Preliminaries on Density Functional Theory (DFT)

1. Introduction:

Basic Hamiltonian for interacting electrons and nuclei.
Simplest approximations for the uniform electron gas:

Hamiltonian of the uniform electron gas.
The free electron gas:

Hartree approximation in a general context (best before see Wavefunctional variational principle and Hamiltonian eigenstates in the Appendix below):

Hartree approximation for the uniform electron gas.

Hartree-Fock approximation in a general context (see before see Functional derivatives and A useful formula in the Appendix below):
- Hartree-Fock approximation for the uniform electron gas (1).
- Hartree-Fock approximation for the uniform electron gas (2).

The first kind of density functional theory: the Thomas-Fermi model.

Electronic density and related magnitudes as key ingredients:

2. The Density Functional Theory:

The Hohenberg-Kohn theorems.

Levy's constrained-search formulation.

The Kohn-Sham method:

The decomposition of the universal functional (1).
The decomposition of the universal functional (2).
The Kohn-Sham equations.
Hohenberg-Kohn and Kohn-Sham schematics.

The exchange energy as a function of the exchange hole.
The correlation energy as a function of the correlation hole (best before see The Hellmann-Feynmann theorem and The adiabatic connection in the Appendix below).

Some approximations for the exchange and correlation functional:

The local-density approximation (LDA).

The generalized-gradient approximation (GGA).

Exchange-correlation functionals beyond the GGA.

Extension to spin density-functional theory.
Practical calculation of Kohn-Sham equations in a basis.

To conclude …

Appendix:

Wavefunctional variational principle and Hamiltonian eigenstates.
Functional derivatives.

A useful formula.

The Hellmann-Feynmann theorem.

The electrostatic force theorem.

The adiabatic connection (best before see The Hellmann-Feynmann theorem above).

Textbooks:

R. Martin, "Electronic Structure: Basic Theory and Practical Methods". Cambridge University Press, 2004.
C. Fiolhais, M. A.L. Marques, F. Nogueira (Eds.), "A Primer in Density Functional Theory". Lecture Notes in Physics; Springer-Verlag Berlin Heidelberg, 2003.
D.S. Sholl and J.A. Steckel, "Density Functional Theory: A practical Introduction". Willey, 2009.
K. Burke, "The ABC of DFT". Department of Chemistry, University of California, Irvine, 2007.
C.A. Ullrich, "Time-Dependent Density-Functional Theory: Concepts and Applications". Oxford University Press, 2011.
N.W. Ashcroft and D. Mermin, "Solid State Physics". Saunders College, 1976.