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 two-body interaction. We take the single-particle approximation, and obtain an eigen-equation for the single-particle states. For the case of no interaction, it is an eigen-equation for the free particles, and solutions are therefore the plane waves. For the case with two-body interaction, however, it is an equation which is the extension of usual Hartre-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 from 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, their practical applications in such fields as studying the temperature effects on the average structure and electronic spectra for macromolecules are discussed.