Generalized Kohn-Sham iteration on Banach spaces

TL;DR

This paper rigorously formulates the Kohn-Sham algorithm within convex analysis on Banach spaces, proving convergence and broadening its applicability beyond traditional quantum chemistry models.

Contribution

It introduces a generalized Kohn-Sham iteration scheme on Banach spaces with a convergence proof, extending the framework to diverse density-functional theories.

Findings

Convergence of the generalized Kohn-Sham iteration is proven.

The formulation applies to a wide range of density-functional theories.

The approach allows for rigorous analysis beyond quantum chemistry models.

Abstract

A detailed account of the Kohn-Sham algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows to rigorously introduce, in contrast to the common unregularized approach, a well-defined Kohn-Sham iteration scheme. Convergence in a weak sense is then proven. This generalized formulation is applicable to a wide range of different density-functional theories and possibly even to models outside of quantum mechanics.