On Kohn-Sham models with LDA and GGA exchange-correlation functionals

TL;DR

This paper provides a rigorous mathematical analysis of Kohn-Sham models with LDA and GGA exchange-correlation functionals, proving existence of solutions for certain systems within these frameworks.

Contribution

It establishes the existence of solutions for extended Kohn-Sham LDA and spin-unpolarized GGA models, advancing the mathematical understanding of these density functional theories.

Findings

Extended Kohn-Sham LDA model has solutions for neutral and positively charged systems.

Existence of solutions for spin-unpolarized GGA model in two-electron systems.

Mathematical derivation from Schrödinger equation to Kohn-Sham models.

Abstract

This article is concerned with the mathematical analysis of the Kohn-Sham and extended Kohn-Sham models, in the local density approximation (LDA) and generalized gradient approximation (GGA) frameworks. After recalling the mathematical derivation of the Kohn-Sham and extended Kohn-Sham LDA and GGA models from the Schr\"odinger equation, we prove that the extended Kohn-Sham LDA model has a solution for neutral and positively charged systems. We then prove a similar result for the spin-unpolarized Kohn-Sham GGA model for two-electron systems, by means of a concentration-compactness argument.