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High Precision Calculations for Helium

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Springer Handbook of Atomic, Molecular, and Optical Physics

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Abstract

Exact analytic solutions to the Schrödinger equation are known only for hydrogen and other equivalent two-body systems (Chap. 9). However, very high precision approximations are now available for helium and other three-body systems that are essentially exact for all practical purposes. This chapter summarizes the computational methods and tabulates numerical results for the ground state and several excited states up to n = 10 and L = 7. General formulas are presented for the evaluation of matrix elements in Hylleraas coordinates. Relativistic and quantum electrodynamic effects are included, along with quantum defect extrapolations to states of higher n and asymptotic expansions to states of higher angular momentum L. Radiative transitions are discussed and oscillator strengths tabulated for a wide variety of transitions.

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References

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

    Google Scholar 

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

    ADS  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

  5. Gottschalk, J.E., Maslen, E.N.: J. Phys. A 20, 2781 (1987) and earlier references therein

    ADS  MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

  7. Leray, J.: In: Ciarlet, P.G., Roseau, M. (eds.) Trends and Applications of Pure Mathematics to Mechanics, Lecture Notes in Physics, vol. 195, pp. 235–247. Springer, Berlin (1984)

    Google Scholar 

  8. Morgan III, J.D.: Theor. Chem. Acta 69, 181 (1986)

    Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  12. Ahlrichs, R., Hoffmann-Ostenhoff, T., Hoffmann-Ostenhoff, M., Morgan III, J.D.: Phys. Rev. A 23, 2107 (1981). and earlier references therein

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  20. Schwartz, C.: Int. J. Mod. Phys. 15, 877 (2006)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  25. Korobov, V.: Phys. Rev. A 69, 054501 (2004). The Bethe logarithms calculated by Korobov include an additional lnZ2 in their definition

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    Google Scholar 

  31. Aznabaev, D.T., Bekbaev, A.K., Korobov, V.I.: Phys. Rev. A 98, 012510 (2018)

    ADS  Google Scholar 

  32. Wang, M., Audi, G., Kondev, F.G., Huang, W.J., Naimi, S., Xu, X.: Chin. Phys. C 41, 030003 (2017)

    ADS  Google Scholar 

  33. Nakashima, H., Nakatsuji, H.: J. Chem. Phys. 128, 154108 (2008)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  36. Araki, H.: Prog. Theor. Phys. 17, 619 (1957)

    ADS  Google Scholar 

  37. Sucher, J.: Phys. Rev. 109, 1010 (1958)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  39. Korobov, V.I.: Phys. Rev. A 100, 012517 (2019)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  43. Stone, A.P.: Proc. Phys. Soc. 77, 786 (1961)

    ADS  Google Scholar 

  44. Yerokhin, V.A., Pachucki, K.: Phys. Rev. A 81, 022507 (2010)

    ADS  Google Scholar 

  45. Pachucki, K., Patkóš, V., Yerokhin, V.A.: Phys. Rev. A 95, 062510 (2017)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  47. Kato, K., Skinner, T.D.G., Hessels, E.A.: Phys. Rev. Lett. 121, 143002 (2018)

    ADS  Google Scholar 

  48. Lu, Z.-T., Mueller, P., Drake, G.W.F., Nortershauser, W., Pieper, S.C., Yan, Z.-C.: Rev. Mod. Phys. 85, 1383 (2013)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  50. Yerokhin, V.A., Patkóš, V., Puchalski, M., Pachucki, K.: Phys. Rev. A 102, 01807 (2020)

    Google Scholar 

  51. Morton, D.C., Wu, Q.X., Drake, G.W.F.: Can. J. Phys. 84, 83 (2006)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  55. Drachman, R.J.: Phys. Rev. A 47, 694 (1993) and earlier references therein

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  57. Wang, X.-F., Yan, Z.-C.: Phys. Rev. A 95, 022505 (2017)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  60. Drake, G.W.F.: Phys. Rev. A 5, 1979 (1972)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    ADS  Google Scholar 

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

    Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    Google Scholar 

  73. Stone, A.P.: Proc. Phys. Soc. 81, 868 (1963)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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Acknowledgements

The author is grateful to R.N. Hill and J.D. Morgan III for suggesting some of the material inSect. 12.1. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

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

    Gordon W. F. Drake

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  1. Gordon W. F. Drake
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Correspondence to Gordon W. F. Drake .

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

    Gordon W. F. Drake

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Drake, G.W.F. (2023). High Precision Calculations for Helium. 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_12

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