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 wave function for strictly correlated electrons of a given density, we turn the search over wave functions 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 maximising density for the bound, if it exists, must have compact support. We also find that, at least for particle numbers N ≤ 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.

GRAPHICAL ABSTRACT

KEYWORDS:

• Density functional theory
• Lieb–Oxford bound
• exchange-correlation functionals
• electronic correlation

Acknowledgments

We are really happy to dedicate this paper to Andreas Savin. His curiosity, deep knowledge, integrity, creativity and generosity are an infinite source of inspiration. We are very grateful to Lukas Schimmer, Mathieu Lewin, Kieron Burke and John Perdew for a critical reading of the manuscript, providing several suggestions to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Financial support was provided by the European Research Council under H2020/ERC Consolidator Grant “corr-DFT” [grant number 648932]; and the Netherlands Organization for Scientific Research (NWO) through an ECHO grant [717.013.004].