Existence of minimizers for Kohn–Sham models in quantum chemistry

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 Schrödinger 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.