A Verbose Note on Density Functional Theory

Abstract

This note is intended for expanding the details on the derivation and properties of density functional theory, in hope to make them more systematic, better motivated, and step-by-step for readers new to the domain. The note starts with basic concepts in quantum mechanics, then takes the step towards many-body systems using the tools of second quantization and Fock space, with some highlights on properties of the Coulomb system. Given these general technical preparations, the Hartree-Fock method is naturally unrolled, with expressions for various cases and quantities. Density functional theory is then presented, with a motivating reasoning of the Kohn-Sham formulation, and a connection and comparison of the expressions with the Hartree-Fock method. Lieb's celebrated 1983 paper on functional analysis on density functionals is also summarized.