The on-shell self-energy of the uniform electron gas in its weak-correlation limit

摘要

The ring-diagram partial summation or random-phase approximation (RPA) for the ground-state energy of the uniform electron gas (with the density parameter r(s)) in its weak-correlation limit r(s)-> 0 is revisited. It is studied, which treatment of the self-energy Sigma(k, omega) is in agreement with the Hugenholtz-van Hove (Luttinger-Ward) theorem mu - mu(o) = Sigma(k(F), mu) and which is not. As known from Macke (1950), Gell-Mann/Brueckner (1957), Onsager/Mittag/Stephen (1966) and using the Seitz theorem (1940), the correlation part of the lhs has the RPA asymptotics a ln r(s) + a'+ O(r(s)) [in atomic units]. The use of renormalized RPA diagrams for the rhs yields the similar expression a ln r(s) + a"+ O(r(s)) with the sum rule a'= a" resulting from three sum rules for the components of a' and a". This includes in the second order of exchange the relation mu(2x) = Sigma(2x) (k(F), k(F)(2)/2) [P. Ziesche, Ann. Phys. (Leipzig) 16(1), 45 (2007)]. (c) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.