Degenerate ground states and nonunique potentials: breakdown and restoration of density functionals

TL;DR

This paper investigates the limitations of the Hohenberg-Kohn theorem in density-functional theory for degenerate systems and introduces weaker theorems that restore the functional framework for practical applications.

Contribution

It identifies a loophole in the HK theorem for systems with degeneracy and proves the joint-degeneracy and internal-energy theorems to address this issue.

Findings

Loophole in HK theorem for degenerate ground states.

Weaker theorems restore the functional framework.

Constraints on degeneracies in many-body systems.

Abstract

The Hohenberg-Kohn (HK) theorem is one of the most fundamental theorems of quantum mechanics, and constitutes the basis for the very successful density-functional approach to inhomogeneous interacting many-particle systems. Here we show that in formulations of density-functional theory (DFT) that employ more than one density variable, applied to systems with a degenerate ground state, there is a subtle loophole in the HK theorem, as all mappings between densities, wave functions and potentials can break down. Two weaker theorems which we prove here, the joint-degeneracy theorem and the internal-energy theorem, restore the internal, total and exchange-correlation energy functionals to the extent needed in applications of DFT to atomic, molecular and solid-state physics and quantum chemistry. The joint-degeneracy theorem constrains the nature of possible degeneracies in general many-body systems.