Closed-form intermediate representations of many-body propagators and resolvent matrices

A new type of ``intermediate'' state, \~{}₉〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, ₍〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈₍^₉₀^N〉, where C^₉ denotes a physical excitation operator and ₀^N〉 is the N-electron ground state. A block QR decomposition G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989) of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.