• weakly-correlated electron gas;
  • self-energy;
  • second order of exchange;
  • Onsager tricks


The on-shell self-energy of the homogeneous electron gas in second order of exchange, Σ2x = Re Σ2x (kF, k2F/2), is given by a certain integral. This integral is treated here in a similar way as Onsager, Mittag, and Stephen [Ann. Physik (Leipzig) 18, 71 (1966)] have obtained their famous analytical expression e2x = equation image(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 Σ2x = e2x. The off-shell self-energy Σ2x (k, o) correctly yields 2e2x (the potential component of e2x) through the Galitskii-Migdal formula. The quantities e2x and Σ2x appear in the high-density limit of the Hugenholtz-van Hove (Luttinger-Ward) theorem.