The Role of Algebraic Formulations of Approximate Green's Functions for Systems With a Finite Number of Electrons

Abstract

Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP (EAGP) in a Brueckner orbital representation can be used to generate a manifold of excited states|K⟩ = OK+|AGP⟩, OK|AGP⟩ = 0 with the excitation operators, O+K, and killers, OK, being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as ⟨⟨X, Y⟩⟩E = - ⟨AGP|XT0(-E + EAGP)Y + YT0(E + EAGP)X|AGP⟩in terms of the resolvent T0(ϵ) = P(ϵ -PHP)-1 with P spanned by the states|K⟩⟨K|, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.