Challenging the Lieb-Oxford Bound in a systematic way

TL;DR

This paper systematically challenges the Lieb-Oxford bound in density functional theory, improving bounds for small electron numbers and providing new insights into the nature of the maximizing densities.

Contribution

It introduces a new approach to test the Lieb-Oxford bound using electron densities, leading to improved bounds and a better understanding of the bound's tightness.

Findings

Improved the Lieb-Oxford bound for two-electron systems.

Established that maximizing densities must have compact support.

Provided a new lower bound for the Lieb-Oxford constant for any particle number.

Abstract

The Lieb-Oxford bound, a nontrivial inequality for the indirect part of the many-body Coulomb repulsion in an electronic system, plays an important role in the construction of approximations in density functional theory. Using the wavefunction for strictly-correlated electrons of a given density, we turn the search over wavefunctions appearing in the original bound into a more manageable search over electron densities. This allows us to challenge the bound in a systematic way. We find that a maximizing density for the bound, if it exists, must have compact support. We also find that, at least for particle numbers $N\le 60$, a uniform density profile is not the most challenging for the bound. With our construction we improve the bound for $N=2$ electrons that was originally found by Lieb and Oxford, we give a new lower bound to the constant appearing in the Lieb-Oxford inequality valid for any $N$, and we provide an improved upper bound for the low-density uniform electron gas indirect energy.