The gradient approximation to the exchange-correlation energy functional: A generalization that works

A complete wave vector analysis has been made of the 2nd order gradient expansion of the exchange-correlation energy for a non-uniform electronic system. When applied to a typical surface, the method shows that the gradient expansion disagrees markedly with the exact small k form, and that the substantial contribution to it comes from the universál region where the wavelength is much longer than the surface healing length. This verifies the substantial correctness of Rasolt and Geldart's RPA calculation for the gradient coefficient. However, it also shows beyond any doubt that the gradient expansion itself is incorrect for a metallic surface. We propose an approximate method, based on an “average gradient” rather than the “local gradient”. When applied to a surface, we find rough agreement with our earlier cruder interpolation scheme, and good agreement with exact results where they exist.