Static dielectric function with exact exchange contribution in the electron liquid

TL;DR

This paper provides an exact analytical evaluation of the exchange contribution to the static dielectric function in the electron liquid, confirming previous results and establishing a rigorous mathematical foundation.

Contribution

It derives a simple, explicit expression for the exchange contribution to the dielectric function and rigorously confirms earlier approximate results by Engel and Vosko.

Findings

Exact expression for exchange contribution derived

Expression verified against previous results

Series expansions at key wavevectors obtained

Abstract

The exchange contribution, $\Pi_1 ({\bf k}, 0)$, to the static dielectric function in the electron liquid is evaluated exactly. Expression for it is derived analytically in terms of one quadrature. The expression, as presented in Eq. (3) in the Introduction, turns out to be very simple. A fully explicit expression (with no more integral in it) for $\Pi_1 ({\bf k}, 0)$ is further developed in terms of series. Equation (3) is proved to be equal to the expression obtained before under some mathematical assumption by Engel and Vosko, thus in the meanwhile putting the latter on a rigorous basis. The expansions of $\Pi_1 ({\bf k}, 0)$ at the wavectors of $k=0$, $k=2k_F$, and at limiting large $k$ are derived. The results all verify those obtained by Engel and Vosko.