Beyond the random-phase approximation in nonlocal-density-functional theory

The previous work by Langreth and Perdew and by Langreth and Mehl for calculation of the exchange-correlation density functional is put on the same sort of basis as standard calculations for the uniform system by the approximate calculation of the following terms which go beyond the random-phase approximation (RPA): the second-order exchange term and self-energy corrections of similar order. It is found when both local local-density approximation (LDA) and nonlocal terms are included, that the net effect of these additional terms is found to be small, so that the RPA previously used is a much better approximation than previously supposed. Evidence is presented that suggests for localized systems that the leading non-RPA terms in the LDA represent mostly a spurious self-interaction error which is removed when the nonlocal beyond-RPA terms are included as well; it is suggested that this error can be most simply avoided by just using the RPA alone for both the local and nonlocal contributions, as done in the simple approximation suggested by Langreth and Mehl Phys. Rev. B 28, 1809 (1983).