The Lieb-Oxford lower bounds on the Coulomb energy, their importance to electron density functional theory, and a conjectured tight bound on exchange (Q2164151)

Summary: \textit{E. H. Lieb} and \textit{S. Oxford} [``Improved lower bound on the indirect Coulomb energy'', J. Quantum Chem. 19, No. 3, 427--439 (1981; \url{doi:10.1002/qua.560190306})] derived rigorous lower bounds, in the form of local functionals of the electron density, on the indirect part of the Coulomb repulsion energy. The greatest lower bound for a given electron number \(N\) depends monotonically upon \(N\), and the \(N\to\infty N\) limit is a bound for all \(N\). These bounds have been shown to apply to the exact density functionals for the exchange- and exchange-correlation energies that must be approximated for an accurate and computationally efficient description of atoms, molecules, and solids. A tight bound on the exact exchange energy has been derived there from for two-electron ground states, and is conjectured to apply to all spin-unpolarized electronic ground states. Some of these and other exact constraints have been used to construct two generations of non-empirical density functionals beyond the local density approximation: the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and the strongly constrained and appropriately normed (SCAN) meta-GGA. For the entire collection see [Zbl 1491.46003].