Improved Lieb-Oxford bound on the indirect and exchange energies

Mathieu Lewin (1, 2) , Elliott H. Lieb (3) , Robert Seiringer (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 PSL - Université Paris sciences et lettres
3 DPPU - Department of Physics, Princeton University
4 IST Austria - Institute of Science and Technology [Klosterneuburg, Austria]

The Lieb-Oxford inequality provides a lower bound on the Coulomb energy of a classical system of $N$ identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree-Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy.

Domaines: Physique mathématique [math-ph]

Mathieu Lewin, Elliott H. Lieb, Robert Seiringer. Improved Lieb-Oxford bound on the indirect and exchange energies. Letters in Mathematical Physics, 2022, 112, pp.92. ⟨10.1007/s11005-022-01584-5⟩. ⟨hal-03618116⟩