Study of the density gradient expansion for the kinetic energy

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Abstract
The density-gradient expansion for the kinetic energy is studied by application of the expansion to an inhomogeneous system of noninteracting fermions, and its convergence demonstrated. The inhomogeneity in the density is created by assuming the electrons move in an effective potential which is linear in the positive half space and constant elsewhere. It is shown that the original von Weisäcker coefficient of the first density-gradient correction is inappropriate for both rapidly and slowly varying densities. The coefficient reduced by a factor of 9, however, is appropriate for all density profiles provided that the second density-gradient correction is included for the rapidly varying case. The applicability of the expansion to the metal-surface problem is discussed, and the inclusion of the second density-gradient correction with its nonlinear response contributions shown to be of major significance in such calculations. A semiempirically determined value of 1.336 for the coefficient of the second gradient correction which leads to results which are essentially exact over a wide range of density profiles including the metallic range is proposed.