Existence of Hartree-Fock excited states for atoms and molecules

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{ö}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.

Final version to appear in Lett. Math. Phys

References in corpus (1)

• Geometric methods for nonlinear many-body quantum systems

Cited by in corpus (6)

• Coupled-Cluster Theory Revisited. Part II: Analysis of the single-reference Coupled-Cluster equations
• Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture
• Differential Equations of Quantum Mechanics
• Coupled-Cluster Theory Revisited. Part I: Discretization
• Structures of sets of solutions to the Hartree-Fock equation
• Finiteness of the number of critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold