A Verbose Note on Density Functional Theory

TL;DR

This paper provides a detailed, systematic introduction to density functional theory, covering foundational quantum mechanics, many-body systems, Hartree-Fock, and Kohn-Sham formulations, aimed at newcomers to the field.

Contribution

It offers a comprehensive, step-by-step pedagogical overview of density functional theory, including derivations, properties, and historical context, enhancing understanding for beginners.

Findings

Clarifies the derivation of density functional theory

Connects DFT with Hartree-Fock method

Summarizes Lieb's 1983 functional analysis

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.