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Improved Lieb–Oxford bound on the indirect and exchange energies

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Abstract

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.

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Acknowledgements

We would like to thank David Gontier for useful advice on the numerical simulations. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreements MDFT No. 725528 of M.L. and AQUAMS No. 694227 of R.S.). We are thankful for the hospitality of the Institut Henri Poincaré in Paris, where part of this work was done.

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Authors and Affiliations

  1. CNRS and CEREMADE, Université Paris-Dauphine, PSL University, 75016, Paris, France

    Mathieu Lewin

  2. Departments of Mathematics and Physics, Jadwin Hall, Princeton University, Washington Rd., Princeton, NJ, 08544, USA

    Elliott H. Lieb

  3. IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400, Klosterneuburg, Austria

    Robert Seiringer

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  1. Mathieu Lewin
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Lewin, M., Lieb, E.H. & Seiringer, R. Improved Lieb–Oxford bound on the indirect and exchange energies. Lett Math Phys 112, 92 (2022). https://doi.org/10.1007/s11005-022-01584-5

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