  We study the Hartree-Fock equation and the Hartree-Fock energy functional
universally used in many-electron problems. We prove that the set of all
critical values of the Hartree-Fock energy functional less than a constant
smaller than the first energy threshold is finite. Since the Hartree-Fock
equation which is the corresponding Euler-Lagrange equation is a system of
nonlinear eigenvalue problems, the spectral theory for linear operators is not
applicable. The present result is obtained establishing the finiteness of the
critical values associated with orbital energies less than a negative constant
and combining the result with the Koopmans' well-known theorem. The main
ingredients are the proof of convergence of the solutions and the analysis of
the Fr\'echet second derivative of the functional at the limit point.
