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The on‐shell self‐energy of the uniform electron gas in its weak‐correlation limit
Author(s) -
Ziesche P.
Publication year - 2007
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.200642474
Subject(s) - physics , sum rule in quantum mechanics , fermi gas , sigma , diagram , order (exchange) , energy (signal processing) , limit (mathematics) , electron , ground state , electronic correlation , omega , quantum mechanics , atomic physics , mathematics , mathematical analysis , quantum chromodynamics , statistics , finance , economics
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)