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The Journal of Chemical Physics

Derivation of coupled cluster equations from general many-body relations

Tom Schork, Peter Fulde

Published Dec 15, 1992• 35 citations
Publisher

Abstract

Recently, general equations for the ground state energy were derived, which are based on the use of cumulants and which are suitable for various approximation schemes, e.g., projection techniques. Here we want to show that one can derive the well-known coupled cluster equations from them. This provides a link between two different methods of treating many-body systems.

DOI
10.1063/1.463295

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Recently, general equations for the ground state energy were derived, which are based on the use of cumulants and which are suitable for various approximation schemes, e.g., projection techniques. Here we want to show that one can derive the well-known coupled cluster equations from them. This provides a link between two different methods of treating many-body systems.
In a series of two articles, we propose a comprehensive mathematical framework for Coupled-Cluster-type methods. These methods aim at accurately solving the many-body Schrodinger equation. In this first part, we rigorously describe the discretization schemes involved in Coupled-Cluster methods using graph-based concepts. This allows us to discuss different methods in a unified and more transparent manner, including multireference methods. Moreover, we derive the single-reference and the Jeziorski-Monkhorst multireference Coupled-Cluster equations in a unified and rigorous manner.
In this paper we review the coupled cluster formulation of quantum many-body theory.Apart from a brief and heuristic, but essentially complete, discussion of 8 the by now rather standard tr-eatments of both ground and excited states within the 2 coupled cluster method (CeM), our main aim is to.give a rather careful critique of the various choices that have to be made in order to implement .themethod.These lS--•divide essentially into two classes.On the one hand we have the inputs to the method, 11 which amount largely to.achoice.ofstarting (or zero-order or model or uncorrelated) wayefunctions; while on the other hand we have the choice of decomposition of the ~equations into a natural hierarchy, and which amounts in practice to a particular approximation or truncati on scheme.We shall show expl tc+t ly .thatwhil e these two 13 ---choices are in principle independent of each other, in practice they are intimately 14
Here, we build on the works of Scuseria et al. [J. Chem. Phys. 129, 231101 (2008)] and Berkelbach [J. Chem. Phys. 149, 041103 (2018)] to show connections between the Bethe–Salpeter equation (BSE) formalism combined with the GW approximation from many-body perturbation theory and coupled-cluster (CC) theory at the ground- and excited-state levels. In particular, we show how to recast the GW and Bethe–Salpeter equations as non-linear CC-like equations. Similitudes between BSE@GW and the similarity-transformed equation-of-motion CC method are also put forward. The present work allows us to easily transfer key developments and the general knowledge gathered in CC theory to many-body perturbation theory. In particular, it may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the GW and BSE frameworks.
Cumulants represent a natural language for expressing macroscopic properties of many-body systems. The most important property of cumulants is that of size consistency, i.e. a cumulant expression for an extensive variable scales with the size of the system, independent of possible further approximations used in the evaluation procedure. Cumulants can be considered as a generalization of linked diagrams known from diagrammatic technique of many-body theory. In this paper we outline a recently introduced method based on cumulants in order to derive expressions for zero-temperature properties of many-particle systems, i.e. the ground-state energy, static expectation values and dynamical correlation functions. This cumulant formalism allows one to describe weakly and strongly correlated systems along the same lines. We show that the coupled-cluster method known from quantum chemistry can be derived from our cumulant approach. Finally, we demonstrate the usefulness of the cumulant method by applying it to examples from solid-state physics and quantum chemistry.