Perturbation theory of relativistic corrections

1. The non-relativistic limit of the Dirac equation and a direct perturbation expansion
  • W. Kutzelnigg


After a discussion of the problems associated with the non-relativistic limit of the Dirac equation and of the expansion of the exact eigenvalues and eigenfunctions of the H atom in powers ofc−2 the traditional approaches for a perturbation theory of relativistic effects are critically reviewed. Then a direct perturbation theory is presented, that is characterized by a change of the metric in 4-component spinor space such that the Lévy-Leblond equation appears as the straightforward non-relativistic limit of the Dirac equation. The various orders in perturbation theory of the energy and the wave function are derived first in a direct way, then in a resolvent formalism. The formulas are very compact and easily generalizeable to arbitrary order. All integrals that arise to any order exist, and no controlled cancellation of divergent terms (as in other approaches) is necessary. In the same philosophy an iterative approach towards the solution of the Dirac equation is derived, in which the solution of the Schrödinger equation is the first iteration step.


11..10.Qr 31.15.+q 31.30.Jv 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dirac, P.A.M.: Proc. R. Soc. (London) A117, 610 (1928); A118, 351 (1928)Google Scholar
  2. 2.
    Darwin, G.G.: Proc. R. Soc. A118, 654 (1928); A120, 621, 631 (1928)Google Scholar
  3. 3.
    Sommerfeld, A., Maue, A.W.: Ann. Phys. (Leipzig)22, 629 (1935)Google Scholar
  4. 4.
    Sewell, G.L.: Proc. Cambr. Phil. Soc.45, 631 (1949)Google Scholar
  5. 5.
    Foldy, L.L., Wouthuysen, S.A.: Phys. Rev.78, 29 (1950)Google Scholar
  6. 6.
    Sucher, J.: Phys. Rev.103, 468 (1956)Google Scholar
  7. 7.
    Ma, S.T.: Nucl. Phys.2, 347 (1956)Google Scholar
  8. 8.
    Pauli, W.: In: Handbuch der Physik, Vol. V1, p. 137–160. Berlin, Göttingen, Heidelberg: Springer 1958Google Scholar
  9. 9.
    Titchmarsh, E.C.: Proc. Roy. Soc. A266, 33 (1962)Google Scholar
  10. 10. (a)
    Stone, A.P.: Proc. Phys. Soc.77, 786 (1961)Google Scholar
  11. 10. (b)
    Stone, A.P.: Proc. Phys. Soc.81, 868 (1963)Google Scholar
  12. 11. (a)
    Lévy-Leblond, J.-M.: Comm. Math. Phys.6, 286 (1967)Google Scholar
  13. 11. (b)
    Lévy-Leblond, J.-M.: Ann. Phys. (NY)57, 481 (1970)Google Scholar
  14. 12.
    Veselić, K.: Comm. Math. Phys.22, 27 (1971)Google Scholar
  15. 13.
    Hunizker, W.: Comm. Math. Phys.40, 215 (1975)Google Scholar
  16. 14.
    Roux, J.F., Van Loc, P.: Nuovo Cimento29, 225 (1975)Google Scholar
  17. 15. (a)
    Moore, R.A.: Can. J. Phys.53, 1240 (1975)Google Scholar
  18. 15. (b)
    Moore, R.A., Lee, S.: Can. J. Phys.59, 614 (1981)Google Scholar
  19. 16.
    Barut, A.O.: J. Phys. B8, L 205 (1975)Google Scholar
  20. 17.
    Osche, G.R.: Phys. Rev. D15, 2181 (1977)Google Scholar
  21. 18.
    Feneuille, S., Luc-Koenig, E.: Comm. At. Mol. Phys.6, 151 (1977)Google Scholar
  22. 19.
    Schoene, A.Y.: J. Math. Ann. Appl.71, 36 (1979)Google Scholar
  23. 20.
    Potvin, J.: J. Phys. A14, 1117 (1981)Google Scholar
  24. 21.
    Morrison, J.D., Moss, R.E.: Mol. Phys.41, 491 (1980)Google Scholar
  25. 22.
    Hostler, L.C.: J. Math. Phys.24, 2366 (1983)Google Scholar
  26. 23. (a)
    Ketley, E.J., Moss, R.E.: Mol. Phys.48, 1131 (1983)Google Scholar
  27. 23. (b)
    Ketley, E.J., Moss, R.E.: Mol. Phys.49, 1289 (1983)Google Scholar
  28. 24.
    Yamada, O.: Proc. Jpn. Acad. Sci.59A, 71 (1983)Google Scholar
  29. 25. (a)
    Gesztesy, S., Grosse, H., Thaller, B.: Phys. Letters116B, 155 (1982)Google Scholar
  30. 25. (b)
    Gesztesy, S., Thaller, B., Grosse, H.: Phys. Rev. Lett.50, 625 (1983)Google Scholar
  31. 25. (c)
    Gesztesy, S., Grosse, H., Thaller, B.: Ann. Inst. Henri Poincaré40, 159 (1984)Google Scholar
  32. 25. (d)
    Gesztesy, S., Grosse, H., Thaller, B.: Adv. Appl. Math.6, 159 (1985)Google Scholar
  33. 26. (a)
    Rutkowski, A.: J. Phys. B19, 149 (1986)Google Scholar
  34. 26. (b)
    Rutkowski, A.: J. Phys. B19, 3431 (1986)Google Scholar
  35. 26. (c)
    Rutkowski, A.: J. Phys. B19, 3443 (1986)Google Scholar
  36. 26. (d)
    Rutkowski, A., Rutkowska, D.: Phys. Scr.36, 397 (1987)Google Scholar
  37. 27.
    Chang, Ch., Pelissier, M., Durand, Ph.: Phys. Scr.34, 394 (1986)Google Scholar
  38. 28.
    Farazdel, A., Smith, V.H. Jr.: Int. J. Quant. Chem.29, 311 (1986)Google Scholar
  39. 29.
    See e.g. Desclaux, J.P.: At. Data Nucl. Data Tables12, 311 (1973)Google Scholar
  40. 30.
    The switch from a point nucleus to an extended one (or vice versa) changes the behaviour of the wave function for very smallr drastically (see e.g. [31]), such that it is not so obvious that a point nucleus is a good approximation to the real physical situationGoogle Scholar
  41. 31.
    Kutzelnigg, W.: In: Aspects of many-body effects in molecules and extended systems. Mukherjee, D. (ed.), Lecture Notes in Chemistry, Vol. 50. Berlin, HeidelbergGoogle Scholar
  42. 32.
    Huzinaga, S., Arnau, C.: Mol. Phys.20, 895 (1971)Google Scholar
  43. 33. (a)
    Chandra, P., Buenker, R.J.: J. Chem. Phys.79, 358 (1983)Google Scholar
  44. 33. (b)
    Buenker, R.J., Chandra, P., Hess, B.A.: J. Chem. Phys.84, 1 (1984)Google Scholar
  45. 33. (c)
    Buenker, R.J., Hess, B.A., Chandra, P.: J. Chem. Phys.80, 6330 (1984)Google Scholar
  46. 34.
    Ahlrichs, R.: Theoret. Chim. Acta41, 7 (1976)Google Scholar
  47. 35.
    Löwdin, P.O.: J. Mol. Spectr.14, 131 (1964)Google Scholar
  48. 36.
    Blinder, S.M.,: J. Mol. Spectr.5, 17 (1960); Adv. Quant. Chem.2, 47 (1965)Google Scholar
  49. 37.
    Harriman, J.E.: Theoretical foundations of electron spin resonance. New York: Academic Press 1978Google Scholar
  50. 38.
    Feshbach, H.: Ann. Phys. (NY)5, 357 (1958)Google Scholar
  51. 39.
    Löwdin, P.O.: J. Math. Phys.3, 969 (1962)Google Scholar
  52. 40.
    Kutzelnigg, W.: Theor. Chim. Acta73, 173 (1988)Google Scholar
  53. 41.
    Pick, S.: Theor. Chim. Acta56, 307 (1980)Google Scholar
  54. 42.
    Rose, E.M.: Relativistic electron theory. New York: Wiley 1961Google Scholar
  55. 43.
    Schwarz, W.H.E., Wallmeier, H.: Mol. Phys.46, 1045 (1982)Google Scholar
  56. 44.
    Kutzelnigg, W.: Int. J. Quantum Chem.25, 107 (1984)Google Scholar
  57. 45.
    Douglas, M., Kroll, N.M.: Ann. Phys.82, 89 (1974)Google Scholar
  58. 46.
    Hess, B.A.: Phys. Rev. A33, 3742 (1986)Google Scholar
  59. 47.
    Chang, Ch., Pelissier, M., Durand, Ph.: Phys. Scr.34, 394 (1987)Google Scholar
  60. 48.
    Hurley, W.J.: Phys. Rev. D3, 2339 (1971); D4, 3605 (1971)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • W. Kutzelnigg
    • 1
  1. 1.Lehrstuhl für Theoretische ChemieRuhr-Universität BochumBochumFederal Republic of Germany

Personalised recommendations