Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Quantum Chemistry

Many‐body Brillouin–Wigner second‐order perturbation theory: A robust and efficient approach to the multireference correlation problem

Péter Papp, P. Mach, Ivan Hubač, Stephen S. Wilson

Published Jan 1, 2007• 15 citations
Publisher

Abstract

Abstract The multireference, state‐specific, second‐order, Brillouin–Wigner perturbation theory is presented as a robust approach to the electron correlation problem for systems demanding the use of a multireference function. A posteriori modifications are made which, in the case of a single reference function, recover the well known formula of second‐order many‐body perturbation theory (MBPT2), i.e. Møller–Plesset (MP2) theory, and in the multireference case can be shown to be equivalent to state‐specific multireference Rayleigh–Schrödinger‐like perturbation theory. It is shown that multireference many‐body Brillouin–Wigner perturbation theory when restricted to second‐order is a true many‐body theory, that is, it is a theory, which has the property of extensivity. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

DOI
10.1002/qua.21514

Related papers

Repartitioning the Hamiltonian in many-body second-order Brillouin–Wigner perturbation theory: Uncovering new size-consistent models

Linus Bjarne Dittmer, Martin Head‐Gordon

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.

The Journal of Chemical Physics•Feb 4, 2025•15 citations

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

Zoltán Rolik, Ágnes Szabados

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

International Journal of Quantum Chemistry•Jan 1, 2009•5 citations

Diagrammatic formulation of the second-order many-body multipartitioning perturbation theory

Andr�i Zaitsevskii, Renzo Cimiraglia

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

International Journal of Quantum Chemistry•Jan 1, 1999•23 citations

How many‐body perturbation theory (MBPT) has changed quantum chemistry

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

International Journal of Quantum Chemistry•Aug 26, 2009•37 citations

Regularized second-order correlation methods for extended systems

Elisabeth Keller, Theodoros Tsatsoulis, Karsten Reuter +1 more

Second-order Møller-Plesset perturbation theory (MP2) constitutes the simplest form of many-body wavefunction theory and often provides a good compromise between efficiency and accuracy. There are, however, well-known limitations to this approach. In particular, MP2 is known to fail or diverge for some prototypical condensed matter systems like the homogeneous electron gas (HEG) and to overestimate dispersion-driven interactions in strongly polarizable systems. In this paper, we explore how the issues of MP2 for metallic, polarizable, and strongly correlated periodic systems can be ameliorated through regularization. To this end, two regularized second-order methods (including a new, size-extensive Brillouin-Wigner approach) are applied to the HEG, the one-dimensional Hubbard model, and the graphene-water interaction. We find that regularization consistently leads to improvements over the MP2 baseline and that different regularizers are appropriate for the various systems.

The Journal of Chemical Physics•Dec 22, 2021•17 citations

On the Brillouin‐zone integrations in second‐order many‐body perturbation calculations for extended systems of one‐dimensional periodicity

Tomomi Shimazaki, So Hirata

Abstract The validity and accuracy of various ways of drastically reducing the number of k‐points in the Brillouin zone integrations occurring in second‐order many‐body perturbation calculations of one‐dimensional solids has been investigated. The most promising approximation can recover correlation energies of polyethylene and polyacetylene within 1% of converged values at less than a tenth of usual computational cost. The quasi‐particle energy bands have also been reproduced quantitatively with the same approximation. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

International Journal of Quantum Chemistry•Apr 30, 2009•26 citations
Abstract The multireference, state‐specific, second‐order, Brillouin–Wigner perturbation theory is presented as a robust approach to the electron correlation problem for systems demanding the use of a multireference function. A posteriori modifications are made which, in the case of a single reference function, recover the well known formula of second‐order many‐body perturbation theory (MBPT2), i.e. Møller–Plesset (MP2) theory, and in the multireference case can be shown to be equivalent to state‐specific multireference Rayleigh–Schrödinger‐like perturbation theory. It is shown that multireference many‐body Brillouin–Wigner perturbation theory when restricted to second‐order is a true many‐body theory, that is, it is a theory, which has the property of extensivity. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
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 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
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
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
Second-order Møller-Plesset perturbation theory (MP2) constitutes the simplest form of many-body wavefunction theory and often provides a good compromise between efficiency and accuracy. There are, however, well-known limitations to this approach. In particular, MP2 is known to fail or diverge for some prototypical condensed matter systems like the homogeneous electron gas (HEG) and to overestimate dispersion-driven interactions in strongly polarizable systems. In this paper, we explore how the issues of MP2 for metallic, polarizable, and strongly correlated periodic systems can be ameliorated through regularization. To this end, two regularized second-order methods (including a new, size-extensive Brillouin-Wigner approach) are applied to the HEG, the one-dimensional Hubbard model, and the graphene-water interaction. We find that regularization consistently leads to improvements over the MP2 baseline and that different regularizers are appropriate for the various systems.
Abstract The validity and accuracy of various ways of drastically reducing the number of k‐points in the Brillouin zone integrations occurring in second‐order many‐body perturbation calculations of one‐dimensional solids has been investigated. The most promising approximation can recover correlation energies of polyethylene and polyacetylene within 1% of converged values at less than a tenth of usual computational cost. The quasi‐particle energy bands have also been reproduced quantitatively with the same approximation. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009