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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

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Atoms in Strong Fields

Part of the book series: NATO ASI 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.

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References

  1. P.H. Bucksbaum, this volume.

    Google Scholar 

  2. R. Damburg, this volume.

    Google Scholar 

  3. A. Bommier, D. Delande, and J.C. Gay, this volume.

    Google Scholar 

  4. M. Nayfeh, D. Humm, and K. Ng, this volume.

    Google Scholar 

  5. J. Pinard, this volume.

    Google Scholar 

  6. H. Walther, this volume.

    Google Scholar 

  7. A. Holle, J. Main, G. Wiebusch, H. Rottke, and K.H. Welge, this volume.

    Google Scholar 

  8. E. Schrödinger, Ann. Phys. (Leipzig) 80, 437 (1926).

    MATH  Google Scholar 

  9. J.R. Oppenheimer, Phys. Rev. 31, 66 (1928).

    ADS  MATH  Google Scholar 

  10. C. Lanczos, Z. Phys. 68, 204 (1931), and earlier papers by the same author.

    ADS  MATH  Google Scholar 

  11. E.C. Titchmarsh, Proc. Roy. Soc. A 210, 30 (1951).

    MathSciNet  ADS  Google Scholar 

  12. E.C. Titchmarsh, “Eigenfunction Expansions Associated with Second Order Differential Equations”, Part II, Oxford University Press, London (1958), Chap. 20.

    MATH  Google Scholar 

  13. T. Kato, “Perturbation Theory for Linear Operators”, 2nd Ed., Springer-Verlag, Berlin (1976), pp. 473–479.

    MATH  Google Scholar 

  14. A.FJ. Siegert, Phys. Rev. 56, 750 (1939).

    ADS  Google Scholar 

  15. T. Berggren, Nucl. Phys. A 109, 265 (1968).

    ADS  Google Scholar 

  16. 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. T.F. O’Malley and S. Geltman, Phys. Rev. 137, A1344 (1965).

    Google Scholar 

  19. J. Aguilar and J.M. Combes, Commun. Math. Phys. 22, 69 (1971).

    MathSciNet  Google Scholar 

  20. E. Balslev and J.M. Combes, Commun. Math. Phys. 22, 280 (1971).

    MathSciNet  ADS  MATH  Google Scholar 

  21. B. Simon, Ann. Math. 97, 247 (1973).

    MATH  Google Scholar 

  22. C.A. Nicolaides, Phys. Rev. A 6, 2078 (1972).

    ADS  Google Scholar 

  23. R.M. More and E. Gerjuoy, Phys. Rev. A 7, 1288 (1973).

    MathSciNet  ADS  Google Scholar 

  24. G.D. Doolen, J. Nuttall, and R.W. Stagat, Phys. Rev. A 10, 1612 (1974).

    ADS  Google Scholar 

  25. G.D. Doolen, J. Phys. B 8, 525 (1975).

    ADS  Google Scholar 

  26. P. Winkler, Z. Phys. A 283, 149 (1977).

    ADS  Google Scholar 

  27. Y.K. Ho, A.K. Bhatia, and T.A. Temkin, Phys. Rev. A 15, 1423(1977).

    ADS  Google Scholar 

  28. B.R. Junker, Adv. At. Mol. Phys. 18, 207 (1982).

    ADS  Google Scholar 

  29. M.H. Rice and R.H. Good, J. Opt. Soc. Am. 52, 239 (1962).

    ADS  Google Scholar 

  30. D.S. Bailey, J.R. Hiskes, and A.C. Riviere, Nucl. Fusion 5, 41 (1965).

    Google Scholar 

  31. M.H. Alexander, Phys. Rev. 178, 34 (1969).

    ADS  Google Scholar 

  32. J.O. Hiischfelder and L.A. Curtiss, J. Chem. Phys. 155, 1395 (1971).

    ADS  Google Scholar 

  33. M. Hehenberger, H.V. Mclntosh, and E. Brändas, Phys. Rev. A 10, 1494 (1974).

    ADS  Google Scholar 

  34. R.J. Damburg and V.V. Kolosov, J. Phys. B 9, 3149 (1976).

    ADS  Google Scholar 

  35. 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. E. Brändas and P. Froelich, Phys. Rev. A 16, 2207 (1977).

    ADS  Google Scholar 

  38. C.A. Nicolaides and D.R. Beck, Int. J. Quantum Chem. 14, 457 (1978).

    Google Scholar 

  39. S. Graffi and V. Grecchi, Commun. Math. Phys. 62, 83 (1978).

    MathSciNet  ADS  Google Scholar 

  40. W. Herbst and B. Simon, Phys. Rev. Lett. 41, 67 (1978).

    MathSciNet  ADS  Google Scholar 

  41. W. Herbst and B. Simon, Phys. Rev. Lett. 41, 1759 (1978).

    MathSciNet  ADS  Google Scholar 

  42. L. Benassi, V. Grecchi, E. Harrell, and B. Simon, Phys. Rev. Lett. 42, 704 (1979).

    ADS  Google Scholar 

  43. Y. Komninos and CA. Nicolaides, Chem. Phys. Lett. 78., 347 (1981).

    ADS  Google Scholar 

  44. C.A. Nicolaides, Y. Komninos, and Th. Mercouris, Int. J. Quantum Chem. Symp. 15, 355 (1981).

    Google Scholar 

  45. L. Benassi and V. Grecchi, J. Phys. B 13, 911 (1980).

    ADS  Google Scholar 

  46. A. Maquet, S.-I. Chu, and W.P. Reinhardt, Phys. Rev. A 27, 2946 (1983).

    ADS  Google Scholar 

  47. E. Luc-Koenig and A. Bachelier, J. Phys. B 13, 1743 (1980).

    MathSciNet  ADS  Google Scholar 

  48. D. Farrelly and W.P. Reinhardt, J. Phys. B 16, 2103 (1983).

    ADS  Google Scholar 

  49. H.J. Korsch and R. Möhlenkamp, Z. Phys. A 314, 267 (1983).

    ADS  Google Scholar 

  50. V.V. Kolosov, J. Phys. B 20, 2359 (1987).

    ADS  Google Scholar 

  51. HJ. Silverstone, this volume.

    Google Scholar 

  52. W.P. Reinhardt, Int. J. Quantum Chem. 21, 133 (1982).

    Google Scholar 

  53. M. Garcia-Sucre and R. Lefebvre, Mol. Phys. 60, 941 (1987).

    ADS  Google Scholar 

  54. V. Franceschine, V. Grecchi, and HJ. Silverstone, Phys. Rev. A 32, 1338 (1985).

    ADS  Google Scholar 

  55. H.J. Silverstone, Phys. Rev. A 18, 1853 (1978).

    ADS  Google Scholar 

  56. R.J. Damburg and V.V. Kolosov, J. Phys. B 11, 1921 (1978).

    ADS  Google Scholar 

  57. T. Yamabe, A. Tachibana, and H.J. Silverstone, Phys. Rev. A 16, 877 (1977).

    ADS  Google Scholar 

  58. H.J. Silverstone, B.G. Adams, J. Cizek, and P. Otto, Phys. Rev. Lett. 43, 1498 (1979).

    ADS  Google Scholar 

  59. H.J. Silverstone, E. Harrell, and C. Grot, Phys. Rev. A 24, 1925(1981).

    ADS  Google Scholar 

  60. H.J. Silverstone and P.M. Koch, J. Phys. B 12, L537 (1979).

    ADS  Google Scholar 

  61. E.J. Austin, Mol. Phys. 40, 893 (1980).

    MathSciNet  ADS  Google Scholar 

  62. 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. J.N. Silverman and Y. Sobouti, Astron. Astrophys. 62, 355 (1978).

    ADS  MATH  Google Scholar 

  65. J.N. Silverman, J. Phys. A 16, 3471(1983).

    ADS  MATH  Google Scholar 

  66. J.N. Silverman, B.S. Sudhindra, and G. Olbrich, Phys. Rev. A 30, 1554 (1984).

    ADS  Google Scholar 

  67. J.N. Silverman, and J. Hinze, Chem. Phys. Lett. 128, 466 (1986).

    ADS  Google Scholar 

  68. J.N. Silverman, and J. Hinze, Phys. Rev. A 37, 1208 (1988).

    ADS  Google Scholar 

  69. J.N. Silverman, D.M. Bishop, and J. Pipin, Phys. Rev. Lett. 56, 1358 (1986).

    ADS  Google Scholar 

  70. J.N. Silverman, D.M. Bishop, and J. Pipin, Phys. Rev. Lett. 57, 274 (1986).

    ADS  Google Scholar 

  71. J.N. Silverman and D.M. Bishop, Phys. Rev. A 34, 5142 (1986).

    ADS  Google Scholar 

  72. J.N. Silverman and D.M. Bishop, Chem. Phys. Lett. 132, 37(1986).

    ADS  Google Scholar 

  73. J.N. Silverman and D.M. Bishop, Chem. Phys. Lett. 130, 132 (1986).

    ADS  Google Scholar 

  74. J.N. Silverman, Chem. Phys. Lett. 142,477 (1987).

    ADS  Google Scholar 

  75. J.N. Silverman and C.A. Nicolaides, Chem. Phys. Lett. 153, 61 (1988).

    ADS  Google Scholar 

  76. I. Motoc, J.N. Silverman, and O.E. Polansky, Phys. Rev. A 28, 3673 (1983).

    ADS  Google Scholar 

  77. I. Motoc, J.N. Silverman, and O.E. Polansky, Chem. Phys. Lett. 103, 285 (1984).

    ADS  Google Scholar 

  78. I. Motoc, J.N. Silverman, O.E. Polansky, and G. Olbrich, Theor. Chim. Acta 67, 63 (1985).

    Google Scholar 

  79. P.M. Koch, Phys. Rev. Lett. 41, 99 (1978).

    ADS  Google Scholar 

  80. J.J. Wendoloski and W.P. Reinhardt, Phys. Rev. A 17, 195 (1978).

    ADS  Google Scholar 

  81. Y.K. Ho, J. Phys. B 12, L543 (1979).

    ADS  Google Scholar 

  82. Y.K. Ho, J. Phys. B 12, 387 (1979).

    ADS  Google Scholar 

  83. B.R. Junker, J. Phys. B 15, 4495 (1982).

    ADS  Google Scholar 

  84. C.A. Nicolaides and Th. Mercouris, this volume.

    Google Scholar 

  85. C.A. Nicolaides and D.R. Beck, Phys. Lett. A 65, 11 (1978).

    ADS  Google Scholar 

  86. C.A. Nicolaides and Th. Mercouris, Phys. Rev. A 36, 390 (1987).

    ADS  Google Scholar 

  87. Y. Komninos, G. Aspromallis, and C.A. Nicolaides, Phys. Rev. A 27, 1865 (1983).

    ADS  Google Scholar 

  88. 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. Th. Mercouris and C.A. Nicolaides, J. Phys. B 21, L285 (1988).

    ADS  Google Scholar 

  90. C.A. Nicolaides and Th. Mercouris, Chem. Phys. Lett. 159, 45 (1989).

    ADS  Google Scholar 

  91. A. Imamura, Mol. Phys. 15 225 (1968).

    ADS  Google Scholar 

  92. A. Dalgarno and G.W.F. Drake, Chem. Phys. Lett. 3, 349 (1969).

    ADS  Google 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. H. Shull and P.-O. Löwdin, J. Chem. Phys. 30, 617 (1959).

    ADS  Google Scholar 

  97. J.P. Connerade, this volume.

    Google Scholar 

  98. 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.

    MATH  Google Scholar 

  99. H.J. Silverstone, J. Chem. Phys. 54, 2325 (1971).

    MathSciNet  ADS  Google Scholar 

  100. 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)].

    ADS  Google Scholar 

  102. K. Helfrich, Theor. Chim. Acta 24, 271 (1972).

    Google Scholar 

  103. G.J. Hatton, Phys. Rev. A 16, 1347 (1977).

    ADS  Google Scholar 

  104. C.A. Mead, J. Chem. Phys. 70, 2276 (1979).

    ADS  Google Scholar 

  105. E. Holoien and J. Midtdal, J. Chem. Phys. 45, 2209 (1966).

    ADS  Google Scholar 

  106. A.U. Hazi and H.S. Taylor, Phys. Rev. A 1, 1109 (1970).

    ADS  Google Scholar 

  107. H.S. Taylor, Adv. Chem. Phys. 18, 91 (1970).

    Google Scholar 

  108. P. Froelich and E. Brändas, Phys. Rev. A 12, 1 (1975).

    ADS  Google Scholar 

  109. A. Macias and A. Riera, J. Phys. B 11, 3827 (1978).

    ADS  Google Scholar 

  110. A. Macias and A. Riera, J. Phys. B 12, 3631 (1979).

    MathSciNet  ADS  Google Scholar 

  111. E.A. Hylleraas and B. Undheim, Z. Phys. 65, 759 (1930).

    ADS  MATH  Google Scholar 

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

    ADS  Google Scholar 

  113. C.A. Baker, Jr., “Essentials of Padé Approximants”, Academic Press, New York (1975), pp. 75–76.

    MATH  Google Scholar 

  114. 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. R.J. Damburg and V.V. Kolosov, J. Phys. B 12, 2637 (1979).

    ADS  Google Scholar 

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Authors and Affiliations

  1. Theoretical and Physical Chemistry Institute, The National Hellenic Research Foundation, 48, Vas. Constantinou Ave, Athens, 116 35, Greece

    Jeremiah N. Silverman & Cleanthes A. Nicolaides

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  1. Jeremiah N. Silverman
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  2. Cleanthes A. Nicolaides
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Editors and Affiliations

  1. Theoretical and Physical Chemistry Institute, Hellenic Research Foundation, Athens, Greece

    Cleanthes A. Nicolaides

  2. National Institute of Standards and Technology, Gaithersburg, Maryland, USA

    Charles W. Clark

  3. University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Munir H. Nayfeh

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Silverman, J.N., Nicolaides, C.A. (1990). Energies and Widths of the Ground and Excited States of Hydrogen in a DC Field via Variationally-Based Large-Order Perturbation Theory. In: Nicolaides, C.A., Clark, C.W., Nayfeh, M.H. (eds) Atoms in Strong Fields. NATO ASI Series, vol 212. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9334-5_17

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  • DOI: https://doi.org/10.1007/978-1-4757-9334-5_17

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