One-Dimensional Lieb-Oxford Bounds

TL;DR

This paper proves Lieb-Oxford bounds in one dimension for convex potentials approximating Coulomb interactions, using logarithmic density expressions, and discusses a conjectured form for the indirect interaction energy.

Contribution

It introduces new Lieb-Oxford bounds for convex potentials in one dimension and explores a conjectured inequality involving the particle density.

Findings

Established Lieb-Oxford bounds for modified and regularized Coulomb potentials.

Utilized logarithmic expressions of particle density in bounds.

Discussed a conjecture relating indirect interaction energy to density squared.

Abstract

We investigate and prove Lieb-Oxford bounds in one dimension by studying convex potentials that approximate the ill-defined Coulomb potential. A Lieb-Oxford inequality establishes a bound of the indirect interaction energy for electrons in terms of the one-body particle density $\rho_\psi$ of a wave function $\psi$. Our results include modified soft Coulomb potential and regularized Coulomb potential. For these potentials, we establish Lieb-Oxford-type bounds utilizing logarithmic expressions of the particle density. Furthermore, a previous conjectured form $I_\mathrm{xc}(\psi)\geq - C_1 \int_{\mathbb R} \rho_\psi(x)^{2} \mathrm{d}x$ is discussed for different convex potentials.