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The Many-Body Perturbation Theory of Brueckner and Goldstone

  • Werner Kutzelnigg

Abstract

After a historical introduction to the many-body problem the landmark papers on the many-body perturbation theory by Brueckner (1955) and by Goldstone (1957) are reviewed in the light of the present knowledge of many fermion systems. Brueckner started from conventional Rayleigh-Schrödinger perturbation theory, and his derivation of what he called the linked-cluster theorem was pedestrian but very tedious and not sufficiently general. Goldstone used the apparatus of quantum electrodynamics for a very elegant (but to some extent criticizable) derivation of the linked cluster expansion, that showed at first glance little resemblance to that of Brueckner. It is shown that most ingredients of Goldstone’s approach are not necessary for a simple and transparent derivation of the linked cluster expansion. It is only compulsory to work in Fock space. A very simple modern derivation is then given. Finally some insight into the time-dependent theory is gained by means of its regularization.

Keywords

Slater Determinant Connected Diagram Energy Denominator Intermediate Normalization Hole Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    P. Nozières, ‘Le problème à N corps’, Dunod, Paris 1963, ’The theory of Interacting Fermi Systems’, Benjamin, New York 1964Google Scholar
  2. 2.
    E.A. Hylleraas, Z. Phys. 54:347 (1929), 65:209 (1930)Google Scholar
  3. 3.
    D.R. Hartree, Proc. Cambridge Phil. Soc. 24:111, 426 (1928); J.C. Slater, Phys. Rev. 25: 210 (1930)Google Scholar
  4. 4.
    V. Fock, Z. Phys. 61: 126 (1930)CrossRefGoogle Scholar
  5. 5.
    C.C.J. Roothaan, Rev. Mod. Phys. 23:69 (1961)Google Scholar
  6. 6.
    A.P. Jucys, J. Exp. Theor. Phys. (USSR) 23:129 (1952) J. Vizbaraité, A. Kancerevinus, A. Jucys, Opt. Spectr. (USSR) 1:9 (1956)Google Scholar
  7. 7.
    H.A. Kramers, ‘Quantenmechanik (in Hand-und Jahrbuch der chemischen Physik Vol 1 (1937), english translation ‘Quantum Mechanics’ North-Holland, Amsterdam 1957 ( Dover, New York 1964 )Google Scholar
  8. 8.
    J.I. Frenkel, ‘Wave mechanics, advanced general theory’ Oxford, Clarendon 1934 ( Dover, New York 1950 )Google Scholar
  9. 9.
    H.D. Ursell, Proc. Cambridge Philos. Soc. 23:685 (1927)Google Scholar
  10. 10.
    J.E. Mayer, J. Chem. Phys. 5:67 (1937)Google Scholar
  11. 11.
    F. Coester, Nucl. Phys. 7:421 (1958) F. Coester, H. Kümmel, Nucl. Phys. 17:477 (1960) H. Kümmel, Nucl. Phys. 22:177 (1961)Google Scholar
  12. 12.
    P.O. Löwdin, Phys. Rev. 97:1474 (1955)Google Scholar
  13. 13.
    R.K. Nesbet, Proc. Roy. Soc. London A230:312 (1955)Google Scholar
  14. 14.
    I. Shavitt, in: ‘Modern Theoretical Chemistry III, Methods of Electronic Structure Theory’, H.F. Schaefer ed., Plenum New York (1977)Google Scholar
  15. 15.
    C. Moller, M.S. Plesset, Phys. Rev. 46:618 (1934)Google Scholar
  16. 16.
    R.J. Bartlett, D.M. Silver, J. Chem. Phys. 62:3258 (1975) J.A. Pople, R. Krishnan, H.B. Schlegel, J.S. Binkley, Int. J. Quant. Chem. Symp. 13:225 (1979)Google Scholar
  17. 17.
    K.A. Brueckner, Phys. Rev. 97:1353 (1955), 100:36 (1955)Google Scholar
  18. 18.
    J. Hubbard, Proc. Roy. Soc. London A 240:539 (1957)Google Scholar
  19. 19.
    N.M. Hugenholtz, Physica 23: 481 (1957)Google Scholar
  20. 20.
    J. Goldstone, Proc. Roy. Soc. London A239:267 (1957)Google Scholar
  21. 21.
    S. Tomonaga, Progr. Theor. Phys. 1:27 (1946), Phys Rev 74:224 (194.8)Google Scholar
  22. 22.
    J. Schwinger, Phys. Rev. 74:1439 (1948), 75:651 (1949)Google Scholar
  23. 23.
    R.P. Feynman, Phys. Rev. 76:749 (1949)Google Scholar
  24. 24.
    M. Gell-Mann, F. Low, Phys. Rev. 84:350 (1951)Google Scholar
  25. 25.
    G.C. Wick, Phys. Rev. 80: 268 (1951)Google Scholar
  26. 26.
    J. Öi2ek, J. Chem. Phys. 45:4256 (1966)Google Scholar
  27. 27.
    H.P. Kelly, Phys. Rev. 131:684 (1963), 134:1450 (1964), 144:39 (1966), Adv. Chem. Phys. 14:129 (1969)Google Scholar
  28. 28.
    A.C. Hurley, E. Lennard-Jones, J.A. Pople, Proc. Roy. Soc. London A220:446 (1953)Google Scholar
  29. 29.
    F.B. Malik, R.H. Richardson, J.Y. Shipiro, paper presented at the symposium on ‘Recent Progress in Many-Body Theories’, Oulu, Finland, August 1987Google Scholar
  30. 30.
    L.M. Frantz, R.L Mills, Nucl. Phys. 15:16 (1960)Google Scholar
  31. 31.
    B.H. Brandow, Rev. Mod. Phys. 39:771 (1976)Google Scholar
  32. 32.
    H. Primas, Helvet. Phys. Acta 34:331 (1961), Rev. Mod. Phys. 35:710 (1963)Google Scholar
  33. 33.
    R.J. Yaris, J. Chem. Phys. 41:2419 (1964), 42:3019 (1965)Google Scholar
  34. 34.
    E.R. Caianiello, in: Combinatorics and Renormalization in Quantum Field Theory, Benjamin, Reading, Mass. (1973)Google Scholar
  35. 35.
    C. Bloch, Nucl. Phys. 6: 329 (1958)CrossRefGoogle Scholar
  36. 36.
    D.J. Klein, J. Chem. Phys. 61:786, (1974) F. Jorgensen, Mol. Phys. 29:1137 (1975)Google Scholar
  37. 37.
    W. Kutzelnigg, Chem. Phys. Letters 83:156 (1981), J. Chem. Phys. 77:3081 (1982), J. Chem. Phys. 80:822 (1981) W. Kutzelnigg and S. Koch J. Chem. Phys. 77:3081 (1982)Google Scholar
  38. 38.
    W. Kutzelnigg in ‘Aspects of Many-Body Effects in Molecules and Extended Systems’ D. Mukherjee ed., Lecture notes in chemistry Vol. 50, p.35 Springer Berlin 1989Google Scholar
  39. 39.
    W. Kutzelnigg, in: ‘Recent Progress in Many-Body Theory’, H. Kümmel and M.L. Ristig ed., Lecture notes in physics Vol. 198 p. 361 (1984)Google Scholar
  40. 40.
    J. Paldus and J. Cizek, Adv. Quantum Chem. 9: 105 (1975)CrossRefGoogle Scholar
  41. 41.
    I. Lindgren, J. Morrison, ‘Atomic Many-Body Theory’, Berlin, Springer 1982CrossRefGoogle Scholar
  42. 42.
    S. Raimes, ‘Many-Electron Theory’, Amsterdam, North-Holland, 1972Google Scholar
  43. 43.
    I. Lindgren, J. Quantum Chem. Symp. 12:33 (1978)Google Scholar
  44. 44.
    D. Mukherjee and S. Pal, Adv. Quantum Chem. 20: 292 (1989)Google Scholar
  45. 45.
    T. Kato, ‘Perturbation Theory of Linear Operators’, Berlin, Springer 1966CrossRefGoogle Scholar
  46. 46.
    J.C. Slater, Phys. Rev. 34:1293 (1929)Google Scholar
  47. 47.
    P.W. Langhoff, S.T. Epstein and M. Karplus, Rev. Mod. Phys. 44:602 (1972)Google Scholar
  48. 48.
    P.W. Langhoff and A.J. Hernandez, Int. J. Quant. Chem. Symp. 10:337 (1975)Google Scholar
  49. 49.
    J. Cizek, J. Chem. Phys. 45:4256 (1966) R.J. Bartlett, J. Phys. Chem. 93:1697 (1989) and references thereinGoogle Scholar
  50. 50.
    K. Bhattacharyya and D. Mukherjee, J. Phys. A 19:67 (1986)Google Scholar
  51. 51.
    J.R. Taylor, ‘Scattering Theory’ New York, Wiley 1972Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Werner Kutzelnigg
    • 1
  1. 1.Lehrstuhl für Theoretische ChemieRuhr-Universität BochumBochumGermany

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