Skip to main content
SpringerLink
Log in

High Precision Calculations for Helium

  • Reference work entry
Springer Handbook of Atomic, Molecular, and Optical Physics

Part of the book series: Springer Handbooks ((SHB))

Abstract

Exact analytic solutions to the Schrödinger equation are known only for atomic hydrogen, and other equivalent two-body systems (see Chapt. 9). However, very high precision approximations are now available for helium, which are essentially exact for all practical purposes. This chapter summarizes the computational methods and tabulates numerical results for the ground state and several singly excited states. Similar methods can be applied to other three-body problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Abbreviations

FS:

fine-structure

HUM:

Hylleraas-Undheim-MacDonald theorem

NFS:

nonfine-structure

QED:

quantum electrodynamics

References

  1. T. Kato: Commun. Pure Appl. Math. 10, 151 (1957)

    Article  MATH  Google Scholar 

  2. M. Hoffman-Ostenhoff, T. Hoffmann-Ostenhoff, H. Stremnitzer: Phys. Rev. Lett. 68, 3857 (1992)

    Article  ADS  Google Scholar 

  3. V. A. Fock: Izv. Akad. Nauk SSSR, Ser. Fiz. 18, 161 (1954)

    MathSciNet  Google Scholar 

  4. V. A. Fock: D. Kngl. Norske Videnskab. Selsk. Forh. 31, 138 (1958)

    Google Scholar 

  5. V. A. Fock: D. Kngl. Norske Videnskab. Selsk. Forh. 31, 145 (1958)

    Google Scholar 

  6. G. B.  Sochilin: Int. J. Quantum Chem. 3, 297 (1969)

    Article  ADS  Google Scholar 

  7. C. R. Myers, C. J. Umriger, J. P. Sethna, J. D. Morgan III: Phys. Rev. A 44, 5537 (1991)

    Article  ADS  Google Scholar 

  8. J. E. Gottschalk, E. N. Maslen: J. Phys. A 20, 2781 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  9. K. McIsaac, E. N. Maslen: Int. J. Quantum Chem. 31, 361 (1987)

    Article  MATH  Google Scholar 

  10. J. H. Macek: Phys. Rev. 160, 170 (1967)

    Article  ADS  Google Scholar 

  11. J. Leray: Trends and Applications of Pure Mathematics to Mechanics, Lect. Notes Phys., Vol. 195, ed. by P. G. Ciarlet, M. Roseau (Springer, Berlin, Heidelberg 1984) pp. 235-247

    Google Scholar 

  12. J. D. Morgan III: Theoret. Chem. Acta 69, 181 (1986)

    Article  Google Scholar 

  13. P. Deift, W. Hunziker, B. Simon, E. Vock: Commun. Math. Phys. 64, 1 (1978)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. J. D. Morgan III: J. Phys. A 10, L91 (1977)

    Article  ADS  Google Scholar 

  15. M. Hoffman-Ostenhoff, T. Hoffmann-Ostenhoff: Phys. Rev. A 16, 1782 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  16. R. Ahlrichs, T. Hoffman-Ostenhoff, M. Hoffmann-Ostenhoff, J. D. Morgan III: Phys. Rev. A 23, 2107 (1981)

    Article  ADS  Google Scholar 

  17. E. A. Hylleraas, B. Undheim: Z. Phys. 65, 759 (1930)

    Article  ADS  Google Scholar 

  18. J. K. L. MacDonald: Phys. Rev. 43, 830 (1933)

    Article  MATH  ADS  Google Scholar 

  19. E. A. Hylleraas: Z. Phys. 48, 469 (1928)

    Article  ADS  Google Scholar 

  20. E. A. Hylleraas: Z. Phys. 54, 347 (1929)

    Article  ADS  Google Scholar 

  21. Y. Accad, C. L. Pekeris, B. Schiff: Phys. Rev. A 4, 516 (1971)

    Article  ADS  Google Scholar 

  22. K. Frankowski, C. L. Pekeris: Phys. Rev. 146, 46 (1966)

    Article  ADS  Google Scholar 

  23. K. Frankowski, C. L. Pekeris: Phys. Rev. 150, 366(E) (1966)

    Article  ADS  Google Scholar 

  24. D. E. Freund, B. D. Huxtable, J. D. Morgan III: Phys. Rev. A 29, 980 (1984)

    Article  ADS  Google Scholar 

  25. A. J. Thakkar, T. Koga: Phys. Rev. A 50, 854 (1994)

    Article  ADS  Google Scholar 

  26. G. W. F. Drake: Nucl. Instrum. Methods Phys. Res. Sect B 31, 7 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  27. G. W. F. Drake, Z.-C. Yan: Phys. Rev. A 46, 2378 (1992)

    Article  ADS  Google Scholar 

  28. G. W. F. Drake: Long Range Casimir Forces: Theory and Recent Experiments in Atomic Systems, ed. by F. S. Levin, D. A. Micha (Plenum Press, New York 1993) p. 107

    Google Scholar 

  29. G. W. F. Drake, Z.-C. Yan: Chem. Phys. Lett. 229, 486 (1994)

    Article  ADS  Google Scholar 

  30. A. M. Frolov, V. H. Smith: J. Phys. B 37, 2917 (2004)

    Article  ADS  Google Scholar 

  31. V. Korobov: Phys. Rev. A 69, 0545012 (2004) The Bethe logarithms calculated by Korobov include an additional lnZ 2 in their definition

    Article  Google Scholar 

  32. G. W. F. Drake: Phys. Rev. A 18, 820 (1978)

    Article  ADS  Google Scholar 

  33. Z.-C. Yan, G. W. F. Drake: Can. J. Phys. 72, 822 (1994)

    Article  ADS  Google Scholar 

  34. J. Ackermann: Phys. Rev. A 52, 1968 (1995) and earlier references therein

    Article  ADS  Google Scholar 

  35. C.-Y. Hu, A. A. Kvitsinsky, J. S. Cohen: J. Phys. B 28, 3629 (1995) and earlier references therein

    Article  ADS  Google Scholar 

  36. R. Krivec, V. B. Mandelzweig, K. Varga: Phys. Rev. 61, 062503 (2000) and earlier references therein

    Article  Google Scholar 

  37. A. H. Wapstra, G. Audi: Nucl. Phys. A 432, 1 (1985)

    Article  ADS  Google Scholar 

  38. A. Dalgarno, N. Lynn: Proc. Phys. Soc. (London) A 70, 802 (1957)

    Article  ADS  Google Scholar 

  39. G. W. F. Drake: Adv. At. Mol. Opt. Phys. 31, 1 (1993)

    Google Scholar 

  40. G. W. F. Drake, S. P. Goldman: Can. J. Phys. 77, 835 (1999)

    Article  ADS  Google Scholar 

  41. S. P. Goldman, G. W. F. Drake: Phys. Rev. Lett. 68, 1683 (1992)

    Article  ADS  Google Scholar 

  42. G. W. F. Drake: Phys. Scr. T95, 22 (2001)

    Article  ADS  Google Scholar 

  43. G. W. F. Drake, R. A. Swainson: Phys. Rev. A 41, 1243 (1990)

    Article  ADS  Google Scholar 

  44. A. P. Stone: Proc. Phys. Soc. (London); 77, 786 (1961)

    Article  ADS  Google Scholar 

  45. A. P. Stone: Proc. Phys. Soc. (London); 81, 868 (1963)

    Article  ADS  Google Scholar 

  46. G. W. F. Drake: Phys. Rev. Lett. 74, 4791 (1995)

    Article  ADS  Google Scholar 

  47. K. Pachucki: Phys. Rev. Lett. 84, 4561 (2000)

    Article  ADS  Google Scholar 

  48. V. Korobov, A. Yelkhovsky: Phys. Rev. Lett. 87, 193003 (2001)

    Article  ADS  Google Scholar 

  49. G. W. F. Drake, I. B. Khriplovich, A. I. Milstein, A. S. Yelkhovsky: Phys. Rev. A 48, R15 (1993)

    Article  ADS  Google Scholar 

  50. J. F. Babb, L. Spruch: Phys. Rev. A 38, 13 (1988)

    Article  ADS  Google Scholar 

  51. C.-K. Au: Phys. Rev. A 39, 2789 (1989)

    Article  ADS  Google Scholar 

  52. C.-K. Au, M. A. Mesa: Phys. Rev. A 41, 2848 (1990)

    Article  ADS  Google Scholar 

  53. C. H. Storry, N. E. Rothery, E. A. Hessels: Phys. Rev. Lett. 75, 3249 (1995) and earlier references therein

    Article  ADS  Google Scholar 

  54. G. W. F. Drake: Adv. At. Mol. Opt. Phys. 32, 93 (1994)

    Google Scholar 

  55. R. J. Drachman: Phys. Rev. A 47, 694 (1993)

    Article  ADS  Google Scholar 

  56. G. W. F. Drake, R. A. Swainson: Phys. Rev. A 42, 1123 (1990)

    Article  ADS  Google Scholar 

  57. E. A. Hessels: Phys. Rev. A 46, 5389 (1992)

    Article  ADS  Google Scholar 

  58. G. W. F. Drake: Phys. Rev. A 45, 70 (1992)

    Article  ADS  Google Scholar 

  59. G. W. F. Drake: Phys. Rev. A; J. Phys. B 5, 1979 (1972)

    Article  ADS  Google Scholar 

  60. G. W. F. Drake: J. Phys. B 9, L169 (1976)

    Article  ADS  Google Scholar 

  61. G. W. F. Drake: Phys. Rev. A 19, 1387 (1979)

    Article  ADS  Google Scholar 

  62. C. J. Sansonetti, J. D. Gillaspy: Phys. Rev. A 45, R1 (1992)

    Article  ADS  Google Scholar 

  63. W. Lichten, D. Shiner, Z.-X. Zhou: Phys. Rev. A 43, 1663 (1991)

    Article  ADS  Google Scholar 

  64. M. C. George, L. D. Lombardi, E. A. Hessels: Phys. Rev. Lett. 87, 173002 (2001)

    Article  ADS  Google Scholar 

  65. G. W. F. Drake: Can. J. Phys. 80, 1195 (2002)

    Article  ADS  Google Scholar 

  66. K. Panchucki, J. Sapirstein: J. Phys. B 36, 803 (2003) and earlier references therein

    Article  ADS  Google Scholar 

  67. L.-B. Wang, P. Müller, K. Bailey, G. W. F. Drake, J. P. Greene, D. Henderson, R. J. Holt, R. V. F. Janssens, c. L. Jiang, Z.-T. Lu, T. P. O'Connor, R. C. Pardo, M. Paul, K. E. Rehm, J. P. Schiffer, X. D. Tang: Phys. Rev. Lett. 93, 142501 (2004)

    Article  ADS  Google Scholar 

  68. G. Ewald, W. Nörtershäuser, A. Dax, S. Göte, R. Kirchner, H.J. Kluge, Th. Kühl, R. Sanchez, A. Wojtaszek, B. A. Bushaw, G. W. F. Drake, Z.-C. Yan, C. Zimmermann: Phys. Rev. Lett. 93, 113002 (2004)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author is grateful to R. N. Hill and J. D. Morgan III for suggesting some of the material at Sect. 11.1. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations

  1. Department of Physics, University of Windsor, 401 Sunset St., N9B 3P4, Windsor, ON, Canada

    Gordon Drake

Authors
  1. Gordon Drake
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gordon Drake .

Editor information

Editors and Affiliations

  1. Department of Physics, University of Windsor, 401 Sunset St., N9B 3P4, Windsor, ON, Canada

    Gordon Drake Prof.

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag

About this entry

Cite this entry

Drake, G. (2006). High Precision Calculations for Helium. In: Drake, G. (eds) Springer Handbook of Atomic, Molecular, and Optical Physics. Springer Handbooks. Springer, New York, NY. https://doi.org/10.1007/978-0-387-26308-3_11

Download citation

Publish with us

Policies and ethics

Search

Navigation