Dielectric function of the electron gas with dynamical-exchange decoupling. I. Analytical treatment

Dynamical-exchange effects in the dielectric function of the electron gas can be described by a frequency-dependent local-field correction G(q,ω). In an earlier derivation, we deduced an explicit expression for G(q,ω) as a sixfold integral, which was obtained from the dynamical-exchange decoupling in the equation of motion for the Wigner distribution function, and a variational treatment of the resulting integro-differential equation. Explicit dynamical calculations were presented. In this paper the formal expression for G(q,ω) is reduced to a double integral in a convenient form for numerical purposes, using various Fourier representations in the integrand. Several limiting cases are evaluated analytically, from which it follows that dynamical-exchange effects drastically influence the structure of the dielectric function. Some numerical results on the frequency-dependent dielectric function with dynamical-exchange decoupling are presented.