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Perturbation Theory

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Springer Handbook of Atomic, Molecular, and Optical Physics

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Abstract

Perturbation theory (PT) represents one of the bridges that takes us from a simpler, exactly solvable (unperturbed) problem to a corresponding real (perturbed) problem by expressing its solutions as a series expansion in a suitably chosen “small” parameter ε in such a way that the problem reduces to the unperturbed problem when ε = 0. It originated in classical mechanics and eventually developed into an important branch of applied mathematics enabling physicists and engineers to obtain approximate solutions of various systems of differential equations 1 ; 2 ; 3 ; 4 ; 5 . For the problems of atomic and molecular structure and dynamics, the perturbed problem is usually given by the time-independent or time-dependent Schrödinger equation 6 ; 7 ; 8 ; 9 ; 10 .

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References

  1. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin, Heidelberg (1966)

    MATH  Google Scholar 

  2. Baumgärtel, H.: Analytic Perturbation Theory for Matrices and Operators. Akademie, Berlin (1984)

    MATH  Google Scholar 

  3. Hinch, E.J.: Perturbation Methods. Cambridge Univ. Press, Cambridge (1991)

    MATH  Google Scholar 

  4. Bogaevski, V.N., Povzner, A.: Algebraic Methods in Nonlinear Perturbation Theory. Springer, Berlin, Heidelberg (1991)

    Google Scholar 

  5. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. Springer, New York (1999)

    MATH  Google Scholar 

  6. Corson, E.M.: Perturbation Methods in the Quantum Mechanics of n-Electron Systems. Blackie & Son, London (1951)

    MATH  Google Scholar 

  7. Lindgren, I., Morrison, J.: Atomic Many-Body Theory. Springer, Berlin, Heidelberg (1982)

    Google Scholar 

  8. Gross, E.K.U., Runge, E., Heinonen, O.: Many-Particle Theory. Hilger, New York (1991)

    MATH  Google Scholar 

  9. Harris, F.E., Monkhorst, H.J., Freeman, D.L.: Algebraic and Diagrammatic Methods in Many-Fermion Theory. Oxford Univ. Press, Oxford (1992)

    Google Scholar 

  10. Shavitt, I., Bartlett, R.J.: Many-Body Methods in Chemistry and Physics: MBPT and Coupled Cluster Theory. Cambridge Univ. Press, Cambridge (2009)

    Google Scholar 

  11. Primas, H.: Helv. Phys. Acta 34, 331 (1961)

    MathSciNet  Google Scholar 

  12. Primas, H.: Rev. Mod. Phys. 35, 710 (1963)

    ADS  MathSciNet  Google Scholar 

  13. Rosenblum, M.: Duke Math. J. 23, 263 (1956)

    MathSciNet  Google Scholar 

  14. Arfken, G.: Mathematical Methods for Physicists. Academic Press, New York, p. 327 (1985)

    MATH  Google Scholar 

  15. Iyanaga, S., Kawada, Y. (eds.): Encyclopedic Dictionary of Mathematics. MIT Press, Cambridge, p. 1494 (1980). Appendix B, Table 3

    Google Scholar 

  16. Paldus, J., Čížek, J.: Adv. Quantum Chem. 9, 105 (1975)

    ADS  Google Scholar 

  17. Paldus, J.: In: Wilson, S., Diercksen, G.H.F. (eds.) Methods in Computational Molecular Physics. NATO ASI Series B, vol. 293, pp. 99–194. Plenum, New York (1992)

    Google Scholar 

  18. Paldus, J.: Chap. 19. In: Wilson, S. (ed.): Handbook of Molecular Physics and Quantum Chemistry, vol. 2, pp. 272–313. Wiley, Chichester (2003)

    Google Scholar 

  19. Silverstone, H.J., Holloway, T.T.: J. Chem. Phys. 52, 1472 (1970)

    ADS  Google Scholar 

  20. Møller, C., Plesset, M.S.: Phys. Rev. 46, 618 (1934)

    ADS  Google Scholar 

  21. Epstein, P.S.: Phys. Rev. 28, 695 (1926)

    ADS  Google Scholar 

  22. Nesbet, R.K.: Proc. R. Soc. Lond. A 250, 312 (1955)

    ADS  Google Scholar 

  23. Goldstone, J.: Proc. R. Soc. Lond. A 239, 267 (1957)

    ADS  Google Scholar 

  24. Hugenholtz, H.M.: Physica (Utrecht) 23, 481 (1957)

    ADS  MathSciNet  Google Scholar 

  25. Brueckner, K.A.: Phys. Rev. 100, 36 (1955)

    ADS  Google Scholar 

  26. Hubbard, J.: Proc. R. Soc. Lond. A 240, 539 (1957)

    ADS  Google Scholar 

  27. Frantz, L.M., Mills, R.L.: Nucl. Phys. 15, 16 (1960)

    Google Scholar 

  28. Coester, F.: Nucl. Phys. 7, 421 (1958)

    Google Scholar 

  29. Coester, F., Kümmel, H.: Nucl. Phys. 17, 477 (1960)

    Google Scholar 

  30. Čížek, J.: J. Chem. Phys. 45, 4256 (1966)

    ADS  Google Scholar 

  31. Čížek, J.: Adv. Chem. Phys. 14, 35 (1969)

    Google Scholar 

  32. Čížek, J., Paldus, J.: Int. J. Quantum Chem. 5, 359 (1971)

    Google Scholar 

  33. Paldus, J., Čížek, J., Shavitt, I.: Phys. Rev. A 5, 50 (1972)

    ADS  Google Scholar 

  34. Bartlett, R.J.: Part I. In: Yarkony, D.R. (ed.) Modern Electronic Structure Theory, pp. 47–108. World Scientific, Singapore (1995)

    Google Scholar 

  35. Bartlett, R.J. (ed.): Recent Advances in Coupled-Cluster Methods. Recent Advances in Computational Chemistry, vol. 3. World Scientific, Singapore (1997)

    MATH  Google Scholar 

  36. Paldus, J., Li, X.: Adv. Chem. Phys. 110, 1 (1999)

    Google Scholar 

  37. Crawford, T.D., Schaefer III, H.F.: In: Lipkowitz, K.B., Boyd, D.B. (eds.) Reviews of Computational Chemistry, vol. 14, pp. 33–136. Wiley, New York (2000)

    Google Scholar 

  38. Bartlett, R.J., Musiał, M.: Rev. Mod. Phys. 79, 291 (2007)

    ADS  Google Scholar 

  39. Čársky, P., Paldus, J., Pittner, J. (eds.): Recent Progress in Coupled Cluster Methods: Theory and Applications. Challenges and Advances in Computational Chemistry and Physics, vol. 11. Springer, Berlin (2010)

    Google Scholar 

  40. Bartlett, R.J.: Mol. Phys. 108, 2905 (2010)

    ADS  Google Scholar 

  41. Bartlett, R.J.: Wiley. Interdiscip. Rev. Comput. Mol. Sci. 2, 126–138 (2012)

    Google Scholar 

  42. Paldus, J.: Chap. 7. In: Dykstra, C.E., Frenking, G., Kim, K.S., Scuseria, G.E. (eds.) Theory and Applications of Computational Chemistry: The First Forty Years, pp. 115–147. Elsevier, Amsterdam (2005)

    Google Scholar 

  43. Bartlett, R.J.: Chap. 42. In: Dykstra, C.E., Frenking, G., Kim, K.S., Scuseria, G.E. (eds.) Theory and Applications of Computational Chemistry: The First Forty Years, pp. 1191–1221. Elsevier, Amsterdam (2005)

    Google Scholar 

  44. Paldus, J.: J. Math. Chem. 55, 477 (2017)

    MathSciNet  Google Scholar 

  45. Bartlett, R.J., Purvis, G.D.: Int. J. Quantum Chem. 14, 561 (1978)

    Google Scholar 

  46. Raghavachari, K., Trucks, G.W., Pople, J.A., Head-Gordon, M.: Chem. Phys. Lett. 157, 479 (1989)

    ADS  Google Scholar 

  47. Lee, T.J., Scuseria, G.E.: In: Langhoff, S.R. (ed.) Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, pp. 47–108. Kluwer, Dordrecht (1995)

    Google Scholar 

  48. Kowalski, K., Piecuch, P.: J. Chem. Phys. 120, 1715 (2004)

    ADS  Google Scholar 

  49. Paldus, J., Čížek, J., Saute, M., Laforgue, A.: Phys. Rev. A 17, 805 (1978)

    ADS  Google Scholar 

  50. Li, X., Paldus, J.: J. Chem. Phys. 101, 8812 (1994)

    ADS  Google Scholar 

  51. Jeziorski, B., Paldus, J., Jankowski, P.: Int. J. Quantum Chem. 56, 129 (1995)

    Google Scholar 

  52. Paldus, J.: In: Malli, G.L. (ed.) Relativistic and Electron Correlation Effects in Molecules and Solids. NATO ASI Series B: Physics, vol. 318, pp. 207–282. Plenum, New York (1994)

    Google Scholar 

  53. Lindgren, I., Mukherjee, D.: Phys. Rep. 151, 93 (1987)

    ADS  Google Scholar 

  54. Jeziorski, B., Monkhorst, H.J.: Phys. Rev. A 24, 1686 (1981)

    ADS  Google Scholar 

  55. Paldus, J., Čársky, P., Pittner, J.: Chap. 17. In: Čársky, P., Paldus, J., Pittner, J. (eds.) Recent Progress in Coupled-Cluster Methods: Theory and Applications, pp. 455–489. Springer, Berlin (2010)

    Google Scholar 

  56. Li, X., Paldus, J.: J. Chem. Phys. 119, 5320 (2003)

    ADS  Google Scholar 

  57. Li, X., Paldus, J.: J. Chem. Phys. 119, 5334 (2003)

    ADS  Google Scholar 

  58. Li, X., Paldus, J.: J. Chem. Phys. 119, 5343 (2003)

    ADS  Google Scholar 

  59. Li, X., Paldus, J.: J. Chem. Phys. 120, 5890 (2004)

    ADS  Google Scholar 

  60. Margoulas, I., Gururangan, K., Piecuch, P., Deustua, J.E., Shen, J.: J. Chem. Theory Comput. 17, 4006 (2021)

    Google Scholar 

  61. Margoulas, I., Shen, J., Piecuch, P.: Addressing Strong Correlation by Approximate Coupled-Pair Methods with Active-Space Full Treatment of Three-Body Clusters, arXiv:2111.13787v1 [physics.chem-ph] (2021)

    Google Scholar 

  62. Chattopadhyay, S., Pahari, D., Mukherjee, D., Mahapatra, U.S.: J. Chem. Phys. 120, 5968 (2004)

    ADS  Google Scholar 

  63. Li, X., Paldus, J.: J. Chem. Phys. 107, 6257 (1997)

    ADS  Google Scholar 

  64. Li, X., Paldus, J.: J. Chem. Phys. 108, 637 (1998)

    ADS  Google Scholar 

  65. Li, X., Paldus, J.: J. Chem. Phys. 129, 054104 (2004)

    ADS  Google Scholar 

  66. Paldus, J.: J. Math. Chem. 59, 1 (2021)

    MathSciNet  Google Scholar 

  67. Paldus, J.: J. Math. Chem. 59, 37 (2021)

    MathSciNet  Google Scholar 

  68. Paldus, J.: J. Math. Chem. 59, 72 (2021)

    MathSciNet  Google Scholar 

  69. Dobrautz, W., Smart, S.D., Alavi, A.: J. Chem. Phys. 151, 94104 (2019)

    Google Scholar 

  70. Dobrautz, W.: Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems. PhD Thesis, University of Stuttgart (2019)

    Google Scholar 

  71. Li Manni, G., Guther, K., Ma, D., Dobrautz, W.: In: González, L., Lindh, R. (eds.): Quantum Chemistry and Dynamics of Excited States: Methods and Applications, Chap. 6, pp. 133–204. Wiley, New York (2021)

    Google Scholar 

  72. Deustua, J.E., Magoulas, I., Shen, J., Piecuch, P.: J. Chem. Phys. 149, 151101 (2018)

    ADS  Google Scholar 

  73. Deustua, J.E., Shen, J., Piecuch, P.: J. Chem. Phys. 154, 124103 (2021)

    ADS  Google Scholar 

  74. Gururangan, K., Deustua, J.E., Shen, J., Piecuch, P.: J. Chem. Phys. 155, 174114 (2021)

    ADS  Google Scholar 

  75. Lee, S., Zhai, H., Sharma, S., Umrigar, C.J., Chan, G.K.-L.: J. Chem. Theory Comput. 17, 3414 (2021)

    Google Scholar 

  76. Chan, G.K.-L., Head-Gordon, M.: J. Chem. Phys. 116, 4462 (2001)

    ADS  Google Scholar 

  77. Chan, G.K.-L.: J. Chem. Phys. 120, 3172 (2003)

    ADS  Google Scholar 

  78. Olivares-Amaya, R., Hu, W., Nakatani, N., Sharma, S., Yang, J., Chan, G.K.-L.: J. Chem. Phys. 142, 034102 (2015)

    ADS  Google Scholar 

  79. Kats, D., Manby, F.R.: J. Chem. Phys. 139, 021102 (2013)

    ADS  Google Scholar 

  80. Kats, D.: J. Chem. Phys. 141, 061101 (2014)

    ADS  Google Scholar 

  81. Kats, D.: J. Chem. Phys. 144, 044102 (2016)

    ADS  Google Scholar 

  82. Kats, D., Kreplin, D., Werner, H.-J., Manby, F.R.: J. Chem. Phys. 142, 064111 (2015)

    ADS  Google Scholar 

  83. Kats, D., Köhn, A.: J. Chem. Phys. 150, 151101 (2019)

    ADS  Google Scholar 

  84. Vitale, E., Alavi, A., Kats, D.: J. Chem. Theory Comput. 16(9), 5621 (2020)

    Google Scholar 

  85. Schraivogel, T., Kats, D.: J. Chem. Phys. 155, 064101 (2021)

    ADS  Google Scholar 

  86. Schraivogel, T., Cohen, A.J., Alavi, A., Kats, D.: J. Chem. Phys. 155, 191101 (2021)

    ADS  Google Scholar 

  87. Thom, A.J.W.: Phys. Rev. Lett. 105, 263004 (2010)

    ADS  Google Scholar 

  88. Deustua, J.E., Shen, J., Piecuch, P.: Phys. Rev. Lett. 119, 223003 (2017)

    ADS  Google Scholar 

  89. Deustua, J.E., Yuwono, S.H., Shen, J., Piecuch, P.: J. Chem. Phys. 150, 111101 (2019)

    ADS  Google Scholar 

  90. Christiansen, O.: J. Chem. Phys. 120, 2149 (2004)

    ADS  Google Scholar 

  91. Dyson, F.J.: Phys. Rev. 75, 486 (1949)

    ADS  MathSciNet  Google Scholar 

  92. Tomonaga, S.: Prog. Theor. Phys. (Kyoto) 1, 27 (1946)

    ADS  Google Scholar 

  93. Schwinger, J.: Phys. Rev. 74, 1439 (1948)

    ADS  MathSciNet  Google Scholar 

  94. Gell-Mann, M., Low, F.: Phys. Rev. 84, 350 (1951)

    ADS  MathSciNet  Google Scholar 

  95. Lippmann, B.A., Schwinger, J.: Phys. Rev. 79, 469 (1950)

    ADS  MathSciNet  Google Scholar 

  96. Dirac, P.A.M.: Proc. R. Soc. Lond. A 112, 661 (1926)

    ADS  Google Scholar 

  97. Dirac, P.A.M.: Proc. R. Soc. Lond. A 114, 243 (1926)

    ADS  Google Scholar 

  98. Joachain, C.J.: Quantum Collision Theory. Elsevier, New York (1975)

    Google Scholar 

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  1. Dept. of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

    Josef Paldus

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Correspondence to Josef Paldus .

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  1. Dept. of Physics, University of Windsor, N9B 3P4, Windsor, ON, Canada

    Gordon W. F. Drake

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Paldus, J. (2023). Perturbation Theory. In: Drake, G.W.F. (eds) Springer Handbook of Atomic, Molecular, and Optical Physics. Springer Handbooks. Springer, Cham. https://doi.org/10.1007/978-3-030-73893-8_5

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