Hartree-Fock-Slater exchange for anisotropic occupation in momentum space

It is shown that the well-known expression of the Hartree-Fock-Slater exchange energy (usually calculated for an occupied Fermi sphere) becomes scaled by a dimensionless factor in the case of a simple (compact) Fermi body and of only one (partially filled) band. Analytical results are presented for the following occupied regions in momentum space: i) ellipsoid of revolution (prolate as well as oblate), ii) ellipsoid with three unequal axes.The authors would like to thank Prof. P. Ziesche, Dr. H. Eschrig, and Dr. W. John for helpful discussions. One of us (M. H.) acknowledges an IREX-fellowship where part of this work has been completed.