Occupation numbers in density-functional calculations

TL;DR

This paper critically examines the role of occupation numbers in density functional theory calculations, revealing that they cannot be variational parameters due to non-differentiability issues, challenging common assumptions.

Contribution

It rigorously demonstrates that occupation numbers are not variational parameters in density functional calculations, clarifying a fundamental aspect of the formalism.

Findings

Occupation numbers are not variational parameters.

Kinetic energy and exchange-correlation potential are not differentiable w.r.t. occupation numbers.

Common assumptions about occupation number variation are invalid.

Abstract

It is the intention of this paper to rigorously clarify the role of the occupation numbers in the current practical applications of the density functional formalism. In these calculations one has to decide how to distribute a given, fixed number of electrons over a set of single-particle orbitals. The conventional choice is to have orbitals below the Fermi level completely occupied and the orbitals above the Fermi level empty. Although there is a certain confusion in literature why this choice is superior to any others, the general belief is that it can justified by treating the occupation numbers as variational parameters and then applying Janak's theorem or similar reasoning. We demonstrate that there is a serious flaw in those arguments,mainly the kinetic energy and therefore the exchange-correlation potential are not differentiable with respect to density for arbitrary occupation numbers. It is rigorously shown that in the present context of the density functional calculations there is no freedom to vary the occupation numbers. The occupation numbers cannot be considered as variational parameters.