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International Journal of Quantum Chemistry

Unitary group approach to spin‐adapted open‐shell coupled cluster theory

Bogumił Jeziorski, Josef Paldus, Piotr Jankowski

Published Nov 5, 1995• 79 citations
Publisher

Abstract

Abstract We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open‐shell, and virtual orbitals. When acting on the closed‐shell reference state with nc doubly occupied and nv unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(nc) ⊗ U(nV) generate all Gelfand‐Tsetlin (GT) states corresponding to appropriate irreducible representation of U(nc + nv). The tensor operators generating the M‐tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group SM. This provides an alternative to the Nagel‐Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(nc) ⊗ U(ns) tensors, can be shown to be connected, the irreducible tensor operators of U(nc) ⊗ U(nv) represent a convenient basis for a spin‐adapted full coupled cluster calculation for closed‐shell systems. For a high‐spin reference determinant with n, singly occupied open‐shell orbitals, the corresponding representation of U(n), n=nc + nv + ns is not simply reducible under the group U(nc) ⊗ U(ns) ⊗ U(nv). The multiplicity problem is resolved using the group chain U(n) ⊃ U(nc + nv) ⊗ U(ns) ⊃ U(nc) ⊗U(ns)⊗ U(nv) ⊗ U(nv). The labeling of the resulting configuration‐state functions (which, in general, are not GT states when nc > 1) by the irreducible representations of the intermediate group U(nc + nv) ⊗U(ns) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first‐, second‐, and third‐order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin‐adapted open‐shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin‐restricted open‐shell many‐body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.

DOI
10.1002/qua.560560302

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International Journal of Quantum Chemistry•Nov 1, 1989•69 citations
Abstract We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open‐shell, and virtual orbitals. When acting on the closed‐shell reference state with nc doubly occupied and nv unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(nc) ⊗ U(nV) generate all Gelfand‐Tsetlin (GT) states corresponding to appropriate irreducible representation of U(nc + nv). The tensor operators generating the M‐tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group SM. This provides an alternative to the Nagel‐Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(nc) ⊗ U(ns) tensors, can be shown to be connected, the irreducible tensor operators of U(nc) ⊗ U(nv) represent a convenient basis for a spin‐adapted full coupled cluster calculation for closed‐shell systems. For a high‐spin reference determinant with n, singly occupied open‐shell orbitals, the corresponding representation of U(n), n=nc + nv + ns is not simply reducible under the group U(nc) ⊗ U(ns) ⊗ U(nv). The multiplicity problem is resolved using the group chain U(n) ⊃ U(nc + nv) ⊗ U(ns) ⊃ U(nc) ⊗U(ns)⊗ U(nv) ⊗ U(nv). The labeling of the resulting configuration‐state functions (which, in general, are not GT states when nc > 1) by the irreducible representations of the intermediate group U(nc + nv) ⊗U(ns) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first‐, second‐, and third‐order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin‐adapted open‐shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin‐restricted open‐shell many‐body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.
The traditional state universal multi-reference coupled cluster (SUMRCC) theory uses the Jeziorski-Monkhorst (JM) based Ansatz of the wave operator: Ω = Σ(μ)Ω(μ)|φ(μ)><φ(μ)|, where Ω(μ) = exp(T(μ)) is the cluster representation of the component of Ω inducing virtual excitations from the model function φ(μ). In the first formulations, φ(μ)s were chosen to be single determinants and T(μ)s were defined in terms of spinorbitals. This leads to spin-contamination for the non-singlet cases. In this paper, we propose and implement an explicitly spin-free realization of the SUMRCC theory. This method uses spin-free unitary generators in defining the cluster operators, {T(μ)}, which even at singles-doubles truncation, generates non-commuting cluster operators. We propose the use of normal-ordered exponential parameterization for Ω:Σ(μ){exp(T(μ))}|φ(μ)><φ(μ)|, where {} denotes the normal ordering with respect to a common closed shell vacuum which makes the "direct term" of the SUMRCC equations terminate at the quartic power. We choose our model functions {φ(μ)} as unitary group adapted (UGA) Gel'fand states which is why we call our theory UGA-SUMRCC. In the spirit of the original SUMRCC, we choose exactly the right number of linearly independent cluster operators in {T(μ)} such that no redundancies in the virtual functions {χ(μ)(l)} are involved. Using example applications for electron detached/attached and h-p excited states relative to a closed shell ground state we discuss how to choose the most compact and non-redundant cluster operators. Although there exists a more elaborate spin-adapted JM-like ansatz of Datta and Mukherjee (known as combinatoric open-shell CC (COS-CC), its working equations are more complex. Results are compared with those from COS-CC, equation of motion coupled cluster methods, restricted open-shell Hartree-Fock coupled cluster, and full configuration interaction. We observe that our results are more accurate with respect to most other theories as a result of the use of the cluster expansion structure for our wave operator. Our results are comparable to those from the more involved COS-CC, indicating that our theory captures the most important aspects of physics with a considerably simpler scheme.
By replacing the usual annihilation and creation operators of second quantization by appropriately normalized fundamental Wigner operators of the unitary group U(n) and by representing the many-electron spin eigenfunctions in terms of the Gelfand–Tsetlin basis for the appropriate irreducible representation, we have succeeded in developing an attractive electron propagator formalism which incorporates closed-shell, open-shell, or multiconfigurational reference states. Matrix element evaluation for the fundamental U(n) Wigner operators is treated, and illustrative three-orbital examples involving reference states of doublet and triplet spin symmetry are presented.
Abstract We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
We introduce a simple coupled cluster approach for multireference and open-shell systems. In the restricted active space coupled cluster framework, the cluster operator only includes excitations from doubly occupied to virtual orbitals, and this ensures the individual operators commute among each other. The ground and excited states are obtained by diagonalisation of the coupled cluster similarity transformed Hamiltonian in a restricted active space. We demonstrate the framework for the RAS-CCSD model, where single and double excitations are included in the cluster operator, and likewise for a complete active space. We consider systems with multireference character and open-shell systems in a spin-adapted manner. The approach captures significantly more dynamical correlation than without the coupled cluster similarity transformation.
Abstract This paper shows that the spin‐shift formalism developed in B. T. Pickup and A. Mukhopadhyay [Int. J. Quantum Chem. 26, 101 (1984)] supports a one‐component diagrammatics which has a systematics akin to that in the spin‐orbital many‐body theory. The diagrams are neither Goldstone nor Yutsis type, and characterize the chain U(2R) ⊃ U(R)⊗SU(2) on which the spin‐shift formalism is based. Accordingly, while the lines in such diagrams are labeled by the orbital indices, the diagram structure adequately reflects the irreducible representation of the group U(R). In this sense the paper presents a unitary group approach to the natural generalization of the usual many‐body theory for the spin‐adapted cases. A set of very simple rules is derived; their similarity with the corresponding rules in the ordinary many‐body theory and practical utility are discussed in connection with (a) matrix elements over many‐electron spin states and (b) closed‐ and open‐shell many‐body perturbation theory. A possibility of integral‐driven many‐body perturbation theory for open‐shells is indicated. Connections of this formalism with others are also discussed.
Abstract A spin‐free symmetry‐adapted valence bond (VB) state, named bonded tableau (BT), is deduced from the classical bonded function and labeled by an at most two‐column Weyl tableau. The complete set, which is composed of the BT basis or canonical bonded tableau (CBT), can be constructed from an overcomplete set of BT states. CI CBT and VB CBT are two kinds of complete sets that are constructed in this paper. They can be used, respectively, in the CI and VB theory. It is shown that there is a one‐to‐one correspondence between the labeling scheme for CI CBT and the Gelfand–Tsetlin (GT) basis. This relationship enables an efficient generation and compact representation of the BT basis if one desires to use the known global representation scheme for the GT basis. Effective algorithms for the matrix element evaluation of unitary group generators and products of generators between BT states are presented. In the formulation, the action of a generator on a BT state yields another BT state times a coefficient, so that the matrix elements of an arbitrary multiple product of generators are reduced to a calculation of the overlaps between BT states. The evaluation of the overlaps leads to a simple factorization into cycle contributions, whose values are given explicitly and only depend on the length parameters of the cycles. It is hoped that the presented formalism can facilitate the procedures for handling of the many‐electron correlation problem.