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, $E_{xc}[n]\ge -C\int d^{3}r{n}^{4/3}$, where $C\le C_{LO}=1.68$. 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 $C\le 1$. 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.

10 pages, 3 figures

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• Lower Bounds on the Exchange-Correlation Energy in Reduced Dimensions
• Systematic investigation of a family of gradient-dependent functionals for solids
• Floating Wigner crystal with no boundary charge fluctuations
• Statistical Mechanics of the Uniform Electron Gas
• Strictly correlated uniform electron droplets
• Non-empirical hyper-generalized-gradient functionals constructed from the Lieb-Oxford bound
• Gradient-dependent upper bound for the exchange-correlation energy and application to density functional theory
• Challenging the Lieb-Oxford Bound in a systematic way
• Spin-dependent gradient correction for more accurate atomization energies of molecules
• Universal Functionals in Density Functional Theory
• Improved Lieb-Oxford bound on the indirect and exchange energies
• Empirical analysis of the Lieb-Oxford bound in ions and molecules
• Hartree potential dependent exchange functional
• On the violation of a local form of the Lieb-Oxford bound
• Testing and Using the Lewin-Lieb Bound in Density Functional Theory
• On the lower bound on the exchange-correlation energy in two dimensions
• One-Dimensional Lieb-Oxford Bounds