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

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 Σ ( k , ω is in agreement with the Hugenholtz–van Hove (Luttinger–Ward) theorem μ – μ 0 = Σ ( k F , μ 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 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 μ 2x = Σ 2x ( k F , $ k^2_{\rm F}/2 $ [P. Ziesche, Ann. Phys. (Leipzig) 16 (1), 45 (2007)]. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)