The optimal Lieb–Oxford constant
For an integer $N \ge 1$, consider symmetric probability measures $P$ on $(\mathbb{R}^3)^N$ whose one-particle marginal has a density, and let $\rho_P$ be $N$ times that density, so that $\int \rho_P = N$. Require $\rho_P \in L^{4/3}(\mathbb{R}^3)$ and all the Coulomb integrals below to be finite, and define the indirect energy $$E_{\mathrm{ind}}(P) = \int \sum_{1 \le i < j \le N} \frac{1}{|x_i - x_j|}\, dP - \frac{1}{2} \iint \frac{\rho_P(x)\, \rho_P(y)}{|x - y|}\, dx\, dy.$$ Lieb proved that $-E_{\mathrm{ind}}(P) \le C \int \rho_P^{4/3}$ for a universal constant $C$, first with $C = 8.52$; Lieb and Oxford obtained $1.68$ and Chan and Handy $1.64$. Lewin, Lieb and Seiringer proved a bound on the best constant by a variational problem that they evaluated numerically at about $1.58$. The best constant is at least $-e_{\mathrm{UEG}}$, the energy per unit volume of the uniform electron gas at unit density with the sign reversed, and $-e_{\mathrm{UEG}} \ge 1.444$ by results of Cotar and Petrache and of Lewin, Lieb and Seiringer. It has been conjectured that the best constant equals $-e_{\mathrm{UEG}}$.
Problem. Determine the value of $$C_{\mathrm{LO}} = \sup_{N, P} \frac{-E_{\mathrm{ind}}(P)}{\int_{\mathbb{R}^3} \rho_P(x)^{4/3}\, dx},$$ the supremum taken over all $N \ge 1$ and all such $P$: the smallest constant in the inequality above, together with admissible $P$ approaching equality.
Source: Lieb (1979) and Lieb and Oxford (1981), who proved the inequality; posed as a challenge by Lewin, Lieb and Seiringer (2018), §2, after Theorem 2.1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, density functional theory, indirect Coulomb energies, Lieb–Oxford inequality
References
@article{lieb1979,
author = {Lieb, Elliott H.},
title = {A lower bound for {C}oulomb energies},
journal = {Phys. Lett. A},
volume = {70},
number = {5-6},
year = {1979},
pages = {444--446},
doi = {10.1016/0375-9601(79)90358-X},
}
@article{lieb1981,
author = {Lieb, Elliott H. and Oxford, Stephen},
title = {Improved lower bound on the indirect {C}oulomb energy},
journal = {Int. J. Quantum Chem.},
volume = {19},
number = {3},
year = {1981},
pages = {427--439},
doi = {10.1002/qua.560190306},
}
@article{lewin2018,
author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert},
title = {Statistical mechanics of the uniform electron gas},
journal = {J. {\'E}c. polytech. Math.},
volume = {5},
year = {2018},
pages = {79--116},
doi = {10.5802/jep.64},
eprint = {1705.10676},
archivePrefix = {arXiv},
}
@article{lewin2022,
author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert},
title = {Improved {L}ieb--{O}xford bound on the indirect and exchange energies},
journal = {Lett. Math. Phys.},
volume = {112},
number = {5},
year = {2022},
pages = {92},
doi = {10.1007/s11005-022-01584-5},
eprint = {2203.12473},
archivePrefix = {arXiv},
}
@article{chan1999,
author = {Chan, Garnet Kin-Lic and Handy, Nicholas C.},
title = {Optimized {L}ieb-{O}xford bound for the exchange-correlation energy},
journal = {Phys. Rev. A},
volume = {59},
number = {4},
year = {1999},
pages = {3075--3077},
doi = {10.1103/PhysRevA.59.3075},
}
@misc{cotar2019,
author = {Cotar, Codina and Petrache, Mircea},
title = {Equality of the jellium and uniform electron gas next-order asymptotic terms for {C}oulomb and {R}iesz potentials},
year = {2019},
eprint = {1707.07664},
archivePrefix = {arXiv},
note = {Version 5, 2019},
}
@article{lewin2019,
author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert},
title = {Floating {W}igner crystal with no boundary charge fluctuations},
journal = {Phys. Rev. B},
volume = {100},
number = {3},
year = {2019},
pages = {035127},
doi = {10.1103/PhysRevB.100.035127},
}
@misc{aim-optimal-lieb-oxford,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The optimal Lieb–Oxford constant}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/198-optimal-lieb-oxford.md},
note = {Entry 198 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00456, "The optimal Lieb–Oxford constant", clopen.solutions.