Generalized gradient-expansion approximation for the exchange energy.

Abstract

In this work we have studied the generalized gradient-expansion approximation (GGA) for the exchange energy proposed by Perdew to each order up to O(${\mathrm{\ensuremath{\nabla}}}^{3}$) for the expansion of the Fermi-hole charge as applied to atoms. We have discovered that the Fermi hole to O(${\mathrm{\ensuremath{\nabla}}}^{3}$) leads to exchange energies less accurate than those of the expansion to O(${\mathrm{\ensuremath{\nabla}}}^{2}$) when compared with the results of Hartree-Fock theory. The results to O(${\mathrm{\ensuremath{\nabla}}}^{2}$), however, are more accurate than those of O(\ensuremath{\nabla}) as expected, but the latter are superior to those of O(${\mathrm{\ensuremath{\nabla}}}^{3}$) for the heavier atoms considered. In addition, the exchange energies in the GGA to O(\ensuremath{\nabla}) and O(${\mathrm{\ensuremath{\nabla}}}^{3}$) are both more accurate than those of the gradient-expansion approximation to O(${\mathrm{\ensuremath{\nabla}}}^{2}$). By an analysis of the structure of the approximate Fermi holes to each order, we explain all these results. We also explain why the GGA to each order cannot lead to the correct asymptotic structure of the effective potential.