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

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

2006-05-08

arxiv.org/abs/cond-mat/0605188v4

Physics

Condensed Matter

Strongly Correlated Electrons

12 pages, 2 figures

Scientific paper

10.1002/andp.200610220

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.

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