The basic idea behind the present work is that an atom is not a linear perturbation of the electron gas. We have thus analyzed the exchange energy of the inhomogeneous electron gas to third order in the deviation from a constant density. We give the symmetry properties obeyed by the corresponding second-order response function L x , and demonstrate how L x gives rise to gradient corrections to the exchange energy. The expansion, which is taken up to sixth order in the density gradient, also includes the Laplacian of the density. In the case of a statically screened Coulomb interaction, we have calculated the coefficients of second-and fourth-order gradient terms both analytically and numerically. In analogy with the corresponding results from linearresponse theory, the fourth-order coefficient is shown to diverge as the screening is made to vanish. For the bare Coulomb interaction we have not succeeded in obtaining analytical results, and, due to numerical problems at small-q vectors, our numerically obtained coefficients have an estimated uncertainty of 20%.

-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.

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

Using a numerical fitting scheme we construct the exact Density-Functional (DF) effective potential, corresponding wave-functions and orbital eigenvalues from an accurate correlated reference density for the neon atom. Our results compare well with those obtained later using a number of different methods, thus demonstrating the utility of the original and relatively simple fitting scheme. The exact DF quantities are then used to test the quality of different improvements (Langreth, Perdew, Mehl, Wang) of the local density approximation (LDA) to DF theory. The tested schemes are seen to considerably reduce the error in the LDA exchange-correlation energy. The density profile is also somewhat improved (particularly close to the core), while the effective potential is only marginally closer to its true DF counterpart.