Theory of nonuniform electronic systems. I. Analysis of the gradient approximation and a generalization that works

A complete wave-vector analysis has been made of the gradient coefficient for the exchange-correlation energy of a nonuniform electronic system. It is shown that the majority of the contribution comes from a very small but universal region of $\overrightarrow{\mathrm{k}}$ space near the origin. From this it can be concluded that random-phase-approximation-like calculations, like the present one or that of Rasolt and Geldart, which treat this region correctly, are likely to provide accurate results for the gradient coefficient and hence for the energy and structure of a system whose density is truly slowly varying. However, it also shows that the criterion for the validity of the gradient approximation itself is much more severe than previously supposed, so that the usual type of application, to say a surface or bulk material, is incorrect. For the surface case this is verified in unequivocal detail. On the other hand, a generalization of the gradient scheme based on an average slope instead of a local slope is proposed. This gives good agreement with limiting cases where they exist, and rough agreement with the interpolation scheme proposed previously by the authors.