The Fermi hole and the exchange parameter in X_α theory

Abstract

In Slater's statistical exchange approximation, the Fermi hole is assumed to be spherical and uniform in density. This leads to an exchange potential that depends on an adjustable parameter α which is different for each atom. The value of this parameter is determined empirically for each atom, usually by equalizing the total energy of the atom calculated by the X_α method to that calculated by Hartree-Fock theory. In this paper an explicit dependence of α on the atomic number Z is developed by considering a linear variation of the Fermi-hole density, with the proper boundary values. This permits the calculation of theoretical α values for all atoms in the periodic table; these are reported. Thesse theoretical α values reproduce the Z dependence of the empirical a values in all details. Further, it is shown that electrons of opposite spin should have different α values-a result that has important consequences in X_α calculations of magnetic properties and hyperfine interactions.