Straightforward gradient approximation for the exchange energy ofs−pbonded solids

Abstract

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. ͓S0163-1829͑96͒01547-0͔

Density gradient expansion of correlation functions

“…Of course, for the development of density functionals for general systems beyond the weakly inhomogeneous regime it is not sufficient to use the straightforward gradient expansion. 23 However, general short-range features, such as the way the exchangecorrelation hole deforms close to the reference electron in the presence of a density gradient can be transferred to more general systems than the weakly inhomogeneous ones. Perhaps, in combination with truly nonlocal functionals for the exchange-correlation hole, 41 this can lead to the development of more accurate density functionals.…”

“…The function γ 2 can be calculated by straightforward differentiation of Eq. (A1) with respect to the potential γ 2 (23,14) = δγ 1 (23,1) δv (4) .…”

“…To do this, we first note that since (ij i) = − (jij) the terms with k = i and k = j in Eq. (23,14).…”

“…For instance, for small densities variations very accurate results for the exchange energy are obtained by summing all terms up to fourth order in the density derivatives. 4,23 Having obtained the general gradient expansion Eq. (14) it remains to find explicit expressions for the coefficient functions N m j .…”

“…This is a problem in application of the formula to general inhomogeneous systems such as molecules or surfaces in which a background density cannot be unambiguously defined (assuming that a low-order gradient expansion makes sense in such systems). 22,23 The usual argument to get rid of the background dependence, is that the replacement B xc (n 0 ) → B xc [n 0 + δn(r)] can be made since the difference between these two quantities is or order δn(r). However, it is not clear that this is consistent with gradient coefficients that arise from terms of order [δn(r)] 3 .…”