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Many-Body Brillouin-Wigner Theories: Development and Prospects

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Abstract

We describe a quantum chemical project focussed on the development of state-specific many-body Brillouin–Wigner methods which was undertaken during the period 1994 to the present day. The Brillouin–Wigner methodology has been shown to provide an approach to the many-body problem which is especially useful when a multireference formulation is required since it completely avoids the ‘intruder state’ problem that often plagues the traditional Rayleigh–Schrödinger expansion. The many-body Brillouin–Wigner approach provides the basis for robust methods which can be applied routinely in situations where the more familiar single-reference formalism is not adequate in Coupled Cluster (cc) and Perturbation Theory (pt) formalisms. It can also be employed in Configuration Interaction (ci) studies. Although the Brillouin–Wigner expansion is not itself a ‘many-body’ theory, it can be subjected to a posteriori adjustments which removes unphysical terms, which in the diagrammatic formalism correspond to unlinked diagrams, and recover a ‘many-body’ method. As well as reviewing progress and prospects of the state-specific Brillouin–Wigner approach, we describe the new methods of communication that were deployed to facilitate effective collaboration between researchers located as geographically distributed sites. A web-based collaborative virtual environment (cve) was designed for research on molecular electronic structure theory which will support the development of quantum chemical methodology for challenging applications. This cve was developed whilst actually carrying out a significant but specific, ‘real life’ quantum chemical project so that those features which were found to be useful in facilitating remote collaboration could be evaluated and incorporated in an emerging framework.

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Notes

  1. 1.

    For a recent account of Einstein’s explanation of Brownian motion see Renn [99].

  2. 2.

    For a recent account of the development of quantum theory see Kumar’s Quantum – Einstein, Bohr and the Great Debate about the Nature of Reality [73]. The six volumes by Mehra and Rechenberg on The Historical Development of Quantum Theory [86] provide a definitive account.

  3. 3.

    See, for example, the Handbook of Molecular Physics and Quantum Chemistry [123] for a recent major reference work.

  4. 4.

    Brillouin–Wigner perturbation theory was originally introduced in independent publications by Lennard-Jones [74] in 1930, by Brillouin [11] in 1932 and by Wigner [107] in 1935.

  5. 5.

    By Moore [87] of Intel. This has been dubbed “Moore’s Law”.

  6. 6.

    gigaflops: 109 floating-point operations per second.

  7. 7.

    petaflops: 1015 floating-point operations per second.

  8. 8.

    exaflops: 1018 floating-point operations per second.

  9. 9.

    For an introduction to the field of collaborative virtual environments see, for example, Churchill et al. [19].

  10. 10.

    Parts of this article are taken from the authors’ book entitled Brillouin–Wigner Methods for Many-Body Systems.

  11. 11.

    See also the work of Das and his coworkers [29].

  12. 12.

    We are not concerned here with questions of computational efficiency.

  13. 13.

    Further details of the Rayleigh–Schrödinger perturbation expansion can be found elsewhere [110].

  14. 14.

    Parts of this digression are taken from the authors’ article entitled A Collaborative Virtual Environment for Molecular Electronic Structure Theory: Prototype for the Study of Many-Body Methods [121] which was published in the volume Frontiers in Quantum Systems in Chemistry and Physics.

  15. 15.

    Project number: D23/0001/01: European Metalaboratory for multireference quantum chemical methods (01/02/2001–18/07/2005). Participants: P. Čársky, J. Pittner (J. Heyrovsky Institute, Prague, Czech Republic), I. Hubač (Comenius University, Slovakia), S. Wilson (Rutherford Appleton Laboratory, UK), W. Wenzel (Universität Dortmund, Germany), L. Meissner (Nicholas Copernicus University, Poland), V. Staemmler (Ruhr Universität Bochum Germany), C. Tsipis (Aristotle University of Thessaloniki, Greece), A. Mavridis (National and Kapodistrian University of Athens, Greece).

  16. 16.

    Loosely speaking, a metalaboratory may be defined as a cluster of geographically distributed resources.

  17. 17.

    eucost Action d9 “Advanced computational chemistry of increasingly complex systems”.

  18. 18.

    For some details see, for example the Wikipedia entry on e-science at http://en.wikipedia.org/wiki/E-Science

    Currently the largest focus in e-science is in the United Kingdom. In the United States similar initiatives are termed cyberinfrastructure projects.

  19. 19.

    These methods are often written as mp2, mp3, mp4, etc., that is Møller-Plesset perturbation theory [88] in second, third, fourth order.

  20. 20.

    These methods are often written as mr-mp2, mr-mp3, mr-mp4, etc., that is multireference Møller-Plesset perturbation theory in second, third, fourth order.

  21. 21.

    See Chap. 1 of reference [51] for a brief discussion of the contribution of Lennard-Jones to Brillouin-Wigner theory.

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Acknowledgements

This work was supported in part by the APVV Grant Agency, Slovakia, under project numbers -0420-10 and -0442-07 and also VEGA Grant Agency, Slovakia, under project number -1/0762/11. IH thanks the Grant Agency of the Czech Republic under project number MSM 4781305903.

Note added in proof: Recent research has shown how the many-body Brillouin-Wigner formalism can be deployed in the study of electron-molecule scattering [57].

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Authors and Affiliations

  1. Department of Nuclear Physics and Biophysics, Division of Chemical Physics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, 84248, Slovakia

    Ivan Hubač

  2. Institute of Physics, Silesian University, 74601, Opava, Czech Republic

    Ivan Hubač

  3. Theoretical Chemistry Group, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ, UK

    Stephen Wilson

  4. Division of Chemical Physics, Faculty of Mathematics, Physics and Informatics Comenius University, Bratislava, 84248, Slovakia

    Stephen Wilson

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  1. Department of Chemistry, Jackson State University, 17910, P.O. Box 17910, 1400 Lynch St., Jackson, 39217, Mississippi, USA

    Jerzy Leszczynski

  2. Dept. Chemistry, Jackson State University, J. R. Lynch St. 1325, Jackson, 39217, Mississippi, USA

    Manoj K. K. Shukla

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Hubač, I., Wilson, S. (2011). Many-Body Brillouin-Wigner Theories: Development and Prospects. In: Leszczynski, J., Shukla, M.K. (eds) Practical Aspects of Computational Chemistry I. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0919-5_2

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