Abstract: We study a family of functionals, derived from a model in quantum electrodynamics in the Hartree-Fock approximation. Each functional describes the energy of systems of electron and positrons in a field of a nucleus of charge Z, and depends on the choice of the one electron subspace. For electrons subspace being the positive spectral projection of a Dirac-Fock-Coulomb operator, we show the infimum of this functional is attained, and that the minima fulfill no-pair Dirac-Fock equations.