A mathematical perspective on density functional perturbation theory

TL;DR

This paper provides a rigorous mathematical analysis of density functional perturbation theory within extended Kohn-Sham models, including cases with degenerate Fermi levels, using the reduced Hartree-Fock framework.

Contribution

It formalizes classical results in non-degenerate cases and extends the analysis to degenerate Fermi levels in the reduced Hartree-Fock model.

Findings

Formalization of classical density functional perturbation theory results

Proof of Wigner's (2n + 1) rule in this context

Extension to degenerate Fermi levels in the reduced Hartree-Fock model

Abstract

This article is concerned with the mathematical analysis of the perturbation method for extended Kohn-Sham models, in which fractional occupation numbers are allowed. All our results are established in the framework of the reduced Hartree-Fock (rHF) model, but our approach can be used to study other kinds of extended Kohn-Sham models, under some assumptions on the mathematical structure of the exchange- correlation functional. The classical results of Density Functional Perturbation Theory in the non-degenerate case (that is when the Fermi level is not a degenerate eigenvalue of the mean-field Hamiltonian) are formalized, and a proof of Wigner's (2n + 1) rule is provided. We then focus on the situation when the Fermi level is a degenerate eigenvalue of the rHF Hamiltonian, which had not been considered so far.