The self-energy of the uniform electron gas in the second order of exchange

TL;DR

This paper derives an analytical expression for the on-shell self-energy of the homogeneous electron gas in second order of exchange, showing it equals the known correlation energy and confirming its consistency with established many-body theory results.

Contribution

It provides a new analytical treatment of the on-shell self-energy in second order exchange, linking it to the correlation energy and validating the off-shell self-energy via the Galitskii-Migdal formula.

Findings

On-shell self-energy equals the second order exchange correlation energy.

Off-shell self-energy yields twice the correlation energy component.

Results are consistent with the high-density limit of many-body theorems.

Abstract

The on-shell self-energy of the homogeneous electron gas in second order of exchange, $\Sigma_{2{\rm x}}= {\rm Re} \Sigma_{2{\rm x}}(k_{\rm F},k_{\rm F}^2/2)$, is given by a certain integral. This integral is treated here in a similar way as Onsager, Mittag, and Stephen [Ann. Physik (Leipzig) {\bf 18}, 71 (1966)] have obtained their famous analytical expression $e_{2{\rm x}}={1/6}\ln 2- 3\frac{\zeta(3)}{(2\pi)^2}$ (in atomic units) for the correlation energy in second order of exchange. Here it is shown that the result for the corresponding on-shell self-energy is $\Sigma_{2{\rm x}}=e_{2{\rm x}}$. The off-shell self-energy $\Sigma_{2{\rm x}}(k,\omega)$ correctly yields $2e_{2{\rm x}}$ (the potential component of $e_{2{\rm x}}$) through the Galitskii-Migdal formula. The quantities $e_{2{\rm x}}$ and $\Sigma_{2{\rm x}}$ appear in the high-density limit of the Hugenholtz-van Hove (Luttinger-Ward) theorem.