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

Abstract

The on-shell self-energy of the homogeneous electron gas in second order of exchange, ${\Sigma }_{2\mathrm{x}}=\mathrm{Re}{\Sigma }_{2\mathrm{x}}(k_{\mathrm{F}},k_{\mathrm{F}}^2/2)$, 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 $e_{2\mathrm{x}}=1/6\ln 2-3\frac{\zeta (3)}{(2\pi {)}^2}$ (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 ${\Sigma }_{2\mathrm{x}}=e_{2\mathrm{x}}$. The off-shell self-energy ${\Sigma }_{2\mathrm{x}}(k,\omega )$ correctly yields $2e_{2\mathrm{x}}$ (the potential component of $e_{2\mathrm{x}}$) through the Galitskii-Migdal formula. The quantities $e_{2\mathrm{x}}$ and ${\Sigma }_{2\mathrm{x}}$ appear in the high-density limit of the Hugenholtz-van Hove (Luttinger-Ward) theorem.

12 pages, 2 figures

Cited by in corpus (7)

• Diagrammatic expansion for positive spectral functions beyond GW: Application to vertex corrections in the electron gas
• Dynamically screened vertex correction to $GW$
• Vertex corrections for positive-definite spectral functions of simple metals
• An optimized twist angle to find the twist-averaged correlation energy applied to the uniform electron gas
• Analysis of the second order exchange self energy of a dense electron gas
• Self-consistent Spectral Function for Non-Degenerate Coulomb Systems and Analytic Scaling Behaviour
• Optical properties and one-particle spectral function in non-ideal plasmas