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Energies and Widths of the Ground and Excited States of Hydrogen in a DC Field via Variationally-Based Large-Order Perturbation Theory

  • Jeremiah N. Silverman
  • Cleanthes A. Nicolaides
Part of the NATO ASI Series book series (NSSB, volume 212)

Abstract

We have recently computed the energies and widths (complex eigenvalues) of the ground and many (more than 100) excited states of hydrogen in a dc field (the Stark effect) for numerous field strengths by combining a procedure for the analytic continuation of real large-order eigenvalue perturbation series with the perturbational-variational Rayleigh-Ritz (PV-RR) method for generating such series. The analytic continuation is accomplished by shifting the origin of the real eigenvalue series into the complex plane where the relevant divergent series are summed by a twofold application of Padé approximants. Our composite PV-RR method fills a gap in theory and results for the ground and highly excited states of the hydrogenic Stark effect; since the procedure implements large-order perturbation theory with the variational method, it is feasible to extend the technique to larger systems such as light atoms. After introducing the problem, this lecture briefly summarizes previous methods of calculating hydrogenic Stark resonances. Next, the theory of our procedure is outlined, including an explanation and illustration of our novel method of relieving zero-order degeneracy of excited Stark states. Our extensive calculations are then described, and selected results are reported and discussed. Finally, we present our conclusions.

Keywords

Excited State Analytic Continuation Complex Eigenvalue Shift Origin Stark State 
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.H. Bucksbaum, this volume.Google Scholar
  2. 2.
    R. Damburg, this volume.Google Scholar
  3. 3.
    A. Bommier, D. Delande, and J.C. Gay, this volume.Google Scholar
  4. 4.
    M. Nayfeh, D. Humm, and K. Ng, this volume.Google Scholar
  5. 5.
    J. Pinard, this volume.Google Scholar
  6. 6.
    H. Walther, this volume.Google Scholar
  7. 7.
    A. Holle, J. Main, G. Wiebusch, H. Rottke, and K.H. Welge, this volume.Google Scholar
  8. 8.
    E. Schrödinger, Ann. Phys. (Leipzig) 80, 437 (1926).zbMATHGoogle Scholar
  9. 9.
    J.R. Oppenheimer, Phys. Rev. 31, 66 (1928).ADSzbMATHGoogle Scholar
  10. 10.
    C. Lanczos, Z. Phys. 68, 204 (1931), and earlier papers by the same author.ADSzbMATHGoogle Scholar
  11. 11.
    E.C. Titchmarsh, Proc. Roy. Soc. A 210, 30 (1951).MathSciNetADSGoogle Scholar
  12. E.C. Titchmarsh, “Eigenfunction Expansions Associated with Second Order Differential Equations”, Part II, Oxford University Press, London (1958), Chap. 20.zbMATHGoogle Scholar
  13. 12.
    T. Kato, “Perturbation Theory for Linear Operators”, 2nd Ed., Springer-Verlag, Berlin (1976), pp. 473–479.zbMATHGoogle Scholar
  14. 13.
    A.FJ. Siegert, Phys. Rev. 56, 750 (1939).ADSGoogle Scholar
  15. T. Berggren, Nucl. Phys. A 109, 265 (1968).ADSGoogle Scholar
  16. 14.
    A.M. Dykhne and A.V. Chaplik, Zh. Eksp. Teor. Fiz. 40, 1427 (1961)Google Scholar
  17. A.M. Dykhne and A.V. Chaplik, [Sov. Phys.-JETP 13, 1002(1961)].Google Scholar
  18. 15.
    T.F. O’Malley and S. Geltman, Phys. Rev. 137, A1344 (1965).Google Scholar
  19. 16.
    J. Aguilar and J.M. Combes, Commun. Math. Phys. 22, 69 (1971).MathSciNetGoogle Scholar
  20. E. Balslev and J.M. Combes, Commun. Math. Phys. 22, 280 (1971).MathSciNetADSzbMATHGoogle Scholar
  21. 17.
    B. Simon, Ann. Math. 97, 247 (1973).zbMATHGoogle Scholar
  22. 18.
    C.A. Nicolaides, Phys. Rev. A 6, 2078 (1972).ADSGoogle Scholar
  23. 19.
    R.M. More and E. Gerjuoy, Phys. Rev. A 7, 1288 (1973).MathSciNetADSGoogle Scholar
  24. 20.
    G.D. Doolen, J. Nuttall, and R.W. Stagat, Phys. Rev. A 10, 1612 (1974).ADSGoogle Scholar
  25. G.D. Doolen, J. Phys. B 8, 525 (1975).ADSGoogle Scholar
  26. 21.
    P. Winkler, Z. Phys. A 283, 149 (1977).ADSGoogle Scholar
  27. Y.K. Ho, A.K. Bhatia, and T.A. Temkin, Phys. Rev. A 15, 1423(1977).ADSGoogle Scholar
  28. 22.
    B.R. Junker, Adv. At. Mol. Phys. 18, 207 (1982).ADSGoogle Scholar
  29. 23.
    M.H. Rice and R.H. Good, J. Opt. Soc. Am. 52, 239 (1962).ADSGoogle Scholar
  30. 24.
    D.S. Bailey, J.R. Hiskes, and A.C. Riviere, Nucl. Fusion 5, 41 (1965).Google Scholar
  31. 25.
    M.H. Alexander, Phys. Rev. 178, 34 (1969).ADSGoogle Scholar
  32. 26.
    J.O. Hiischfelder and L.A. Curtiss, J. Chem. Phys. 155, 1395 (1971).ADSGoogle Scholar
  33. 27.
    M. Hehenberger, H.V. Mclntosh, and E. Brändas, Phys. Rev. A 10, 1494 (1974).ADSGoogle Scholar
  34. 28.
    R.J. Damburg and V.V. Kolosov, J. Phys. B 9, 3149 (1976).ADSGoogle Scholar
  35. 29.
    W.P. Reinhardt, Int. J.Quantum Chem. Symp. 10, 359 (1976).Google Scholar
  36. C. Cerjan, R. Hedges, C. Holt, W.P. Reinhardt, K. Scheibner, and J.J. Wendolowski, Int. J. Quantum Chem. 14, 393(1978).Google Scholar
  37. 30.
    E. Brändas and P. Froelich, Phys. Rev. A 16, 2207 (1977).ADSGoogle Scholar
  38. 31.
    C.A. Nicolaides and D.R. Beck, Int. J. Quantum Chem. 14, 457 (1978).Google Scholar
  39. 32.
    S. Graffi and V. Grecchi, Commun. Math. Phys. 62, 83 (1978).MathSciNetADSGoogle Scholar
  40. 33.
    W. Herbst and B. Simon, Phys. Rev. Lett. 41, 67 (1978).MathSciNetADSGoogle Scholar
  41. W. Herbst and B. Simon, Phys. Rev. Lett. 41, 1759 (1978).MathSciNetADSGoogle Scholar
  42. L. Benassi, V. Grecchi, E. Harrell, and B. Simon, Phys. Rev. Lett. 42, 704 (1979).ADSGoogle Scholar
  43. 34.
    Y. Komninos and CA. Nicolaides, Chem. Phys. Lett. 78., 347 (1981).ADSGoogle Scholar
  44. C.A. Nicolaides, Y. Komninos, and Th. Mercouris, Int. J. Quantum Chem. Symp. 15, 355 (1981).Google Scholar
  45. 35.
    L. Benassi and V. Grecchi, J. Phys. B 13, 911 (1980).ADSGoogle Scholar
  46. 36.
    A. Maquet, S.-I. Chu, and W.P. Reinhardt, Phys. Rev. A 27, 2946 (1983).ADSGoogle Scholar
  47. 37.
    E. Luc-Koenig and A. Bachelier, J. Phys. B 13, 1743 (1980).MathSciNetADSGoogle Scholar
  48. 38.
    D. Farrelly and W.P. Reinhardt, J. Phys. B 16, 2103 (1983).ADSGoogle Scholar
  49. 39.
    H.J. Korsch and R. Möhlenkamp, Z. Phys. A 314, 267 (1983).ADSGoogle Scholar
  50. 40.
    V.V. Kolosov, J. Phys. B 20, 2359 (1987).ADSGoogle Scholar
  51. 41.
    HJ. Silverstone, this volume.Google Scholar
  52. 42.
    W.P. Reinhardt, Int. J. Quantum Chem. 21, 133 (1982).Google Scholar
  53. 43.
    M. Garcia-Sucre and R. Lefebvre, Mol. Phys. 60, 941 (1987).ADSGoogle Scholar
  54. 44.
    V. Franceschine, V. Grecchi, and HJ. Silverstone, Phys. Rev. A 32, 1338 (1985).ADSGoogle Scholar
  55. 45.
    H.J. Silverstone, Phys. Rev. A 18, 1853 (1978).ADSGoogle Scholar
  56. 46.
    R.J. Damburg and V.V. Kolosov, J. Phys. B 11, 1921 (1978).ADSGoogle Scholar
  57. 47.
    T. Yamabe, A. Tachibana, and H.J. Silverstone, Phys. Rev. A 16, 877 (1977).ADSGoogle Scholar
  58. H.J. Silverstone, B.G. Adams, J. Cizek, and P. Otto, Phys. Rev. Lett. 43, 1498 (1979).ADSGoogle Scholar
  59. H.J. Silverstone, E. Harrell, and C. Grot, Phys. Rev. A 24, 1925(1981).ADSGoogle Scholar
  60. 48.(a)
    H.J. Silverstone and P.M. Koch, J. Phys. B 12, L537 (1979).ADSGoogle Scholar
  61. (b).
    E.J. Austin, Mol. Phys. 40, 893 (1980).MathSciNetADSGoogle Scholar
  62. 49.
    For comprehensive surveys of LOPT until 1981, see B. Simon, Int. J. Quantum Chem. 21, 3 (1982).Google Scholar
  63. J. Cizek and E. R. Vrscay, Int. J. Quantum Chem. 21, 27 (1982).Google Scholar
  64. 50.
    J.N. Silverman and Y. Sobouti, Astron. Astrophys. 62, 355 (1978).ADSzbMATHGoogle Scholar
  65. 51.
    J.N. Silverman, J. Phys. A 16, 3471(1983).ADSzbMATHGoogle Scholar
  66. 52.
    J.N. Silverman, B.S. Sudhindra, and G. Olbrich, Phys. Rev. A 30, 1554 (1984).ADSGoogle Scholar
  67. 53.
    J.N. Silverman, and J. Hinze, Chem. Phys. Lett. 128, 466 (1986).ADSGoogle Scholar
  68. J.N. Silverman, and J. Hinze, Phys. Rev. A 37, 1208 (1988).ADSGoogle Scholar
  69. 54.
    J.N. Silverman, D.M. Bishop, and J. Pipin, Phys. Rev. Lett. 56, 1358 (1986).ADSGoogle Scholar
  70. J.N. Silverman, D.M. Bishop, and J. Pipin, Phys. Rev. Lett. 57, 274 (1986).ADSGoogle Scholar
  71. J.N. Silverman and D.M. Bishop, Phys. Rev. A 34, 5142 (1986).ADSGoogle Scholar
  72. J.N. Silverman and D.M. Bishop, Chem. Phys. Lett. 132, 37(1986).ADSGoogle Scholar
  73. 55.
    J.N. Silverman and D.M. Bishop, Chem. Phys. Lett. 130, 132 (1986).ADSGoogle Scholar
  74. J.N. Silverman, Chem. Phys. Lett. 142,477 (1987).ADSGoogle Scholar
  75. 56.
    J.N. Silverman and C.A. Nicolaides, Chem. Phys. Lett. 153, 61 (1988).ADSGoogle Scholar
  76. 57.
    I. Motoc, J.N. Silverman, and O.E. Polansky, Phys. Rev. A 28, 3673 (1983).ADSGoogle Scholar
  77. I. Motoc, J.N. Silverman, and O.E. Polansky, Chem. Phys. Lett. 103, 285 (1984).ADSGoogle Scholar
  78. I. Motoc, J.N. Silverman, O.E. Polansky, and G. Olbrich, Theor. Chim. Acta 67, 63 (1985).Google Scholar
  79. 58.
    P.M. Koch, Phys. Rev. Lett. 41, 99 (1978).ADSGoogle Scholar
  80. 59.
    J.J. Wendoloski and W.P. Reinhardt, Phys. Rev. A 17, 195 (1978).ADSGoogle Scholar
  81. Y.K. Ho, J. Phys. B 12, L543 (1979).ADSGoogle Scholar
  82. Y.K. Ho, J. Phys. B 12, 387 (1979).ADSGoogle Scholar
  83. 60.
    B.R. Junker, J. Phys. B 15, 4495 (1982).ADSGoogle Scholar
  84. 61.
    C.A. Nicolaides and Th. Mercouris, this volume.Google Scholar
  85. 62.
    C.A. Nicolaides and D.R. Beck, Phys. Lett. A 65, 11 (1978).ADSGoogle Scholar
  86. 63.
    C.A. Nicolaides and Th. Mercouris, Phys. Rev. A 36, 390 (1987).ADSGoogle Scholar
  87. 64.
    Y. Komninos, G. Aspromallis, and C.A. Nicolaides, Phys. Rev. A 27, 1865 (1983).ADSGoogle Scholar
  88. 65.
    C.A. Nicolaides in “Advanced Theories and Computational Approaches to the Electronic Structure of Molecules”, ed. C.E. Dykstra, Reidel, Dordrecht (1984), pp. 161–184.Google Scholar
  89. 66.
    Th. Mercouris and C.A. Nicolaides, J. Phys. B 21, L285 (1988).ADSGoogle Scholar
  90. C.A. Nicolaides and Th. Mercouris, Chem. Phys. Lett. 159, 45 (1989).ADSGoogle Scholar
  91. 67.
    A. Imamura, Mol. Phys. 15 225 (1968).ADSGoogle Scholar
  92. A. Dalgarno and G.W.F. Drake, Chem. Phys. Lett. 3, 349 (1969).ADSGoogle Scholar
  93. R. Carbo, Theor. Chim. Acta 17, 74 (1970).Google Scholar
  94. R. Carbo, Int. J. Quantum Chem. 6, 609 (1972).Google Scholar
  95. R. Carbo and R. Gallifa, Nuovo Cimento B 10, 576 (1972).Google Scholar
  96. 68.
    H. Shull and P.-O. Löwdin, J. Chem. Phys. 30, 617 (1959).ADSGoogle Scholar
  97. 69.
    J.P. Connerade, this volume.Google Scholar
  98. 70.
    H.A. Bethe and E.E. Salpeter, “Quantum Mechanics of One-and Two-Electron Atoms”, Academic Press, New York (1957), pp. 27–29, 228-238 incl.zbMATHGoogle Scholar
  99. 71.
    H.J. Silverstone, J. Chem. Phys. 54, 2325 (1971).MathSciNetADSGoogle Scholar
  100. 72.
    S.P. Alliluev and I.A. Malkin, Zh. Eksp. Teor. Fiz. 66, 1283 (1974).Google Scholar
  101. S.P. Alliluev and I.A. Malkin, [Sov. Phys.-JETP 39, 627 (1974)].ADSGoogle Scholar
  102. 73.
    K. Helfrich, Theor. Chim. Acta 24, 271 (1972).Google Scholar
  103. 74.
    G.J. Hatton, Phys. Rev. A 16, 1347 (1977).ADSGoogle Scholar
  104. 75.
    C.A. Mead, J. Chem. Phys. 70, 2276 (1979).ADSGoogle Scholar
  105. 76.
    E. Holoien and J. Midtdal, J. Chem. Phys. 45, 2209 (1966).ADSGoogle Scholar
  106. 77.
    A.U. Hazi and H.S. Taylor, Phys. Rev. A 1, 1109 (1970).ADSGoogle Scholar
  107. H.S. Taylor, Adv. Chem. Phys. 18, 91 (1970).Google Scholar
  108. 78.
    P. Froelich and E. Brändas, Phys. Rev. A 12, 1 (1975).ADSGoogle Scholar
  109. 79.
    A. Macias and A. Riera, J. Phys. B 11, 3827 (1978).ADSGoogle Scholar
  110. A. Macias and A. Riera, J. Phys. B 12, 3631 (1979).MathSciNetADSGoogle Scholar
  111. 80.
    E.A. Hylleraas and B. Undheim, Z. Phys. 65, 759 (1930).ADSzbMATHGoogle Scholar
  112. J.K.L. MacDonald, Phys. Rev. 43, 830 (1933).ADSGoogle Scholar
  113. 81.
    C.A. Baker, Jr., “Essentials of Padé Approximants”, Academic Press, New York (1975), pp. 75–76.zbMATHGoogle Scholar
  114. 82.
    R.J. Damburg and V.V. Kolosov in “Rydberg States of Atoms and Molecules”, ed. R.F. Stebbings and F.B. Dunning, Cambridge University Press, London (1983), pp. 31–71.Google Scholar
  115. 83.
    R.J. Damburg and V.V. Kolosov, J. Phys. B 12, 2637 (1979).ADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Jeremiah N. Silverman
    • 1
  • Cleanthes A. Nicolaides
    • 1
  1. 1.Theoretical and Physical Chemistry InstituteThe National Hellenic Research FoundationAthensGreece

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