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Coupled-cluster approach to the many-body perturbation theory for open-shell systems. [Bloch equation]

Ingvar Lindgren

Published Mar 1, 1978• 0 citations
OpenAlex

Abstract

A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh--Schroedinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kuemmel for closed-shell systems. Self-consistent coupled-cluster equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Cizek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed. 56 references.

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Brillouin–Wigner coupled cluster theory. Fock-space approach

Nicholas D. K. Petraco, Ľuboš Horný, Henry F. Schaefer +1 more

The generalization of coupled cluster (CC) theory is far from being a standard method to account for correlation effects of ubiquitous open-shell systems. This dilemma is largely due to three problems. The first concerns the incorporation of multiple reference “guess” wave functions into nonvariational theories. Next is the size-extensivity issue. Finally, and perhaps most importantly is the notorious intruder state problem. Brillouin–Wigner (BW) and generalized Brillouin–Wigner (gBW) perturbation theories are used to aid in the development of new Fock-space coupled cluster theories in an attempt to alleviate some of these problems. Bloch equations are derived which can be used to formulate BWCC and gBWCC theories of arbitrary dimension in all sectors of Fock space. Since this is our first study of Brillouin–Wigner coupled cluster theories in Fock space we have chosen to keep our approach very simple. Explicit effective Hamiltonian and amplitude equations for up to two-body S-amplitudes in the (0,1) and (1,0) sectors are given for the single reference case. Aspects concerning the connectivity of the amplitude equations are addressed.

The Journal of Chemical Physics•Nov 14, 2002•19 citations
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh--Schroedinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kuemmel for closed-shell systems. Self-consistent coupled-cluster equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Cizek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed. 56 references.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh-Schrödinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kümmel for closed-shell systems. Self-consistent “coupled-cluster” equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Čížek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed.
The graphical techniques of spin algebras are combined with a diagrammatic approach based on the time independent Wick theorem to yield the spin-adapted form of the coupled cluster theory. The general rules for the implementation of this formalism are formulated and illustrated on the basic coupled-pair many-electron theory pertaining to closed shell ground states, for which case the explicit spin-adapted equations are derived. The advantages of the spin-adapted form of the theory are discussed along with the new insights it affords.
In these lectures I shall give a review of the non‐relativistic many‐body perturbation theory (MBPT). In the first section a brief summary will be given of the Rayleigh‐Schrödinger perturbation theory for non‐degenerate and degenerate systems, and in the second section the second quantization and the graphical representation and introduced. In the third section the diagrammatic expansions are developed for closed‐ and open‐shell systems, and in the fourth section, finally, the procedure is generalized to the all‐order and coupled‐cluster approaches.
In this paper, we develop a time-dependent approach to many-body perturbation theory for open shells based on the resolvent of the Schrödinger equation. We introduce, analogous to the closed-shell case, quantities Sij(t) = i<φi‖e−i(H−Ec)t ‖ψj>/<ψj0‖ψj>, where ψj0 and ψj are the jth unperturbed and exact functions, respectively. The ψj0’s can be expressed in terms of ’’model space’’ functions φi = Ωi+‖φ>, where the Ωi+ are appropriate creation/annihilation operator products acting on a conveniently chosen closed-shell vacuum φ. These φi’s are not necessarily degenerate with respect to the unperturbed Hamiltonian H0. Ec is the exact (correlated) energy of the vacuum φ. The Fourier transforms Sij(ω) of Sij(t) have the form Sij (ω)=<φi‖(ω +Ec−H)−1‖ψj>/<ψj0‖ψ j> and thus have poles at energy differences (Ej−Ec), i.e., relative to the exact vacuum energy. Using the time-dependent perturbation expansion of S(t), we obtain a Dyson-like equation N̄−1(ω)=N̄0–1(ω)+Σ, where N̄ is defined as N̄ij (ω) = <φi‖(ω+Ec−H)−1‖φj> and N̄0 is the corresponding unperturbed component. Knowledge of the combining coefficients Cji in ψj0=ΣJCjiφi is thus not required for finding the poles. We arrive at the Dyson-like equation by first eliminating closed diagrams and then regrouping the remaining terms in the perturbation series for S into ’’top’’ and ’’bottom’’ parts. Regrouping appropriate to the Brillouin–Wigner (BW) case together with an associated time-integration procedure yields ΣBW which consists of disconnected and ω-dependent diagrams. This is shown to yield the open-shell BW series in the Bloch–Horowitz form. An alternative regrouping procedure and use of the ’’folding technique’’ of Johnson and Baranger leads to a ΣRS which is ω-independent, Hermitian, contains connected diagrams only, and is, thus, size-consistent.
A particle-hole formalism is defined which is consistent with the tensor notation (and the index pairing) of the generators of the unitary group and their many-particle generalization. The corresponding generalized Wick theorem is formulated. The classification of basis operators as ‘‘closed,’’ ‘‘open,’’ ‘‘closed from above,’’ and ‘‘closed from below’’ given previously is extended to the present context. Both the perturbative and nonperturbative construction of effective Fock space Hamiltonians is discussed in the particle-hole formalism, especially for the case that the ‘‘physical vacuum’’ satisfies a Hartree–Fock condition. For the calculations of the correlation energy of the physical vacuum Cizek’s CP-MET is compared formally with the unitary variant, the UNICEP (unitary coupled electron pair) approximation. A simple derivation of Goldstone’s ‘‘linked-cluster expansion’’ is also given.
The generalization of coupled cluster (CC) theory is far from being a standard method to account for correlation effects of ubiquitous open-shell systems. This dilemma is largely due to three problems. The first concerns the incorporation of multiple reference “guess” wave functions into nonvariational theories. Next is the size-extensivity issue. Finally, and perhaps most importantly is the notorious intruder state problem. Brillouin–Wigner (BW) and generalized Brillouin–Wigner (gBW) perturbation theories are used to aid in the development of new Fock-space coupled cluster theories in an attempt to alleviate some of these problems. Bloch equations are derived which can be used to formulate BWCC and gBWCC theories of arbitrary dimension in all sectors of Fock space. Since this is our first study of Brillouin–Wigner coupled cluster theories in Fock space we have chosen to keep our approach very simple. Explicit effective Hamiltonian and amplitude equations for up to two-body S-amplitudes in the (0,1) and (1,0) sectors are given for the single reference case. Aspects concerning the connectivity of the amplitude equations are addressed.