Comments on the Static Dielectric Function of Jellium

The equation of motion for the Wigner distribution function in a homogeneous electron gas, linearized to first order in the applied external field, and decoupled using the Hartree‐Fock prescription, rigorously describes the dynamical exchange effects in the dielectric response to a ‐weak perturbation. This equation of motion can approximately be solved with a variational principle, and a trial solution for the Wigner distribution function can be found, yielding a density that exactly satisfies its equation of motion. The dielectric function ϵ var obtained in this way thus includes dynamical exchange interactions. Comparing ϵ var with the dielectric function ϵ it obtained from an iterative procedure, where exchange effects are supposed to be small compared to the Hartree terms, it turns out that ϵ it provides the terms to order e 4 in the geometric progression of ϵ var in powers of e 2 . In the static limit ϵ it is the dielectric function calculated by Geldart and Taylor from a diagrammatic expansion. Thus a theoretical justification is given for Sham's approximation that the static susceptibility including exchange is obtained by considering the susceptibility from the diagrammatic expansion as the first two terms in a geometric progression.