Letters in Mathematical Physics

, Volume 108, Issue 4, pp 985–1006 | Cite as

Existence of Hartree–Fock excited states for atoms and molecules



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.


Hartree-Fock theory Excited states Palais-Smale property Min-max methods Atoms and molecules HVZ theorem 

Mathematics Subject Classification

81V55 35Q40 



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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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.CNRS and CEREMADE, Université Paris-DauphinePSL Research UniversityParisFrance

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