finite element approximations for solving the Kohn–Sham equation. We also address the high performance issues for the electronic structure calculations in Sect. 10.6. Finally, we conclude with some remarks.

10.2 Preliminaries

Let $$\Omega = \mathbb {R}^3$$ or $$\Omega \subset \mathbb {R}^3$$ be a polyhedral domain. We shall use the standard notation for Sobolev spaces Ws, p( Ω) and their associated norms and seminorms (s = 1, 2, ⋯ , 1 ≤ p ≤∞), see, e.g., [3]. For p = 2, we denote by Hs( Ω) = Ws, 2( Ω) with the norm ∥⋅∥s,Ω = ∥⋅∥s,2,Ω. For s = 1, $$|| \cdot ||{ }_{s,\Omega } = \displaystyle \left (\int _\Omega |v|{ }^2 + |\nabla v|{ }^2\right )^{1/2}$$ and $$H^1_0(\Omega )=\{v\in H^1(\Omega ): v\mid _{\partial \Omega }=0\}$$ , where v∣∂ Ω = 0 is understood in the sense of trace. The space H⁻¹( Ω), the dual of $$H^1_0(\Omega )$$ , will also be used. We consider real-valued functions throughout.

For convenience, the symbol $$\lesssim $$ will be used throughout this paper, and $$A\lesssim B$$ means that $$A\leqslant C B$$ for some constant C that is independent of mesh parameters. To address the mathematical theory of the finite element method, we need to introduce several function spaces. Let $$\mathscr {P}(p,(c_1,c_2))$$ be a class of functions satisfying some growth conditions:

with $$c_1\in \mathbb {R}$$ and c₂, p ∈ [0, ∞). Let $$\mathcal {H}=(H_0^1(\Omega ))^N$$ be the Hilbert space with H¹ inner product

Let $$\mathbb {Q}$$ be a subspace with orthonormality constraints:

where $$\displaystyle \Phi ^T\Psi =\left ( \int _{\Omega }\varphi _i\psi _j \right )^N_{i,j=1}\in \mathbb {R}^{N\times N}$$ . For $$\Phi \in \mathcal {H}$$ and a subdomain ω ⊂ Ω, we define

where $$||\varphi _i||{ }_{0,p,\omega } = \displaystyle \left (\int _\omega |\varphi _i| ^p\right )^{1/p}$$ ||φi||s,ω = ||φi||s,2,ω and $$||\varphi _i||{ }_{1,2,\omega }=\left (||\varphi _i||{ }^2_{0,2,\omega }+||\nabla \varphi _i||{ }^2_{0,2,\omega }\right )^{1/2}$$ . In our discussions, we will use the following sets:

For any $$\Phi \in \mathbb {Q}$$ , we may decompose $$\mathcal {H}$$ into a direct sum of three subspaces (see, e.g., [12, 35]):

where $$\mathcal {S}_{\Phi }=\Phi \mathcal {S}^{N\times N}$$ , $$\mathcal {A}_{\Phi }=\Phi \mathcal {A}^{N\times N}$$ , and $$ \mathcal {T}_{\Phi }=\left \{\widetilde {\Phi }\in \mathcal {H}:\widetilde {\Phi }^T\Phi =0\in \mathbb {R}^{N\times N}\right \}. $$

10.2.1 Kohn–Sham Models

Based on the Kohn–Sham DFT, the total energy of the system is defined by the following Kohn–Sham energy functional

where Φ = (φ₁, φ₂, ⋯ , φN), $$\rho _{\Phi }(\mathbf {r})=\sum \limits _{i=1}^{N} |\varphi _i(\mathbf {r})|{ }^2$$ is the electron density (sometimes it is convenient to abuse the notation Φ), exc(ρΦ) is the exchange-correlation energy, and vext is the external potential. For simplicity, we use the assumption that each Kohn–Sham orbital is occupied by one electron, and therefore the number of occupied Kohn–Sham orbitals is also N. Again for simplicity, we will not introduce a new notation but still use N to denote the number of occupied Kohn–Sham orbitals. In most cases, we consider vext to be the Coulomb potential vne, which is defined by

The ground state charge density of the system is achieved by solving the minimization problem

(10.2)

We refer to [5, 18, 83] for a discussion on the existence of a minimizer of problem (10.2). If the energy functional is differentiable, then a minimizer Φ = (φ₁, φ₂, ⋯ , φN) of (10.2) should satisfy the following equation (i.e. the Euler–Lagrange equation associated with the minimization problem): Find $$\varphi _i(\mathbf {r}) \in H_0^1(\mathbb {R}^3)(i=1,2,\ldots ,N)$$ such that

(10.3)

where

is the Lagrange multiplier, and $$H_{\Phi }: H_0^1(\mathbb {R}^3)\rightarrow H^{-1}(\mathbb {R}^3)$$ is the Kohn–Sham Hamiltonian operator defined by

for all $$u,v\in H_0^1(\mathbb {R}^3)$$ . Here,

(10.4)

is the Hartree potential, which is the electrostatic potential generated by the electronic density ρ(r), and $$v_{\mathrm {xc}}(\rho ) = e^{\prime }_{\mathrm {xc}}(\rho _{\Phi })$$ is the exchange-correlation potential.

Note that

(10.5)

where $$\mathcal {O}^{N\times N}$$ is the set of orthogonal matrices. We see from (10.5) that if Φ is a minimizer of (10.2), then ΦU is also a minimizer for any orthogonal matrix U.

For any $$\Phi \in \mathcal {H}$$ , we define

and the set of ground state solutions

Since the electron density ρΦ and operator veff(ρΦ) are also invariant under any unitary transform of Φ, we may diagonalize the Lagrange multiplier Λ. More precisely, there exists a unitary matrix U such that the Lagrange multiplier is diagonal for $$\widetilde {\Phi } =\Phi U=(\psi _1,\psi _2,\cdots ,\psi _N)$$ :

Consequently, instead of (10.3), we obtain

which is the well-known Kohn–Sham equation. It is a nonlinear eigenvalue problem defined in $$\mathbb {R}^3$$ .

The exact exchange-correlation energy density exc(ρ) and exchange-correlation potential vxc(ρ) are unknown and must be approximated in computations. The commonly used approximations include local density approximation (LDA), local spin density approximation (LSDA), and generalized gradient approximation (GGA) and so on (see, e.g., [60] or Chap. 1 by Toulouse in this volume for more details). We refer to [50] for a comparison of different approximations. Since $$e_{\mathrm {xc}}:[0,\infty )\rightarrow \mathbb {R}$$ does not have an analytical expression, we need to consider some approximations and may mathematically assume throughout this chapter that

(10.6)

which is satisfied by many LDAs used in computation.

For convenience, we focus on the finite systems in our following discussion. We mention that similar conclusions may be applied to periodic systems.

Physically, the Kohn–Sham model is set in $$\mathbb {R}^3$$ . However, due to the exponential decay of the ground state wavefunction of the Schrödinger equation for finite systems (cf., e.g., [4, 93]), the solutions of the Kohn–Sham models for finite systems exhibit exponential decay [98, 99], too. Therefore, $$\mathbb {R}^3$$ is usually replaced by some polyhedral domain $$\Omega \subset \mathbb {R}^3$$ in practical computations. Consequently, we shall focus on the following Kohn–Sham equation

(10.7)

or

(10.8)

As usual, the ground state charge density can be obtained by solving the lowest N eigenpairs of (10.8) in practice [60].

Although the Kohn–Sham equation is tractable, the solution of the Kohn–Sham equation is still full of difficulties and challenges. For instance, the Kohn–Sham equation is a nonlinear eigenvalue problem, the Coulomb potential vext(r) is singular, the number of eigenpairs required is large, some of the eigenvalues are degenerate, and the gaps between two eigenvalues are of multiscale.

10.2.1.1 Hartree Potential

The Hartree potential (10.4) represents the Coulomb interaction between electrons. Instead of using (10.4), we usually prefer to solve the following Poisson equation

(10.9)

To get a highly accurate approximation, an appropriate boundary value condition is required. Since the Hartree potential vH(ρ)(r) decays to 0 with speed of $$\mathcal {O}(\frac {1}{|\mathbf {r}|})$$ only, it is not a good choice to simply take 0 as the boundary value. To avoid the long range effect, we introduce a neutralizing charge ρc(r) which cancels out ρ(r) in the computational domain so that

(10.10)

Denote vc the potential generated by ρc(r), the solution of (10.9) is then replaced by solving the following equivalent equation

and $$v_H = \widetilde {v}-v_{c}$$ . We refer to [25, 92] for some other choices of the boundary value conditions.

10.2.1.2 Pseudopotential

The Coulomb potential vne is singular, which makes the eigenfunctions oscillate rapidly near the core regions. Since the states in the core region in many cases have a negligible contribution to the electronic properties of matter, in computations, we replace the Coulomb potential in the core regions by a pseudopotential which is constructed to reproduce the atomic scattering properties: it has a Coulombic form outside the core region but is smoother inside. The remaining states, called valence states, are described by pseudo-wavefunctions which are significantly smoother without loss of accuracy [24, 68].

The pseudopotential consists of two parts: one local part vloc and one nonlocal part vnl. In the pseudopotential setting, the Kohn–Sham equation is still formulated as (10.8), but vext now becomes vloc + vnl, N is the number of valence electrons, and $$\{\psi \}_{i=1}^{N}$$ is the set of the pseudo-wavefunctions of the valence electrons.

10.2.1.3 Self-consistent Field Iteration

The Kohn–Sham equation is a nonlinear eigenvalue problem. It is usually solved by using a self-consistent field (SCF) iteration approach [56, 60]. Typically one starts with an initial guess for ρ, then calculates the corresponding veff(ρ) and solves the Kohn–Sham equation for ψi(i = 1, 2, ⋯ , N), with which one updates the density and repeats the calculations. This procedure is then repeated until convergence is reached. The following is the general procedure for SCF iteration.

1.

Give an initial input charge density ρin.

2.

Compute the effective potential veff(ρin).

3.

Solve the following linear eigenvalue problem

4.

Compute the new output charge density ρout.

5.

Convergence check: if not converged, use some density mixing method to get a new input charge density ρin, go to step 2; else, stop.

The variation of the charge density is often used as the criterion for the convergence of the SCF iteration in a quantum chemistry calculation. For the density mixing method in step 5, if we simply use Picard iteration: take ρout as the initial density of the next iteration, the iteration sequence converges too slowly or even does not converge. Therefore, it is very important to choose a proper density mixing method. Many such density mixing methods have been proposed so far. The most widely used ones include simple mixing [53], Pulay mixing [69, 70], Broyden’s mixing method [77, 79], and a modified Broyden mixing method [53, 56]. We refer to [90] for the most recent progress on designing SCF iteration schemes.

10.2.2 Weak Form and Functional Assumption

To state the numerical theory of finite element approximations to the Kohn–Sham equation, we need some notation and assumptions.

Define the trilinear form a(⋅;⋅, ⋅) by:

We then have the weak form of (10.7) as follows

(10.11)

We call ( Λ,  Φ) with Λ = (λij) and Φ = (φ₁, ⋯ , φN) satisfying (10.11) a Kohn–Sham ground state. Since the Kohn–Sham energy functional is nonconvex and invariant with respect to any unitary transform, we define the set of Kohn–Sham ground states by

We also define the set of states that satisfy (10.11) as:

Note that $$\Theta \subsetneq \mathcal {W}$$ . Sometimes we assume that there is a gap between the energies of corresponding to states in Θ and other states in $$\mathcal {W}$$ :

(10.12)

which is reasonable [15].

We see that (10.8) can be transferred to the following weak formulation: Find $$(\varepsilon _i, \psi _i) \in \mathbb {R}\times H^1_0(\Omega )(i=1, \cdots , N)$$ such that

(10.13)

where $$\widetilde {\Phi } = (\psi _1, \cdots , \psi _N)$$ .

Note that any solution of (10.11) can be obtained from a unitary transform of some solution of (10.13). That is, once we get all solutions of (10.13), we then obtain all solutions of (10.11).

We define the operator $$\mathcal {F}:\mathbb {R}^{N\times N}\times \mathcal {H}\rightarrow \mathcal {H}^*$$ as

The Fréchet derivative of $$\mathcal {F}$$ with respect to Φ at ( Λ,  Φ), denoted by $$\mathcal {F}_{\Phi }^{\prime }(\Lambda ,\Phi ):\mathcal {H}\rightarrow \mathcal {H}^{\ast }$$ , is as follows

To study the convergence and complexity, we use the following assumptions [18].

Assumptions 10.1

(A1)

$$|e_{\mathrm {xc}}^{\prime }(t)|+|te_{\mathrm {xc}}^{\prime \prime }(t)|\in \mathscr {P}(p_1,(c_1,c_2))$$ for some p₁ ∈ [0, 2].

(A2)

There exists a constant α ∈ (0, 1] such that

(A3)

( Λ,  Φ) is a solution of (10.3) and there exists a constant β > 0 depending on ( Λ,  Φ) such that

(10.14)

We remark here that Assumption (A2) implies Assumption (A1) and the commonly used Xα and LDA exchange-correlation energy functionals satisfy Assumption (A2). Assumption (A3) is equivalent to $$\mathcal {F}_{\Phi }^{\prime }(\Lambda ,\Phi )$$ being an isomorphism from $$\mathcal {T}_{\Phi }$$ to $$\mathcal {T}_{\Phi }$$ . We observe that if Assumption (A3) is satisfied for $$\Phi \in \mathbb {Q}$$ , then Assumption (A3) is satisfied for any $$\widetilde {\Phi }\in [\Phi ]$$ with the same constant β, too. A stronger condition than (10.14), namely that

is used in [12, 74], which is satisfied for a linear self-adjoint operator when there is a gap between the Nth lowest eigenvalue and (N + 1)th eigenvalue [74].

10.3 Finite Element Discretization

Let $$d_{{ }_\Omega }$$ be the diameter of Ω and $$\{\mathcal {T}_h\}$$ be a shape regular family of nested conforming meshes over Ω with size $$h\in (0,d_{{ }_\Omega })$$ : there exists a constant γ∗ such that

(10.15)

where hτ is the diameter of τ for each $$\tau \in \mathcal {T}_h$$ , ρτ is the diameter of the biggest ball contained in τ, and $$h=\max \{h_{\tau }: \tau \in \mathcal {T}_h\}$$ . Let $$\mathcal {E}_h$$ denote the set of interior faces of $$\mathcal {T}_h$$ .

Let Sh, k( Ω) be a subspace of continuous functions on Ω such that

where $$P^k_{\tau }$$ is the space of polynomials of degrees no greater than k over τ. Let $$S^{h,k}_0(\Omega )=S^{h,k}(\Omega )\cap H^1_0(\Omega )$$ . To simplify the notation we shall denote $$S^{h,k}_0(\Omega )$$ by $$S^{h}_0(\Omega )$$ and let $$V_h=(S^h_0(\Omega ))^N$$ .

Then the finite element approximation of (10.2) is stated as:

(10.16)

We see from [5, 18, 83] that the minimizer of (10.16) exists under condition (10.6). Similarly, the finite element approximation of (10.11) reads

(10.17)

with the Lagrange multiplier

Define the set of finite element approximations for Kohn–Sham ground states:

StartLayout 1st Row upper Theta Subscript h Baseline equals left brace left parenthesis upper Lamda Subscript h Baseline comma upper Phi Subscript h Baseline right parenthesis element of double struck upper R Superscript upper N times upper N Baseline times left parenthesis double struck upper Q intersection upper V Subscript h Baseline right parenthesis colon upper E left parenthesis upper Phi Subscript h Baseline right parenthesis 2nd Row equals min Underscript upper Psi element of double struck upper Q intersection upper V Subscript h Baseline Endscripts upper E left parenthesis upper Psi right parenthesis comma left parenthesis upper Lamda Subscript h Baseline comma upper Phi Subscript h Baseline right parenthesis solves left parenthesis question mark question mark question mark right parenthesis right brace EndLayout

and the set of ground state finite element solutions:

We have from [18] that the finite element approximations are uniformly bounded, i.e., there exists a constant C such that

To carry out the numerical analysis, we consider the model (10.17). To get the numerical solution of the Kohn–Sham model, we use the following discretized Kohn–Sham equation

(10.18)

where $$\widetilde {\Phi }_h = (\psi _{1,h}, \cdots , \psi _{N,h})$$ .

Similarly, any solution of (10.17) can be obtained from a unitary transform of a solution of (10.18). That is, once we get solutions of (10.18), we then obtain solutions of (10.17).

By using the SCF iteration addressed in Sect. 10.2.1.3, the solution of (10.18) may be obtained by the repeated solution of the following linear eigenvalue problem

(10.19)

where ρin,h is an input charge density approximation.

Assume $$\{\varphi _{1, h}, \varphi _{2, h}, \cdots , \varphi _{i, h}, \cdots , \varphi _{N_b, h}\}$$ is a basis of Sh, k( Ω), namely,

where Nb is the dimension of space Sh, k( Ω). We then see that each ψ ∈ Sh, k( Ω) can be expressed as

(10.20)

where $$c_{j}\in \mathbb {R}(j=1,2,\cdots ,N_b)$$ .

Inserting (10.20) into (10.19) and choosing φh to be φl,h(l = 1, ⋯ , Nb), respectively, we obtain the following algebraic eigenvalue problem

(10.21)

with H and B being the stiffness and mass matrices in the finite element bases, respectively. That is,

with Hkl = a(ρin,h;φl,h, φk,h) and Bkl = (φl,h, φk,h).

We then call an existing eigensolver, e.g., the Lanczos method [11], the LOBPCG method [54], the Jacobi–Davidson method [78], to solve the algebraic eigenvalue problem (10.21). We refer to [71] for various eigensolvers. Note that the algebraic version of (10.9) under the finite element discretization becomes

(10.22)

where A is symmetric and positive definite. We may apply a preconditioned conjugate gradient method (PCG) [94] or an algebraic multigrid method [8, 82] to solve (10.22) (cf., e.g., [72] for more details).

10.4 The A Priori and A Posteriori Analysis

The numerical analysis of Kohn–Sham models plays an important part in our understanding of the efficiency of the numerical methods used in electronic structure calculations. However, significant results in this area have only been obtained very recently. The main difficulties involved in the numerical analysis of Kohn–Sham models are that we have to either handle nonconvex minimization problems with orthogonality constraints or deal with nonlinear eigenvalue problems with a large number of (possibly degenerate) eigenvalues.

We point out that a systematic a priori analysis for a general finite-dimensional discretization is presented in [18], where the following results are obtained: all the limit points of finite-dimensional approximations are ground state solutions of the system, and some ground state solution can be well approximated by finite-dimensional solutions if the associated local isomorphism condition is satisfied. Note that not only convergence of ground state energy approximations but also convergence rates of both eigenvalue and eigenfunction approximations are provided in [18].

The a posteriori analysis for finite element discretizations of the Kohn–Sham equation was later provided in [15]. It is proved in [15] that the error of the finite element approximations for the Kohn–Sham equation is bounded from below and above by the residual type a posteriori error estimators which are also proposed in [15]. This is the first result on the a posteriori error estimate for numerical discretizations of the Kohn–Sham equation.

10.4.1 The a priori Analysis

In this subsection, we show the a priori error estimates for the finite element approximations, whose proofs can be found in [18]. Indeed, all the analyses in [18] are focused on finite-dimensional approximations in a class of finite-dimensional subspaces $$S_n \in H_0^1(\Omega )$$ satisfying the following assumption (see also (3.1) of [18])

(10.23)

Assumption (10.23) is a very mild assumption satisfied by many typical finite-dimensional spaces used in practice, including the piecewise polynomial finite element spaces [22].

The following convergence conclusions are proved in [18].

Theorem 10.2

The following convergences hold:

where Eh = E( Φh) for any $$\Phi \in \mathcal {G}_h$$ , and the distance between sets $$X,Y\subset \mathbb {R}^{N\times N}\times \mathcal {H}$$ is defined by

Theorem 10.2 says that all the finite element approximations for eigenvalues and eigenfunctions together with the total energy converge and the limit points are ground state solutions of the system.

It is also shown in [18] that the quadratic convergence rate of ground state energy approximations can be further achieved.

Theorem 10.3

Let E₀be the ground state energy of (10.2) and Ehbe the ground state energy of (10.16), namely, E₀ = E( Φ) for all $$\Phi \in \mathcal {G}$$ , and Eh = E( Φh) for all $$\Phi _h \in \mathcal {G}_h$$ . If Assumption(A1)holds, then

where the distance between sets $$X,Y\subset \mathcal {H}$$ is defined by

We observe that Theorem 10.3 is a generalization and improvement of results in [12, 83]. Furthermore, we see that under certain assumptions every ground state solution can be approximated with some approximation rate by finite-dimensional solutions.

For any $$(\Lambda ,\Phi ) \in \mathbb {R}^{N \times N} \times \mathcal {H}$$ , we define Bδ( Λ,  Φ) as

Let

we then have the following conclusion (see [18] for the proof.)

Theorem 10.4

Let ( Λ,  Φ) ∈ Θ. If Assumption(A2)is true and ( Λ,  Φ) satisfies Assumption(A3), then there exists a δ > 0 such that for sufficiently small h, (10.17) has a unique local solution ( Λh, Φh) ∈ XΦ,h ∩ Bδ(( Λ,  Φ)). Moreover,

(10.24)

(10.25)

(10.26)

with r(h) → 0 as $$h\rightarrow 0$$ .

10.4.2 The a posteriori Analysis

In this subsection, we turn to the residual type-based a posteriori analysis.

The residual type a posteriori error estimators for the finite element approximation of the Kohn–Sham equation consist of two parts: the residuals and the jumps [15]. We define the element residual $$\mathcal {R}_\tau (\Phi _h)$$ and the jump Je( Φh) by

where e is the common face of elements τ₁ and τ₂ with unit outward normals $$\overrightarrow {n_1}$$ and $$\overrightarrow {n_2}$$ , respectively.

Let ωh(e) be the union of elements that share the face e. For $$\tau \in \mathcal {T}_h$$ , define the local error indicator ηh( Φh, τ) and oscillation osch( Φh, τ) by

(10.27)

(10.28)

where $$\overline {w}$$ is the L²-projection of w ∈ L²( Ω) to polynomials of some fixed degree k on τ or e. Given a subset ω ⊂ Ω, define the error estimator ηh( Φh, ω) and oscillation osch( Φh, ω) by

It is proved in [15] that the errors of the finite element approximations for the Kohn–Sham equation are bounded from below and above by the residual type a posteriori error estimators designed above. Indeed, we have (see [15] for details)

Theorem 10.5

Suppose h₀ ≪ 1 and h ∈ (0, h₀]. Let ( Λ,  Φ) ∈ Θ. If Assumptions(A2)and(A3)are satisfied, then there exist positive constants C₁, C₂and C₃depending on the shape regularity constant γ∗(in (10.15)), such that

where ( Λh, Φh) ∈ XΦ,his the finite element approximation satisfying (10.24)–(10.26).

Further, we obtain (see, also, [15])

Theorem 10.6

Suppose h₀ ≪ 1 and h ∈ (0, h₀]. Let $$({\mathbbm {\mu }}_h, \widetilde {\Phi }_h)$$ be a solution of (10.18). If Assumptions(A2)and(A3)are satisfied, then

here

and $$\Theta _{\widetilde {\Phi }_h} \subseteq \Theta _h$$ .

10.5 Adaptive Finite Element Approximation

Even in the pseudopotential setting, the eigenfunctions of (10.8) still vary rapidly around nuclei and chemical bonds [7, 26, 47, 99]. Hence it is quite natural and reasonable to apply adaptive finite element approaches to improve the approximation accuracy and reduce the computational cost. Although there are many works on implementing adaptive finite element methods for electronic structure calculations, as we have seen in Sect. 10.1, there was no work on numerical analysis for an adaptive finite element discretization of the Kohn–Sham models until [15] appeared. We refer to [19, 20] for the adaptive finite element analysis of orbital-free models.

We note that an adaptive mesh-refining algorithm usually consists of the following loop [13, 30]:

Let us address each step of the loop in more detail.

Solve

This step computes the piecewise polynomial finite element approximation with respect to a given mesh.

Estimate

Given a partition $$\mathcal {T}_h$$ and the corresponding output ( Λh, Φh) from the “Solve” step, “Estimate” computes the a posteriori error estimators $$\{\eta _h(\Phi _h,\tau )\}_{\tau \in \mathcal {T}_h}$$ , e.g., the ones defined in (10.27).

Mark

Based on the a posteriori error indicators $$\{\eta _k(\Phi _k,\tau )\}_{\tau \in \mathcal {T}_k}$$ , “Mark” gives a strategy to choose a subset of elements $$\mathcal {M}_k$$ of $$\mathcal {T}_k$$ for refinement. Here and hereafter, we replace the subscript h (or hk) by an iteration counter k whenever convenient.

One of the most widely used marking strategies to enforce error reduction is the so-called Dörfler strategy [33], which is stated as follows:

Dörfler Strategy: Given a marking parameter 0 < θ < 1:

1.

Construct a subset $$\mathcal {M}_k \subset \mathcal {T}_k$$ such that

(10.29)

2.

Mark all the elements in $$\mathcal {M}_k$$ .

Another commonly used marking strategy, which is called the “Maximum strategy”, only requires that the set of marked elements $$\mathcal {M}_k$$ contains at least one element of $$\mathcal {T}_k$$ holding the largest value estimator [45, 46]. Namely, there exists at least one element $$\tau ^{\max }_k\in \mathcal {M}_k$$ such that

(10.30)

The Maximum strategy is weaker than the Dörfler strategy. We observe that the most commonly used marking strategies, e.g., the Dörfler strategy and the equidistribution strategy, satisfy (10.30).

Refine

Given the partition $$\mathcal {T}_k$$ and the set of marked elements $$\mathcal {M}_k$$ , “Refine” produces a new partition $$\mathcal {T}_{k+1}$$ by refining all elements in $$\mathcal {M}_k$$ at least once. Define

as the set of refined elements, we have $$\mathcal {M}_k\subset \mathcal {R}_{\mathcal {T}_k\rightarrow \mathcal {T}_{k+1}}$$ . Note that usually more than just the marked elements in $$\mathcal {M}_k$$ are refined in order to keep the mesh conforming.

Combining the basic loop of an adaptive element algorithm and the residual type a posteriori error estimators (10.27), two adaptive finite element algorithms for the Kohn–Sham equation, the adaptive finite element algorithm with the Maximum strategy and the adaptive finite element algorithm with the Dörfler strategy, are designed in [15], which are stated as follows.

Algorithm 1

An algorithm of steps from 1 to 6. The steps are to pick an initial mesh T subscript 0 and let k equals 0, solve equations to get discrete solutions, compute the local error indicators, construct equations by maximum strategy, refine T subscript k, and let k = k + 1. Again the process repeats from step 2.

Algorithm 2

An algorithm of steps from 1 to 6. The steps are to pick an initial mesh T subscript 0 and let k equals 0, solve equations to get discrete solutions, compute the local error indicators, construct equations by Dorfler strategy, refine T subscript k, and let k = k + 1. Again the process repeats from step 2.

10.5.1 Convergence and Convergence Rate

The numerical analysis for Algorithm 1 has been carried out in [15], for instance, the following theorem has been proved there.

Theorem 10.7

Let $$\{\Theta _k\}_{k\in \mathbb {N}_0}$$ be the sequence generated by Algorithm 1 with a sufficiently fine initial mesh $$\mathcal {T}_0$$ . If (10.12) and Assumption(A1)are satisfied, then

It is shown by Theorem 10.7 that all limits of the adaptive finite element approximations for the Kohn–Sham equation are Kohn–Sham ground state solutions when the initial mesh is sufficiently fine. Indeed, the initial mesh is not necessary fine enough for the convergence, we refer to [91] for more details.

Thanks to the use of a stronger Dörfler marking strategy, not only the convergence but also the convergence rate can be obtained for Algorithm 2 [15]. Let us first do some preparation for addressing the results.

For ( Λ,  Φ) ∈ Θ and Φh ∈ Vh, we say the equivalence class [ Φh] approximates the equivalence class [ Φ] if

A sequence $$\{[\widetilde {\Phi }_{k}]\}_{k\in \mathbb {N}_0}$$ is said to converge to an equivalence class [ Φ] if there exists a sequence of unitary matrices $$\{U_{k}\}_{k\in \mathbb {N}_0}$$ , such that

It is proved in [15] that the error of the adaptive finite element approximations produced by Algorithm 2 is reduced. Indeed, we have (see [15] for the proof)

Theorem 10.8

Let θ ∈ (0, 1) and h₀ ≪ 1. Let $$\{\widetilde {\Phi }_k\}_{k\in \mathbb {N}_0}$$ be a sequence of finite element approximations obtained by Algorithm 2 and $$\{[\widetilde {\Phi }_{k_i}]\}_{i\in \mathbb {N}_0}$$ be the subsequence that converges to some [ Φ] with ( Λ,  Φ) ∈ Θ. If Assumption(A2)is true and ( Λ,  Φ) satisfies Assumptions(A3), then

where $$\Phi _{k_{i+1}}\in X_{\Phi ,k_{i+1}}$$ and $$\Phi _{k_{i}}\in X_{\Phi ,k_i}$$ satisfy (10.24)–(10.26) with h being replaced by $$h_{k_{i+1}}$$ and $$h_{k_i}$$ , respectively, γ > 0 and ξ ∈ (0, 1) are constants depending only on the coercivity constant ca, the shape regularity constant γ∗, and the marking parameter θ. Therefore, the km-th iteration solution of Algorithm 2 satisfies

and

Furthermore, there holds

It is shown by Theorem 10.8 that the Kohn–Sham ground state solutions can be well approximated by the adaptive finite element approximations with a certain convergence rate when the initial mesh is fine enough.

10.5.2 Complexity

The quasi-optimal complexity of adaptive finite element approximations may be derived, for which we define [15].

where γ > 0 is some constant and

and $$\mathcal {T}\subset \mathcal {T}_0$$ means $$\mathcal {T}$$ is a refinement of $$\mathcal {T}_0$$ . We see that, for all γ > 0, $$\mathcal {A}_{\gamma }^s = \mathcal {A}_1^s$$ . So we may use $$\mathcal {A}^s$$ to stand for $$\mathcal {A}_1^s$$ and use | Φ|s to denote | Φ|s,γ. We note that $$\mathcal {A}^s$$ is the class of functions that can be approximated within a given tolerance ε by continuous piecewise polynomial functions over a partition $$\mathcal {T}_k$$ with the number of degrees of freedom satisfying $$\#\mathcal {T}_k-\# \mathcal {T}_0 \lesssim \varepsilon ^{-1/s} |\Phi |{ }_{s}^{1/s}$$ , where $$\#\mathcal {T}$$ denotes the number of elements in a mesh $$\mathcal {T}$$ .

To investigate the computational complexity of Algorithm 2, we need some assumptions as follows: Let

where C∗ is a positive constant depending on the shape regularity constant γ∗ (see [15] for details).

Assumptions 10.9

1.

The marking parameter θ satisfies θ ∈ (0, θ∗).

2.

The marked $$\mathcal {M}_{h_k}$$ satisfies (10.29) with minimal cardinality.

3.

The distribution of refinement edges on $$\mathcal {T}_{h_0}$$ satisfies condition (b) of Section 4 in [81].

With Assumptions 10.9, we then have the optimal complexity [15].

Theorem 10.10

Suppose θ ∈ (0, 1) and h₀ ≪ 1. Assume that Assumption(A2)is satisfied and (10.11) has m solutions in Θ (up to invariance under unitary transforms), which are denoted as [ Φ(l)](l = 1, 2, …, m) where $$m \in \mathbb {N}$$ or can be chosen to be ∞. Let $$\{\Psi _k\}_{k\in \mathbb {N}_0}$$ be a sequence of finite element solutions produced by Algorithm 2 with Assumptions10.9being satisfied. Then the following quasi-optimal bound is valid

(10.31)

where $$\Phi ^{l} \in [\Phi ^{(l)}] \cap \mathcal {A}^s$$ satisfies (10.14), $$\Phi _{k_{n_l}}^l\in X_{\Phi ^l,k_{n_l}}$$ satisfy (10.24) and (10.26) with h being replaced by $$h_{k_{n_l}}$$ , and the hidden constant depends on the exact solution Φland the discrepancy between θ and θ∗(l = 1, 2, …, m). Here, nland $$k_{n_{l}}$$ are the total number and the maximal index of iterations which approximate [ Φ(l)](l = 1, 2, …, m) among the n iterations, respectively.

It is shown by Theorem 10.10 that under some mild and reasonable assumptions, the computational complexity of the adaptive finite element algorithm with the Dörfler strategy is quasi-optimal when the initial mesh is fine enough.

Applying the non-polynomial behavior of the eigenfunctions, we see from [91] that the adaptive finite element method has an asymptotic linear convergence rate and an asymptotic optimal complexity from any initial mesh. More precisely, the adaptive finite element method has a linear convergence rate and an optimal complexity after finite iteration steps.

10.6 High Performance Computing

Due to the needs of science and engineering and the availability of high performance computers, there is a great demand for highly efficient and scalable algorithms for electronic structure calculations. There have been many contributions toward this effort to design high performance algorithms. We mention here, for instance, the symmetry-based decomposition approach [36, 40], the parallel orbital-updating approach [27, 28, 63], the local and high order finite element discretization [29], the three-scale schemes [31, 44], and the hexahedron-grid-based two-scale higher order finite element scheme [21]. In the following, we focus on a brief introduction to the first two approaches that involve a two-level parallelization.

10.6.1 Symmetry-Based Decomposition Approach

A symmetry-based decomposition approach is proposed to deal with linear differential eigenvalue problems as well as the Kohn–Sham equation in [40] by making use of the Abelian or non-Abelian symmetries of the systems. By this approach, for instance, the solution of the associated algebraic eigenvalue problem is then replaced by the solutions of some independent eigenvalue subproblems that are of small scale. Due to the independence, as we see, the computation can be carried out in parallel.

The group theory-based decomposition approach can be seamlessly incorporated with grid-based discretizations such as finite difference, finite element, or finite volume methods. We may place the approach into a two-level parallelization setting, which consequently reduces the CPU time notably.

To show the main idea of the group theory-based decomposition approach, we consider a model problem

(10.32)

where L is a Hermitian operator on a Hilbert space V ⊂ L²( Ω).

A group G is said to be a symmetry group associated with eigenvalue problem (10.32) if

and the imposed boundary condition is also invariant under {PR : R ∈ G}, which is defined as follows: for each R ∈ G and f ∈ V , there holds

We see that {PR : R ∈ G} forms a group isomorphic to G[34].

Associated with the model (10.32), we have [40]

Theorem 10.11

Suppose a finite group G = {R} is a symmetry group associated with eigenvalue problem (10.32). Denote all the inequivalent, irreducible, unitary representations of G as { Γ(ν) : ν = 1, 2, …, nc}. Then the eigenvalue problem can be decomposed into $$\displaystyle \sum _{\nu =1}^{n_c}d_{\nu }$$ subproblems. For each ν ∈{1, 2, …, nc}, the corresponding dνsubproblems are

where k is any chosen number in {1, 2, …, dν}.

Suppose the finite group G = {R} of order n is the symmetric group associated with the problem (10.32). Denote all the inequivalent, irreducible, unitary representations of group G by { Γ(ν), ν = 1, 2, …, nc}, with dν being the dimension of Γν. Then the original eigenvalue problem (10.32) can be decomposed into $$\displaystyle \sum _{\nu =1}^{n} d_{\nu }$$ subproblems. Define $$n_{\mathrm {sub}} = \displaystyle \sum _{\nu =1}^{n} d_{\nu }$$ . For any ν, the corresponding dν subproblems can be expressed as

(10.33)

For illustration, we provide an analysis of this approach for electronic structure calculations of symmetric cluster systems based on the finite element discretizations. Assume the number of degrees of freedom for discretizing (10.32) is Ng and the number of eigenpairs required is N. By the traditional approaches, the computational cost scales as N²Ng. By applying the above symmetric-based decomposition approach, the solution of (10.32) is transferred to the solutions of nsub subproblems (10.33), with each subproblem being discretized by Ng∕n degrees of freedom and only requiring the smallest N∕nsub eigenvalues and their corresponding eigenfunctions to be computed. Therefore, the total computational cost is significantly reduced. More importantly, since the subproblems (10.33) are independent of each other, they can be solved in parallel intrinsically. Thus the algorithm allows two level parallelization: one level is the solution of these N independent subproblems in parallel intrinsically, the other level is to solve each boundary problem in parallel by traditional algebraic parallel strategies or domain decomposition approaches. We refer to [36, 40] for the detailed implementation issues.

10.6.2 Parallel Orbital-Updating Approach

Now we turn to introduce the parallel orbital-updating approach, which is first proposed in [27]. The basic version is stated as follows [27].

Algorithm 3

An algorithm of steps from 1 to 5 for the parallel orbital updating approach. The steps are to give the initial guess, enlarge the finite-dimensional space, update psi superscript k subscript i values for i = 1 to N, solve equations, and let k = k + 1. Again the process repeats from step 2.

Here, $${\widetilde V}_{k+1}=~\mbox{span}~ \{{\widetilde \psi }^{(k+1)}_1,{\widetilde \psi }^{(k+1)}_2,\ldots ,{\widetilde \psi }^{(k+1)}_N\}$$ .

With the parallel orbital-updating approach, the solution of the eigenvalue problem is replaced by solutions of a series of independent source problems and some small scale eigenvalue problems. Because of the independence of the source problems, these source problems can be solved in parallel essentially. For each source problem, traditional parallel strategies (for example, domain decomposition or parallelization in matrix-vector multiplication) can be used. Therefore, it allows for a two-level parallelization: one level of parallelization is obtained by partitioning these source problems into different groups of processors, another level of parallelization is obtained by assigning each source problem to several processors contained in each group. This two-level parallelization makes the parallel orbital-updating approach more competitive for large scale calculations.

The approach is then extended to solutions of the constrained minimization problem for the Kohn–Sham energy functional [28]. Some extensions to plane wave discretizations for the Kohn–Sham equation as well as modified versions are investigated in [63].

10.7 Concluding Remarks

In this chapter, we have introduced the finite element method to DFT calculations, together with its mathematical theory. We have presented the a priori and a posteriori error estimations of the finite element approximations to the Kohn–Sham equation. We have also provided the convergence rate for adaptive finite element approximations and described its complexity theory. Consequently, we have shown the efficiency and reliability of finite element approximations of the Kohn–Sham DFT models, together with a mathematical theory of the error control during the calculations. In addition, we have presented two high performance algorithms, the symmetry-based decomposition approach and the parallel orbital-updating approach, which allow for the two-level parallelization and may have the potential for exascale computing. Finally, we mention that the emphasis of this chapter has been on the work of our group, and related research.

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_11

11. Flexibilities of Wavelets as a Computational Basis Set for Large-Scale Electronic Structure Calculations

Luigi Genovese1   and Thierry Deutsch1  

(1)

Univ. Grenoble Alpes, CEA, IRIG-MEM, Grenoble, France

Luigi Genovese

Email: luigi.genovese@cea.fr

Thierry Deutsch (Corresponding author)

Email: thierry.deutsch@cea.fr

Abstract

The BigDFT project started in 2005 with the aim of testing the advantages of using a Daubechies wavelet basis set for Kohn–Sham density functional theory with pseudopotentials. This project led to the creation of the BigDFT code, which employs a computational approach with optimal features for flexibility, performance and precision of the results. In particular, the employed formalism has enabled the implementation of an algorithm able to tackle DFT calculations of large systems, up to many thousands of atoms, with a computational effort which scales linearly with the number of atoms. In this work we show how the localised description of the Kohn–Sham problem, emerging from the features of the basis set, are helpful in providing a simplified description of large-scale electronic structure calculations. We recall some of the features that have been made possible by the peculiar mathematical properties of Daubechies wavelets and also interpolating scaling functions.

11.1 Introduction

Since their foundation, disciplines like computational physics and quantum chemistry have had to deal with the question of the computational reliability of results. The reliability of a given approach can be defined in terms of two key concepts, namely “accuracy”, i.e. the ability of the model to predict quantities which can be externally verified, e.g. through experiment, and “precision”, i.e. the ability of the employed numerical approach to find the solution to a given physical model. A precise approach should therefore reduce the computational uncertainties of quantities extracted from a well-defined model, and provide reference results which can be compared to other computer codes employing the same model. For theoretical approaches wherein no analytic solution exists, reducing the computational uncertainty is the only way to shed light on the predictive power of the model. The accuracy of a result with respect to experimental data may therefore only be reliably quantified if the computational uncertainty is significantly lower than the observed discrepancy.

Density functional theory (DFT) [38, 43] has had widespread success in simulating a range of materials, from molecules to solids, and has therefore become the most popular approach to electronic structure simulations. While the accuracy of DFT is dominated by the approximations made to the exchange-correlation (XC) functional, the precision of a given simulation depends on a number of factors, in particular the choice of the basis set. Thus two different DFT codes might use the same physical formalism (including the same XC functional), but differ in results due to the use of different numerical approaches. In order to compare results across DFT codes, careful attention must therefore be paid to the precision of the results, as seen for example in the DeltaCode project, in which a systematic comparison of a number of periodic DFT codes was undertaken [45].

In this context, an important distinction should be made between codes which use systematic and non-systematic basis sets. A systematic basis set allows one to calculate the exact solution of the Kohn–Sham (KS) equations with arbitrarily high precision by increasing the number of basis functions. In other terms, the numerical precision of the results is related to the number of basis functions used to expand the KS orbitals. With such a basis set it is thus possible to obtain results which are free of errors related to the choice of the basis, eliminating a source of uncertainty. As such, it is highly desirable to have at hand a computational formalism which is able to provide, at the same time:

a set of reliable results, which can be systematically improved by the end-user, in view of increasing—when needed—the precision of the calculations;

a flexible approach, in which the desired models can be explicitly implemented without having to deal with correction terms and intrinsic approximations;

an efficient computer program, which enables the optimal use of computational resources, especially in the context of high performance computing;

the ability to connect together different levels of theory, where various approaches might be linked within a given computational setup.

In 2005, the EU FP6-STREP-NEST BigDFT project funded a consortium of four European laboratories (L_Sim, CEA-Grenoble, France; Basel University, Switzerland; Louvain-la-Neuve University, Belgium; and Kiel University, Germany), with the aim of developing a novel approach for DFT calculations based on Daubechies wavelets [16]. Beyond building a DFT code from scratch, the objective of this 3-year project was to test the potential benefit of a new formalism in the context of electronic structure calculations.

This project was motivated by the fact that Daubechies wavelets exhibit a set of properties which make them ideal for a precise and optimized DFT approach. In particular, their systematicity provides a reliable basis set for high-precision results, whereas their locality (both in real and reciprocal space) is highly desirable to improve the efficiency and the flexibility of the treatment. Indeed, a localized basis set allows the optimization of the number of degrees of freedom for a required accuracy, which is highly desirable given the complexity and inhomogeneity of systems under investigation nowadays. Moreover, an approach based on localized functions makes it possible to explicitly control the nature of the boundaries of the simulation domain, allowing complex environments like mixed boundary conditions and/or systems with a net charge.

We organize this chapter as follows. We first develop shortly the KS formalism and the concepts of operators for any basis set. We then present some basic illustrations of the properties of Daubechies wavelets associated to the multi-resolution analysis, and their peculiarities in the context of computational structure electronic calculations. We briefly outline, based on the interpolating scaling functions, the main features of the Poisson Solver implemented in the code. Finally, we will then explain in more detail how the solution of the KS problem is implemented in the code and how the properties of wavelets enable the realization of a computational algorithm whose time-to-solution is linearly scaling with the number of atoms in the system.

11.1.1 Basis Set of Coordinate Representation

We start the chapter by illustrating the notation employed in the following. We represent the state associated to a particle coordinate by a label with r labelling a point in real space and the value of α = 1, 2 corresponding to the spinorial state up or down, respectively. The coordinate representation might be used to express orbitals and operators according to the conventions

vertical bar psi right angle bracket equals integral normal d bold x left angle bracket bold x vertical bar bold italic psi vertical bar bold x vertical bar bold italic psi right angle bracket vertical bar bold x right angle bracket equals sigma summation Underscript alpha Endscripts integral normal d bold r psi Superscript alpha Baseline left parenthesis bold r right parenthesis vertical bar bold r right angle bracket circled times vertical bar alpha right angle bracket period

The completeness relation in the coordinate representation might be written as follows

ModifyingAbove double struck upper I With caret equals sigma summation Underscript alpha Endscripts integral normal d bold r vertical bar bold r right angle bracket left angle bracket bold r vertical bar circled times vertical bar alpha right angle bracket left angle bracket alpha vertical bar equals integral normal d bold x vertical bar bold x right angle bracket left angle bracket bold x vertical bar equals sigma summation Underscript alpha Endscripts integral normal d bold r vertical bar bold r comma alpha right angle bracket left angle bracket bold r comma alpha vertical bar period

We may also express as a shorthand.

The electronic charge density ρ(r) might be defined from the above equations as follows:

rho left parenthesis bold r right parenthesis equals trace left parenthesis sigma summation Underscript alpha Endscripts ModifyingAbove upper F With caret vertical bar bold r comma alpha right angle bracket left angle bracket bold r comma alpha vertical bar right parenthesis comma

where the spinorial degrees of freedom have been traced out. The Electronic Density Matrix is an operator that can be implicitly defined via its kernel

ModifyingAbove upper F With caret equals integral normal d bold x normal d bold x prime vertical bar bold x prime right angle bracket upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis left angle bracket bold x vertical bar period

With the above notation we imply that the kernel F(x, x′) is expressed in the basis of the position representation. Let us point out the notable relations:

StartLayout 1st Row 1st Column StartFraction delta upper F left parenthesis bold y comma bold y Superscript prime Baseline right parenthesis Over delta upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis EndFraction 2nd Column equals StartFraction delta left angle bracket bold y prime vertical bar ModifyingAbove upper F With caret vertical bar bold y right angle bracket Over delta left angle bracket bold x prime vertical bar ModifyingAbove upper F With caret vertical bar bold x right angle bracket EndFraction equals delta left parenthesis bold r Subscript x Baseline minus bold r Subscript y Baseline right parenthesis delta left parenthesis bold r prime Subscript x Baseline minus bold r prime Subscript y right parenthesis delta Subscript alpha Sub Subscript x Subscript alpha Sub Subscript y Baseline delta Subscript beta Sub Subscript x Subscript beta Sub Subscript y Baseline 2nd Row 1st Column StartFraction delta ModifyingAbove upper F With caret Over delta upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis EndFraction 2nd Column equals integral normal d bold y normal d bold y prime vertical bar bold y prime right angle bracket left angle bracket bold y vertical bar StartFraction delta upper F left parenthesis bold y comma bold y Superscript prime Baseline right parenthesis Over delta upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis EndFraction equals vertical bar bold x prime right angle bracket left angle bracket bold x vertical bar 3rd Row 1st Column integral normal d bold x normal d bold x prime StartFraction delta trace left parenthesis ModifyingAbove upper F With caret ModifyingAbove upper O With caret right parenthesis Over delta upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis EndFraction 2nd Column equals integral normal d bold x normal d bold x prime trace left parenthesis vertical bar bold x prime right angle bracket left angle bracket bold x vertical bar ModifyingAbove upper O With caret right parenthesis equals integral normal d bold x normal d bold x prime ModifyingAbove upper O With caret left parenthesis bold x prime comma bold x right parenthesis 4th Row 1st Column StartFraction delta rho left parenthesis bold r right parenthesis Over delta upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis EndFraction 2nd Column equals delta left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis equals delta left parenthesis bold r Subscript x Baseline minus bold r right parenthesis delta left parenthesis bold r prime Subscript x Baseline minus bold r right parenthesis delta Subscript alpha Sub Subscript x Subscript beta Sub Subscript x prime Baseline 5th Row 1st Column integral normal d bold x normal d bold x prime StartFraction delta trace left parenthesis ModifyingAbove upper F With caret ModifyingAbove upper O With caret right parenthesis Over delta upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis EndFraction vertical bar bold x right angle bracket left angle bracket bold x prime vertical bar 2nd Column equals ModifyingAbove upper O With caret comma EndLayout

which would hold for any one-body operator $$\hat {O}$$ .

11.1.2 Definition of the KS Hamiltonian from the Total Energy

In the constrained search formulation we minimize the Hohenberg–Kohn functional E[ρ] with respect to the internal degrees of freedom of the density operator , by imposing the constraint $$\rho [\hat {F}] = \rho $$ . In this way, we can generalize the dependency of the functional to the density matrix, and the terms which explicitly depend on non-diagonal terms of $$\hat {F}$$ can be included in this way. We might therefore consider, with more generality, that the Kohn–Sham energy functional is defined in terms of the density matrix, namely .

It is interesting to consider the linearization of the energy functional with respect to the density operator. We define the one-body operator

ModifyingAbove upper H With caret Subscript KS Baseline left bracket ModifyingAbove upper F With caret right bracket identical to integral normal d bold x normal d bold x prime StartFraction delta upper E left bracket ModifyingAbove upper F With caret right bracket Over delta upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis EndFraction vertical bar bold x right angle bracket left angle bracket bold x prime vertical bar comma

which is self-adjoint (as upper F left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis identical to left angle bracket bold x prime vertical bar upper F vertical bar bold x right angle bracket )). We may split the total energy into a band structure and a double counting term. The band structure term is defined as

upper E Subscript BS Baseline left bracket ModifyingAbove upper F With caret right bracket equals trace left parenthesis ModifyingAbove upper H With caret Subscript KS Baseline left bracket ModifyingAbove upper F With caret right bracket ModifyingAbove upper F With caret right parenthesis comma

and it is therefore thought of as a functional of the sole density matrix $$\hat {F}$$ .

The double counting term would then express the difference between the band structure term and the total energy, this time interpreted as a function of the charge density of the system:

(11.1)

The above definitions are valid for functionals which are a linear combination of each other, namely if L[ρ] = E[ρ] + F[ρ], then LBS,DC = EBS,DC + FBS,DC, and the same is valid for the respective KS Hamiltonians. Yet, obviously the ground-state (GS) density of the functional L will not be in relation with the GS densities of E and F.

Therefore we might formally write

where with the above notation we indicate the minimum of the functional over all the density matrices that have ρ as a charge density.

The GS energy of the system is found by minimizing the functional E[ρ] over all N-representable densities, with N the number of electrons. Practically, such N-representability is achieved by representing the density in terms of the Fermi (super-) operator, defined by

applied onto the KS Hamiltonian operator. The chemical potential μ is fixed by imposing that the total number of electrons corresponds to N, namely by . The constrained search minimization can therefore be expressed in the following way:

where the GS energy functional is, as pointed out by Hohenberg and Kohn in their seminal work, written in terms of the total number of electron N and of the external potential Vext.

11.2 Operators of Generalized Kohn–Sham Formalism

As we discussed before, each of the terms contributing to the total energy creates a contribution to the KS Hamiltonian, to the double counting term and their respective derivatives. We list here such contributions:

Kinetic Term

The single particle term for the kinetic energy depends only on the density matrix:

upper T Subscript s Baseline left bracket ModifyingAbove upper F With caret right bracket equals minus one half integral normal d bold x trace left parenthesis vertical bar bold x right angle bracket nabla squared left angle bracket bold x vertical bar ModifyingAbove upper F With caret right parenthesis period

This term gives rise to the kinetic operator of $$\hat {H}_{\text{KS}}$$ :

ModifyingAbove upper T With caret Subscript s Baseline identical to minus one half integral normal d bold x vertical bar bold x right angle bracket nabla squared left angle bracket bold x vertical bar comma

and as, for this term EBS = Ts, we have EDC = 0. Once again, we have employed this notation to highlight that the Laplacian operator is associated to the kinetic operator in the position representation.

External Potential

Another important term for the energy comes from the external potential. Let us assume a generic non-local form for a one-body potential:

where the external potential term is

ModifyingAbove upper V With caret Subscript ext Baseline equals integral normal d bold x normal d bold x prime vertical bar bold x prime right angle bracket upper V Subscript ext Baseline left parenthesis bold x comma bold x Superscript prime Baseline right parenthesis left angle bracket bold x vertical bar period

The double-counting term is also zero in this case.

Exchange and Correlation Energy

In the traditional semi-local approximations to the XC terms, the functional Exc[ρ] is considered as a function of the spin density rho Superscript alpha beta Baseline left parenthesis bold r right parenthesis identical to left angle bracket bold r comma beta vertical bar ModifyingAbove upper F With caret vertical bar bold r comma alpha right angle bracket ). The one-body exchange and correlation potential is the functional derivative

ModifyingAbove upper V With caret Subscript xc Baseline equals sigma summation Underscript alpha comma beta Endscripts ModifyingAbove upper V With caret Subscript xc Superscript alpha beta Baseline comma ModifyingAbove upper V With caret Subscript xc Superscript alpha beta Baseline equals integral normal d bold r vertical bar bold r comma alpha right angle bracket StartFraction delta upper E Subscript xc Baseline left bracket rho left bracket ModifyingAbove upper F With caret right bracket right bracket Over delta rho Superscript alpha beta Baseline left parenthesis bold r right parenthesis EndFraction left angle bracket bold r comma beta vertical bar comma

evaluated for the density ρ. This potential will appear in the $$\hat {H}_{\text{KS}}$$ operator, and the corresponding band structure energy will be $$E_{\text{BS}}=\mathrm {tr}\left (\hat {V}_{\text{xc}} \hat {F}\right )$$ , which will be expressed as a function of ραβ(r). The double counting term will thus be a functional of the spin density alone:

Hartree Potential from the Electrostatic Energy

Another important term is related to the electrostatic energy. To generalize the treatment let us suppose that such term is provided by

where $$\mathcal V_{\text{H}}[\epsilon ]$$ denotes the Green’s function of the Poisson operator $$\mathcal P[\epsilon ]$$ with a generic dielectric function 𝜖. We assume that this function can also be dependent on the charge density ρ. Such operators are implicitly defined by the following relations:

(11.2)

The Hartree term that has to be included in the KS Hamiltonian is therefore

ModifyingAbove upper V With caret Subscript upper H Baseline left bracket rho right bracket equals sigma summation Underscript alpha Endscripts integral normal d bold r vertical bar bold r comma alpha right angle bracket upper V Subscript upper H Baseline left bracket epsilon comma rho right bracket left parenthesis bold r right parenthesis left angle bracket bold r comma alpha vertical bar

with the Hartree potential being

The band structure term in this case is EBS = 2EH and the double counting term is thus EDC = −EH.

11.2.1 KS-DFT Formalism: Nearsightedness and Support Functions

An equivalent expression of this functional can be provided by giving the expression of the electronic density in term of the Kohn–Sham (KS) orbitals . In our approach the KS orbitals are expressed as a linear combination of intermediate, possibly minimal, basis functions , also referred to as support functions (SFs):

StartLayout 1st Row vertical bar upper Psi Subscript i Baseline right angle bracket equals sigma summation Underscript alpha Endscripts c Subscript i Superscript alpha Baseline vertical bar phi Subscript alpha Baseline right angle bracket period EndLayout

(11.3)

In other terms, we assume that the density matrix of the system can be defined from a set of localized SFs as follows:

ModifyingAbove upper F With caret identical to sigma summation Underscript i Endscripts f left parenthesis epsilon Subscript i Baseline right parenthesis vertical bar upper Psi Subscript i Baseline right angle bracket left angle bracket upper Psi Subscript i Baseline vertical bar equals sigma summation Underscript alpha comma beta Endscripts vertical bar phi Subscript alpha Baseline right angle bracket upper K Superscript alpha beta Baseline left angle bracket phi Subscript beta Baseline vertical bar comma

(11.4)

with an SF overlap matrix upper S Subscript alpha beta Baseline equals left angle bracket phi Subscript alpha Baseline vertical bar phi Subscript beta Baseline vertical bar phi Subscript alpha Baseline vertical bar phi Subscript beta Baseline right angle bracket ), which can be chosen to have a unit diagonal and where Kαβ is the so-called density kernel. Here the fi denote the occupation numbers associated with the KS orbitals , which determine the density ρ(x) = F(x, x). Such occupation numbers are dependent on the KS eigenvalues 𝜖i, fi = f(𝜖i), as by definition the KS orbitals satisfy the eigenvalue problem of HKS.

This kernel is related to the density matrix formulation of Hernández and Gillan [37], and has to be thought of as functionally dependent on the KS Hamiltonian, namely K = K[HKS]. The density matrix F(r, r′) decays exponentially with respect to the distance |r −r′| for systems with a finite gap or for metals at finite temperature [3, 11, 12, 26, 36, 39, 42]; for metals at zero temperature it decays algebraically [47]. Therefore in these cases it can be represented by strictly localized basis functions. A natural and exact choice for these would be the maximally localized Wannier functions (MLWFs) which have the same exponential decay [48]. In our case, the localized functions are constructed in situ during the self-consistency cycle in terms of a underlying wavelet basis set.

The energy of the system in the KS formalism can then be defined by employing the external potential Vext, which contains local and non-local pseudopotential (PSP) terms, and depends on a set of electron-independent parameters λ which label the system, for instance by indicating the positions of atoms. We also consider the core charge density ρc[λ], which depends on such a label, but it is assumed to be independent of the KS orbitals. We also include the exact exchange term EX[F] and the associated Fock operator already defined above, multiplied by a parameter αX which quantifies the fraction of exact exchange introduced in the formalism. We then obtain:

StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column upper E left bracket lamda comma rho Subscript c Baseline comma bold upper K comma StartSet phi Subscript alpha Baseline EndSet right bracket equals minus one half integral normal d bold r trace left parenthesis nabla squared vertical bar bold r right angle bracket left angle bracket bold r vertical bar upper F right parenthesis 2nd Row 1st Column Blank 2nd Column Blank 3rd Column plus trace left parenthesis upper F upper V Subscript ext Baseline left bracket lamda right bracket right parenthesis plus upper E Subscript xc Baseline left bracket rho plus rho Subscript c Baseline left bracket lamda right bracket right bracket plus upper E Subscript upper H Baseline left bracket rho right bracket plus alpha Subscript upper X Baseline upper E Subscript upper X Baseline left bracket upper F right bracket 3rd Row 1st Column Blank 2nd Column Blank 3rd Column equals sigma summation Underscript alpha beta Endscripts upper H Subscript KS Baseline Subscript alpha beta Baseline upper K Superscript alpha beta Baseline 4th Row 1st Column Blank 2nd Column Blank 3rd Column minus upper E Subscript upper H Baseline left bracket rho right bracket minus alpha Subscript upper X Baseline upper E Subscript upper X Baseline left bracket upper F right bracket plus upper E Subscript xc Baseline left bracket rho plus rho Subscript c Baseline right bracket minus integral normal d bold r rho left parenthesis bold r right parenthesis upper V Subscript xc Baseline left bracket rho plus rho Subscript c Baseline right bracket left parenthesis bold r right parenthesis comma EndLayout

(11.5)

The KS Hamiltonian is defined as

upper H Subscript KS Baseline left bracket lamda comma rho Subscript c Baseline comma bold upper K comma StartSet phi Subscript alpha Baseline EndSet right bracket identical to negative one half nabla squared plus upper V Subscript upper H Baseline left bracket rho right bracket plus upper V Subscript xc Baseline left bracket rho plus rho Subscript c Baseline right bracket plus alpha Subscript upper X Baseline ModifyingAbove upper D With caret Subscript upper X Baseline plus upper V Subscript ext Baseline left bracket lamda right bracket period

(11.6)

BigDFT efficiently treats Gaussian pseudopotentials of the Goedecker–Teter–Hutter (GTH) [29] and Hartwigsen–Goedecker–Hutter (HGH) [35] types (see also [44, 64]), since the intrinsic separability of both the basis set and Gaussian pseudopotentials allows for the simplification of several 3D operations into a sum of 1D products. The approximation of the all-electron KS quantities induced by the PSP terms has been shown to be much less severe than the exchange and correlation terms. Such PSP terms have proven to yield all-electron precision for most of the quantities of interest in ground-state DFT calculations, as can be seen in the DeltaTest initiative [45], where an accuracy of 0.1 meV/atom—the best among the PSP calculations—can be obtained for the set of atoms belonging to the first three rows of the periodic table, or in Willand et al. [64], where we show that the accuracy of the G2-1 and S22 test sets is comparable with all-electron calculations made by highly precise Gaussian basis sets.

11.2.2 Atomic Forces

The atomic forces are, by definition, the opposite of the derivative of the total energy with respect to the atom position. In this notation we should thus calculate

(11.7)

Clearly, numerically, the set of is expressed in a finite basis set. This means that the action of $$\hat {H}_{\text{KS}}$$ can in principle lie outside the span of the . We can define therefore a residual function

StartFraction normal d upper E Over normal d lamda EndFraction equals sigma summation Underscript i Endscripts f Subscript i Baseline left angle bracket upper Psi Subscript i Baseline StartAbsoluteValue StartFraction normal d upper V Subscript ext Baseline Over normal d lamda EndFraction EndAbsoluteValue upper Psi Subscript i Baseline right angle bracket plus sigma summation Underscript sigma Endscripts integral normal d bold r StartFraction normal d rho Subscript c Superscript sigma Baseline left parenthesis bold r right parenthesis Over normal d lamda EndFraction upper V Subscript xc Superscript sigma Baseline left bracket rho plus rho Subscript c Baseline right bracket left parenthesis bold r right parenthesis period

(11.8)

which represents the deviation of the numerical KS orbital from being the exact KS orbital. By definition $$\left \langle \Psi _j | \chi _i \right \rangle =0\ \forall i,j$$ . The norm of this vector, once projected onto the basis set used to express , is often used as a convergence criterion for the ground state energy. However, even though the basis set is finite, the orthogonality of KS orbitals holds exactly. It is thus easy to show that the numerical atomic forces are defined as follows:

StartLayout 1st Row 1st Column minus StartFraction normal d upper E Subscript BS Baseline Over normal d bold upper R Subscript a Baseline EndFraction equals 2nd Column minus 3rd Column sigma summation Underscript i Endscripts f Subscript i Baseline left angle bracket upper Psi Subscript i Baseline vertical bar StartFraction partial differential ModifyingAbove upper H With caret Subscript KS Baseline Over partial differential bold upper R Subscript a Baseline EndFraction vertical bar upper Psi Subscript i Baseline right angle bracket 2nd Row 1st Column Blank 2nd Column minus 3rd Column 2 sigma summation Underscript i Endscripts upper R e left parenthesis left angle bracket chi Subscript i Baseline vertical bar StartFraction partial differential upper Psi Subscript i Baseline Over partial differential bold upper R Subscript a Baseline EndFraction right angle bracket right parenthesis comma EndLayout

(11.9)

where the first term of the right-hand side of the above equation is the Hellman–Feynman contribution to the forces. The second term should not exist in a exact representation and only originates from the fact that the residual vector is non-zero.

Let us now suppose that the KS Hamiltonian and orbitals are expressed in a basis set which is complete enough to describing the orbitals and their derivatives within a targeted error. In this case the norm of can be reduced within the same basis set so as to meet this targeted precision. Therefore the projection of onto the basis set used for the calculation can be safely neglected as it is associated to the same numerical precision. When the basis set is complete enough to express also , then the atomic forces can be evaluated by the Hellmann–Feynman term only, as the remaining part is proportional to the desired precision. We can then neglect the second term of Eq. (11.9); this is an example of how having a systematic basis set, where we have estimators of the error for generic quantities, may help to drastically simplify the calculation of GS-derived quantities.

11.3 Wavelets as a Computational Basis Set

In the first two decades of the development of density functional theory, wavelet basis sets have rarely been used for electronic structure calculations, with most efforts having been devoted to their use in all-electron calculations, e.g. in MRChem (see [40, 53]) and most applications of MADNESS [32]. Since such a basis is therefore rather uncommon, we explain its use in the context of KS-DFT calculations. While referring the reader to Goedecker [30] for an exhaustive presentation of how wavelet basis sets can be used for numerical simulations, here we summarize the main properties of Daubechies wavelets, with a special focus on the representation of the objects (wavefunctions and operators) involved in the KS-DFT formalism. We will start by illustrating the principles of one-dimensional Daubechies wavelets basis.

Wavelet theory [15, 17] comes from multi-resolution analysis (MRA), which has the goal of properly defining how we can enlarge a given basis set for a given accuracy. MRA defines nested sets and their complements with the idea to have, for a given function, a suite of different approximated functions in order to control the accuracy. Before formally defining MRA, we prefer to first define the scaling function.

11.3.1 Introduction to Wavelet Theory

The key concept of wavelet theory is the scaling relation which links two levels of resolution. A mother scaling function Φ(x) is defined by the scaling relation

(11.10)

where the Φ(x − j) functions are centred on an equidistant grid. The integer m gives the length 2m + 1 of the compact support where the coefficients are non-zero. This scaling relation with the filter $$\left \{h_j\right \}_{j=-m,m}$$ fully defines the mother scaling function as a sum of the same function but at a twice finer resolution. To apply any operation, we need to manipulate this filter $$\left \{h_j\right \}$$ by means of convolutions.

The simplest scaling function, called the Haar function, is a simple rectangle function which is a sum of two smaller rectangle functions. Graphically we can express the scaling relation for the Haar functions as

A squarewave graph illustrates the Haar function. A half-cycle square wave is equal to 2 narrow half-cycle square waves. The flat lines of the pulses are connected by a number of dots.

For the inverse operation, which consists in going from a fine resolution to a coarse one, we need to introduce a new function, called the wavelet function Ψ(x), which gives the details allowing us to avoid a loss of information. For the Haar scaling function, we have

A squarewave graph illustrates the Haar function. A narrow half-cycle squarewave is equal to half of a half-cycle squarewave plus half of a narrow full-cycle squarewave. The flat lines of the pulses are connected by a number of dots.

In this figure, we point out that a fine resolution (the grid with black and grey points) can be expressed using a coarse resolution (the grid with only black points) with a double grid step and two functions per node, the scaling function and the wavelet function. In the case of the BigDFT code, the mesh is uniform on the simulation domain but the number of basis functions is different per node depending on the resolution we want.

Like the scaling relation (11.10) for the mother scaling function, the wavelet function Ψ(x) can also be expressed as a sum of scaling functions at a finer resolution

(11.11)

which we sketch for the Haar wavelet function as

A square wavelet illustrates the Haar function. A full-cycle square wave is equal to a half-cycle square wave minus a half-cycle square wave. The flat lines of the pulses are connected by a number of dots.

The scaling functions Φ(2x − j) come from the mother scaling function Φ(2x) centred on the points of abscissa j∕2 which are on a finer mesh. The wavelet families that we use have a compact support, they have non-zero values only in a given interval [−m, m], but non-compact wavelet families also exist [46]. This is particularly interesting because all the operations can be applied only in this interval. This is the reason why we call this basis set a localized basis set in real space.

Using orthogonal wavelets, as we did, means that we only manipulate the two filters {hj}, {gj} which are linked by the scaling relations. These filters can decompose any function into scaling and wavelet functions at different levels of resolution.

The Daubechies wavelets [15, 17] are an important family of orthogonal wavelets. To develop any function f(x) into Daubechies scaling functions, we apply the scalar product

(11.12)

to obtain an approximation f₀(x) of the given function.

This property of orthogonality is especially interesting when expressing the electronic orbitals for each electron state avoiding the use of an overlap matrix. In this case we don’t have to solve a generalized eigenvalue problem HX = ESX to find the Kohn–Sham orbitals.

The number of continuous derivatives of the Daubechies functions, i.e. their smoothness, which is well described by the Sobolev spaces, depends strongly on the wavelet order. The Haar function is a Daubechies function of order 1 which is piecewise continuous.

BigDFT uses the least asymmetric Daubechies wavelet family of order m = 8 (see Fig. 11.1) for the electronic orbitals with two levels of resolution.

A homogeneous grid is created with a constant step grid h. For each atom type, two radii are defined: one, rc, gives the extension of the mesh around each atom of the same nature, the second one, rf, gives the extension where the electronic orbitals have one coefficient for the scaling function per point but also other ones for the corresponding wavelet functions.

To express the electronic density and the potentials, we prefer to use biorthogonal scaling functions (see Sect. 11.4.1). After some clues about wavelets, we come back to a formal definition of multi-resolution analysis following the notation from the book of S. Goedecker [31].

11.3.2 Definition of Multi-Resolution Analysis

We can introduce many levels of resolution using the full possibility of multi-resolution analysis (MRA). We here sketch some notions of MRA. A k ( $$k \in \mathbb {N}$$ ) factor is used to specify the resolution level and i ( $$i \in \mathbb {Z}$$ ) corresponds to the centre of the Daubechies basis function. We define

(11.13)

Using the bra and ket notation, we can write the orthogonality conditions as

left angle bracket upper Phi Subscript i Superscript left parenthesis k right parenthesis Baseline vertical bar upper Phi Subscript j Superscript left parenthesis k right parenthesis Baseline vertical bar upper Phi Subscript i Superscript left parenthesis k right parenthesis Baseline vertical bar upper Phi Subscript j Superscript left parenthesis k right parenthesis Baseline right angle bracket equals delta Subscript i j Baseline semicolon left angle bracket upper Phi Subscript i Superscript left parenthesis k right parenthesis Baseline vertical bar upper Psi Subscript j Superscript left parenthesis k Super Superscript prime Superscript right parenthesis Baseline vertical bar upper Phi Subscript i Superscript left parenthesis k right parenthesis Baseline vertical bar upper Psi Subscript j Superscript left parenthesis k Super Superscript prime Superscript right parenthesis Baseline right angle bracket equals 0 semicolon left angle bracket upper Psi Subscript i Superscript left parenthesis k right parenthesis Baseline vertical bar upper Psi Subscript j Superscript left parenthesis k Super Superscript prime Superscript right parenthesis Baseline vertical bar upper Psi Subscript i Superscript left parenthesis k right parenthesis Baseline vertical bar upper Psi Subscript j Superscript left parenthesis k Super Superscript prime Superscript right parenthesis Baseline right angle bracket equals delta Subscript i j Baseline delta Subscript k k Sub Superscript prime Subscript Baseline comma

(11.14)

We then introduce the vector spaces spanned by Φ(k) and Ψ(k), respectively:

script upper V Subscript k Baseline identical to span left brace vertical bar upper Phi Subscript i Superscript left parenthesis k right parenthesis Baseline right angle bracket right brace comma script upper W Subscript k Baseline identical to span left brace vertical bar upper Psi Subscript i Superscript left parenthesis k right parenthesis Baseline right angle bracket right brace comma

(11.15)

with $${\mathcal V}_k \cap {\mathcal W}_k = \{0\}$$ , from the orthogonality conditions. The scaling relations are valid for any level k

script upper V Subscript k Baseline identical to span left brace vertical bar upper Phi Subscript i Superscript left parenthesis k right parenthesis Baseline right angle bracket right brace comma script upper W Subscript k Baseline identical to span left brace vertical bar upper Psi Subscript i Superscript left parenthesis k right parenthesis Baseline right angle bracket right brace comma

(11.16)

which means that $${\mathcal V}_{k} = {\mathcal V}_{k-1} \oplus {\mathcal W}_{k-1}$$ , or stated otherwise, that the wavelet spaces $$\mathcal {W}_k$$ are the orthogonal complement of $${\mathcal V}_k$$ in the space $${\mathcal V}_{k+1}$$ . Thus we define a sequence of nested sets

(11.17)

For all $$k \geqslant 0$$ , the space of square integrable functions $$L^2(\mathbb {R})$$ can be decomposed into an infinite sum of spaces:

(11.18)

This can be generalized into biorthogonal wavelet families with two nested sets, one for the direct space and another one for the dual space.

The relation (11.18) shows that any square integrable function can be uniquely decomposed as

(11.19)

In contrast to BigDFT which uses only two levels of resolutions, two groups have developed multi-resolution basis sets based on multiwavelets [33, 34, 40]. They can achieve very accurate results which are very useful for reference calculations to decouple the different approximations coming from the basis sets, the choice of algorithms, or which are inherited from the used methods such as the DFT functional. The high level of resolution is only used near the nuclei to describe the oscillations and the nuclei cusp of each electronic orbital.

11.3.3 Daubechies Wavelets

In the BigDFT code, the least asymmetric Daubechies wavelet family of order m = 8 is the basis set used to describe the KS orbitals as shown in Fig. 11.1. These functions feature a compact support [−m + 1, m] and are also localized in Fourier space. The use of Daubechies wavelets families is guided by different criteria. Daubechies wavelets represent the best compromise between compact support, continuity and orthogonality for a wavelet family. We chose the family of order 16 as it is the most compact one which has a degree of continuity of at least two, thereby enabling a unbiased evaluation of the kinetic operator. Nonetheless, such a family exhibits polynomial exactness of degree 8, which means that it is able to represent exactly the Taylor expansion of a Kohn Sham orbital up to the eighth order. This observation, combined with the Magic Filter method (see [23]), enables an accurate and efficient approach for the evaluation of the potential energy in KS-DFT calculations.

The coefficients of the filters hj and gj = (−1)jh−j of the scaling relation (or “refinement relations”) for the Daubechies families are called low- and high-pass filters, respectively. A wavelet family is therefore completely defined by its low-pass filter. In the case of Daubechies-2m wavelets, j ∈ [1 − m, m].

A line graph of lease asymmetric Daubechies 16 for scaling function and wavelet. Both lines begin and end at the same points. They fluctuate between negative 3 and 3 on the x-axis. The scaling function has a maximum peak at (0, 1.2). The wavelet function has 2 peaks and 1 trough.

Least asymmetric Daubechies wavelet family of order m = 8. Note that both the scaling function Φ(x) and the wavelet Ψ(x) are different from zero only within the interval [1 − m, m]

11.3.4 One-Dimensional Operators with Daubechies Wavelets

The multiresolution property also plays a fundamental role in the wavelet representation of differential operators. For example, it can be shown that the exact matrix elements of the kinetic operator can be written in the form of a circulant matrix, namely:

(11.20)

and are equal to the entries of an eigenvector of a matrix which solely depends on the low-pass filter (see e.g. [30]).

Daubechies-2m wavelets exhibit m vanishing moments, thus any polynomial of degree less than m can be represented exactly by an expansion over the sole scaling functions of order m. For higher order polynomials the error is $$\mathcal {O}(h^m)$$ , i.e. vanishingly small as soon as the grid is sufficiently fine. Hence, the difference between the discretized representation and the exact function f decreases as hm. The discretization error due to Daubechies-2m wavelets is therefore controlled by the grid spacing. Among all the orthogonal wavelet families, Daubechies wavelets feature the minimum support length for a given number of vanishing moments.

Given a potential V  known numerically on the points {xk} of a uniform grid, it is possible to identify an effective approximation for the potential matrix elements upper V Subscript i j Baseline identical to left angle bracket upper Phi Subscript j Baseline vertical bar upper V vertical bar upper Phi Subscript i Baseline right angle bracket ). It has been shown [23, 55] that a quadrature filter {ωk} can be defined such that the matrix elements given by

upper V Subscript i j Baseline identical to left angle bracket upper Phi Subscript j Baseline vertical bar upper V vertical bar upper Phi Subscript i Baseline right angle bracket equals sigma summation Underscript k Endscripts omega Subscript k minus i Baseline upper V left parenthesis x Subscript k Baseline right parenthesis omega Subscript k minus j

(11.21)

yield excellent accuracy with the optimal convergence rate $$\mathcal O(h^{2m})$$ for the potential energy. The same quadrature filter can be used to express the grid point values of a (wave)function given its expansion coefficients in terms of scaling functions:

As a result, the potential energy can equivalently be computed either in real space or in the wavelet space, i.e. left angle bracket f vertical bar upper V vertical bar f right angle bracket asymptotically equals sigma summation Underscript k Endscripts f left parenthesis x Subscript k Baseline right parenthesis upper V left parenthesis x Subscript k Baseline right parenthesis f left parenthesis x Subscript k Baseline right parenthesis asymptotically equals sigma summation Underscript i j Endscripts c Subscript i Baseline upper V Subscript i j Baseline c Subscript j ). The quadrature filter elements can therefore be considered as the most reliable transformation between grid point values f(xk) and scaling function coefficients ci, as they provide exact results for polynomials of order up to m − 1 and do not alter the convergence properties of the basis set discretization. The filter {ωk} is of length 2m and is defined unambiguously by the moments of the scaling functions (which in turn depend only on the low-pass filter) [30].

Using the above formulae, the (so far one-dimensional) Hamiltonian matrix Hij = Tij + Vij can be constructed. Note that, in contrast to other discretization schemes (finite differences, plane waves etc.), in the wavelet basis set neither the potential nor the kinetic terms have diagonal representations. Instead, is represented by a band matrix of width 2m. We will discuss the mathematical considerations beyond these arguments in the forthcoming section. First, we will introduce the discretization we will employ for the three-dimensional domain.

11.3.5 Three-Dimensional Wavelet Basis

A support function φα(r) can thus be expanded in the wavelet basis as follows:

StartLayout 1st Row 1st Column phi Subscript alpha Baseline left parenthesis bold r right parenthesis 2nd Column equals sigma summation Underscript i 1 comma i 2 comma i 3 Endscripts s Subscript i 1 comma i 2 comma i 3 Baseline upper Phi Subscript i 1 comma i 2 comma i 3 Superscript left parenthesis 0 right parenthesis Baseline left parenthesis bold r right parenthesis 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript j 1 comma j 2 comma j 3 Endscripts sigma summation Underscript l equals 1 Overscript 7 Endscripts d Subscript j 1 comma j 2 comma j 3 Superscript left parenthesis script l right parenthesis Baseline upper Psi Subscript i 1 comma i 2 comma i 3 Superscript left parenthesis script l right parenthesis Baseline left parenthesis bold r right parenthesis period EndLayout

(11.22)

Here $$\Phi ^{(0)}_{i_1,i_2,i_3}(\mathbf {r})=\Phi (x-i_1) \Phi (y-i_2) \Phi (z-i_3)$$ denotes the tensor product of three one-dimensional scaling functions, whereas $$\Psi ^{(\ell )}_{j_1,j_2,j_3}(\mathbf {r})$$ are the seven tensor products containing at least one one-dimensional wavelet. The sums over i₁, i₂, i₃ (j₁, j₂, j₃) run over all grid points where the scaling functions (wavelets) are centered.

In other words, the three-dimensional basis functions are a tensor product of one-dimensional basis functions, a mixed basis set of scaling functions augmented by a set of 7 wavelets. These points are associated with regions of low and high resolution levels, respectively. Note that we are using a cubic grid, where the grid spacing is the same in all directions, but the following description can be straightforwardly applied to general orthorhombic and non-orthorhombic grids.

In a simulation domain, there are therefore three regions: those which are closest to the atoms (“fine region”) carry one (three-dimensional) scaling function and seven (three-dimensional) wavelets; those which are further from the atoms (“coarse region”) carry only one scaling function, corresponding to a resolution which is half that of the fine region; and those which are even further away (“empty region”) carry neither scaling functions nor wavelets. To determine these regions of different resolution, we construct two spheres around each atom a; a small one with radius $$R_a^f=\lambda ^f\cdot r_a^f$$ and a large one with radius $$R_a^c=\lambda ^c\cdot r_a^c$$ ( $$R_a^c&gt;R_a^f$$ ). The values of $$r_a^f$$ and $$r_a^c$$ are fixed for each atom type, whereas λf and λc can be specified by the user in order to control the accuracy of the calculation. The fine (coarse) region is then given by the union of all the small (large) spheres, as shown in Fig. 11.2. Hence in BigDFT the basis set is controlled by three user specified parameters; systematic convergence of the total energy is achieved by increasing the values of λc and λf while reducing the value of h.

A chemical bond illustration of a simulation grid. Two hexagonal bonds are joined together with an atom attached to the other 8 free atoms. Each atom is covered by small and large circular areas to form a floral pattern.

Fig. 11.2

Example simulation grid for a molecule with coarse (fine) grid points depicted in (gold) blue

We have all the tools to express any 3D functions as Kohn–Sham orbitals in a 3D wavelet basis set. Our goal is to use the properties of the compactness of the wavelets to build a linear-scaling structure electronic code. To do this, we need to introduce the concept of localization regions.

11.3.6 Localization Regions

Thanks to the nearsightedness principle, it is possible to define an approach in which the computational cost is linear scaling (LS) with respect to the number of atoms, N, rather than the cubic scaling (CS) which arises when extended KS orbitals are used. Such approaches allow one to go beyond the treatment of a few hundred atoms as is typically seen with $$\mathcal O(N^3)$$ DFT approaches and instead treat systems containing several thousands of atoms. This has the benefit of also opening up the treatment of new types of materials and simulations using a pure quantum mechanical approach, as discussed, for example, in Ratcliff et al. [60] and Zaccaria et al. [65]. The $$\mathcal O(N)$$ formalism implemented in the BigDFT code exploits the possibility that, for systems with suitable electronic structure, the support functions φ can be optimized while preserving their strict locality, namely so that their support is within a pre-defined localization region. A similar approach is for example used in the ONETEP [61] and Conquest [9] codes.

For large systems where the nearsightedness principle guarantees that a local description of the orbitals is possible, the large number of degrees of freedom offered by the wavelet basis is a waste. It is therefore advantageous to build a minimal basis formed of localized (e.g. atom centered) functions. Of course, these functions will also be expanded in terms of the underlying wavelet basis, but to strictly impose locality, they will be expressed only in a subset of this global basis set. To do so we set to zero all scaling function and wavelet coefficients if they lie outside of a sphere with radius Rloc around the point Rα on which the function is centered:

(11.23)

Here $${\mathbf {R}}_{(i_1,i_2,i_3)}$$ is the position of the grid point (i₁, i₂, i₃) and Rα that of the atom on which the minimal basis function φα(r) is centered. These localization regions can still contain various resolution levels, as they are constructed on top of the global simulation domain. The index α is instead used in the following formulae to label kets that are associated to SFs.

In other terms, instead of working directly with the function |φα〉, we work with the localized function $$\mathcal L^{(\alpha )} | \varphi _\alpha \rangle $$ , where the definition of the localization projector operator in the Daubechies basis space is, as described:

(11.24)

from which it becomes apparent that such a projection operator $$\mathcal L^{(\alpha )}$$ explicitly depends on the localization radius Rloc and the localization region center Rα. Clearly, if |φα〉 is localized around Rα and Rloc is large enough, $$\mathcal L^{(\alpha )}$$ leaves |φα〉 unchanged and no approximation is introduced to the KS equations.

It is important to emphasize that, since the Daubechies basis set is independent of Rα, |φα〉 depends on the center of the localization region by the introduction of the projector $$\mathcal L^{(\alpha )}$$ :

(11.25)

By taking the derivative of this equation with respect to Rβ it is easy to find

(11.26)

Let us now employ this result in the calculation of the atomic forces. When the KS orbitals are expressed in terms of the SFs, the non-Hellmann–Feynman term can be written as follows:

(11.27)

where the SF residue is

StartLayout 1st Row 1st Column phi Subscript alpha Baseline left parenthesis bold r right parenthesis 2nd Column equals sigma summation Underscript i 1 comma i 2 comma i 3 Endscripts s Subscript i 1 comma i 2 comma i 3 Baseline upper Phi Subscript i 1 comma i 2 comma i 3 Superscript left parenthesis 0 right parenthesis Baseline left parenthesis bold r right parenthesis 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript j 1 comma j 2 comma j 3 Endscripts sigma summation Underscript l equals 1 Overscript 7 Endscripts d Subscript j 1 comma j 2 comma j 3 Superscript left parenthesis script l right parenthesis Baseline upper Psi Subscript i 1 comma i 2 comma i 3 Superscript left parenthesis script l right parenthesis Baseline left parenthesis bold r right parenthesis period EndLayout

(11.28)

This result would be completely identical to Eq. (11.9) when no localization projectors are applied on the SF. Therefore the only term of the forces which cannot be captured by the localization regions is the part which is projected outside the localization regions (but still inside the computational domain of the CS approach). The extra Pulay term due to the localization constraint is therefore

(11.29)

From Eq. (11.26) we obtain

(11.30)

and from Eq. (11.24) we derive:

(11.31)

Therefore if the support functions are zero at the border of the localization region, there is no Pulay term in the atomic forces.

The Hellman-Feynman force, given by the expression

(11.32)

involves only the functional derivative of the Hamiltonian operator, which is independent of the localization regions. The CS and the LS implementations of the atomic forces are therefore identical.

Daubechies wavelets are used to describe all terms related to KS orbitals. For the electronic density and potential, the orthogonality property is not the key parameter, but efficiently discretizing a function in a uniform real-space mesh is very important. The next section is devoted to this problem and develops tools related to the interpolating scaling function used in BigDFT.

11.4 Collocation Problems for Discretized Functions

Discretizing an analytic function on a uniform real-space grid is often done via a straightforward collocation method. This is ubiquitous in all areas of computational physics and quantum chemistry. An example in DFT is given by the external potential describing the interaction between ions and electrons. Notable examples are also given by the analytic functions defining compensation charges for range-separated electrostatic treatments.

A real space approach is mandatory in the solution of complex partial differential equation problems, as well as for the treatment of complex environments and non-trivial boundary conditions. The solution of the Poisson equation in vacuum and in the presence of continuum solvents is a notable example. In this framework, the collocation method is a straightforward procedure that is used to discretize a known function, to express its values in the real-space domain.

The accuracy of the collocation method used is therefore very important for the reliability of subsequent treatments like self-consistent field solutions of electronic structure problems. When the real-space grid is too coarse, the collocation method introduces numerical artifacts, spoiling the numerical stability of the description.

We present in this section a new quadrature scheme that is able to exactly preserve the multipoles of the original function for a wide range of grid spacings, in the spirit of the so-called “Magic-Filter” method, which has been used to identify a passage matrix between Daubechies wavelets and real-space grid meshes used to express the electronic density and potential.

For discretization on uniform grid spacings, the collocation method is well-justified when the original function can be reasonably approximated by an interpolation of its values on the grid mesh points. Let us consider a one-dimensional function f. Suppose we want to discretize this function on a uniform grid of spacing h and coordinates xk = hk. Given a family of interpolating functions {Lk(t)}, if the approximation

(11.33)

is reasonably accurate, the collocation method can be applied. This fact stems from the interpolating property of the family {Lk(t)}. Indeed, an interpolating family comprises a set of functions Lk, each one associated to a point k of the grid, such that Lk(j) = δkj. Given Eq. (11.33), then fL(xk) = f(xk) and the continuous representation of f(x) may be given by fL(x).

Given the interpolating property, it is also said that an interpolating function family is dual to the Dirac deltas. In other terms, denoting the above function by the bra-ket notations, we have

where |δk〉 represents the Dirac distribution centered at point xk, i.e. 〈δk|f〉 = f(xk). The above defined interpolating property implies that the duality relation δkℓ = 〈δℓ|Lk〉 holds.

11.4.1 Bi-orthogonal Wavelets

Examples of interpolating functions are the interpolating scaling functions used in BigDFT to represent the electronic density and potential and solve the Poisson equation in order to calculate the Hartree potential.

The interpolating scaling functions, which are an important biorthogonal wavelet family, have the delta function as dual functions, which considerably simplifies the expansion of a given function f(x):

(11.34)

There are different families of interpolating wavelets. In BigDFT, we use the Deslauriers–Dubuc scaling functions, which are the auto-correlation scaling functions of the Daubechies wavelet [4]:

(11.35)

In Fig. 11.3 we show the interpolating scaling Deslauriers–Dubuc function and its associated wavelet. Unlike the Daubechies wavelets, these functions are symmetric.

A line graph of interpolating Deslauriers Dubuc 8 for scaling function and wavelet. Both lines begin and end at the same points. They fluctuate between negative 3 and 3 on the x-axis. The scaling function has a maximum peak at (0, 1). The wavelet function has a trough at (0.5, negative 1). Values are estimated.

Fig. 11.3

Eighth order scaling and wavelet interpolating functions

We use these interpolating scaling functions in order to express the electronic density, the different potentials and to solve the Poisson equation. Because BigDFT uses the pseudopotential approximation, two levels of resolution are enough and BigDFT uses only interpolating scaling functions expressed in the same mesh as the one for the electronic orbitals but with the fine resolution in the whole simulation domain.

11.4.2 Polynomial Exactness and Discrete Multipoles

The collocation method is therefore meaningful for the functions for which the action of the projector operator ∑k|Lk〉〈δk| approaches the action of the identity operator. It is easy to understand that this condition is valid only when the grid spacing size h is considerably smaller than the typical oscillations of the function |f〉 we want to represent. As soon as this is not the case, the function |fL〉 becomes so different from |f〉 that the numerical accuracy of the approximation is severely affected. To have an idea of how rapidly the accuracy of this approximation is spoiled, in Fig. 11.4 we consider the collocation of a Gaussian function centered in x₀ and with standard deviation σ. When the ratio h∕σ becomes bigger than one, the interpolated function fL given by Eq. (11.33) becomes too crude an approximation. This is particularly visible for a localized function centered between two grid points. As the function becomes too sharp, the collocated values are nearly zero and the function fL is not representative of the original f. This loosening of the accuracy can be easily quantified by having a look at the multipoles of the discretized functions.

Two line graphs of Gaussians function for different sigma values and relative errors. Graph 1 has 5 single peak curves for sigma values 0.1, 0.3, 0.5, 0.7, and 1. The width becomes narrow as the values decrease. Graph 2 of relative error versus h over sigma plots 9 concave-down increasing and decreasing trends for p = 0 to 8.

Fig. 11.4

Collocation of Gaussians of different standard deviation σ on a grid of spacing h = 1. All Gaussians are centered in points which lie between two grid spacings. In the left panel, it is easy to see that the collocated values are no longer reliable when the ratio h∕σ grows above 1. This fact can also be confirmed by the discrete multipoles of the collocated function, whose error is presented in the right panel

This situation seems unavoidable: as the expansion coefficients of the function |fL〉 are given in terms of the scalar products 〈δj|f〉, the grid has to provide a reasonable sampling of the function f.

11.4.3 Polynomial Exactness

The accuracy of the approximation (11.33) is of great importance for a reliable computational treatment. Clearly, such accuracy is intimately related to the family of interpolating functions chosen.

Interpolating function families are normally constructed using families of polynomial functions. An interpolating family {Lk(x)} is said to be of order m if any monomial function xp, with 0 < p < m, is exactly expressed by the interpolated function fL of Eq. (11.33). This is the concept of polynomial exactness. This concept is important in determining the accuracy of the interpolation: a smooth function can reasonably be approximated by its Taylor polynomial around a given point. The higher the order of the polynomial exactness of the functions Lk, the better the Taylor expansion of the original function would be approximated by the function fL(x), therefore the difference

will be reduced. Let us now demonstrate that polynomial exactness of order m implies the moment preserving property. Polynomial exactness implies that, for a monomial function xp, indicated by |p〉,

Therefore for all positive integers p, q such that p + q < m we have

and this is valid only if

(11.36)

The above quantity is the q-moment (or multipole) of the interpolating function Lj(x). Polynomial exactness therefore implies that the first m multipoles of the interpolated function |fL〉 can actually be calculated exactly with their discrete multipoles: Indeed, by Eq. (11.36), we can express the moments of the function fL in terms of discrete moments of the collocated function:

As |f〉 is an analytic function, the accuracy of the multipoles Mp[fL] provides a quantitative evaluation of the accuracy of fL. In Fig. 11.4 we see that the collocation method becomes unstable as soon as the grid spacing is bigger than the typical oscillations of the original function.

Let us now see if it is possible to define an alternative set of dual functions, such that the multipoles of the original functions are preserved. In other terms, we search for a family of dual functions such that

It is easy to see that this condition can be obtained by imposing the polynomial exactness of the dual set $$\widetilde L_k$$ . In other terms, if the set of $$\widetilde L_k$$ is such that

then the multipole preserving property is guaranteed.

A particular family of interpolating functions is given by the Interpolating Scaling Function (ISF):

Indeed the ISF basis $$\{\varphi _j\}_{j \in \mathbb Z}$$ contains precisely the polynomials up to order m − 1. This means that this basis is, in principle, able to express the Taylor expansion of the original function f. However, this does not happen when extracting the coefficients with the collocation method, as can be seen from Fig. 11.4. Therefore, there should exists a set of coefficients fj such that

In the above discussion we have proceeded as if we have defined the basis of |φj〉 as a self-dual basis, modulo an error of $$\mathcal O(h^m)$$ . This is indeed a general property of all interpolating functions. In the following we will demonstrate that any interpolating function L(t) which has a Taylor expansion is orthogonal to the ISF family up to $$\mathcal O(h^{m})$$ . To demonstrate this point, let us define the Taylor expansion of L(t) =∑kαktk. We have

This results stems from three factors: the interpolating property of L(t), the fact that L can be defined (at least up to $$\mathcal O(h^m)$$ ) by its Taylor expansion on its support χ(φ) and the vanishing moment property of φ. Therefore, if L can be exactly defined in terms of polynomials of order less than m, the duality relation is preserved exactly. Under these hypotheses, let us now consider what would happen to the evaluation of the discrete moments Mp[f] when using such basis $$\left \{L_k\right \}$$ as a dual basis. In order to do that, let us redefine the polynomial expansion of the function as $$L_j(x)= \sum _{p=0}^m \beta _{j,p} x^p$$ on its support. This can easily be done by considering

In this way we obtain

where we have split the integral over the support of Lj in chunks of size of one grid spacing. With this relation we have

If the condition

(11.37)

held for all i, then we would have

and the discrete and the continuous moments would coincide.

11.5 Multipole-Preserving Collocations for Daubechies Wavelets: The Magic Filter

Let us now apply the concepts of the above section to functions which are discretized in this basis set. Given a set of 2m-family Daubechies scaling functions Φ centered in a uniform mesh of spacing h, the expansion coefficients of a given function f(x) in this set are defined as

(11.38)

Here we have defined the normalisation factor of the basis set in order to preserve the equivalence $$|| f ||{ }_2 = \sum _i c_i^2$$ , if the function f is exactly represented. Given the momentum-preserving property of Daubechies scaling functions, the discretization of the function f in this basis set has an algebraic hm convergence rate. In other terms

A wavelet quadrature in this context is based on the idea of approximating the expression above by a collocation formula. In other terms, we should define some coefficients wi, where i = 1 − m, …, m such that

(11.39)

This will be possible only if the above formula gives the exact result when f(x) = xp, p = 0, …, 2m − 1. By comparing (11.38) and (11.39) in the case of polynomials we thus find the equation defining the Magic Filter:

which is solved for all i if and only if

(11.40)

which can be written in matricial form A ⋅w = M, where the matrix Ak,j = jk is the Vandermonde matrix, which has an inverse written in terms of Lagrange polynomial coefficients in the basis of monomials:

therefore, given $$\sum _p \bar A_{k,j} A_{k,i} = \delta _{i,j}$$ , the Magic Filters are given by Neelov and Goedecker [55, Eq. (10)]:

(11.41)

This equation shows that the Magic Filter can be viewed as the expansion coefficients of Daubechies scaling functions in the basis of Lagrange polynomials. Johnson et al. [41] can be used as a reference in this regard.

11.5.1 From Magic Filters to a Passage Matrix

We proceed as if we had found a recipe to interpolate the values of the scaling function $$\Phi (j)\simeq \widetilde \Phi (j) = w_j$$ such that the collocation formula

would apply exactly for functions f which can be expressed by polynomials of order less than 2m. This is a very interesting feature to exploit when it is desirable to calculate the collocation values of a functional of f, for instance the charge density, or the exchange and correlation potential.

Let us explore this concept in more detail and start with the following definition of equivalence class. Two functions f and g are considered as equivalent (denoted f ⇔ g) if they have the same expansion coefficients in the given Daubechies scaling function basis:

It follows from the above properties that $$f-g = \mathcal O (h^m)$$ . These functions therefore have the same representation in the scaling function basis set, provided by the coefficients {ci} defined in (11.38). The function

is by definition equivalent to f. Let us now consider the moments of the function Cup to order 2m − 1, and suppose we would like to find a set of quadrature coefficients Cj so that the following holds:

The values of the coefficients Cj can be easily related to the Magic Filters. Indeed

and by using Eq. (11.40) we have

Therefore we have found $$C_j=\sum _{j'} c_{j'} w_{j-j'}$$ . The same reasoning could have been applied if the moment had been centered at another grid point.

Let us now consider the polynomial function which has Cj coefficient as collocation values. This function is (locally) expressed as a linear combination of Lagrange polynomials, which have the well-known interpolating property.

By definition, the moments of this function written in collocation coincide with $$M^C_p$$ . Let us now consider the expansion coefficients of D(x) in the scaling function basis set. For simplicity we omit the grid spacing, i.e.

Moreover, by using the result of Eq. (11.40), together with the orthogonality of Lagrange polynomials, the coefficients di can be calculated from Ci by applying the following convolution:

It can be shown, see Johnson et al. [41] and Neelov and Goedecker [55, Appendix A], that for Daubechies scaling functions, $$\sum _j^{\prime } w_{i-j^{\prime }} w_{j^{\prime }-j}=\delta _{i,j}$$ . Therefore in this case we have proved that dj = cj.

This not only means that D(x) :⇔ C(x), but also that we have defined a prescription that is more precise than the collocation of C, as the difference between D and C is of order h2m. It is as if the Magic Filter prescription chooses among the functions belonging to the same equivalency class as C(x), the collocation values of D(x), which is the smoothest possible function preserving the moments of C.

The convolution with the Magic Filters, direct or transposed, acts therefore as a Passage Matrix between the Daubechies scaling function basis and the “collocation basis” provided by the Lagrange interpolating polynomials. The interpolating property of the Deslauriers–Dubuc interpolating scaling function (which is the autocorrelation of two Daubechies scaling functions) plays a key role in this property.

11.6 Representing the Charge Density on Real Space: An ISF Poisson Solver

The Magic Filter method makes it possible to deal with real-space quantities with a very high precision and without the need to calculate complicated numerical quantities. The multipole-preserving property of this method makes it ideal for the treatment of electrostatic problems where the preservation of moments (e.g. multipoles) are important.

We illustrate the basis of our approach for the calculation of terms related to the Poisson equation. In Eq. (11.6), the Hartree potential VH[ρ] depends on the charge density from the Poisson equation, which in atomic units, in vacuum, reads ∇²V = −4πρ. Having efficient algorithms to solve the Poisson equation is therefore essential. The large variety of situations in which this equation can be found requires us to face this problem with different choices of boundary conditions (BCs) in mind. The long-range behaviour of the inverse Laplacian operator makes this problem strongly dependent on the BCs of the system.

In Genovese et al. [24, 25] and Cerioni et al. [10], a novel method for solving the screened and unscreened Poisson equation in vacuum with free, fully periodic, surface-like and wire-like BCs was presented, including non-orthorhombic cells. Such a method is direct (rather than iterative) in that the solution along the isolated directions is found in its integral form using the Green’s function method. For instance, in the case of a fully isolated (or “cluster-like”) system,

(11.42)

with the Green’s function being $$G(\mu _0; r) \equiv \frac {\mathrm{e} ^{-\mu _0 r}}{4 \pi r}$$ . Homogeneous Dirichlet BCs (V = 0 at $$|\mathbf {r}|\rightarrow \infty $$ ) along the isolated directions are explicitly enforced by the selection of the Green’s function.

The method has been in use for a few years in a number of ab initio codes (see the references cited in Cerioni et al. [10]) and has proven to be highly efficient and accurate in every application attempted to date. It is based on a representation of ρ and V  in terms of interpolating scaling functions (ISFs), which allows any sort of periodicity to be modelled in the most natural, clean and mathematically rigorous way. ISFs—arising in wavelet theory [30]—enjoy several properties which make them superior to other basis sets. For instance, the representation in terms of m-th order ISFs make the first m moments of the continuous and discrete charge distributions coincide [23]. As a consequence the representation is definitely faithful (more than just convenient), since the different moments of the charge distribution capture the major features of the potential. Moreover, ISFs are genuinely localized due to their compact support (the length of which is equal to 2m) and endowed with the refinement relations which easily allow for switching from a representation on a grid with spacing h to a doubly refined grid with spacing h∕2.

The inclusion of such functionalities is motivated by the strong theoretical, experimental and technological interest in the characterization of nanostructured materials, since solving the Poisson equation is only one of the many steps involved in state-of-the-art computer simulations and is repeated several times. Moreover, in the context of KS-DFT and extensions thereof, there are quantities which are computed via convolution integrals very similar to that in Eq. (11.42): for instance, the exact exchange term arising within those generalizations of KS-DFT employing orbital-dependent or hybrid functionals (see Ratcliff et al. [58] and references therein), or the coupling-matrix in time-dependent DFT (TDDFT) [54]. In this respect, the electrostatic problem of concern here provides the paradigm for many other computations, even well beyond the scope of electrostatics.

11.6.1 The Soft-Sphere Implicit Solvation Model

This high-degree of flexibility makes the BigDFT Poisson solver library optimal for calculations of polarized systems or systems with non-Born–von Karman boundary conditions, such as material surfaces and isolated molecules. The computational study of matter in various environments is a continuously growing field in solid state physics and chemistry. Systems of interest are, for instance, molecules, clusters or surfaces in contact with solvents [13]. An alternative to the explicit inclusion of a wet environment is to give an implicit description, while still treating the other parts of the system explicitly on an atomic quantum level [2]. Such an explicit/implicit treatment requires three main ingredients:

1.

a dielectric cavity represented by a proper function 𝜖(r) mimicking the surrounding solvent of a solute as a continuum dielectric;

2.

a solver for the generalized Poisson equation [21]

(11.43)

where φ(r) is the potential generated by a given charge density ρ(r);

3.

a model for the non-electrostatic terms to the total free energy of solvation.

The dielectric function 𝜖(r) has to take the value of 𝜖₀ = 1 where the solute is placed to solve a vacuum-like quantum problem, and the bulk dielectric constant 𝜖 outside.

The “soft-sphere” model developed by Fisicaro et al. [20] and implemented in BigDFT improves upon previous solvation approaches (see e.g. Tomasi’s method [63]). Model features are: accurate forces and a numerical cost comparable to standard vacuum calculations; feasible extensive potential energy surface (PES) explorations; a small number of model parameters; exact treatment of molecular or slab-like geometries; and the ability to treat neutral and charged molecules simultaneously in order to tackle complex interfaces (e.g. a double layer).

The interface between the quantum-mechanical solute and the surrounding environment is described by a fully continuous permittivity built up with atomic-centered “soft” spheres. This approach combines many of the advantages of the self-consistent continuum solvation model [1] in handling solutes and surfaces in contact with complex dielectric environments or electrolytes in electronic-structure calculations. In addition, it is able to describe accurately both neutral and charged systems.

We developed, tested and implemented within the BigDFT suite a solver for the generalized Poisson (Eq. 11.43) and the Poisson–Boltzmann equations to treat neutral and ionic solutions, respectively [21]. The solver for the solution of the generalized Poisson equation and the linear regime of the Poisson–Boltzmann is based on a preconditioned conjugate gradient scheme. It allows for the iterative solution of the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure solves the non-linear Poisson–Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of free, slab and wire-like boundary conditions.

The continuous function, describing the variation of the permittivity, allows for the analytic computation of the non-electrostatic contributions to the solvation free energy that are described in terms of the quantum surface. The capability of treating arbitrary molecular or slab-like geometries as well as charged molecules is key to tackling electrolytes within mixed explicit/implicit frameworks. Within the soft-sphere model two parameters are sufficient to give a mean absolute error of only 1.12 kcal/mol with respect to the experimental aqueous solvation energies for a set of 274 neutral solutes. For charged systems, the same set of parameters provides solvation energies for a set of 60 anions and 52 cations with an error of 2.96 and 2.13 kcal/mol, respectively, improving upon previous values in the literature.

The soft-sphere model has already been applied to the study of molecular doping of silicon [57], the interface of fluorite terminations with water [22] and the investigation of wet environment effects for ethanol and water adsorption on anatase TiO₂ (101) surfaces [19].

11.6.2 (Exact) Exchange and Correlation Terms

For a collinear-spin formalism, the calculation of the exact exchange energy EX requires a double summation over all the N occupied orbitals

(11.44)

where we have defined $$\rho _{ij}^\sigma (\mathbf {r}) = \psi _{j,\sigma }^*(\mathbf {r}) \: \psi _{i,\sigma }(\mathbf {r})$$ . The diagonal (i = j) contribution to EX exactly cancels out the Hartree electrostatic energy EH[ρ]. The action of the Fock operator to be added to the KS Hamiltonian directly stems from the EX definition:

(11.45)

where we have defined

(11.46)

which is the solution of the Poisson equation $$\nabla ^2 V_{ij}^\sigma = -4 \pi \rho _{ij}^\sigma $$ . In a KS-DFT code which searches for the ground state orbitals, during the SCF procedure, for a given set of ψi,σ(r), one has to repeatedly evaluate the value of EX as well as the action of the corresponding Fock operator on the entire set of occupied orbitals.

11.7 The BigDFT Code Approach for Ground-State

The BigDFT code may therefore express the solution of the KS problem in two ways. The traditional approach, which has a computational overhead that scales cubically with the number of atoms in the system and therefore called the cubic scaling algorithm, expresses the KS orbitals Ψi directly in a wavelet basis. In this case only the KS optimization loop is needed, and no localization projection operator $$\mathcal L_\alpha $$ is considered. The orbitals are directly labelled by their index i.

The linear scaling (LS) approach in BigDFT instead consists of two optimization loops, as depicted by the flowchart in Fig. 11.5. The SF and kernel optimization loops are independent of each other, with the number of iterations, convergence criteria etc. specified independently.

A flow chart of the high-level algorithm. The atomic orbitals optimize S Fs and K. After optimization of k, it either returns to optimize S Fs or generates energy and forces.

Fig. 11.5

Flowchart summarizing the high-level algorithm used in LS-BigDFT

Although additional approximations are introduced in the LS approach compared to the extended KS orbitals used in the CS approach, excellent agreement between total energies and forces calculated with the LS and CS approaches has been demonstrated for a range of materials and system sizes [51, 52].

Furthermore, systematic convergence remains possible—as the value of Rloc is increased, both the total energy and forces converge towards the CS result. For example for a fullerene molecule it was shown that for SF radii of 7.4 Å the total energies agree to within 0.1 meV, while the forces show better than 1 meV/Å agreement [51]. Thus, in addition to the wavelet basis parameters, the user should take care to ensure the localization radii (which may be varied independently for different atomic species) are large enough for the required accuracy. In some cases it may also be desirable to increase the number of SFs per atom, although in the majority of cases a minimal basis is sufficient, e.g. 4 SFs per C/N/O atom, 1 SF per H atom.

Aside from the basic SF parameters, a number of additional options are available in LS-BigDFT to allow for additional flexibility, such as whether or not to impose orthogonality on the SFs. Given the number of additional parameter choices compared to the CS approach, it is worth asking to what extent the choice of computational parameters depends on the system in question. Where a very high accuracy or optimal parallel performance are required it is important to carefully converge with respect to parameters such as the SF radii. Nonetheless we have demonstrated that a common set of parameters may be used to achieve consistent accuracy and robust performance across a wide range of systems [52]. Such parameters might easily be accessed by using the appropriate input profile. These profiles have been defined to give a suitable set of parameters for a number of common use cases.

11.7.1 The Direct Minimization Approach

We have seen that the support functions or the KS orbitals are represented in a Daubechies wavelet basis and are therefore susceptible to be optimized in the LS and CS algorithm respectively. We present in the following the main algorithm which is employed for their optimization.

Let us now impose that the wavefunctions have to be orthonormal with respect to a certain Hermitian metric operator (not to be confused with the SF overlap matrix), that is . For norm-conserving PSPs, frozen core and/or all-electron calculations is the identity operator, whereas it is a non-trivial quantity for projector augmented wave (PAW) [8] and ultrasoft PSPs. As in the previous sections, we denote by λ a multi-index labelling the system (indicating for instance the coordinates of the atomic positions).

Orthogonality is imposed via the following Lagrangian

As always for Lagrange multiplier techniques, criticality with respect to Λ leads to the orthogonality constraint. The coefficients Λij form a Hermitian matrix, due to the Hermiticity of . Criticality of this Lagrangian under variation of the KS orbitals 〈ψi| leads to the equation:

(11.47)

Multiplying Eq. (11.47) by 〈ψk| leads to the condition on the Hermitian part of upper Lamda Subscript i k Baseline equals one half left parenthesis f Subscript i Baseline plus f Subscript k Baseline right parenthesis left parenthesis left angle bracket psi Subscript k Baseline vertical bar ModifyingAbove upper H With caret Subscript KS Baseline psi Subscript i Baseline right angle bracket plus left angle bracket psi Subscript i Baseline vertical bar upper H Subscript KS Baseline psi Subscript k Baseline vertical bar psi Subscript i Baseline vertical bar upper H Subscript KS Baseline psi Subscript k Baseline right angle bracket right parenthesis ). Convergence is achieved when the mean of the norms of the Lagrangian derivatives is below a user-defined numerical tolerance.

11.7.2 Support Function Optimization

We illustrate the main guidelines for the optimization of the support functions. In principle, the optimization of KS orbitals in the basis of the support functions should minimize the total energy. This is in fact equivalent to minimizing the band structure energy, assuming a fixed density matrix and Hamiltonian, i.e. the following functional

StartLayout 1st Row min Underscript phi Subscript alpha Baseline Endscripts sigma summation Underscript alpha comma beta Endscripts upper K Superscript alpha beta Baseline left angle bracket phi Subscript alpha Baseline StartAbsoluteValue ModifyingAbove upper H With caret Subscript KS Baseline EndAbsoluteValue phi Subscript beta Baseline right angle bracket period EndLayout

(11.48)

As discussed, the SFs used in LS-BigDFT are strictly localized (numerical) functions which are expressed in a Daubechies wavelet basis. Starting from an atomic orbital (AO) input guess, they are optimized by minimizing the target function , subject to the orthonormality condition of the KS orbitals. The operator is the sum of the density-dependent KS Hamiltonian plus a confining operator such that

StartLayout 1st Row 1st Column left angle bracket phi Subscript alpha Baseline vertical bar ModifyingAbove upper V With caret Subscript c Baseline vertical bar phi Subscript beta Baseline right angle bracket 2nd Column equals 3rd Column delta Subscript alpha beta Baseline left angle bracket phi Subscript alpha Baseline vertical bar ModifyingAbove upper V With caret Subscript c Superscript alpha Baseline vertical bar phi Subscript alpha Baseline right angle bracket comma EndLayout

(11.49)

(11.50)

We therefore have to minimize the following functional:

upper Omega equals trace left parenthesis ModifyingAbove upper F With caret ModifyingAbove upper H With caret Subscript c Baseline right parenthesis minus sigma summation Underscript alpha comma beta comma i comma j Endscripts upper Lamda Subscript i j Baseline left parenthesis c Subscript i Superscript alpha Baseline c Subscript j Superscript beta Baseline left angle bracket phi Subscript alpha Baseline vertical bar phi Subscript beta Baseline right angle bracket minus delta Subscript i j Baseline right parenthesis

(11.51)

where the coefficients Λij are determined by the relation:

sigma summation Underscript i comma j Endscripts c Subscript i Superscript alpha Baseline c Subscript j Superscript beta Baseline upper Lamda Subscript i j Baseline equals sigma summation Underscript rho sigma Endscripts upper K Superscript alpha rho Baseline left angle bracket phi Subscript sigma Baseline StartAbsoluteValue ModifyingAbove upper H With caret Subscript KS Baseline EndAbsoluteValue phi Subscript rho Baseline right angle bracket upper S Subscript sigma beta Superscript negative 1 Baseline period

(11.52)

The KS Hamiltonian $$\hat {H}_{\text{KS}}$$ does not commute with the $$\mathcal L^{(\alpha )}$$ operators. Therefore when calculating $$ \mathcal L^{(\alpha )} | \hat {H}_{\text{KS}} \varphi _\beta \rangle $$ the localization constraint has to be relaxed before applying the Hamiltonian operator. Practically, this is done as follows. When applying the Hamiltonian, the value of the cutoff radius Rloc must be increased by half of the convolution filter length times the grid spacing, corresponding to a buffer region of eight grid points around the localization region. These buffers are initialized to zero, but the convolution will result in non-zero values in those regions. When the scalar product with another basis function is evaluated, it is therefore important to keep this buffer zone. Therefore, given a set of truncation radii Rloc, the KS Hamiltonian can be explicitly evaluated within the applied truncation scheme, preserving the variationality of the result.

We impose therefore the localization condition $$|\varphi _\alpha \rangle = \mathcal L^{(\alpha )} | \varphi _\alpha \rangle $$ on the SFs. The functional to be minimized then becomes

(11.53)

where the components of the vector are the Lagrange multipliers of the constraints. The stationary condition on the functional provides the following gradient:

sigma summation Underscript i comma j Endscripts c Subscript i Superscript alpha Baseline c Subscript j Superscript beta Baseline upper Lamda Subscript i j Baseline equals sigma summation Underscript rho sigma Endscripts upper K Superscript alpha rho Baseline left angle bracket phi Subscript sigma Baseline StartAbsoluteValue ModifyingAbove upper H With caret Subscript KS Baseline EndAbsoluteValue phi Subscript rho Baseline right angle bracket upper S Subscript sigma beta Superscript negative 1 Baseline period

(11.54)

which is explicitly localized ( ). Here the gradient is expressed in terms of the orthogonalized support functions . The localization condition can therefore be imposed more easily by applying the constraint on (quasi-) orthogonal support functions, i.e. Sαβ = δαβ. This further simplifies the evaluation of the gradient. To ensure a good compromise between locality and flexibility, in general the orthogonality is not ensured strictly for the support functions, but it is inserted into the gradient to provide a search direction which optimizes the diagonality of the overlap matrix.

Such a minimization proceeds by applying the same guidelines as the direct minimization approach of Sect. 11.7.1, assuming unit values for the occupation numbers. The coefficient cα is dynamically adjusted during the basis set optimization procedure. This approach has the effect of keeping the SFs confined in their localization regions, centered on the position Rα, while reducing the KS band structure energy. Usually the position Rα of the support function α coincides with the position Ra of the atom a where φα is initially centered at the beginning of the SCF optimization procedure. To some extent this enables one to associate φα to a particular atom a.

As illustrated in Fig. 11.6, this procedure results in a set of SFs which have adapted to their local chemical environments. For a molecular calculation, we therefore obtain a minimal set of molecular orbitals that, by construction, exactly represent the occupied KS orbitals. The SF basis also has a non-zero projection into the unoccupied orbitals subspace, although in general the unoccupied KS orbitals are not expected to be well represented. Although the SFs resulting from LS-BigDFT are entirely numerical and are therefore not constrained to any particular form, even in extended systems they generally retain some resemblance to AOs, particularly by preserving the orbital quantum number, and are thus referred to as s, or p-like SFs.

A schematic representation illustrates the process flow in big D F T. An extended Kohn-Sham orbitals cubic scaling of high accuracy is localized to support functions of linear scaling of low accuracy to optimize localized adaptive support functions of linear scaling of high accuracy.

Fig. 11.6

Illustration of the different approaches in BigDFT for the example of anthracene. Shown are an extended KS orbital from the CS approach (left), selected unoptimized AOs used as a starting guess in the LS approach (centre) and selected resulting optimized SFs which constitute an accurate minimal basis in the LS approach. Note that the colours are used only to highlight different SFs, while the localization regions are spherical

11.7.3 Density Kernel Optimization

Given a recipe for optimizing the SFs, the question remains of how to find the density kernel for a given set of SFs. Three options are available in BigDFT: diagonalization, direct minimization and the Fermi Operator Expansion (FOE). The first two approaches retain explicit reference to the KS wavefunctions, while FOE works directly with the density kernel. As such, FOE is the preferred approach when strict linear scaling behaviour is required.

The first approach is straightforward and uses standard linear algebra routines in LAPACK, or optionally SCALAPACK, to solve the generalized eigenproblem defined by the SF Hamiltonian and overlap matrices. Although such an approach is of course not linear scaling, it can be useful as a benchmark approach, while the minimal size of the SF basis means that the computational cost is low compared to diagonalizing in the full wavelet basis.

The direct minimization approach works directly with the KS wavefunctions, but avoids explicit diagonalization by instead minimizing the band structure energy, subject to appropriate orthogonality constraints, as described in Mohr et al. [51]. The direct minimization approach does not scale as well with respect to the system size (see [51]), however it may nonetheless be preferred to FOE in certain cases. Notably, a few unoccupied states may be straightforwardly included in both the SF and kernel optimization steps, and so direct minimization is typically used for cases where it is important to have an SF basis which is capable of accurately representing the lowest unoccupied molecular orbital (LUMO).

Finally, in the FOE approach [27, 28] the density kernel is expressed as a function of the SF Hamiltonian matrix, i.e. K = f(H), where f is the Fermi function. The Fermi function is written as an expansion of Chebyshev polynomials, in such a way as to allow the K to be constructed using only matrix vector multiplications. This is combined with sparse matrix algebra, as implemented in the CheSS library [50], resulting in LS behaviour, while it can also be used to treat metallic systems at a (small) finite temperature [49]. The use of a finite temperature can also be used to ensure robust convergence even when the gap of a system closes due to a poor initial guess, bond breaking, or when computing charged systems.

The LS behaviour of BigDFT when using the FOE approach has been demonstrated for a number of materials, for systems containing up to tens of thousands of atoms [49, 51, 52, 59].

11.7.4 Suitability of the Linear Scaling Approach

The SF basis of BigDFT offers numerous benefits for linear algebra-based code bottlenecks. The in situ optimized approach allows for the accuracy of a large basis, while keeping the number of basis functions similar to the size of a minimal basis, leading to small matrices even for large systems. The use of strictly localized, quasi-orthogonal basis functions further ensures that the matrices used are sparse and well conditioned. In Table 11.1, we report the matrix dimensions and sparsities for four different systems: a 1CRN protein [62] in gas phase, a pentacene cluster, a 1L2Y protein [56] in solution, and a cluster of water molecules.

Table 11.1

Matrix properties of four example systems. NNZ refers to the percentage of non-zero elements in the Hamiltonian (H), overlap (S), and density kernel (K)

System

Natoms

NSFs

H NNZ

S NNZ

K NNZ

1CRN

642

1623

22.09

9.40

37.20

Pentacene

6876

19,482

2.89

1.04

5.70

1L2Y

1942

4045

5.90

2.13

11.57

Water

1719

3438

9.43

3.43

18.30

The benefits of this basis set are further reflected in the spectral quantities of these matrices, as shown in Table 11.2. We see that the spectral width of the overlap matrix is quite low, reflecting how well conditioned the basis is. The ratio of the band gap to the spectral width of the Hamiltonian is also relatively high, which leads to huge efficiency gains for diagonalization-free methods. The lower this ratio, the more polynomials that are required in order to approximate the Fermi function of the Hamiltonian. BigDFT’s matrices require few polynomials, similar to what would be needed for minimal basis calculations with Gaussians or tight binding calculations, as was shown in a head-to-head comparison when using density matrix purification techniques [18].

Table 11.2

Spectral properties of four example systems. The “Gap To Width” is the ratio of the band gap to the spectral width of the Hamiltonian

System

S width

H width (eV)

Gap (eV)

Gap to width

1CRN

0.9557

47.0783

1.9977

0.0424

Pentacene

0.9852

42.2971

1.0323

0.0244

1L2Y

0.8936

47.9860

1.3682

0.0285

Water

0.4496

40.8961

7.7297

0.1890

Sparse matrices are stored in a custom Segment Storage Format, which groups together consecutive non-zero values in a matrix row. This format not only reduces the storage overhead of a matrix, but also can improve the performance of matrix-vector multiplication by using calls to dense operations. The Hamiltonian matrix is replicated across processes and columns of the density matrix are distributed. This data distribution allows each column to be computed independently to improve parallel performance.

As mentioned in Sect. 11.7.3, the diagonalization free method of choice in BigDFT is the Fermi operator expansion based on matrix vector multiplication. This approach is usually far more expensive than those based on recursive polynomial expansions such as density matrix purification [14]. However, the high sparsity, relatively low dimension, and good conditioning of BigDFT’s matrices enables a more tailored choice of algorithm and parallelization scheme. This novel approach is made possible by the unique properties of the Daubechies wavelet basis set employed in BigDFT.

11.8 Perspective

Daubechies wavelets have a number of favourable properties which make them an ideal basis set for electronic structure calculations. In this work we have outlined the use of such a basis set for density functional theory calculations, as implemented in the open source BigDFT code. Through examples presented here and referenced within the text, we have shown how the combination of a wavelet basis set with an implementation designed for massively parallel machines allows for efficient and accurate calculations of hundreds of atoms, even within a traditional cubic scaling approach. Such a treatment also allows for the simulation of relatively large systems using hybrid functionals, particularly where GPUs are available, while the availability of different boundary conditions allows for the straightforward treatment of molecules, surfaces and solids. A number of functionalities are available in BigDFT, including dynamics, explicit charges and electric fields and implicit environments, while, for example, there are ongoing developments in the treatment of excited states.

Going beyond the cubic scaling approach, the localized nature of wavelets is also highly suitable for a linear scaling approach, wherein the nearsightedness of matter is exploited by imposing localization on the system via the use of a minimal set of localized support functions. Such an approach further expands the applicability of BigDFT to systems containing several thousand atoms, and has been shown to converge reliably for a range of materials. The localized support function-based approach may also be further exploited to define a fragment approach, in which the computational cost may be significantly reduced by exploiting repetition in molecular or periodic systems.

The treatment of such large systems brings new types of problems within the reach of first principles simulations. However, such simulations also bring new challenges, for example the increased size of the configuration space associated with complex materials containing large numbers of atoms. Furthermore, as well as treating large lengthscales it is desirable to treat also long timescales, which remains unfeasible within a purely quantum mechanical approach, so that QM/MM approaches are required. To this end, the support function approach allows not only the treatment of large enough systems to test and validate QM/MM approaches, but can also be used to analyze and fragment a system without requiring any a priori knowledge of the system. This offers a route to reduce the complexity of QM calculations of large systems and thereby inform the setup of multiscale simulations.

In the context of large scale electronic structure simulations, a wavelet-based approach therefore offers another significant advantage in that it facilitates the implementation of a range of approaches designed to treat different system sizes, as illustrated in Fig. 11.7. The existence of such a comprehensive framework with a single underlying formalism means that each successive approximation is applied in a systematic and controlled manner. Therefore, one can easily test and validate the approximations between different levels of theory. For example, QM/MM simulations may be benchmarked with respect to fragment calculations, which may be compared with full linear scaling calculations and so on. Furthermore, we have also introduced a number of indicators which can be used to predict whether or not a particular approximation is appropriate for a given system.

An atoms graph of the level of approximation versus the number of atoms. The graph illustrates the approaches of localized orbitals, fixed basis, and implicit solvent from O of N cube D F T to Q M over M M via O of N D F T and fragments, respectively.

Fig. 11.7

Illustration of the different approaches available in BigDFT, the approximations introduced and approximate applicable lengthscales

The development and improvement of a comprehensive multi-scale framework is an ongoing priority for BigDFT. As with other developments in BigDFT, a key aspect of this is a module-based approach which aims towards sustainable software development. In tandem with the implementation of new approaches, we also aim to prioritize both accessibility and reproducibility of new functionalities, by providing Jupyter-notebooks demonstrating how such functionalities may be used via the PyBigDFT interface. Further information on BigDFT may be found on the website [6] and in the documentation [5], while the code may be downloaded from GitLab [7].

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

E. Cancès, G. Friesecke (eds.)Density Functional TheoryMathematics and Molecular Modelinghttps://doi.org/10.1007/978-3-031-22340-2_12

12. Accurate Approximations of Density Functional Theory for Large Systems with Applications to Defects in Crystalline Solids

Kaushik Bhattacharya1  , Vikram Gavini2  , Michael Ortiz1  , Mauricio Ponga3   and Phanish Suryanarayana4  

(1)

California Institute of Technology, Pasadena, CA, USA

(2)

University of Michigan, Ann Arbor, MI, USA

(3)

University of British Columbia, Vancouver, BC, Canada

(4)

Georgia Institute of Technology, Atlanta, GA, USA

Kaushik Bhattacharya (Corresponding author)

Email: bhatta@caltech.edu

Vikram Gavini

Email: vikramg@umich.edu

Michael Ortiz

Email: ortiz@aero.caltech.edu

Mauricio Ponga

Email: mponga@mech.ubc.ca

Phanish Suryanarayana

Email: phanish.suryanarayana@ce.gatech.edu

Abstract

This chapter presents controlled approximations of Kohn–Sham density functional theory (DFT) that enable very large scale simulations. The work is motivated by the study of defects in crystalline solids, though the ideas can be used in other applications. The key idea is to formulate DFT as a minimization problem over the density operator, and to cast spatial and spectral discretization as systematically convergent approximations. This enables efficient and adaptive algorithms that solve the equations of DFT with no additional modeling, and up to desired accuracy, for very large systems, with linear and sublinear scaling. Various approaches based on such approximations are presented, and their numerical performance is demonstrated through selected examples. These examples also provide important insights into the mechanics and physics of defects in crystalline solids.

12.1 Introduction

Defects are common in crystalline solids [33, 65]. These include point defects (vacancies with missing atoms, substitutional elements where an atom of an impurity (solute) replaces an atom of the actual material, or interstitial atoms where an extra atom is inserted into the solid), cluster defects (vacancy cluster or prismatic dislocation loop with a missing disc of atoms), line defects (dislocations where an extra plane of atoms terminates along a line) or planar defects (twin or grain boundaries across which the crystal orientation changes or phase boundaries across which the crystal structure changes).

Defects play a critical role in determining important properties of materials. Vacancies mediate creep, solutes strengthen solids, vacancy clusters lead to void nucleation and dislocations mediate plasticity. Remarkably, they do so at extremely dilute concentrations. Vacancies affect creep at parts per million, and dislocations densities are of the order of one in a million amongst atomic columns during plastic deformation.

The reason why defects can have such a profound effect on properties at dilute concentrations is because they trigger physics at multiple length and time scales [65, 80]. In this review, we are interested in the equilibrium structure, and therefore focus only on the length scales. The defects cause an imbalance of forces on the neighboring atoms which in turn lead to deformations. Even though electronic interactions decay quickly, displacement of the atoms from their periodic equilibrium positions lead to imbalanced forces on their neighbors, and so on, leading to extremely slow decay of the displacement field. The complex quantum mechanical or chemical interactions at the defect core lead to a complex atomistic and electronic structure that need an electronic structure theory for its description. As we move away, the displacements from the periodic structure are less complex and may be understood through atomistic interactions. Even farther away, the displacements are smaller and may be described by continuum elasticity theory.

Crucially, these scales interact intimately and one scale does not dominate over others. We consider two examples. The first example is a vacancy, where a single atom is missing from an otherwise perfect lattice. In the far field, elasticity theory tells us that the displacement decays as r⁻² so that the stress and the strain decay as r⁻³ [58]. This is a relatively fast decay and the energy is summable in the far field. The divergence at the origin is regularized by atomistic and electronic interactions. One can estimate the energy due to the chemistry of the core by the cohesive energy of a solid (the energy difference between an isolated atom and an atom in the crystal), and this is typically a few electron volts in metals. One can also estimate the energy due to the elastic field far away, and this is of the order of a few tenths of electron volts for a typical metal. While one is smaller than the other, it is not negligibly so; therefore these fields do interact with each other even in this simple defect. Further, the elastic fields generated by the vacancy are large enough to interact with macroscopic stress due to boundary conditions resulting in a stress-induced driving force on the vacancy. The second example is that of a dislocation. The stress and strain decay as r⁻¹ away from the line, which means that the elastic energy density is logarithmic, and thus divergent at both the origin and the far field [33, 58]. In other words, dislocations provide a direct link between continuum scale boundary conditions and electronic scale interactions at the core. In short, defects connect the far field to the electronic scale and it is this ability to bridge scales that result in defects having a profound effect on macroscopic properties. This also makes defects extremely difficult to study.

Kohn–Sham DFT [34, 43] has emerged as the method of choice for the study of electronic structure in condensed matter [11]. It converts the many particle Schrödinger equation to a single particle problem with an effective single-electron potential. While one can show the existence of such a theory, the functional that gives rise to the potential (or even the locality or lack thereof) remains unknown, and is modeled. Thus, while DFT is nominally ab initio in that it is agnostic about the material (other than the atomic number), it does require a constitutive model of the universal exchange-correlation functional. It has proved to be an extremely useful compromise between practical application of quantum mechanics and fidelity.

Given the importance of electronic structure in determining the structure and energetics of defects, it is desirable to study defects using DFT. Such studies require large domains including large numbers of electrons due to the slow decay and low concentrations that are typical in materials; unfortunately such large domains are beyond the capability of brute force (full-resolution) DFT calculations using existing widely-used methods with reasonable computational resources. However, the complexity and details of the electronic structure are important near the core, and less so in the large regions of slow decay. This has motivated multiscale modeling of materials, where one builds a cascade of models (DFT, atomistic, continuum) to study the phenomena at different scales [1, 6, 91, 97]. However, the interaction between the scales means that one has to link them, which requires further modeling. Much of this modeling is empirical, taking us away from the ab initio point of view. Moreover, such a cascade of models linked empirically do not have an inherent or quantifiable notion of error. A notion of error is important since the study of defects requires comparing the energies of different configurations, and therefore evaluating a small difference between two large numbers. Therefore an estimate of the accuracy and an ability to control the accuracy is important.

This chapter presents a line of work that seeks to solve the equations of DFT, and only the equations of DFT, with no further modeling on large domains relevant to defects by introducing approximations where the error can be controlled. The idea is to formulate DFT as a well-defined minimum problem over spaces of operators, and then introduce systematically convergent approximations. Specifically, there are two approximations: (1) Spatial discretization, resulting in finite-dimensional approximate problems obtained by constrained minimization of the DFT energy functional; and (2) Spectral discretization, based on approximations of the DFT energy functional itself, or variational crimes in the parlance of approximation theory.

A related issue is the fact that widely used DFT methods for condensed matter are limited to periodic systems. This is motivated by crystals that involve a periodic arrangement of atoms. The periodicity enables one to work in Fourier space using a plane wave basis, and this has proven to be extremely efficient on moderate computational resources. One can extend this approach to defects using “super-cells”, i.e., studying a periodic arrangement of defects and the resulting unit cell. However, since defects interact over long distances, these can lead to artifacts. Further, dislocations are topological defects and therefore not amenable to periodic arrangement unless one studies defect complexes (dislocation dipoles or quadrapoles) with zero topological content, further leading to potential artifacts. Therefore, it is desirable to move away from periodic arrangements. Finally, defects are interesting because they interact with far-field stimuli. Therefore, it is desirable to study defects under arbitrary boundary conditions. These motivate the need to solve DFT in real space.

The systematically convergent approximations lead to various algorithms that enable the solution of DFT with controlled error on large systems in real space. This chapter describes three in some detail. The first involves variable spatial discretization by exploiting adaptive higher finite elements. The second involves spectral discretization using quadratures. The final method combines spectral and spatial discretization.

The chapter is organized as follows. Section 12.2 provides a variational formulation of DFT. An important result is the reformulation (12.4) which presents DFT as a nested variational problem. We then use spectral theory to rewrite the inner variational problem. This reformulation enables us to introduce spatial discretization and spectral quadratures as convergent (Rayleigh–Ritz) approximations. Section 12.3 introduces three ideas that are useful for the efficient practical implementation of the methods. Section 12.4 introduces spatial discretization using (higher order) finite-elements, and describes how this can be used for spatial adaptivity. This section also presents a series of examples that describe the efficacy of such an approach in studying defects, and the overall performance of the method. We turn to spectral quadratures in Sect. 12.5. We discuss the relation of this method to other approaches, including the recursion method (widely used in tight binding), Padé approximations and Fermi-operator expansion. We discuss convergence and demonstrate the performance of the method using various examples. We turn to combined spatial and spectral coarse-graining in Sect. 12.6. We describe a sub-linear scaling method for the study of defects, and its application to study vacancy clusters and dislocation cores in magnesium.

12.2 Variational Formulation of Density Functional Theory

In this section, we proceed formally following the notation and presentation of Refs. [3, 90] to formulate Kohn–Sham DFT as a well-defined minimum problem over spaces of operators, and the approximation schemes—spatial and spectral—that it suggests.

12.2.1 Kohn–Sham Density Functional Theory as a Minimum Problem

We consider a closed shell spin-unpolarized system in an insulated, bounded, open and Lipschitz domain $$\Omega \subset \mathbb {R}^3$$ for simplicity. The presentation may be extended to spin-polarized systems and unbounded domains [3]. Let $$\mathcal {V} = H^1_0(\Omega )$$ , $$\mathcal {H} = L^2(\Omega ;\mathbb {C})$$ , $$\mathcal {D}(h) = H^1_0(\Omega ; \mathbb {C})$$ , and $$\mathcal {X} = \{\gamma \in \mathcal {S}(\mathcal {H}),\ \mathcal {R}(\gamma ) \subset \mathcal {D}(h)\}$$ , where h denotes the single-particle Hamiltonian, γ represents the single-particle density matrix, and $$\mathcal {S}(\mathcal {H})$$ denotes the vector space of bounded self-adjoint operators on $$\mathcal {H}$$ . Let

be a constraint set defining the admissible density operators. Define the Kohn–Sham energy functional $$F: \mathcal {X} \to \bar {\mathbb {R}}$$ as

which can be written as

where

is the electron density. In addition,

where v is an external potential, and

(12.1)

is the classical electrostatic energy [37]. A formal connection between (12.1) and the oft-used equivalent expression (up to an inessential constant) based on the Coulombic interaction formula $$\frac 12 \int _\Omega \int _\Omega \frac { \rho (\mathbf {r}) \rho (\mathbf {r}')} {|\mathbf {r} - \mathbf {r}'|} \mathrm {d}\mathbf {r} \mathrm {d}\mathbf {r}'$$ can be established simply by writing out the Euler–Lagrange equation of (12.1) and solving for ϕ using the Green’s function for the Laplacian. The expression (12.1) simply recognizes the fact that J(ρ) is the dual of the Dirichlet functional. Representation (12.1) is advantageous over the Coulombic representation from the standpoint of approximation, which only requires local conforming interpolation of the electrostatic field ϕ. Finally, Exc(ρ) is the exchange-correlation energy functional. It must necessarily be modeled. Here, for simplicity, we choose the local density approximation (LDA) [60]. The Kohn–Sham DFT problem is to find the ground state energy

(12.2)

and attendant energy-minimizing states. For subsequent purposes, we use duality to reformulate the energy in trace form. To this end, assume for definiteness that the exchange correlation $$E_{\mathrm {xc}} : \mathcal {V} \to \bar {\mathbb {R}}$$ is convex with dual $$E_{\mathrm {xc}}^* : \mathcal {V} \to \bar {\mathbb {R}}$$ , so that [90]

Now, define the Hamiltonian

where the electrostatic potential operator Φ, the exchange-correlation potential operator Vxc, and the external potential operator V  are bounded self-adjoint operators over $$\mathcal {H}$$ defined by the properties

Then,

and the Kohn–Sham DFT problem (12.2) becomes

(12.3)

It is possible to exchange the order of the $$\inf $$ and $$\sup $$ operations in the above equation [90] to arrive at the reformulated Kohn–Sham DFT problem,

(12.4)

This reformulation offers various advantages and serves as the basis for the approximations to follow. First, in the same spirit as in (12.1), the representation (12.4) only involves local operators and requires local or conforming interpolation of ϕ and vxc. Second, the functional is expressed in terms of linear operators acting on γ only, thus paving the way for a spectral treatment of the problem, as we do presently.

We now focus on the inner $$\inf $$ operation that yields the energy-minimizing density matrix for fixed (ϕ, u):

(12.5)

where, here and subsequently, we omit the dependence of h on the fixed fields (ϕ, vxc) for simplicity of notation. Note that the quantity U is commonly referred to as the band structure energy in the physics literature.

It follows from spectral theory (cf., e.g., [70]) that the minimizing density matrix operator γ of (12.5) shares the same spectral measure as the Hamiltonian h, i.e., we may write

(12.6)

for 0 ≤ f ≤ 1, where $$\mathcal {E}$$ is a resolution of the identity over the Borel sets of the real line. In addition, γ and h have the same spectral measure if and only if they commute, i.e.,

Finally, we can show that there is a minimizer f ∈{0, 1} so that γ² = γ. Therefore, the minimum problem is

(12.7a)

(12.7b)

The variational problem (12.4) is often solved by a fixed point iteration of self-consistent field (SCF) iteration where (12.7) is solved for γ with a fixed ϕ, vxc and the outer sup problem in (12.4) is used to update ϕ and vxc for fixed γ.

12.2.2 Approximations Resulting from Spatial Discretization

We proceed to discretize problem (12.4) à la Rayleigh–Ritz, i.e., by restriction to finite-dimensional spaces. To this end, let $$\mathcal {V}_h$$ be a nested sequence of finite-dimensional subspaces of $$\mathcal {V}$$ spanned by orthonormal bases $$\{e_1, \dots , e_{N_g}\}$$ , e.g., corresponding to a finite element discretization.¹ Let $$\mathcal {H}_h = \mathcal {V}_h$$ be the corresponding sequence of subspaces of $$\mathcal {H}$$ . Then, the discrete wave function, electrostatic field and exchange-correlation potential field are of the form

Likewise, the discrete Hamiltonian is

with

and

We note the additional linear structure

where

Finally, the discrete density matrix is of the form

and the discrete electron density follows as

This sequence $$\{ {\mathcal V}_k \}$$ of finite-dimensional subspaces of $${\mathcal V}$$ defines a nested sequence of subspaces of $$\mathcal {X}$$ of density matrices $$\mathcal {X}_h = \{\gamma _h \in \mathcal {S}(\mathcal {H}_h)\}$$ , where $$\mathcal {S}(\mathcal {H}_h)$$ denotes the vector space of symmetric linear operators on $$\mathcal {H}_h$$ . This in turn defines a sequence of discrete constraint sets $$\mathcal {K}_h = \{\gamma _h\in \mathcal {X}_h \,:\, 0 \leqslant \gamma _h \leqslant 1, 2 \mathrm {tr}(\gamma _h) = N\}$$ , where $$0 \leqslant \gamma _h \leqslant 1$$ expresses the requirement that $$0 \leqslant (\varphi _h|\gamma _h|\varphi _h) \leqslant 1$$ for all $$\varphi _h\in \mathcal {H}_h$$ . We note that, if the spaces $$\mathcal {H}_h$$ are nested, then $$\mathcal {K}_h$$ defines a decreasing sequence of sets in $$\mathcal {X}$$ and that $$\mathcal {K}\subset \mathcal {K}_h$$ . Then, the corresponding sequence of discrete energies $$F_h:\mathcal {X}_h\to \bar {\mathbb {R}}$$ follows as

(12.8)

where $$I_{\mathcal {K}_h}$$ is the indicator function of $$\mathcal {K}_h$$ , and $$B_h = D_h^* D_h$$ with Dh the discrete gradient operator. The discrete Kohn–Sham DFT problem becomes

(12.9)

where we have again exchanged the order of the inf and sup operations [90].

As before, we may rewrite the inner inf problem as

(12.10a)

(12.10b)

12.2.3 Spectral Reformulation of the Discrete Kohn–Sham Problem

By the spectral decomposition theorem (cf., e.g., [70]), we can write

where $${\mathcal E}_h$$ is an operator-valued measure. In this representation, we have

and

where

is a spectral measure with:

where δ is the Dirac delta function. Given the spectral measure $$\mathcal {M}_h$$ , the calculation of the energy-minimizing discrete density matrix γh at fixed (ϕh, uh) reduces to the scalar problem

(12.11)

where Xh denotes the space of bounded real-valued Borel functions over the real line.

12.2.4 Approximation by Numerical Quadrature

We proceed to reduce problem (12.11) by recourse to numerical quadrature. Let

be a sequence of quadrature rules, parameterized by $$k\in \mathbb {N}$$ , with weights Aj and nodes εj. Here,

(12.12)

where

for j = 0, …, k are the Lagrange polynomials.

Define the sequence of approximate energies

and the sequence of approximate masses

Then, we have a corresponding sequence of discretized problems

(12.13)

The solution of these approximate problems then follows from the algorithm:

(i)

Set f₀(ε) = 0, i = 0, …, k, I₀ = {0, …, k}, N₀ = 0, n = 1.

(ii)

Let in ∈argmin{ε, i ∈ In−1}, $$N_n = N_{n-1} + A_{i_n}$$ .

(iii)

If Nn < N, set $$f_n(\varepsilon _{i_n})=1$$ , In = In−1∖{in}, n ← n + 1, go to (ii).

(iv)

Otherwise, set $$f_n(\varepsilon _{i_n})=(N-N_{n-1})/A_{i_n}$$ , fh = fn, exit.

12.2.5 Rayleigh–Ritz Interpretation

The numerical-quadrature reduction can again be given an appealing Rayleigh–Ritz interpretation. Begin by noting the identity

From this identity and (12.12) we have

where

Likewise,

Define now the sequence of spaces

where {li, i = 0, …, k} are the Lagrange polynomials defined by the roots of the orthogonal polynomial pk+1 generated by H. Define, in addition, the sequence of relaxed constraint sets

Then, the reduced problem (12.13) is equivalent to solving

which corresponds to a Rayleigh–Ritz reduction of problem (12.5) to the subspaces of density matrices $$\mathcal {X}_k$$ generated by numerical quadrature, and to the corresponding relaxed constraint sets $$\mathcal {K}_k$$ .

12.2.6 Convexification and Thermalization

The DFT problem (12.4) and the inner minimization problem (12.7) are not convex due to the constraint that f(ε) takes values in {0, 1}. We convexify the problems by allowing the function to take values in the entire interval [0, 1], the resulting function henceforth referred to as fβ(ε). We expect the minimizers to take extreme values only and thus the convexified and original problems to yield the same minimizers and the same minimum energy. We can now enforce the constraint $$0 \leqslant f_{\beta }(\varepsilon ) \leqslant 1$$ by entropic penalization. We present it for the infinite-dimensional version (12.4), though it can readily be extended to the versions with spatial and spectral discretization. Introduce the entropy

and the thermalized problem

StartLayout 1st Row 1st Column Blank 2nd Column minimize colon upper U Subscript beta Baseline left parenthesis gamma right parenthesis equals upper U left parenthesis gamma right parenthesis plus StartFraction 1 Over beta EndFraction upper S left parenthesis gamma right parenthesis 2nd Row 1st Column Blank 2nd Column equals 2 trace left parenthesis h gamma right parenthesis plus StartFraction 2 Over beta EndFraction trace left bracket gamma log left parenthesis gamma right parenthesis plus left parenthesis script upper I minus gamma right parenthesis log left parenthesis script upper I minus gamma right parenthesis right bracket comma EndLayout

(12.14a)

(12.14b)

where β is an inverse temperature. The minimizer of Uβ(γ) is

(12.15)

where μ is a chemical potential introduced to enforce the number constraint. μ and fβ are commonly referred to as the Fermi level and Fermi–Dirac distribution, respectively, with $$\beta \rightarrow \infty $$ representing the zero-temperature limit.

The thermalized total energy

Finally, we may estimate the zero temperature ground state energy as

(12.16)

12.2.7 Spatial Densities

For later use, we note that the quantities in (12.4) and (12.16) have associated spatial densities and can be rewritten in terms of volume integrals. Following (12.6), and explicitly introducing the spatial variables

(12.17)

Therefore, the number of electrons, the band structure energy and the entropy may be written as

in terms of the charge or number densityρ, band structure energy densityu and entropic densitys. Indeed, recall that ρ(r) = 2γβ(r, r). These densities play a key role in later sections.

12.2.8 Eigenvalue Problem

We close our formulation by connecting it to the way DFT is usually presented as an eigenvalue problem. The direct solution of problem (12.9) entails the computation of N∕2 eigenvalues and eigenvectors. To see this, consider the inner $$\inf $$ operation in (12.9). Write

$$Q_h\in \mathcal {L}(\mathcal {H}_h,\mathbb {R}^{N/2})$$ , with

where IN∕2 denotes the identity in $$\mathcal {L}(\mathbb {R}^{N/2})$$ . Here and subsequently, $$\mathcal {L}(\mathcal {A},\mathcal {B})$$ denotes the space of linear transformations between two linear spaces $$\mathcal {A}$$ and $$\mathcal {B}$$ , and $$\mathcal {L}(\mathcal {A})$$ the space of linear transformations from a linear space $$\mathcal {A}$$ to itself. Then, PT = P, $$0 \leqslant P \leqslant 1$$ and tr(P) = N∕2, hence $$\gamma _h\in \mathcal {K}_h$$ . The problem under consideration thus becomes

The Euler–Lagrange equations of this problem are

where $$\Lambda _h \in \mathcal {L}(\mathbb {R}^{N/2})$$ , $$\Lambda _h^{\mathrm {T}} = \Lambda _h$$ , is a Lagrange multiplier. Clearly, these Euler–Lagrange equations are solved if the columns of Qh consist of eigenvectors of H and Λh stores the corresponding eigenvalues in its diagonal. In addition, if $$\{\varepsilon _1,\dots ,\varepsilon _{N_g}\}$$ are the ordered eigenvalues of H is ascending order and $$\{\varphi _1,\dots ,\varphi _{N_g}\}$$ are the corresponding eigenvectors, then the minimum problem is solved by Qh = {φ₁, …, φN∕2} and Λh = diag{ε₁, …, εN∕2}. Finally, the energy follows as

Clearly, this computation becomes intractable for large material samples containing a large number electrons N. Therefore, computational tractability of large samples requires an additional reduction (beyond spatial discretization) that we refer to as spectral reduction above.

12.3 Filtering, Spectrum Splitting and Pseudopotentials

This section introduces three ideas that enable faster calculations. The first two, filtering and spectrum splitting, are convergent approaches and take advantage of the spectral formulation. The third, pseudopotentials, involves modeling.

12.3.1 Filtering

The discrete DFT problem (12.9) is posed as a problem in an Ng-dimensional subspace $${\mathcal V}_h$$ of $${\mathcal V} = H_0^1(\Omega )$$ . In practice, the accurate solution of the equations requires that Ng ≫ N. However, the solution to our problem, the density matrix γh, has rank N (in the thermalized problem, the thermalized density matrix γβ,h has rank larger than but close to N). Therefore, one can obtain significant savings in computational effort by identifying a priori a sub-space $${\mathcal V}_h^f$$ such that $$ \text{range} (\gamma _h) \subset {\mathcal V}^f_h \subset \subset {\mathcal V}_h$$ , and restricting the problem (12.10) and specifically the Hamiltonian hh to the sub-subspace $${\mathcal V}_h^f$$ . This can be achieved using filtering. While many approaches have been proposed based on filtering such as purification (cf. e.g. [31, 48, 59, 76]) and approximations to the Fermi–Dirac functions (cf. e.g. [4, 24, 49]), the Chebyshev filtering technique [98, 99] is adopted in many recent DFT codes [21, 22, 51, 53]. The main idea in Chebyshev filtering is to approximate the subspace $${\mathcal V}^f_h$$ as

where $$X_h\subset \mathcal {V}_h$$ with $$\text{dim}(X_h)=\text{dim}(\mathcal {V}^f_h)$$ , Tm is a Chebyshev polynomial of order m, and

with $$b=\max \sigma (h_h) &gt;&gt; 1$$ (σ denoting the spectrum) and $$a=\max \sigma (\gamma _h h_h)+\mathcal {O}(1)$$ . In particular, $$a\approx \mu +\mathcal {O}(1)$$ is a reasonable choice. We note that g transforms the spectrum of hh such that σ(g(hh)) < 1 and σ(g(γhhh))) ∈ (−∞, −1). Thus, as Tm(x) > 1 for x ∈ (−∞, −1), Tm(g(hh))Xh provides a good approximation to $${\mathcal V}^f_h$$ . We note that a suitable choice of m depends on the value (b − a), a larger m being needed for larger values of (b − a). For instance, based on numerical studies, if $$(b-a)\sim \mathcal {O}(10^2)$$ , values of m ∼ 10 − 30 are sufficient to construct a good approximation to $$\mathcal {V}^f_h$$ [53, 99]. However, if $$(b-a)\sim \mathcal {O}(10^6)$$ , values of m ∼ 1000 are needed [71].

If $$\mathcal {P}_f: \mathcal {V}_h\to \mathcal {V}^f_h$$ denotes the projection operator onto the filtered subspace, then the solution to the DFT problem can be obtained by replacing hh in (12.10) with $$\mathcal {P}_f h_h \mathcal {P}_f$$ . As the spectral width $$\Sigma (\mathcal {P}_f h_h \mathcal {P}_f)\ll \Sigma (h_h)$$ , it enables faster numerical solution of the DFT problem, and has been the basis for subspace projection methods (cf. e.g. [16, 54]).

12.3.2 Spectrum Splitting

The next idea combines filtering with a feature of the solution of a typical problem. Here, we assume that the DFT problem has already been projected onto $${\mathcal V}_h^f$$ , and denote $$\mathcal {P}_f h_h \mathcal {P}_f$$ by hf. We denote by σh = σ(hf) the spectrum of hf, and assume in the following that $$\max \sigma _h &lt;0$$ (i.e., hf is appropriately shifted such that this condition is satisfied). It has long been recognized that the spectrum of hf has a gap that separates the so-called core, or deeply bound states at the lower end, from the rest. In other words, the spectrum $$ \sigma _h = \sigma _h^c \cup \sigma _h^r$$ with $$\varepsilon ' + E_g \leqslant \varepsilon '' \ \forall \ \varepsilon ' \in \sigma _h^c, \varepsilon '' \in \sigma _h^r$$ for a gap Eg > 0. We can therefore split the Hamiltonian hf and the density operator γh (corresponding to hf) into

(12.18)

where the spectrum of $$h_f^c$$ is $$\sigma _h^c\cup \{0\}$$ . It follows that we can divide $${\mathcal V}_h^f$$ into two orthogonal subspaces,

where $${\mathcal V}_h^{c,r}$$ is the range of $$h_f^{c,r}$$ . Further, since $$\sigma _h^c$$ is the lower end of the spectrum, it follows that

(12.19)

is the projection operator from $${\mathcal V}^f_h$$ to $${\mathcal V}_h^c$$ .

Now, in light of the spectral gap, we can again use filtering on hf, and then readily identify $${\mathcal V}_h^c $$ as the range of $$h^c_f$$ . Therefore, we can use (12.19) to easily compute $$\gamma _h^c$$ . Further, using the orthogonality of the subspaces,

Since the spectra $$\sigma _h^c$$ and $$\sigma _h^r$$ are disjoint, it follows that

(12.20)

We may now reduce (12.10) as

(12.21a)

(12.21b)

where Nc denotes the number of core electrons. This approach of spectrum splitting provides a number of advantages. First, the computation of $$\gamma _h^c$$ , the core part of the density matrix, is relatively simple as described above. Second, in practice, the width of the spectrum of $$h_f^r$$ ( $$\Sigma (h_f^r)$$ ) is significantly smaller than that of hf (Σ(hf)), and this allows for a more efficient numerical solution. Finally, the core subspace $${\mathcal V}_h^c$$ consists of functions which have a compact support close to the nuclei. In other words, this is the subspace spanned by the orbitals of the core electrons. This can be further exploited to gain numerical efficiency. Further, its complement, $${\mathcal V}_h^r$$ , which contains so-called valance and conduction electrons, consists of functions that vary smoothly outside a core region around the nucleus. Therefore, we can use a spatially adaptive resolution to discretize it.

We may proceed similarly in the thermalized problem to find that (12.19) and (12.20) still hold, and

(12.22)

It is common to compute this by expanding it in a polynomial basis (Fermi operator expansion [24, 25]), which we shall show later in Sect. 12.5 is related to the spectral quadratures. Therefore, the advantages of spectrum splitting carry over to the thermalized setting.

The accuracy and efficacy of this approach for large-scale all-electron DFT calculations has been demonstrated in [56]. Here, we present some representative results on Si and Au nanoclusters. Figure 12.1 shows the results from ground-state energies computed using two approaches: (1) SubPJ-FE: A subspace projection approach via filtering (Sect. 12.3.1) implemented in a finite-element basis, where γh,β = fβ(hf) is computed via Fermi-operator expansion using Chebyshev polynomials for various orders; (2) Spectrum-splitting method: In addition to the subspace projection via filtering, spectrum splitting is used, where $$\gamma _{h,\beta }=\gamma ^c_h+ f_{\beta }(h_f^r)$$ and $$f_{\beta }(h_f^r)$$ is evaluated via Fermi-operator expansion using Chebyshev polynomials for various orders. The results for Si₉₅ are provided for two values of β corresponding to T = 500 and 1000 K, and results for the Au₆ cluster are shown for T = 500 K. As is evident, spectrum splitting not only provides computationally efficiency—due to a substantial reduction in the polynomial order required in Fermi operator expansion—it is indispensable to obtain the desired accuracy for systems with large atomic numbers, like Au.

2 line graphs of a b s error in energy versus fermi operator expansion degree. A plots 4 decreasing lines for spectrum splitting method for T= 500 and 1000 Kelvin and sub P J F E for T= 500 and 1000 Kelvin. The line for sub P J F E for T= 500 Kelvin is at the top, followed by sub P J F E for T= 1000 Kelvin. B plots 2 decreasing lines for the spectrum splitting method and sub P J F E which is at the top.

Fig. 12.1

Accuracy and computational efficiency afforded by spectrum splitting in all-electron calculations via Fermi operator expansion. (Left) Si₉₅ nanocluster; (Right) Au₆ nanocluster. Adapted from [56]

We conclude this subsection by noting that spectrum splitting is also closely related to the so-called enrichment methods. Note that the identity (12.19) means that we can use any basis set to represent $${\mathcal V}_h^c$$ . Therefore, picking ∼ Nc∕2 functions that are computationally convenient and approximate the span of $${\mathcal V}_h^c$$ provides a good starting point. Subsequently, choosing a spatial discretization sufficient to span $${\mathcal V}_h^f$$ provides the desired accuracy. This is computationally effective since the spatial discretization does not have to be so fine as to represent the core electrons. The basis set approximately spanning $${\mathcal V}_h^c$$ can be iteratively updated as the calculation proceeds. These ideas lead to augmented plane wave (APW), linearized augmented plane wave (LAPW) [73], and enriched finite basis [40, 71, 95]. We refer the reader to Chap. 9 by Chen and Schneider in this volume for a detailed discussion of these methods.

12.3.3 Frozen Core Approximation and Pseudopotentials

The formulations discussed till now have considered all electrons in the system. However, it is a long-held observation in the field that core electrons play a minimal role in the bonding between atoms. Specifically, it is observed that the $$\gamma _h^c$$ is relatively independent of the external potentials v that arise in molecules and crystals. This motivates the desire to exclude these electrons from the calculations, and to focus on the valance and conduction electrons.

One approach to doing so is the so-called frozen core approximation. Here, a high resolution all-electron calculation for a single atom is conducted to obtain the core density matrix, $$\gamma _{h,Z}^c$$ for a single atom (the subscript Z here refers to the single atom of atomic number Z with the nucleus located at the origin). Subsequently, this is used as an ansatz for the core electrons for any given problem. Specifically, for a problem with Na atoms with atomic numbers {Zi} located at {ri},

is used as an ansatz for

(12.23)

in (12.10) to solve for $$\gamma _h^r$$ . Note that the computational complexity of the problem is now reduced from N electrons to N − Nc electrons. Further, as noted above, the range of $$\gamma _h^r$$ is spanned by relatively smooth functions outside the core, and therefore one can use a spatially adaptive discretization to represent this problem.

Note that this is an uncontrolled approximation since it is based on an ansatz. Table 12.1 from Ref. [52] shows the errors from the frozen core approximation for a range of systems. In particular, the two metrics used to measure the approximation are: (1) the relative error in the core electron density at the ground-state $$||{\rho }_{0}^c-\bar {\rho }^c||{ }_{L^2}/||{\rho }_{0}^c||{ }_{L^2}$$ , where $${\rho }_{0}^c$$ is the core electron density at the ground-state from the all-electron calculation and $$\bar {\rho }^c = \bar {\gamma }_h^c({\mathbf r},{\mathbf r})$$ ; (2) the relative error in the total electron density at the ground-state $$||{\rho }_{0}-\bar {\rho }_{0}||{ }_{L^2}/||{\rho }_{0}||{ }_{L^2}$$ , where ρ₀ is the total electron density at the ground-state from the all-electron calculation, and $$\bar {\rho }_0$$ is the total electron density at the ground-state from the frozen core approximation. As evident, while the approximation is good for some systems, it can incur larger errors for others (such as Si nanoclusters).

Table 12.1

Error incurred from the frozen core approximation for various systems [52]

System

Li₂

0.00703

0.00787

O₂

0.00102

0.00128

CO

0.00181

0.00129

Si₁₈

0.01272

0.0130

Si₃₁

0.01273

0.0134

A closely related idea is that of a pseudopotential. Here, the objective is to fully exclude the core states by using a fictitious potentials, namely pseudopotentials, thus replacing hh with hPS. The pseudopotentials are generated such that $$\gamma _{PS}^r=f_{\beta }(h_{PS})$$ closely approximates $$\gamma _{h}^r$$ outside a core radius around each atom, but the range of $$\gamma _{PS}^r$$ is smooth all through the simulation domain. Thus, this alleviates the need for a spatially refined basis to resolve the core states. Various pseudopotentials have been proposed and are widely used (cf. e.g. [8, 29, 88]). Despite the errors and the uncontrolled nature of these approximations, it is often the only practical route to proceed in large systems of interest.

12.4 Spatial Coarse-Graining: Finite-Element Discretization

Spatial discretization (cf. Sect. 12.2.2) plays a central role in the practical aspects of computing the solution to the Kohn–Sham DFT problem in an efficient manner. Many discretization schemes have been adopted by the scientific community in solving the Kohn–Sham problem, and besides the algorithms employed, the discretization schemes have been the main differentiator for the various DFT codes and their performance based on computational efficiency and scalability. The widely used discretization methods include the plane-wave basis (cf. e.g. [23, 27, 44]) and atomic orbital type basis functions (cf. e.g. [9, 32, 36, 86]). While the plane-wave basis offers spectral convergence, it is primarily efficient for periodic problems owing to the lack of spatial adaptivity, and is constrained by limited parallel scalability of numerical implementations. The atomic orbital type basis functions present a reduced order basis, but in practice may not guarantee a robust and systematically convergent solution, especially for metallic systems. Also, they suffer from limited parallel scalability owing to the global nature of the basis functions. Although the finite-element and finite difference discretization schemes were explored over two decades ago [45, 62, 63, 83, 84], they have only recently started to gain traction as efficient and scalable approaches for solving the Kohn–Sham problem [21, 22, 53, 57].

The finite-element discretization in particular offers many attractive features, including the following: (1) Systematic convergence. Piecewise polynomials of a fixed degree p are dense in H¹( Ω) as the finite-element mesh-size h becomes small. Further, polynomials of increasing p are dense for a fixed h. (2) Flexibility. It has the ability to easily handle complex geometries and mixed boundary conditions, which is especially important when treating defects where periodicity may not be appropriate. (3) Spatial adaptivity. The discretization can be exploited to provide desired basis resolution in regions of interest and coarse-graining elsewhere. (4) Parallel scalability. The locality of the FE basis provides for efficient parallel scalability of numerical implementation. We also refer the reader to Chap. 10 by Dai and Zhou in the present volume for a broad discussion of the application of finite element discretization to DFT.

12.4.1 Higher-Order Spectral Finite-Elements

Despite the aforementioned advantages of the finite-element basis, and many prior efforts that explored the use of a finite-element basis for electronic structure calculations, they have not been competitive with widely used plane-wave and atomic orbital basis sets until recently. The two main issues limiting the performance of a finite-element basis in Kohn–Sham DFT had been: (1) the significant degree of freedom disadvantage of commonly used linear finite-elements in comparison to a plane-wave basis that affects the computational efficiency in practical DFT calculations; (2) the non-orthogonality of the finite-element basis that either limits the available solution schemes or requires an additional evaluation of the inverse of the overlap matrix.

Figure 12.2 provides insights into the lack of computational efficiency of linear finite-elements observed in prior studies. The figure shows the error in the ground-state energy for various finite element discretizations of different finite-element orders for two materials systems The higher order finite-elements employed in the study are hexahedral finite-elements, where the finite-element basis functions are constructed as a tensor product of basis functions in each dimension. The hexahedral finite-element basis functions in the isoparametric formulation are constructed from polynomial basis functions in the reference domain [−1, 1]³ as

(12.24)

2 scatterplots of relative error in energy versus 1 by N subscript e whole power 1 by 3 plot 5 legends each, in which T E T 4 of p = 1 in A and H E X 8 of p = 1 in B and 4 others of H E X 27 with p = 2, H E X 125 S P E C T of p = 4, H E X 343 S P E C T of p = 6, and H E X 729 S P E C T of p = 8 are in an increasing trend. Their respective q values in A are 1.86, 3.65, 7.69, and 11.72, and in B are 1.76, 3.61, 7.70, and 11.61.

Fig. 12.2

Convergence of finite-element discretization for various finite-element orders. Ne denotes the number of elements with $$( \frac {1}{N_e})^{1/3}$$ providing a measure of the finite-element mesh size h; TET4 denotes a linear tetrahedral element; HEX27 denotes a quadratic hexahedral element (p = 2); HEX125SPECT denotes a quartic spectral hexahedral element (p = 4); HEX343SPECT denotes a sixth order spectral hexahedral element (p = 6); HEX729SPECT denotes an eighth order spectral hexahedral element (p = 8). The benchmark systems comprise of (Left) Barium cluster, non-periodic system; (Right) Face-centered cubic Calcium unit cell, periodic system. Adapted from [57]

where li(ξ) is a Lagrange polynomial of degree p constructed based on the p + 1 nodes of the finite-element. Conventionally, the finite-element nodes are chosen to be equidistant, however the conditioning of basis functions is known to deteriorate with increasing order [10]. Instead, spectral finite-elements, where the finite-element nodes are chosen to be the roots of the Chebyshev polynomial, or the roots of the derivative of the Legendre polynomial, are known to provide a better conditioned basis for higher-order discretizations. From the results in Fig. 12.2, we note that for all orders of finite-element discretizations, the relative error in ground-state energy $$|\frac {E_{\text{KS},h}-E_{\text{KS}}}{E_{\text{KS}}}|\sim Ch^{q}$$ , where EKS,h is the discrete ground-state energy, EKS is the converged ground-state energy, and h is a measure of the finite-element mesh size, chosen to be $$(\frac {1}{{N_{e}}})^{1/3}$$ where Ne is the number of elements. The results show that q is close to 2p with p denoting the finite-element order (degree of the Lagrange polynomial li). These results also show that the faster convergence of higher-order finite element approximations also provide a substantial reduction in the number of finite-elements required to achieve chemical accuracy (∼10⁻⁵ relative errors in energy). This suggests the use of higher-order finite-element discretization as a potential path to bridging the significant degree of freedom disadvantage with a plane-wave basis.

Figure 12.3 (left) shows the degrees of freedom needed to solve two benchmark systems—a copper nanocluster with 55 atoms (non-periodic systems) and Mo supercell with a monovacancy containing 53 atoms—to chemical accuracy (0.1 mHa/atom in energy and 0.1 mHa/Bohr in force) with various orders of hexahedral spectral finite elements. It is evident that by using a fourth order finite element in comparison to a linear finite element, the basis function requirement can be reduced by ∼1000 ×. This subsequently translates into a ∼1000 × improvement in computational efficiency, as shown in Fig. 12.3 (right), which provides the corresponding computational times in CPU-Hrs. While the gap between the number of basis functions required to achieve chemical accuracy is substantially reduced between plane-wave and higher-order finite-element discretization, the number of basis functions using finite-element discretization is still ∼5-fold larger than plane-waves. However, computational cost per basis function is typically lower compared to plane-waves, and, given the better parallel scalability, finite-element discretization is emerging as an alternative to plane-waves for systematically convergent, fast and scalable DFT calculations.

Two grouped bar graphs of the number of F E degrees of freedom and C P U time in C P U hours versus F E orders 1, 2, 4, and 6 plot 2 bars for C u nanoparticles of 55 atoms and M o periodic supercell of 53 atoms. The highest values of the 2 bars are with F E order 1 in both graphs.

Fig. 12.3

(Left) The number of finite element degrees of freedom required to achieve chemical accuracy for the various orders of finite elements for two benchmark systems involving a Cu nanoparticle (non-periodic system) and a Mo supercell with a mono-vacancy (periodic system). (Right) The corresponding computational efficiency afforded by the various orders of finite elements. Results obtained using DFT-FE code [53]

The use of spectral higher-order finite-elements, while improving the conditioning of the basis, provides a path to addressing the non-orthogonality of the finite element basis. In particular, the Löwdin orthonormalized finite element basis ( $$\{e_1, \dots , e_{N_b}\}$$ ) can be constructed from the Lagrange finite element basis ( $$\{q_1, \dots , q_{N_b}\}$$ ) as

However, such a transformation requires the computation of M−1∕2, which can be prohibitively expensive for large Nb. We note that by using spectral finite elements with the nodes located at the derivative of the Legendre polynomial (in addition to nodes at the end points) in conjunction with Gauss–Lobatto–Legendre (GLL) quadrature rules, M is rendered diagonal and the transformation is trivial. In particular,

where Ωel, el = 1, 2, …, Ne denote the domains corresponding to each finite-element, and $$J_{\Omega _{el}}$$ is the Jacobian of the transformation from Ωel to [−1, 1]³. Pα(ξ, η, κ) is the Lagrange polynomial defined on [−1, 1]³ (Eq. 12.24) with α = (i, j, k) denoting a composite index corresponding to a node in the element. The integral in the above expression for Mαβ is evaluated using quadrature rules as

where Nq denotes the number of quadrature points and wq are the weights associated with the quadrature points $$(\bar {\xi }_{q},\bar {\eta }_{q},\bar {\kappa }_{q})$$ for q = 1, 2, …, Nq. In particular, while using spectral finite elements (Legendre) in conjunction with the GLL quadrature rule, the quadrature points are coincident with the nodes, i.e., $$(\bar {\xi }_{q},\bar {\eta }_{q},\bar {\kappa }_{q}) = ({\xi }_{i},{\eta }_{j},{\kappa }_{k})$$ with q = (i, j, k) denoting a composite index, i, j, k = 0, 1, …, p. Further, noting the Kronecker delta property of Lagrange polynomials, $$P_{(i,j,k)}(\xi _{i'}, \eta _{j'}, \kappa _{k'}) = \delta _{ii'}\delta _{jj'}\delta _{kk'}$$ , it is easy to infer PαPβ = δαβ. Thus, for spectral finite elements (Legendre) with the GLL quadrature rule,

Thus, the evaluation of M−1∕2, and subsequently the construction of Löwdin orthonormalized finite element basis, is rendered trivial. We note that numerical results show that the use of a reduced order quadrature rule for the evaluation of M−1∕2 does not affect the convergence rates or limit the accuracy of the calculation [57]. This can be rationalized by noting that the quadrature error for the GLL quadrature rule is $$\mathcal {O}(h^{2p})$$ , which is also the order of discretization error. Further, the GLL quadrature is needed for the aforementioned simplification only in the evaluation of M, whereas all other integrals are evaluated using Gauss quadrature.

Thus, by addressing the two main limitations of the finite element discretization—the degree of freedom disadvantage when using higher-order spectral finite element discretizations and the nonorthogonality of the basis when using spectral finite elements in conjunction with GLL quadrature—the finite-element discretization has emerged as a competing basis to plane-waves in practical DFT calculations (cf. Sect. 12.4.3), especially owing to the benefits derived from it being a real-space basis, the locality of the basis functions, and its potential for excellent parallel scalability.

12.4.2 Spatial Adaptivity

Spatial adaptivity can naturally be realized in finite-element discretization by using a spatially refined mesh in regions of interest and coarsening elsewhere. Figure 12.4 shows a spatially adaptive mesh for a Cu nanoparticle with spatial refinement around the Cu atoms and coarse-graining away from the atoms. In addition to higher-order finite-elements, spatial adaptivity can be leveraged to further reduce the dimensionality of the finite-element subspace to achieve the desired accuracy. In particular, spatial adaptivity can significantly aid computational efficiency of all-electron DFT calculations where the solution to the Kohn–Sham problem can be sharply varying. Pseudopotential calculations involving transition metals, where electrons in the penultimate shell are also treated as valence electrons, can also benefit from spatial adaptivity of finite-element discretization. Further, spatial adaptivity can provide a substantial benefit in reducing the number of basis functions for non-periodic problems such as clusters of atoms as evidenced by the results in Table 12.2—8 × reduction in the basis functions, in comparison to a uniform mesh—which, in turn, translates to improved computational efficiency. The spatial adaptivity is realized via a priori and a posteriori mesh adaption strategies based on error estimates obtained from numerical analysis of the finite-element discretization of the Kohn–Sham problem. We refer to Chap. 10 by Dai and Zhou in this volume for a detailed discussion on the finite-element error estimates for the Kohn–Sham DFT problem, and refer to [12, 14, 53, 57] for the mesh adaption strategies proposed in the context of the Kohn–Sham problem.

2 parts. A, there are overlapping layers of grid planes with different-sized cells. The size of each cell is half the size of the cell of the next bigger grid. B has a dot surrounded by 10 dots and multiple dots are clustered to form a circular pattern. The far-end dots are blurred and the central dot is focused.

Fig. 12.4

(Left) Spatially adaptive mesh on the mid-plane of a Cu icosahedral nanoparticle (309 atoms, 5871 e-); (Right) Electron-density contours for the nanoparticle. Results obtained using DFT-FE code [53]

Table 12.2

Comparison of uniform and spatially adaptive finite-element discretizations for Cu icosahedral nanoparticle (309 atoms, 5871 e-). The discretizations are chosen such that basis discretization errors in ground-state energy and forces are under 0.1 mHa/atom and 0.1 mHa/Bohr, respectively. The simulations are performed on NERSC-Cori. Results obtained using DFT-FE code [53]

FE mesh

# Basis functions

Energy (Ha/atom)

CPU-time

Uniform (FE order = 6)

81,182,737

−1.82590939e+02

16.33 node-hrs

Adaptive (FE order = 6)

9,804,717

−1.82590932e+02

1.94 node-hrs

12.4.3 DFT-FE: A Massively Parallel Code for Real-Space Finite-Element DFT Calculations

In addition to systematic convergence and being amenable to spatial adaptivity, the finite-element basis also has potential for excellent parallel scalability owing to the locality of the basis. Further, the data structures inherent to the finite-element basis make it amenable to GPU acceleration to take advantage of the hybrid CPU-GPU computing architectures. The recent development of DFT-FE [53], a massively parallel open-source code for Kohn–Sham DFT calculations using adaptive higher order finite-element discretization, is an effort in the direction of enabling fast and accurate large-scale DFT calculations. The ionic forces and stresses in DFT-FE are computed via configurational forces corresponding to inner variations of the Kohn–Sham variational problem [55]. Recent benchmark studies [53] have shown that DFT-FE outperforms state-of-the-art plane-wave codes in computational efficiency for systems containing a few thousand electrons, and beyond. Further, the parallel scalability of DFT-FE and the GPU acceleration [13] have enabled fast DFT calculations with wall-times of a few seconds per self-consistent field (SCF) iteration—the eigenvalue problem corresponding to the inner minimization problem in Eq. (12.9) (cf. Sect. 12.2.8)—on systems containing ∼30, 000 electrons. Figure 12.5a shows the comparison of minimum wall-times for an SCF iteration achieved² using DFT-FE and Quantum Espresso (QE)—a widely used state-of-the-art plane-wave DFT code—on the NERSC Cori supercomputer, for a benchmark system containing Mo supercells with a monovacancy (periodic calculation). In addition, the minimum wall-times for DFT-FE on the Summit supercomputer using GPUs are also provided. These benchmark results suggest that, by exploiting the parallel scalability and the GPU acceleration, DFT-FE can provide a ∼100 × boost over QE. Figure 12.5b shows the electron density contours of the pyramidal II dislocation in Mg computed using DFT-FE, with the calculation representing a fully resolved defect core containing ∼6000 atoms (∼60, 000 electrons). These recent developments have provided the capability to conduct fast and accurate fully resolved DFT calculations containing 10,000s of electrons that enables an efficient and accurate treatment of the defect core.

2 parts. A, a grouped bar graph of minimum wall time per S C F iteration versus 3 numbers of electrons, 6034, 14322, and 27986 plots 3 bars for quantum espresso of C P U, N E R S C cori, D F T F E of C P U, N E R S C cori, and D F T F E of G P U, O L C F summit. Quantum espresso has the highest value in all 3 numbers of electrons. B, color contoured square with multiple tiny dots of different sizes has the lowest value in the center. The dots are scattered evenly. The electron density scale is on the right.

Fig. 12.5

(Left) Wall-time comparison of DFT-FE and Quantum Espresso on NERSC-Cori and Summit supercomputers for benchmark systems comprising of a mono-vacancy in Molybdenum supercells with varying system sizes. The wall-time for DFT-FE on Summit is using GPUs. All benchmark calculations have been conducted using ONCV pseudopotentials with discretization errors commensurate with chemical accuracy, 0.1 mHa/atom in ground-state energy and 0.1 mHa/Bohr in forces. (Right) Electron density contour of pyramidal II screw dislocation system in Mg, with the fully resolved defect core containing 6164 Mg atoms (61,640 electrons) [13]

The spatial adaptivity of the finite-element basis in DFT-FE has enabled systematically convergent pseudopotential and all-electron calculations in the same framework. While pseudopotential calculations have been the workhorse of DFT calculations owing to their computational efficiency, there are many scenarios where all-electron calculations are indispensable—such as material properties under extreme environments, prediction of ionization potentials, magnetizability and spectroscopic properties. In particular, a systematically convergent approach for computing the spin Hamiltonian parameters that are crucial to understanding the properties of spin defects in semiconductors—promising quantum bits for quantum computing—was lacking, until recently. The systematic convergence of the finite-element basis for all-electron calculations in DFT-FE has filled this gap [18]. Further, as pseudopotential and all-electron calculations are treated using the same framework, this has opened the possibility of mixed all-electron and pseudopotential calculations, where only a subset of atoms are treated using all-electron accuracy, while other atoms are treated using a pseudopotential approximation. These mixed calculations have enabled the calculation of spin Hamiltonian parameters of spin defects with all-electron accuracy using simulation domains that provide cell-size converged properties [19].

12.4.4 Enriched Finite-Element Bases

The finite-element basis with spatial adaptivity provides a systematically convergent approach for conducting all-electron calculations. However, numerical studies have shown that, despite using higher order adaptive finite-elements, they require a substantially larger number of basis functions than atomic orbital type basis functions or approaches such as APW, LAPW and LAPW+lo (see Chap. 9 of this volume, by Chen and Schneider). This limitation of the finite-element basis can be mitigated by using augmentation techniques in the finite element basis (similar to augmentation in the plane-wave basis), where the finite element basis is enriched with compactly supported atomic orbitals [40, 71], or via the partition of unity finite element method [2, 61]. We note that these augmentation techniques are in the spirit of coarse-graining presented in this chapter, where enrichment functions capturing the known oscillatory behavior of wavefunctions near the atom are used to numerically coarsegrain the higher-order adaptive finite element basis. Table 12.3 shows the comparison of the (classical) finite-element basis with an enriched finite-element basis in terms of basis functions required to achieve chemical accuracy, and the respective computational times, for all-electron calculations of Si nanoclusters. As is evident, there is ∼30 × reduction in the finite-element basis functions using enrichments, and this translates to a staggering ∼100 × improvement in computational efficiency. Table 12.4 shows the comparison with Elk code—a state-of-the-art LAPW code—on the accuracy and computational efficiency afforded by an enriched finite-element basis for all-electron periodic calculations on supercells of NV Diamond. For a more comprehensive discussion on the convergence properties of enriched finite-element bases, accuracy, computational efficiency and scalability of parallel implementation, we refer to recent works [40, 71] where benchmark all-electron calculations on systems containing up to ∼10, 000 electrons are reported.

Table 12.3

Comparison of classical and enriched finite element (FE) bases: Energy per atom (E in Ha), degrees of freedom per atom (DoF), and the computational CPU time (in node-hours) for various silicon nanoclusters [40]

Classical FE

Enriched FE

Si 1 × 1 × 1 (252 e-)

E

− 288.320035

− 288.319450

DoF

402, 112

14, 728

CPU Hrs

66.63

1.03

Si 2 × 2 × 2 (1, 330 e-)

E

− 288.359459

− 288.359266

DoF

360, 467

10, 642

CPU Hrs

3164

23.1

Table 12.4

Comparison of the ground-state energy per atom (E) in Ha and computational CPU time (C) in node-hours of various NV-diamond supercells, using enriched FE (EFE) and LAPW+lo bases. All reported energies are evaluated at Γ-point [71]

Supercell

Atoms (Electrons)

EFE (E)

LAPW+lo (E)

EFE (C)

LAPW+lo (C)

2 × 2 × 2

63 (379)

− 38.0520

− 38.0522

0.19

0.32

3 × 3 × 3

215 (1291)

− 37.8716

− 37.8720

1.6

15.1

4 × 4 × 4

511 (3067)

− 37.8276

–

16.1

–

12.5 Spectral Coarse-Graining: The Spectral Quadrature Method

In the previous section, we exploited spatial coarse-graining for numerical efficiency and consideration of large-scale materials systems. In this section, we discuss another aspect of coarse-graining that concerns the eigenspectrum to enable even larger systems. In particular, the quantities of interest in the Kohn–Sham problem can be directly evaluated without having to calculate all the occupied eigenvalues and corresponding orbitals of the Hamiltonian, a strategy that we refer to as spectral coarse-graining. One such technique is the recently proposed Spectral Quadrature (SQ) method [68, 75, 78, 79], which is the focus here. Notably, the SQ method allows the development of the infinite-cell approach [75, 78], which enables non-traditional boundary conditions [20], an important aspect for the study of crystal defects.

For simplicity, let us suppose that the Hamiltonian operator is discretized using an orthogonal basis that satisfies the Kronecker-delta property. Though we make this assumption, the discussion that follows can be easily generalized. In particular, we describe the calculation of the diagonal components of the density matrix, band structure energy, and electronic entropy—quantities that need to be determined from the inner variational problem (12.14) or linear eigenvalue problem arising in each SCF iteration—using the Gauss and Clenshaw–Curtis variants of the SQ method. Indeed, the electronic ground state energy can be determined using the knowledge of these quantities. The off-diagonal components of the density matrix are also available, and these are needed to calculate the Hellmann–Feynman atomic forces [68, 79] and stress tensor [72].

12.5.1 Spectral Integrals and Quadrature

We start by rewriting the expression for the density matrix:

where the shifted and scaled quantities are as follows:

Above, I denotes the identity matrix of size indicated by the subscript, and the shift and scale factors are:

Next, analogous to their continuous versions in Sect. 12.2.7, the constraint on the number of electrons, electron density, band structure energy, and electronic entropy can be written in the discrete setting as [26, 75, 78]:

(12.25)

(12.26)

(12.27)

(12.28)

where ρn denotes the nth component of ρ and en denotes the standard basis vector. Note that it is from (12.25) that the scaled chemical potential $$\widehat {\mu }$$ is determined, which can then be used for the calculation of the electron density (12.26), band structure energy (12.27), and electronic entropy energy (12.28). Also, note that we have dropped the subscript β in the band structure energy U, for simplicity of notation.

The key idea underlying the SQ method is the approximation of the integrals arising in the definition of the above quantities using a quadrature rule:

(12.29)

where g is any one of the functions arising in the integrals presented in (12.25–12.28), and $$\{ \widehat {\varepsilon }^{n}_j \}_{j=1}^{k}$$ and $$\{\widehat {w}_j^n\}_{j=1}^{k}$$ are the nodes and weights of the quadrature rule, respectively. Among the various quadrature schemes possible, Gauss and Clenshaw–Curtis quadrature present themselves as attractive choices [81, 82], whose evaluation in the current spectral setting is described in Sects. 12.5.3 and 12.5.4, respectively. In order to evaluate these quadrature rules efficiently and make them more amenable to spatial coarse graining, it is common to employ spatial localization, as described in Sect. 12.5.2.

Remark

The SQ method does not require computation of the eigenvalues and eigenvectors of the Hamiltonian $$\widehat {H}$$ and uses (12.29) instead, for which we note the connection. The measure $$\mathcal {E}_{n}$$ may be written as:

script upper E Subscript n Baseline left parenthesis ModifyingAbove epsilon With caret right parenthesis equals StartLayout Enlarged left brace 1st Row 1st Column 0 comma 2nd Column if ModifyingAbove epsilon With caret less than ModifyingAbove epsilon With caret Subscript 1 Baseline equals negative 1 2nd Row 1st Column sigma summation Underscript i equals 1 Overscript m Endscripts phi Subscript i comma n Superscript 2 Baseline comma 2nd Column if ModifyingAbove epsilon With caret Subscript m Baseline less than or slanted equals ModifyingAbove epsilon With caret less than ModifyingAbove epsilon With caret Subscript m plus 1 Baseline comma 3rd Row 1st Column sigma summation Underscript i equals 1 Overscript upper N Subscript g Baseline Endscripts phi Subscript i comma n Superscript 2 Baseline comma 2nd Column if 1 equals ModifyingAbove epsilon With caret Subscript upper N Sub Subscript g Subscript Baseline less than ModifyingAbove epsilon With caret EndLayout

(12.30)

where φi,n denotes the nth component of φi. In using (12.29), the SQ method avoids the calculation of the eigenvalues and eigenvectors of the Hamiltonian, thereby circumventing the bottleneck encountered in traditional diagonalization-based Kohn–Sham DFT calculations.

12.5.2 Spectral Integrals and Quadrature with Spatial Localization

To significantly reduce the computational cost as well as make the quantities amenable to coarse-graining, we now introduce spatial localization by taking advantage of the nearsightedness of electronic correlations, i.e., exponential decay of the density matrix for metals at nonzero smearing values as well as insulators [5, 69, 77]. To do so, we introduce the ‘nodal’ density matrices [68, 79]

where

with

Above, Hn is the submatrix of the Hamiltonian H formed by spatially localizing it around the point of interest, i.e., a matrix formed by the $$N_g^n$$ rows and columns of H that are ‘near’ the nth row and column. In addition, $$\varepsilon _m^n \, , \, m=1, \ldots , N_g^n \, ,$$ are the eigenvalues of Hn. Thereafter, we approximate the constraint on the number of electrons, electron density, band structure energy, and electronic entropy given in Eqs. 12.25–12.28 as:

(12.31)

(12.32)

(12.33)

StartLayout 4th Row 1st Column upper S 2nd Column almost equals 2 sigma summation Underscript n equals 1 Overscript upper N Subscript g Baseline Endscripts e Subscript s Superscript normal upper T Baseline left bracket upper P Superscript n Baseline log upper P Superscript n Baseline plus left parenthesis upper I Subscript upper N Sub Subscript g Sub Superscript n Subscript Baseline minus upper P Superscript n Baseline right parenthesis log left parenthesis upper I Subscript upper N Sub Subscript g Sub Superscript n Subscript Baseline minus upper P Superscript n Baseline right parenthesis right bracket e Subscript s Baseline 5th Row 1st Column Blank 2nd Column equals 2 sigma summation Underscript i equals 1 Overscript upper N Subscript g Baseline Endscripts integral Subscript negative 1 Superscript 1 Baseline left bracket f Subscript ModifyingAbove beta Sub Superscript n Subscript With caret Baseline left parenthesis ModifyingAbove epsilon With caret comma ModifyingAbove mu With caret right parenthesis log f Subscript ModifyingAbove beta With caret Sub Superscript n Subscript Baseline left parenthesis ModifyingAbove epsilon With caret comma ModifyingAbove mu With caret right parenthesis plus left parenthesis 1 minus f Subscript ModifyingAbove beta Sub Superscript n Subscript With caret Baseline left parenthesis ModifyingAbove epsilon With caret comma ModifyingAbove mu With caret right parenthesis right parenthesis log left parenthesis 1 minus f Subscript ModifyingAbove beta Sub Superscript n Subscript With caret Baseline left parenthesis ModifyingAbove epsilon With caret comma ModifyingAbove mu With caret right parenthesis right parenthesis right bracket normal d script upper E Subscript s Superscript n Baseline left parenthesis ModifyingAbove epsilon With caret right parenthesis comma EndLayout

(12.34)

where es denotes the standard basis vector corresponding to the node of interest in the truncated Hamiltonian, i.e., the row and column corresponding to the node around which spatial truncation has been performed.

We now proceed to approximate the integrals arising in the definition of the above quantities using a quadrature rule:

(12.35)

where g is any one of the functions arising in the integrals presented in (12.32–12.34), and $$\{ \widehat {\varepsilon }^{n}_j \}_{j=1}^{k}$$ and $$\{\widehat {w}_j^n\}_{j=1}^{k}$$ are the nodes and weights of the quadrature rule (dropped index s, for simplicity of notation). Specifically, we describe the evaluation of the Gauss and Clenshaw–Curtis spectral quadrature rules in Sects. 12.5.3 and 12.5.4, respectively.

Remark

The measure $$\mathcal {E}^{n}_s$$ can be written as:

script upper E Subscript s Superscript n Baseline left parenthesis ModifyingAbove epsilon With caret right parenthesis equals StartLayout Enlarged left brace 1st Row 1st Column 0 comma 2nd Column if ModifyingAbove epsilon With caret less than ModifyingAbove epsilon With caret Subscript 1 Superscript n Baseline equals negative 1 2nd Row 1st Column sigma summation Underscript i equals 1 Overscript m Endscripts left parenthesis phi Subscript i comma s Superscript n Baseline right parenthesis squared comma 2nd Column if ModifyingAbove epsilon With caret Subscript m Superscript n Baseline less than or slanted equals ModifyingAbove epsilon With caret less than ModifyingAbove epsilon With caret Subscript m plus 1 Superscript n Baseline comma 3rd Row 1st Column sigma summation Underscript i equals 1 Overscript upper N Subscript g Superscript n Baseline Endscripts left parenthesis phi Subscript i comma s Superscript n Baseline right parenthesis squared comma 2nd Column if 1 equals ModifyingAbove epsilon With caret Subscript upper N Sub Subscript g Sub Superscript n Subscript Superscript n Baseline less than ModifyingAbove epsilon With caret EndLayout

where $$\varphi _{i}^n$$ denote the eigenvectors of the truncated Hamiltonian Hn. As stated previously, the SQ method does not require the calculation of the measure $$\mathcal {E}^{n}_s$$ explicitly, thereby avoiding the need to calculate the eigenvalues and eigenvectors of the truncated Hamiltonians Hn, n = 1, …, Ng, resulting in significant computational savings.

12.5.3 Gauss Spectral Quadrature

To generate the Gauss SQ rule for the integral in (12.35), we use the Lanczos type iteration [26, 75, 78]

(12.36)

where

and $$b_j^n$$ is computed such that $$ (v_j^{n})^{\mathrm {T}} v_j^{n} =1 \, , \, j=0, \ldots , k-1$$ . Subsequently, we form the symmetric tridiagonal Jacobi matrix:

(12.37)

whose eigenvalues and squares of the first elements of the normalized eigenvectors are the nodes $$\{ \widehat {\varepsilon }^{n}_j \}_{j=1}^{k}$$ and weights $$\{\widehat {w}_j^n\}_{j=1}^{k}$$ of the quadrature rule, respectively. To show this result, the above procedure can be viewed as first performing the following decomposition of the nodal Hamiltonian:

where $$V_k^n$$ is a matrix with the j + 1 column being the vector $$v_j^n$$ generated during the Lanczos iteration in (12.36). Thereafter,

where $$\{ \widehat {\varepsilon }^{n}_j \}_{j=1}^{k}$$ and $$\{\widehat {w}_j^n\}_{j=1}^{k}$$ are the eigenvalues and squares of the first elements of the normalized eigenvectors of $$J_k^n$$ , respectively. Note that the nodes and weights are independent of the function being integrated within the above scheme.

In Gauss SQ, the constraint on the number of electrons, electron density, band structure energy, and electronic entropy can then be written as:

Since the nodes and weights are independent of the Fermi level, they do not need to be recomputed for the different quantities above, nor do they need to be recomputed for the different guesses for the Fermi level in solving for the constraint on the number of electrons.

In cases where the off-diagonal components of the density matrix are required, e.g., the computation of Hellmann–Feynman atomic forces and stress tensor, the nth column of the density matrix can be obtained using the relation:

Indeed, all these quantities are already computed as part of the above procedure, and so do not introduce any additional cost.

Relation to the Recursion Method and Padé Approximation

The spectral Gauss SQ method bears a resemblance to the recursion method [30] that had been developed in the context of the tight binding method. To see this, we note the relation [28]:

where $$i=\sqrt {-1}$$ , and ∮C represents a contour that encloses the spectrum of $$\widehat {H}^n$$ in the complex plane, from which it follows:

(12.38)

In the current framework, the recursion method involves using the following approximation:

StartLayout 1st Row e Subscript s Superscript normal upper T Baseline left parenthesis z upper I Subscript upper N Sub Subscript g Sub Superscript n Subscript Baseline minus ModifyingAbove upper H With caret Superscript n Baseline right parenthesis Superscript negative 1 Baseline e Subscript s Baseline almost equals StartStartStartFraction 1 OverOverOver z minus a 1 Superscript n Baseline minus StartStartFraction left parenthesis b 1 Superscript n Baseline right parenthesis squared OverOver z minus a 2 Superscript n Baseline minus midline horizontal ellipsis minus StartFraction left parenthesis b Subscript k minus 1 Superscript n Baseline right parenthesis squared Over z minus a Subscript k Baseline EndFraction EndEndFraction EndEndEndFraction equals StartFraction q Subscript k Superscript n Baseline left parenthesis z right parenthesis Over p Subscript k Superscript n Baseline left parenthesis z right parenthesis EndFraction period EndLayout

In particular, the continued fraction above is used within the integral of (12.38) to evaluate the quantity of interest. Since the rational function has zeros of $$p_k^n(z)$$ , a number of techniques to smoothen it have been developed [30]. It can however be shown that [78]:

which when substituted into (12.38) along with the spectral theorem recovers the Gauss SQ quadrature rule:

Note that the rational functions satisfy the following best approximation property [74]:

which make them the Padé approximants. Indeed, it can be shown from the above equation—multiplying both sides by a polynomial of degree 2k − 1 and integrating along a contour encircling the real line [87]—that polynomials of degree 2k − 1 are integrated exactly using the above quadrature rule, as is the property of Gauss quadrature.

12.5.4 Clenshaw–Curtis Spectral Quadrature

In Clenshaw–Curtis SQ [68, 75, 79], rather than determining quadrature weights corresponding to the quadrature nodes (zeros of the Chebyshev polynomials), it is advantageous to perform the following expansion in terms of Chebyshev polynomials:

where the summation with a prime indicates that the first term is halved, and the Chebyshev coefficients are given by

(12.39)

We can then write

where $$t_j^i$$ are evaluated from the three-term recurrence relation:

(12.40)

In Clenshaw–Curtis SQ, the constraints on the number of electrons, electron density, band structure energy, and electronic entropy take the form:

Note that in cases where the off-diagonal components of the density matrix are required, the nth column of the density matrix can be obtained using the relation:

Indeed, all these quantities are already computed as part of the above procedure, and so do not incur any additional cost.

Relation to Fermi Operator Expansion (FOE)

The Clenshaw–Curtis quadrature bears a resemblance to the classical Fermi Operator Expansion (FOE) [24, 25]. In particular, the FOE method employs the following expansion of the density matrix in terms of Chebyshev polynomials:

where the matrices $$T_j(\widehat {H})$$ are evaluated using the three-term recurrence relation:

In order to achieve linear scaling with system size, truncation is introduced into the matrix-matrix multiplication routines. In spite of the similarity of this approach with Clenshaw–Curtis SQ, there are a number of key differences. First, compared to the sparse matrix-vector routines in Clenshaw–Curtis SQ, the operations involved in FOE are sparse matrix-matrix routines, which are challenging to write, particularly for efficient scaling to a large number of processors. Second, the effect of truncation is not automatically incorporated into FOE, as it is done in Clenshaw–Curtis SQ. Third and finally, since the Chebshev matrices cannot be generally stored, an outer loop on the Fermi level is required, which makes the FOE significantly more costly as well.

12.5.5 Convergence Rates

In the SQ method, the error with respect to the quadrature order decays as [75]:

where

(12.41)

is the rate of convergence. Here, nq = 1 and nq = 2 for the Clenshaw–Curtis and Gauss SQ methods, respectively. In addition, r is the sum of the semi-major and semi-minor axes for the largest ellipse in the complex plane where the function g is analytic. In the current context, the closest singularity of the Fermi–Dirac function $$f_{\widehat {\beta }^n}$$ to the interval [−1, 1] is at

The corresponding ellipse is as shown in Fig. 12.6, for which we have:

(12.42)

A 4-gradient graph plots an ellipse for z = x + i y, at the center. 2 lines, d subscripts 1 and 2 from (negative 1, 0) and (1, 0) meet at the point (mu hat, pi over beta hat power n) on the ellipse in the second quadrant.

Fig. 12.6

Largest ellipse in the complex plane where the Fermi–Dirac function $$f_{\widehat {\beta }^n}$$ is analytic

Performing a first-order Taylor series expansion for the convergence rate α about $$1/\widehat {\beta }^n \rightarrow 0 $$ , we obtain

(12.43)

This expression represents a very good approximation for practical DFT calculations, since the spectral width of the Hamiltonian (2ξ) is generally large and the smearing ( $$1/\widehat {\beta }^n$$ ) used for ambient conditions is typically small.

Although the above error estimates are also valid for insulating systems, the bounds are not expected to be tight, especially as the smearing becomes smaller. In fact, it is common to not use any smearing for insulators, i.e., $$1/\widehat {\beta }^n \rightarrow 0$$ . It has been predicted that for an insulating system with band-gap Eg and smearing $$1/\widehat {\beta }^n \rightarrow 0$$ [75]:

(12.44)

Above, the Fermi level has been assumed to be in the middle of the band-gap and $$\widehat {E}_g = E_g/\xi $$ .

We now compare the predicted convergence rate with that obtained numerically within a DFT calculation. Specifically, we consider a 107-atom system consisting of a vacancy in face-centered cubic (FCC) aluminum. We choose a smearing of 1 eV, commensurate with that adopted for metallic systems in practical Kohn–Sham calculations. In Fig. 12.7, we plot the convergence in electron density with quadrature order for a specific point in space, while choosing a large enough truncation radius, so as to put associated errors well below the quadrature errors of interest. All simulations are performed using the real-space Kohn–Sham DFT code SPARC [21, 22, 94], in which the SQ method has been recently implemented.

A connected line graph of error in electron density versus Gauss S Q order plots 2 decreasing parallel lines that are long and short. The alpha value is 0.0671.

Fig. 12.7

Convergence in electron density with Gauss SQ order at a spatial point. The thick red line represents the theoretically predicted convergence rate. The system under consideration is a 107-atom system consisting of a vacancy in FCC aluminum, with smearing of 1 eV

12.5.6 Scaling Estimates

The computational cost of the Gauss and Clenshaw–Curtis SQ methods is dictated by the cost of the matrix-vector products appearing in the iteration described by Eqs. 12.36 and 12.40, respectively. Given the sparse nature of $$\widehat {H}^{n}$$ , the cost of each matrix-vector product scales as $$\mathcal {O}(N_g^n)$$ . Since there are k such matrix-vector products in the iteration and n ranges from 1 to Ng, the total computational cost scales as $$\mathcal {O}(N_g^n k N_g)$$ . As can be seen from the theoretical results presented above, the quadrature order k required for a certain accuracy is independent of the number of grid points Ng. Moreover, for large enough system sizes, $$N_g^n$$ is also independent of Ng. Therefore, the scaling of the SQ method is $$\mathcal {O}(N_g)$$ , which makes it $$\mathcal {O}(N)$$ with the number of electrons in the system, i.e., linear scaling with system size. Therefore, the cubic scaling bottleneck inherent to traditional diagonalization approaches can be overcome using the SQ method, enabling the study of large system sizes that were previously intractable. Note that the unlike orbital-based diagonalization and linear scaling approaches, the cost of the SQ method decreases with increasing temperature [68, 77], making it ideal for the study of materials under extreme conditions [7, 92, 96].

12.5.7 Numerical Results

We now study the accuracy and efficiency of the aforedescribed Gauss and Clenshaw–Curtis SQ methods. As a representative example, we choose an unrelaxed vacancy in FCC aluminum, which is modeled by removing a single atom within a supercell of FCC aluminum.

In Fig. 12.8, considering a 107-atom system, we plot the convergence of the ground state energy, Hellmann–Feynman atomic forces, and Hellman–Feynman stress tensor with quadrature order and truncation radius, which are the two new parameters introduced within the SQ method. Note that we employ Gauss SQ for the calculation of the electron density and energy in each SCF iteration, and Clenshaw–Curtis SQ for the atomic forces and stress tensor. It is clear that there is systematic geometric convergence in all quantities, demonstrating the accuracy of the SQ method.

2 connected line graphs of error in energy and force and error in stress axis versus quadrature order and truncation radius plot 3 decreasing lines for energy, force, and stress. In both, the line for stress is above the other lines, followed by force.

Fig. 12.8

Convergence of the energy, atomic forces, and stress tensor with quadrature order and truncation radius for the SQ method as implemented in the SPARC code. The system under consideration is a 107-atom system containing a vacancy with smearing of 1 eV. Gauss SQ has been used for the energy, and Clenshaw–Curtis SQ is used for the force and stress. The error in force and stress correspond to the maximum difference in any component. (a) Convergence with quadrature order. (b) Convergence with truncation radius

In Fig. 12.9, we plot the strong and weak parallel scaling of the SQ method, as implemented in the SPARC code [21, 22, 94]. All parameters, including mesh-size, quadrature order and truncation radius have been chosen so that the error in energy and force are within 0.001 Ha/atom and 0.001 Ha/Bohr, numbers that are representative of the accuracy targeted in typical DFT simulations. For the strong scaling, we use a 107-atom system, while increasing the number of processors from 24 to 960. For the weak scaling, we increase the system size from 107 to 10,975, while proportionally increasing the processors from 27 to 2744. It is clear that the SQ method demonstrates excellent strong and weak scaling, enabling the study of large systems needed in the study of crystal defects.

2 connected line graphs of wall time versus the number of processors and atoms. A plots a decreasing slant line. In B, the line is almost horizontal with minor fluctuations.

Fig. 12.9

Strong and weak scaling of the SQ method, with timings reported for a single SCF iteration. The system under consideration for strong scaling is a 107-atom system representing a vacancy in aluminum. The systems for weak scaling are larger and larger supercells, each with a vacancy. (a) Strong scaling. (b) Weak scaling

12.6 Spatial and Spectral Coarse-Graining

In this section, we combine spatial and spectral coarse-graining to enable very large scale studies of defects in crystalline materials. It exploits the nature of defects where the electronic and atomistic fields decay away from the defect to those associated with a periodic crystal to construct a controlled approximation to DFT.

12.6.1 Periodic Systems

The presentation in Sect. 12.2 can be adapted to the periodic setting. The complete basis consists not only of periodic functions but Bloch–Floquet waves. Consequently the orbitals, the operator γ and the partition of unity $${\mathcal E}$$ are not periodic (i.e., (r, r′)↦γ(r, r′) is not periodic). This leads to plane waves and k-point sampling. However, the measure $$\mathcal {M} = \text{tr }{\mathcal E}$$ is periodic.

However, and this is the key observation, the densities ρ, u, s are in fact periodic since they depend on the trace of $$\mathcal E$$ . It also follows that the dual variables, the electrostatic potential ϕ and exchange correlation potential Vxc are also periodic.

12.6.2 Coarse-Grained Representation

We consider a Bravais lattice first, and then describe the extension to other lattices.

12.6.2.1 Atoms

Consider a crystalline solid whose crystal structure is given by a Bravais lattice. Introduce a defect at the origin (e.g., a vacancy cluster by removing a cluster of atoms at the origin) and consider the restriction of the lattice (with a defect) to a simply connected domain $${\mathcal D}$$ . Let $$\{x_m\}_{m=1}^M$$ denote the positions of the atoms and pick these to be the reference configuration. There are unbalanced forces on the atoms near the core and they deform. We are interested in finding the deformed positions $$\{r_m\}_{m=1}^M$$ of these atoms. We can find a smooth deformation $$y: {\mathcal D} \to {\mathbb R}^3$$ such that rm = y(xm), m = 1, …, M. We expect the displacements y(x) − x to be large and oscillate on a fine scale (that of the lattice) near the core (origin), but vary smoothly on the scale of the lattice and decay as we go away from the defect. Thus, we need a fine discretization near the defect core, but can coarsen as we move away.

Therefore, we use a quasi-continuum approximation [17, 41, 66, 67, 78, 80] to represent the positions of the atoms. We consider a subset of atoms $${\mathcal P}_a$$ we call the representative atoms, and introduce a Lagrangian triangulation $${\mathcal T}_a$$ with the representative atoms as nodes. We track the position of the representative atoms $$\{r_a\}_{a=1}^A$$ and represent the positions of the remaining atoms using the interpolation $$\Gamma ^a_{an}$$ induced by the triangulation $${\mathcal T}_a$$ :

(12.45)

We pick $${\mathcal P}_a$$ to be dense near the core and gradually coarsen away from it.

12.6.2.2 Electronic Fields

We now turn to the electronic fields—electron density, electrostatic potential—for the specimen of a crystalline solid with a defect at its center. Consider a region distant from the defect where the deformation is smooth and the deformation gradient F is uniform on a scale large compared to the lattice: i.e., F = O(1) and ∇F = O(a∕L) where a is a typical lattice spacing and L is the radius of the computational domain. The atomic positions are periodic to a good approximation, and we expect the electronic fields to be periodic to a good approximation in that region due to the short-sightedness of electronic matter [42]. In other words, for an electronic field $${\mathcal Q}$$ of interest, we expect

where $$\widetilde {\mathcal Q} (\cdot , z)$$ is periodic with the periodicity of the reference unit cell. In other words, we expect

where $${\mathcal Q}^p (r) = \widetilde {\mathcal Q} \left (\frac {a}{L}r, F^{-1}(r) \right )$$ and $${\mathcal Q}^c(r)$$ decays smoothly for large r. The idea then is to represent $${\mathcal Q}^p$$ (the projection onto continuous functions of piecewise periodic functions) and $${\mathcal Q}^c$$ on a grid that is fine near the defect core and coarsens away from it. We call $${\mathcal Q}^p$$ the predictor and $${\mathcal Q}^c$$ the corrector.

We achieve this representation using two spatial meshes. The first is the fine electronic mesh $${\mathcal P}_f$$ , which is a uniform finite difference mesh. We use this to represent the Hamiltonian and in our Lanczos algorithm. The second is the coarse electronic mesh $${\mathcal P}_c$$ , which is a subset of the fine electronic mesh $${\mathcal P}_f$$ . We compute the electronic quantities on this mesh and therefore call the elements of $${\mathcal P}_c$$ the electronic sampling points (ESPs). As with the atomistic grid, the coarse grid $${\mathcal P}_c$$ is fine (includes all points in $${\mathcal P}_c$$ ) close to the defect but gradually coarsens away.

We represent an electronic field $${\mathcal Q}$$ as follows. First, we define the predictor. Recall that the deformation (12.45) is affine in each element of the Lagrangian atomistic triangulation $${\mathcal T}_a$$ , and that it convects the reference lattice to a deformed periodic lattice. We perform a unit cell calculation based on this deformed periodic lattice in each element Ωe of the $${\mathcal T}_a$$ to obtain the electron density $${\mathcal Q}_e(y)$$ on the image $$\bar {y}(\Omega _e)$$ of the element, and define the predictor as the L² → H¹ projection of this piecewise periodic function

where yf is the position of the fth node of $${\mathcal P}_f$$ and $$\chi _{\mathcal A}$$ is the characteristic function of a set $${\mathcal A}$$ .

We now turn our attention to the corrector. Let $${\mathcal Q}_c$$ the quantity of interest at an ESP labelled c. We define the corrector at the ESP as the difference between the computed electron density and predictor:

We then extend the definition of the corrector to the fine grid $${\mathcal P}_f$$ through interpolation:

where Γc is the interpolation associated with the triangulation induced by $${\mathcal P}_c$$ . In summary, we represent the electron density as

(12.46)

While we have the representation on the fine grid, we do not need to evaluate the quantities on the fine grid. Since we seek to perform the Lanczos procedure only at the ESPs, we need the Hamiltonian in a sufficiently large neighborhood of each ESP. Therefore, we create clusters of fine grid points around each ESP, and collect these points into the set $${\mathcal P}_f^{\text{eval}}$$ . We evaluate the electronic quantities only on $${\mathcal P}_f^{\text{eval}}$$ . Note that $${\mathcal P}_f^{\text{eval}}$$ is fully dense near the core, but becomes sparse as we go away.

Finally, to compute global quantities like energy, we have to compute sums like

We do so following the cluster summation approach of Knap and Ortiz [41] using $${\mathcal P}_c$$ and $${\mathcal P}_f^{\text{eval}}$$ (see [66] for details).

The overall approach is summarized in Algorithm 1.

Algorithm 1: Spatial and spectral coarse-grained approach

A 17-lined algorithm for spatial and spectral coarse-grained approach. Line 1 is blank. 2, given an initial configuration of atoms. Lines 3 to 16 include commands within nested loops of while, do, and end. Line 17 is blank.

12.6.2.3 Crystals

In the case of a crystal where one has more than one atom per unit cell, we limit the representative atoms to belong to the skeletal lattice as we coarsen, and use the periodic calculation within each element of $${\mathcal T}_a$$ to determine the positions of the other atoms in the unit cell.

12.6.3 Selected Results

We now demonstrate the approach using a few selected examples from magnesium, which forms a hexagonal close-packed (HCP) crystal structure. Magnesium and its alloys have received recent interest due to their high strength to weight ratio (with a density of 1.8 g/cm³ and yield strength exceeding 100 MPa), and have been explored for automotive, biomedical and other engineering applications. However, these alloys often have limited ductility and suffer sudden, almost brittle, failure. We refer the reader to recent reviews [39, 46, 47, 93]. Therefore the study of defects in magnesium and its alloys have been a topic of much recent interest.

These examples are drawn from [66, 67]. We take the exchange-correlation function to be the parametrized form of Perdew and Wang [64], and a local pseudopotential proposed by Huang and Carter [35]. We take σ = 0.8 eV corresponding to a temperature of 10, 000 K. We use a sixth-order finite difference stencil adopted to hexagonal symmetry that combines a triangular stencil on the basal plane with a normal stencil normal to it [15]. The energy and force convergence thresholds are 10⁻⁵ and 10⁻³ eV Å⁻¹ respectively.

Figure 12.10, adapted from [66], shows the capabilities of the proposed approach using a vacancy. Figure 12.10a shows the computed total energy of a series of calculations with various amounts of coarse-graining. The computational domain in each of these calculations consists of 93,312 atoms discretized with 8.1 × 10⁷ nodes. The six calculations have a progressively larger number of electronic sampling points: we see that the total energy converges at about 1.8 × 10⁵ electronic sampling points. In other words, a calculations with 1.8 × 10⁵ degrees of freedom is able to correctly reproduce the energy of a calculation with 8.1 × 10⁷ degrees of freedom, a saving factor of 440. Remarkably, this factor increases as the size of the computational domain increases since larger domains have larger regions of coarser discretization.

4 parts. A, a line graph of formation energy versus the number of E S Ps plots a logarithmically decreasing line for full resolution. B, a graph of log base 10 of time versus atoms plots dots in an increasing slant trend for macro D F T, a line over it for t = b M power A, and a steeply increasing line for linear scaling. C, a line graph of relative error versus index of spatial point plots horizontally fluctuating line for error %. D has a color-contoured concentric circle diagram on a color gradient grid plane. The inner circle has the lowest value and the outer circle has the highest value.

Fig. 12.10

Computational results from the study of a vacancy in HCP magnesium. (a) The computed total energy of a specimen with 93, 312 atoms discretized with 8.1 × 10⁷ nodes computed with various levels of coarse-graining. (b) Computational time vs domain size shows dramatic sublinear scaling. (c) Relative error between the electron density computed with and without coarse-graining at 1600 grid points. (d) The corrector electron density on the basal plane shows fine oscillations close to the vacancy. Reprinted from [66] with permission from Elsevier

Consequently one obtains dramatic sub-linear performance as shown in Fig. 12.10b. In this example, also with a vacancy, we see that the computational time t scales as a power law of the number of atoms M with an exponent 0.05 (t = bMa, a = 0.05) up to a billion atoms. Of course simplicity of the example where the defect is confined to a small area contributes to the remarkable sublinearity, but we expect at least square-root scaling in all examples of defects.

Importantly, this saving in computational cost does not come at the cost of accuracy. This is demonstrated in Fig. 12.10c. This shows the relative error $$(\rho _n^{\text{CG}} - \rho _n^{\text{full}})/\rho _n^{\text{full}}$$ at the nth grid point where $$\rho _n^{\text{CG}}$$ is the electron density computed by the coarse-grained approximation (by recourse or (12.46)) and $$\rho _n^{\text{full}}$$ is the electron density computed without any coarse-graining over about 1600 grid points. We observe that the relative error is less than 0.8% in any of these grid points. In fact the average and root-mean-square errors are 10⁻⁵ times the mean density.

This efficacy of the coarse-graining method shows that subgrid sampling can be effective away from the defects. However, the details are complex and important near the core and require full resolution. Figure 12.10d shows the corrector electron density on the basal plane in the vicinity of the vacancy. We see oscillations on a scale finer than the atomic spacing—these are the analogs of the Friedel oscillations on interfaces and contribute to the electronic character of the defects. Therefore, it is important to resolve these carefully. Further, they interact with the far field stresses, and one reason why the decay length of defects tend to be high and why defects require large computational cells.

We now turn to the importance of sufficiently large computational unit cells in accurately calculating the binding energy of a divacancy. The binding energy is the energy difference between two isolated vacancies and a divacancy complex. This is illustrated in Fig. 12.11, adapted from [66]: it shows the divacancy binding energy of various divacancy complexes computed with computational domains of varying sizes. We see that we need a sufficiently large computational domain with >10³ atoms to accurately predict the divacancy binding energy. Importantly, the result leads to qualitative differences: calculations with small computational domains incorrectly predict that some vacancies barely bind, while the large computational domains predict strong binding consistent with experimental observations [38, 50, 85, 89].

2 parts. A is a ball and stick diagram of a hexagonal prism. A triangle is present inside it. Some of the nodes are numbered and the node in the top base is labeled V. B, a connected line graph of divacancy binding energy versus the size of computational domain plots 6 logarithmically increasing lines for E subscript d b comma 1 to 6. The line for E subscript d b comma 1 is at the top.

Fig. 12.11

Computational results from the study of divacancies in HCP magnesium. (a) Various divacancy complexes: one vacancy is located at the site labeled V while the others are at sites labeled with numbers. (b) Computed divacancy binding energy of various divacancy complexes for computational domains of varying sizes. Reprinted from [66] with permission from Elsevier

The final example is adapted from Ref. [67] and concerns the study of dislocations. Recall that the elastic energy of a dislocation scales logarithmically with the size of the domain. Figure 12.12 shows the computed excess energy—the difference in total energy between a domain with a dislocation and a domain without for two types of screw dislocations for domains of various sizes. It shows that our coarse-grained DFT approach correctly predicts this elastic scaling. The details (see [67]) describe the core structure, and the intercept at r = r₀ provides the “core energy”.

2 graphs of excess energy versus log base 10 of r by r naught plot dots in an increasing slant trend for M D F T prediction and a line over it for an elastic solution. A is for screw basal and B is for screw prismatic. They include 2 diagrams of a cluster of tiny cylindrical objects forming a circle of radius R and a dot and stick diagram of a hexagonal prism with a dot in the center of the base.

Fig. 12.12

Computational results from the study of screw dislocations in HCP magnesium. Scaling of the excess energy computed with various domain sizes for (a) basal $$ \frac {a_0}{3} {}[{}11 \bar {2}0{}]{} \{ 0001 \}$$ screw and (b) $$ \frac {a_0}{3} {}[{} 11 \bar {2}0 {}]{} \{ 10 \bar {1} 0 \}$$ prismatic dislocations. Reprinted from [67] with permission from Elsevier

We end by noting the excellent numerical performance with respect to parallelization in Fig. 12.13 in a benchmark problem of a seven vacancy cluster obtained on MIRA, an IBM BG/Q 1.6 GHz PowerPC A2 supercomputer at Argonne National Laboratory.

A line graph of speed up versus threads and cores plots 2 increasing slant lines for ideal and M a c l subscript actual, ideal. The lines are overlapping and begin from the origin. The line for M a c l subscript actual, ideal is linear and at the top.

Fig. 12.13

Parallel performance in a benchmark problem of a seven vacancy cluster. Reprinted from [67] with permission from Elsevier

Acknowledgements

We are grateful to Phani Motamarri for sharing the unpublished results shown in Table 12.1. We acknowledge the help of Arpit Bhardwaj, Sambit Das and Xin Jing in running some of the DFT-FE and SQ simulations, and generating the corresponding figures. KB, MO and MP acknowledge the support of the Army Research Laboratory under Cooperative Agreement Number W911NF-12-2-0022. VG acknowledges the support of the U.S. Department of Energy, Office of Science through grants DE-SC0008637 and DE-SC0017380. VG also gratefully acknowledges the support of the Army Research Office through the DURIP grant W911NF1810242. PS acknowledges support of the U.S. Department of Energy, Office of Science through grant DE-SC0019410. The computations presented here were conducted on the Resnick High Performance Cluster at Caltech, the Great Lakes High Performance Cluster at the University of Michigan, the Oak Ridge Leadership Computing Facility, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory under contract DE-AC05-00OR22725, and the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory, Department of Energy, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

Appendix: Crystalline solids and the Cauchy–Born Rule

A Bravais lattice is a lattice with a single atom in its unit cell:

where a set of linearly independent vectors or lattice vectors $$\{a_i\}_{i=1}^3$$ describes the unit cell, or translational symmetry, and o signifies the presence of an atom at the origin. A crystal (also called lattice with a basis) is a periodic arrangement of atoms (points) in $${\mathbb R}^3$$ with a finite number M of atoms in the unit cell. It may be regarded as a union of P congruent Bravais lattices which are displaced from each other:

where $$\{a_i\}_{i=1}^3$$ are the lattice vectors and the shift vectors pα, α = 1, …, M describe the relative positions of the atoms within the unit cell. It is conventional to take p₁ = o, but this is not necessary. The underlying Bravais lattice is often referred to as the skeletal lattice.

A crystalline solid is a restriction of a lattice to a domain $$\Omega \subset {\mathbb R}^3$$ . Let $$\{r_a\}_{a=1}^A$$ denote the positions of the atoms in a crystalline solid $${\mathcal L} (a_i^0,p_\alpha ^0) \cap \Omega $$ in the reference domain $$\Omega \subset {\mathbb R}^3$$ . As the solid deforms, the current position of the atoms are given by $$\{y_a\}_{a=1}^A \subset {\mathbb R}^3$$ . Let $$y: \Omega \to {\mathbb R}^3$$ denote a smooth deformation that maps the positions of the underlying skeletal lattice, i.e., $$y_a = y (r_a) \ \forall \ r_a \in {\mathcal L}_B(a_i^0, p_1^0)$$ . We call y the macroscopic deformation. Now, if the scale of the lattice is small compared to the size of the domain, and if the deformation y varies slowly on the scale of the lattice, i.e., it may be approximated by an affine map of a scale large compared to that of ai, then at any r₀ ∈ Ω, the current positions of the atoms in the neighborhood of y(r₀) are arranged in a lattice $${\mathcal L} (a_i, q_\alpha )$$ where

In other words, for moderate macroscopic deformations, the deformation gradient convects the lattice vectors. This is known as the Cauchy–Born rule. Note that the macroscopic deformation only constrains the skeletal Bravais lattice and the atoms are free to “shuffle” within the unit cell.

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Footnotes

1

Note that we use the subscript h to index the nested spaces following the typical notation in computational science, and not to signify a relationship with the Hamiltonian.

2

Minimum wall-times computed using a metric of 40% parallel efficiency.

Glossary

DFT

Density functional theory. Has its origin in the work of Thomas, Fermi, and Dirac in the 1920s. In its modern form, it was introduced by Hohenberg, Kohn and Sham in 1964/1965. Encompasses both exact DFT, which re-formulates the quantum mechanical ground state problem for many-electron systems in terms of the electron density, and Kohn–Sham DFT, which is concerned with computationally tractable approximations of the ground state problem in terms of the electron density.

Exchange-correlation functional

Functional giving the exchange-correlation energy or an approximation thereof in terms of the single-particle density. Often denoted Exc.

Exchange energy functional

Exchange contribution to the exchange-correlation functional. Often denoted Ex.

Correlation functional

Correlation contribution to the exchange-correlation functional. Often denoted Ec.

Exchange-correlation potential

Functional derivative of the exchange-correlation functional with respect to the density. Often denoted vxc[ρ]. Together with the Hartree potential and the external potential, constitutes the effective one-body potential in the Kohn–Sham equations.

Exchange potential

Exchange contribution to the exchange-correlation potential. Often denoted vx.

Correlation potential

Correlation contribution to the exchange-correlation potential. Often denoted vc.

Hartree functional

Classical Coulomb self-repulsion energy of the electron density as a functional of the latter. Important ingredient in most Kohn–Sham DFT models. Often denoted EH.

Hartree potential

Electrostatic potential of the electron density. Also arises as the functional derivative of the Hartree functional with respect to the density. Often denoted vH.

Hartree-exchange-correlation functional

Sum of the Hartree and exchange-correlation functional. Often denoted EHxc.

Hartree-exchange-correlation potential

Sum of the Hartree and exchange-correlation potential. Together with the external potential, constitutes the effective one-body potential in the Kohn–Sham equations. Often denoted vHxc.

LDA

Local density approximation to the exchange-correlation functional and potential, obtained by approximating the exchange-correlation energy density of an inhomogeneous system at a point by the exchange-correlation energy density of the uniform electron gas of the same density. Often denoted, respectively, $$E_{\text{xc}}^{\text{LDA}}$$ and $$v_{\text{xc}}^{\text{LDA}}$$ .

SCE functional

Strictly correlated electrons functional. Minimum Coulomb interaction energy of a many-electron system as a functional of its density. Corresponds to the exact Hartree-exchange-correlation functional in the strong-interaction limit.