One-Dimensional Lieb-Oxford Bounds

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 $I_{\mathrm{xc}}(\psi )\ge -C_1{\int }_{\mathbb{R}}{\rho }_{\psi }(x{)}^2\mathrm{d}x$ is discussed for different convex potentials.