  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.
