Convergence study of the density-gradient expansion for the screened-Coulomb exchange energy

Abstract

In this work we have studied the convergence properties of the density-gradient expansion for the exchange energy for screened-Coulomb interaction as applied to surfaces. The expansion examined is that which is valid within the Hartree-Fock approximation, and the densities for which the convergence is tested are those derived from the step-potential model of a surface. Surprisingly, contrary to intuitive expectations for a short-ranged interaction, it is observed that both the gradient expansion as well as the local-density approximations are poor. Furthermore, the former approximation does not reduce the error of the latter to any significant degree even for the most slowly varying of the densities considered. The results of calculations for the gradient expansion for the bare-Coulomb exchange energy, together with consideration of various convergence conditions, indicate that this lack of convergence is principally due to the limitations of the step-potential model. The arguments and different conclusions arrived at in previous work on this expansion by others using the step-potential model are also examined, and suggestions for future work made.