Density Functional Theory

1. Introduction.- 2. Basic Formalism for Stationary Non-Relativistic Systems.- 2.1 The Hohenberg-Kohn Theorem.- 2.2 Degenerate Groundstates.- 2.3 v-Representability and Related Questions.- 2.4 Fractional Particle Number, Chemical Potential, and Derivative Discontinuities.- 3. Extensions.- 3.1 Spin-Polarised Systems.- 3.2 Finite Temperature Ensembles.- 3.3 Multicomponent Systems.- 3.4 Hartree-Fock Limit.- 3.5 Excited States.- 3.6 Density Matrix Functionals.- 3.7 Momentum Space.- 3.8 Bose Systems.- 3.9 Superconducting Systems.- 4. The Kohn-Sham Scheme.- 4.1 The Basic Kohn-Sham Equations.- 4.2 Degenerate Kohn-Sham Groundstates and the Question of v-Representability.- 4.3 Spin-Polarised Systems.- 4.4 Fractional Occupation, Janak's Theorem, and the Slater Transition State.- 4.5 Excited States: The Kohn-Sham Scheme for Ensembles.- 4.6 Schrödinger Equation for the Square Root of the Groundstate Density.- 4.7 Hellmann-Feynman, Virial, and Scaling Properties of Density Functionals.- 4.8 Single-Particle Equations for Superconductors: A Generalized Bogoliubov-deGennes Scheme.- 5. Explicit Functionals I: Kinetic and Exchange Energy Functionals Derived from the One-Particle Density Matrix.- 5.1 Density-Gradient Expansions from Semiclassical Expansions: A Survey.- 5.2 The Kirzhnits Method.- 5.2.1 Semiclassical Expansion.- 5.2.2 Density-Gradient Expansion.- 5.3 The Wigner-Kirkwood Approach and Partial Resummation of the Gradient Expansion.- 5.4 Empirical Convergence Studies of the Gradient Expansion.- 5.5 Original von Weizsäcker Functional Versus Gradient Expansion.- 5.6 Padé Approximants and Other Parametrisations.- 5.7 Phase-Space Approach Based on Local Thermodynamics.- 5.8 The Classical Density Functional Models of Thomas, Fermi, Dirac, and von Weizsäcker.- 5.8.1 Definition of the Models.- 5.8.2 The Thomas-Fermi Variational Equation.- 5.8.3 Numerical Comparison and Some Applications.- 6. Many-Body Perturbation Theory.- 6.1 Diagrammatic Approach to the Inhomogeneous Electron Gas.- 6.2 The Exchange-Correlation Functional Expressed in Terms of the Irreducible Self-Energy.- 6.3 The Band Gap in Insulators and Semiconductors.- 6.4 The Fermi Surface in Metals.- 6.5 The Homogeneous Electron Gas.- 6.5.1 Basic Relations.- 6.5.2 The Groundstate Energy.- 7. Explicit Functionals II: The Local Density Approximation and Beyond.- 7.1 The Local Density Approximation.- 7.2 Discussion of the Local Density Approximation.- 7.3 Nonlocal Density Schemes.- 7.4 Self-Interaction Corrections.- 7.5 Wave Vector Analysis.- 7.6 Gradient Corrections.- 7.6.1 General Considerations.- 7.6.2 Low-Order Perturbative Results.- 7.6.3 Hartree-Fock Results.- 7.6.4 Results Based on Random Phase and Related Approximations.- 7.6.5 Results Derived from the Local Field Correction.- 7.6.6 Results Based on Wave Vector Analysis.- 7.7 Kohn-Sham Results for Atoms and Molecules.- 7.7.1 Atomic Systems.- 7.7.2 Molecular Systems.- 8. Density Functional Theory of Relativistic Systems.- 8.1 Introduction.- 8.2 Existence Theorems.- 8.3 Explicit Functionals I: The Relativistic Kirzhnits Expansion.- 8.3.1 The Dirac Orbital Picture.- 8.3.2 The Quantum Electrodynamical Picture.- 8.4 The Homogeneous Relativistic Electron Gas.- 8.5 Explicit Functionals II: The Local Density Approximation.- 8.5.1 Unpolarised Systems.- 8.5.2 Polarised Systems.- 8.6 Remarks and Applications.- A. Definition of Density Matrices, Green's Functions, and Correlation Functions.- B. Compilation of Literature on Atomic and Molecular Kohn-Sham Results.- References.