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International Journal of Quantum Chemistry

Multipartitioning Møller–Plesset perturbation theory: Size‐extensivity at third order and symmetry conservation

Zoltán Rolik, Ágnes Szabados

Published Jan 1, 2009• 5 citations
Publisher

Abstract

Abstract Multipartitioning multireference many‐body perturbation theory (Zaitevskii and Malrieu, Chem. Phys. Lett. 1995, 233, 597) is investigated with regard to symmetry and size‐extensivity. We show that the spin‐adapted formulation suffers from spatial symmetry breaking and propose a general symmetry‐conserving zero‐order Hamiltonian. We analyze size‐extensivity of various partitionings at the third order and find that extensivity holds if one‐particle quantities in the zero‐order Hamiltonian are properly chosen. In particular, third order of the spin‐adapted and general symmetry‐adapted theory prove to be extensive. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

DOI
10.1002/qua.22059

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Abstract Multipartitioning multireference many‐body perturbation theory (Zaitevskii and Malrieu, Chem. Phys. Lett. 1995, 233, 597) is investigated with regard to symmetry and size‐extensivity. We show that the spin‐adapted formulation suffers from spatial symmetry breaking and propose a general symmetry‐conserving zero‐order Hamiltonian. We analyze size‐extensivity of various partitionings at the third order and find that extensivity holds if one‐particle quantities in the zero‐order Hamiltonian are properly chosen. In particular, third order of the spin‐adapted and general symmetry‐adapted theory prove to be extensive. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
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The second-order multireference perturbation theory employing multiple partitioning of the many-electron Hamiltonian into a zero-order part and a perturbation is formulated in terms of many-body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two-particle terms in zero-order operators. The novel formulation naturally gives rise to a “sum-over-orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one-electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999
Second-order Møller-Plesset perturbation theory is well-known as a computationally inexpensive approach to the electron correlation problem that is size-consistent with a size-consistent reference but fails to be regular. On the other hand, the less well-known many-body version of Brillouin-Wigner perturbation theory has the reverse properties: it is regular but fails to be size-consistent when used with the standard MP partitioning. Consequently, its widespread use remains limited. In this work, we analyze the ways in which it is possible to use alternative non-MP partitions of the Hamiltonian to yield variants of BW2 that are size-consistent as well as regular. We show that there is a vast space of such BW2 theories and also show that it is possible to define a repartitioned BW2 theory from the ground state density alone, which regenerates the exact correlation energy. We also provide a general recipe for deriving regular, size-consistent, and size-extensive partitions from physically meaningful components, and we apply the result to small model systems. The scope of these results appears to further set the stage for a revival of BW2 in quantum chemistry.
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
Abstract In connection with spin adaptation in many‐body perturbation theory, this paper reexamines the use of spin graphs in view of a Hugenholtz‐like representation where both the orbital and the spin parts coexist. Together with the idea of essentially distinct diagrams, this representation leads to an economic handling of spin adaptation. As a side issue, the appropriate generalization of the Epstein–Nesbet partitioning for spin‐adapted perturbation theory is obtained. Compact formulas up to fourth order of the ground‐state energy, and up to third order for excitation energies and ionization potentials are given.