Challenging the Lieb-Oxford Bound in a systematic way

Abstract

The Lieb-Oxford bound, a nontrivial inequality for the indirect part of the many-body Coulomb repulsion in an electronic system, plays an important role in the construction of approximations in density functional theory. Using the wavefunction for strictly-correlated electrons of a given density, we turn the search over wavefunctions appearing in the original bound into a more manageable search over electron densities. This allows us to challenge the bound in a systematic way. We find that a maximizing density for the bound, if it exists, must have compact support. We also find that, at least for particle numbers $N\le 60$, a uniform density profile is not the most challenging for the bound. With our construction we improve the bound for $N=2$ electrons that was originally found by Lieb and Oxford, we give a new lower bound to the constant appearing in the Lieb-Oxford inequality valid for any $N$, and we provide an improved upper bound for the low-density uniform electron gas indirect energy.

accepted in Mol. Phys. in the special issue in honour of Andreas Savin; revised version with new calculations

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• Optimal-transport formulation of electronic density-functional theory
• Strong correlation in Kohn-Sham density functional theory
• Gedanken Densities and Exact Constraints in Density Functional Theory
• How tight is the Lieb-Oxford bound?
• Kohn-Sham density functional theory for quantum wires in arbitrary correlation regimes
• Improved Lieb-Oxford exchange-correlation inequality with gradient correction
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Cited by in corpus (6)

• Interpolated energy densities, correlation indicators and lower bounds from approximations to the strong coupling limit of DFT
• Statistical Mechanics of the Uniform Electron Gas
• Augmented potential, energy densities, and virial relations in the weak- and strong-interaction limits of DFT
• Hartree potential dependent exchange functional
• Seven Useful Questions in Density Functional Theory
• On the validity of power functionals for the homogeneous electron gas in reduced.density-matrix-functional theory