Variational derivation of density functional theory

TL;DR

This paper presents a variational derivation of density functional theory that addresses limitations of traditional approaches, emphasizing the importance of orbital-based derivatives for consistency with quantum principles.

Contribution

It introduces a mathematically consistent variational framework for Kohn-Sham theory using orbital densities, moving beyond the local density approximation.

Findings

Traditional derivations can violate the exclusion principle.

Orbital-based functional derivatives are explicitly constructible.

Progress beyond LDA requires nonlocal exchange-correlation methods.

Abstract

It is shown here that Kohn-Sham equations cannot be derived from Hohenberg-Kohn theory without an additional postulate. Assuming that a functional derivative with respect to total electron density exists leads in general to a theory inconsistent with the exclusion principle. A mathematically and physically correct variational theory of the Kohn-Sham model can be developed using functional derivatives with respect to orbital densities. These partial (G\^ateaux) derivatives can be constructed explicitly from general N-electron theory and are defined throughout the orbital Hilbert space. This theory is consistent with the local density approximation (LDA), but does not in general imply multiplicative local exchange-correlation potentials. Progress beyond the LDA in condensed-matter physics requires development of methodology for nonlocal exchange and correlation potentials.