Orbital approximation for the reduced Bloch equations: Fermi–Dirac distribution for interacting fermions and Hartree–Fock equation at finite temperature

In this paper, we solve a set of hierarchy equations for the reduced statistical density operator in a grand canonical ensemble for an identical many-body fermion system without or with a two-body interaction. We take the single-particle approximation, and obtain an eigenequation for the single-particle states. For the case of no interaction, it is an eigenequation for the free particles, and the solutions are therefore the plane waves. For the case with a two-body interaction, however, it is an equation which is the extension of the usual Hartree–Fock equation at zero temperature to the case of any finite temperature. The average occupation number for the single-particle states with mean field interaction is also obtained, which has the same Fermi–Dirac distribution form as that for the free fermion gas. The derivation demonstrates that even for an interacting fermion system, only the lowest N-orbitals, where N is the number of particles, are occupied at zero temperature. In addition, we discuss their practical applications in such fields as studying the temperature effects on the average structure and electronic spectra for macromolecules.