In the present work we perform a straightforward gradient expansion of the exchange energy of a perturbed electron gas. Studied perturbations range from very weak to those that produce, e.g., a siliconlike band structure with a band gap. The expansions involve density gradients up to fourth degree and we include all terms originating in linear- and second-order response theory. The expansion reproduces our exactly calculated exchange energies with an accuracy of the order of a few mRy per electron for metallic systems. For systems with a bandgap the accuracy is reduced by an order of magnitude. When the coefficient of the fourth-degree gradient originating in second-order response theory is used as a variable parameter, we find a best fit to calculated exchange energies when the coefficient agrees with that obtained in previous work on second-order response theory. Thus, the present results corroborate our previous analytical work. We emphasize the possibility of obtaining very accurate exchange energies for $s-p$ bonded solids and we discuss the possibility of also including correlation energies within the same simple scheme.

• Received 28 May 1996

DOI:https://doi.org/10.1103/PhysRevB.54.17392