How tight is the Lieb-Oxford bound?

Abstract

Density-functional theory requires ever better exchange-correlation (xc) functionals for the ever more precise description of many-body effects on electronic structure. Universal constraints on the xc energy are important ingredients in the construction of improved functionals. Here we investigate one such universal property of xc functionals: the Lieb-Oxford lower bound on the exchange-correlation energy, $E_{xc}[n] \ge -C \int d^3r n^{4/3}$, where $C\leq C_{LO}=1.68$. To this end, we perform a survey of available exact or near-exact data on xc energies of atoms, ions, molecules, solids, and some model Hamiltonians (the electron liquid, Hooke's atom and the Hubbard model). All physically realistic density distributions investigated are consistent with the tighter limit $C \leq 1$. For large classes of systems one can obtain class-specific (but not fully universal) similar bounds. The Lieb-Oxford bound with $C_{LO}=1.68$ is a key ingredient in the construction of modern xc functionals, and a substantial change in the prefactor $C$ will have consequences for the performance of these functionals.