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It is shown how the smooth part of the Wigner transform of ρ corresponding to a Woods-Saxon potential can be calculated semiclassically. We find that its surface thickness in phase space is appreciable but always much smaller than the Fermi energy thus justifying the Wigner- Kirkwood _ expansion. Using this fact we propose and perform a new semiclassical Hartree-Fock method directly based on the Wigner-Kirkwood expansion. No functional τ [ρ] is therefore needed

SEMICLASSICAL CALCULATION OF THE WIGNER TRANSFORM OF THE SINGLE PARTICLE DENSITY MATRIX AND SEMICLASSICAL HARTREE-FOCK