Existence of Hartree-Fock excited states for atoms and molecules

TL;DR

This paper proves the existence of infinitely many Hartree-Fock excited states for atoms and molecules below the first energy threshold, extending previous results and employing minimax methods in N-particle space.

Contribution

It establishes the Palais-Smale property for the Hartree-Fock functional below the first energy threshold, enabling the use of minimax methods to find critical points.

Findings

Proves existence of infinitely many Hartree-Fock critical points below the first energy threshold.

Improves upon Lions' 1987 theorem by constructing lower-energy critical points.

Employs minimax methods in N-particle space to achieve results.

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 Schr{\"o}dinger 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.