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Existence of Hartree–Fock excited states for atoms and molecules

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Abstract

For neutral and positively charged atoms and molecules, we prove the existence of infinitely many Hartree–Fock critical points below the first energy threshold (that is, the lowest energy of the same system with one electron removed). This is the equivalent, in Hartree–Fock theory, of the famous Zhislin–Sigalov theorem which states the existence of infinitely many eigenvalues below the bottom of the essential spectrum of the N-particle linear Schrödinger operator. Our result improves a theorem of Lions in 1987 who already constructed infinitely many Hartree–Fock critical points, but with much higher energy. Our main contribution is the proof that the Hartree–Fock functional satisfies the Palais–Smale property below the first energy threshold. We then use minimax methods in the N-particle space, instead of working in the one-particle space.

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Notes

  1. The notation \(f\in L^{3/2}+L^\infty _\varepsilon \) means that for any \(\varepsilon >0\) we can write \(f=f_{3/2}+f_\infty \) with \(\Vert f_\infty \Vert _{L^\infty }\leqslant \varepsilon \), see [45]. Such potentials are relative form-compact (hence infinitesimal form-bounded) perturbations of \(-\Delta \) by [44, Sec. X.2] and [45, Sec. XIII.4].

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Acknowledgements

I thank Éric Séré for useful comments. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement MDFT No. 725528).

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

    Mathieu Lewin

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Lewin, M. Existence of Hartree–Fock excited states for atoms and molecules. Lett Math Phys 108, 985–1006 (2018). https://doi.org/10.1007/s11005-017-1019-y

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