Perturbation theory of a many-fermion system: (II). Expansions in reaction matrices

Abstract

Currently, the most satisfactory version of Brueckner theory from the formal point of view is the generalized-time-ordered series of Brandow. In this formalism the Brueckner-Hartree-Fock (BHF) self-energy for normally empty (“particle”) states does not factorize, and for this reason it is not included in the single-particle potential, U. This feature of the theory leads to differences: (i) between single-particle and separation energies, and (ii) between U and the real part of the optical potential. In article I the time-dependent Green function formulation was used to achieve more extensive factorization, with the consequence that U could be defined to include the leading terms of the on-energy-shell BHF potential for “particles”. In the present paper this earlier expansion is written in terms of Brueckner reaction matrices, and the low orders of the resulting series are investigated. For this task, it has been found more convenient to employ the Fourier-transformed formalism of energy-dependent Green functions, G (ω). First, diagram rules for expansions in the bare interaction, V, and an arbitrary U are given in sufficient generality to apply to finite systems, for which U and f(ω) are not diagonal matrices. The binding energy is expressed in terms of closed, propagator-renormalized, ω-dependent, ground state diagrams including over-counting corrections. Then U is chosen, in correspondence with article I and a paper of Klein, in terms of the real part of the proper self-energy. In the framework employed here U may be defined conveniently to be hermitian. A prescription, stated with remarkable simplicity in the G(ω) formalism, is obtained for converting an expansion in V to one in a general transition or reaction matrix. Our U is the quasiparticle separation-energy potential, containing on-energy-shell self-energies of all topologies. For example, when the reaction matrix is that of Brueckner, the first-order (BHF) and secondorder (“rearrangement energy”) insertions are shown to be included in U both for “particles” and holes. Truncation schemes based on a stationary property of the propagator-renormalized series are explored. The simplest is a renormalized BHF approximation similar to that for Brandow's expansion but differing from it by (a) containing the RBHF potential for “particle” states, which implies a smaller energy gap and greater diffuseness at the Fermi surface; and (b) propagator renormalization (“line weighting”) factors which contain single-particle density matrices (“true occupation probabilities”) as in Brandow's expansion, plus “folded” terms. The present formulation allows a close connection between “virtual” and “valence” single-particle states, while staying close to the computationally successful framework of the Brueckner theory and avoiding complex energies.