The Many-Body Perturbation Theory of Brueckner and Goldstone

Leonard S. Rodberg

Peter W. Langhoff, A. J. Hernández

Connections between the Brueckner and Goldstone forms of the linked-cluster theorem are clarified in an elementary fashion employing a finite many-body system of electrons interacting via two-body forces as an illustrative example. It is shown that the order-by-order cancellations of Brueckner unlinked terms in the many-body wave function and energy occur in the Goldstone development upon the additions of the different time orderings of Feynman-Goldstone diagrams having disconnected vacuum parts. The factoring of the vacuum amplitude from the wave function, upon which customary presentations of the theorem focus attention, is seen to be irrelevant to the cancellations of Brueckner unlinked terms, and has to do, rather, with the presence of secular and normalization terms in the time-dependent perturbation functions for an adiabatically switched static perturbation. Similarly, the equivalence of the vacuum amplitude with the exponential of all connected vacuum diagrams, originally demonstrated by Feynman in the case of hole theory, is seen to be irrelevant to the cancellations of Brueckner unlinked terms in the energy, which occur in the Goldstone development upon the summations over all time orderings of disconnected vacuum diagrams. The distinction between Brueckner unlinked terms on the one hand, and secular and normalization terms on the other, is confused in customary presentations of the linked-cluster theorem by the use of exponential switching, following Gell-Mann and Low, and is clarified in the present development by considering the Friedrichs limit of ideal adiabatic switching.

Reiner M. Dreizler, Eberhard K. U. Gross

Andrei Derevianko

Manipulating expressions in many-body perturbation theory becomes unwieldily with increasing order of the perturbation theory. Here I derive a set of theorems for efficient simplification of such expressions. The derived rules are specifically designed for implementing with symbolic algebra tools. As an illustration, we count the numbers of Brueckner-Goldstone diagrams in the first several orders of many-body perturbation theory for matrix elements between two states of a mono-valent system.

Werner Kutzelnigg

Abstract The history of many‐body perturbation theory (MBPT) and its impact on Quantum Chemistry is reviewed, starting with Brueckner's conjecture of a linked‐cluster expansion and the time‐dependent derivation by Goldstone of such an expansion. A central part of this article is the time‐independent formulation of quantum chemistry in Fock space and its diagrammatic representation including the particle‐hole picture and the inversion of a commutator. The results of the time‐independent derivation of MBPT are compared with those of Goldstone. It is analyzed which ingredients of Goldstone's approach are decisive. The connected diagram theorem is derived both in a constructive way based on a Lie‐algebraic formulation and a nonconstructive way making use of the separation theorem. It is discussed why the Goldstone derivation starting from a unitary time‐evolution operator, ends up with a wave operator in intermediate normalization. The Møller–Plesset perturbation expansions of Bartlett and Pople are compared. Examples of complete summations of certain classes of diagrams are discussed, for example, that which leads to the Bethe‐Goldstone expansion. MBPT for energy differences is analyzed. The paper ends with recent developments and challenges, such as the generalization of normal ordering to arbitrary reference states, contracted Schrödinger k‐particle equations and Brillouin conditions, and finally the Nakatsuji theorem and the Nooijen conjecture. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009