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We study the Hartree-Fock equation and the Hartree-Fock energy functionaluniversally used in many-electron problems. We prove that the set of allcritical values of the Hartree-Fock energy functional less than a constantsmaller than the first energy threshold is finite. Since the Hartree-Fockequation which is the corresponding Euler-Lagrange equation is a system ofnonlinear eigenvalue problems, the spectral theory for linear operators is notapplicable. The present result is obtained establishing the finiteness of thecritical values associated with orbital energies less than a negative constantand combining the result with the Koopmans' well-known theorem. The mainingredients are the proof of convergence of the solutions and the analysis ofthe Fr\'echet second derivative of the functional at the limit point.

Finiteness of the number of critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold