Gradient expansion in the density functional approach to an inhomogeneous electron system

Abstract

The linearized integral equation obeyed by the irreducible vertex function associated with the density fluctuations is solved exactly up to second power in the wave vector. This is used to compute the static polarizability of the homogeneous electron system up to this order. It determines the gradient expansion coefficient in the density-functional formalism for the inhomogeneous electron system. The result is compared with existing approximate calculations. The method of solution is applicable to irreducible vertex functions which appear in the determination of other correlation functions of the homogeneous systems. Our gradient term vanishes for both extreme high- and low-density regions, unlike the results of Kleinman and Sham.