On the gradient expansion of the exchange energy within linear-response theory and beyond

In the present work, we reexamined the gradient expansion of the exchange energy of an electron gas with a slowly varying charge density. We stay within the exchange-only approximation of Sharp, Horton, Talman, and Shadwick but go to second order in the deviation from the homogeneous limit. The coefficient of the lowest-order gradient correction is obtained analytically both for a bare and a screened Coulomb interaction-the former yielding the value previously obtained by Kleinman numerically and by Engel and Vosko analytically. A screened Coulomb interaction gives Sham's coefficient in the limit of infinite screening length. The cause of the difference between the coefficients of Kleinman and Sham is clearly exhibited. The coefficients of the two next highest-order gradient corrections, one of which originates in second-order response theory, is shown to diverge as the screening length becomes large. The bare Coulomb interaction gives finite coefficients of which the one originating from linear response is obtained analytically and differs from the presumably correct result obtained by Engel and Vosko. This discrepancy demonstrates the extreme sensitivity of the analytical expressions to different regularization procedures. We suggest that coefficients should rather be chosen according to the performance of the resulting gradient approximations in weakly perturbed electron gases.