Density Functional Theory: An Advanced Course (Theoretical and Mathematical Physics)

✍ Scribed by Eberhard Engel, Reiner M. Dreizler

• Publisher: Springer
• Year: 2011
• Tongue: English
• Leaves: 543
• Category: Library

✦ Synopsis

Density Functional Theory (DFT) has firmly established itself as the workhorse for atomic-level simulations of condensed phases, pure or composite materials and quantum chemical systems. This work offers a rigorous and detailed introduction to the foundations of this theory, up to and including such advanced topics as orbital-dependent functionals as well as both time-dependent and relativistic DFT. Given the many ramifications of contemporary DFT, the text concentrates on the self-contained presentation of the basics of the most widely used DFT variants: this implies a thorough discussion of the corresponding existence theorems and effective single particle equations, as well as of key approximations utilized in implementations. The formal results are complemented by selected quantitative results, which primarily aim at illustrating the strengths and weaknesses of particular approaches or functionals. The structure and content of this book allow a tutorial and modular self-study approach: the reader will find that all concepts of many-body theory which are indispensable for the discussion of DFT - such as the single-particle Green's function or response functions - are introduced step by step, along with the actual DFT material. The same applies to basic notions of solid state theory, such as the Fermi surface of inhomogeneous, interacting systems. In fact, even the language of second quantization is introduced systematically in an Appendix for readers without formal training in many-body theory.

✦ Table of Contents

Preface
Contents
Acronyms
Chapter 1 Introduction
Chapter 2 Foundations of Density Functional Theory: Existence Theorems
2.1 Hohenberg-Kohn Theorem
2.2 Degenerate Ground States
2.3 Variational Equation, Interacting v-Representability, Functional Differentiability
(a) Admissible densities and potentials
(b) Definition of Lieb functional and basic properties
(c) Functional differentiability of Lieb functional
(d) Representation of Lieb functional in terms of density matrices
2.4 Fractional Particle Numbers, Derivative Discontinuity
2.5 Spin-Polarized Systems
2.6 Current Density Functional Theory
2.7 Excited States: Part 1
Chapter 3 Effective Single-Particle Equations
3.1 Kohn-Sham Equations
3.2 Noninteracting v-Representability
3.3 Degenerate Kohn-Sham Ground States
3.4 Janak’s Theorem, Fractional Particle Numbers
3.5 Kohn-Sham Equations for Spin-Polarized Systems
3.6 Interpretation of Kohn-Sham Eigenvalues: Relation to Ionization Potential, Fermi Surface and Band Gap
3.6.1 Ionization Potential
3.6.2 Fermi Surface
3.6.3 Band Gap
3.7 Kohn-Sham Equations of Current Density Functional Theory
Chapter 4 Exchange-Correlation Energy Functional
4.1 Definition of Exact Exchange within DFT
4.2 Exact Representations of Exc[n]
4.2.1 Variant (a): Kohn-Sham Perturbation Theory
4.2.2 Variant (b): Adiabatic Connection
4.3 Local Density Approximation (LDA)
4.3.1 Exchange
4.3.2 Correlation: High-Density Limit
4.3.3 Correlation: Low-Density Limit
4.3.4 Correlation: Interpolation Between High- and Low-Density Regime
4.3.5 Density Functional: Local Density Approximation (LDA)
4.3.6 Spin-Polarized Electron Gas: Local Spin-Density Approximation (LSDA)
4.4 Nonlocal Corrections to the LDA
4.4.1 Weakly Inhomogeneous Electron Gas
4.4.2 Complete Linear Response
4.4.3 Gradient Expansion
4.5 Generalized Gradient Approximation (GGA)
4.5.1 Momentum Space Variant
4.5.2 Real Space Variant
4.5.3 Combination of Momentum and Real Space Variants
4.5.4 Semi-Empirical Construction of GGAs
4.5.5 Merits and Limitations of GGAs
4.6 Weighted Density Approximation (WDA)
4.7 Self-Interaction Corrections (SIC)
4.8 Meta-GGA (MGGA)
4.9 LDA+U
Chapter 5 Virial Theorems
5.1 Scaling Behavior of Energy Contributions
5.2 Conventional Virial Theorem
5.3 DFT Virial Theorem
5.4 Hellmann-Feynman Theorem
Chapter 6 Orbital Functionals: Optimized Potential Method
6.1 Motivation
6.1.1 Atomic Negative Ions
6.1.2 Dispersion Forces
6.1.3 Strongly Correlated Systems
6.1.4 Third Generation of DFT
6.2 Derivation of OPM Integral Equation
6.2.1 Compact Notation
6.2.2 Direct Functional Derivative
6.2.3 Total Energy Minimization
6.2.4 Invariance of Density
6.2.5 Exact Relations Based on OPM Integral Equation
6.2.6 Krieger-Li-Iafrate Approximation (KLI)
6.2.7 OPM in Case of Degeneracy
6.3 Exchange-Only Results
6.4 First-Principles Implicit Correlation Functionals
6.4.1 Kohn-Sham Perturbation Theory
6.4.2 Kohn-Sham-Based Random Phase Approximation
6.4.3 Interaction Strength Interpolation (ISI)
6.5 Model-Based Orbital-Dependent Exchange-Correlation Functionals
6.5.1 Self-Interaction Corrected LDA
6.5.2 Colle-Salvetti Functional
6.5.3 Meta-GGA
6.5.4 Global, Screened and Local Hybrid Functionals
6.6 Analysis of Orbital-Dependent Correlation Functionals
6.6.1 Dispersion Force
6.6.2 Correlation Energy
6.6.3 Correlation Potential
6.7 Orbital-Dependent Representation of 2-Particle Density
Chapter 7 Time-Dependent Density Functional Theory
7.1 Runge-Gross Theorem
7.2 Time-Dependent Kohn-Sham Equations
7.3 Exchange-Correlation Action: Adiabatic Local Density Approximation and Beyond
7.4 Time-Dependent Linear Response
7.5 Spin-Polarized Time-Dependent Density Functional Theory
7.6 Excited States: Part II
Chapter 8 Relativistic Density Functional Theory
8.1 Notation
8.2 Field Theoretical Background
8.3 Existence Theorem
8.4 Relativistic Kohn-Sham Equations
8.5 Towards a Workable RDFT Scheme: No-pair Approximation
8.6 No-pair RDFT
8.7 Variants of RDFT
8.8 Relativistic Exchange-Correlation Functional: Concepts and Illustrative Results
8.8.1 Relativistic Implicit Functionals: Optimized Potential Method
8.8.2 Role of Relativistic Corrections in Exc: I. Prototype Results for Atoms
8.8.3 Relativistic Local Density Approximation
8.8.4 Relativistic Generalized Gradient Approximation
8.8.5 Role of Relativistic Corrections in Exc: II. Prototype Results for Molecules and Solids
Chapter 9 Further Reading
Erratum to: Introduction
Erratum to: Density Functional Theory
Erratum to: Density Functional Theory
Appendix A Functionals and the Functional Derivative
A.1 Definition of the Functional
A.2 Functional Derivative
A.3 Calculational Rules
A.4 Variational Principle
Appendix B Second Quantization in Many-Body Theory
B.1 N-Particle Hilbert Space
B.1.1 Realization in First Quantized Form
B.1.2 Formal Representation
B.2 Fock Space
B.2.1 Creation and Annihilation Operators
B.2.2 1-Particle Operators
B.2.3 2-Particle Operators
Appendix C Scaling Behavior of Many-Body Methods
Appendix D Explicit Density Functionals for the Kinetic Energy: Thomas-Fermi Models and Beyond
Appendix E Asymptotic Behavior of Quasi-Particle Amplitudes
Appendix F Quantization of Noninteracting Fermions in Relativistic Quantum Field Theory
Appendix G Renormalization Scheme of Vacuum QED
Appendix H Relativistic Homogeneous Electron Gas
H.1 Basic Propagators
H.2 Response Functions
H.3 Ground State Energy
H.4 Ground State Four Current
Appendix I Renormalization of Inhomogeneous Electron Gas
Appendix J Gradient Corrections to the Relativistic LDA
Appendix K Gordon Decomposition
Appendix L Some Useful Formulae
References
Index