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Journal of Mathematical Physics -- October 1998 -- Volume 39, Issue 10, pp. 5112-5122

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Improving the convergence and estimating the accuracy of summation approximants of 1/D expansions for Coulombic systems

Melchior O. Elout, David Z. Goodson, Carl D. Elliston, Shi-Wei Huang, and Alexei V. Sergeev
Department of Chemistry, Southern Methodist University, Dallas, Texas 75275
Deborah K. Watson
Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019

(Received 30 March 1998; accepted 4 May 1998)

The convergence of large-order expansions in  delta = 1/D, where D is the dimensionality of coordinate space, for energies E( delta ) of Coulomb systems is strongly affected by singularities at  delta = 1 and  delta = 0. Padé–Borel approximants with modifications that completely remove the singularities at  delta = 1 and remove the dominant singularity at  delta = 0 are demonstrated. A renormalization of the interelectron repulsion is found to move the dominant singularity of the Borel function F( delta ) = (summation)jE[sub j][sup (prime)]/j!, where E[sub j][sup (prime)] are the the expansion coefficients of the energy with singularity structure removed at  delta = 1, farther from the origin and thereby accelerate summation convergence. The ground-state energies of He and H[sub 2][sup +] are used as test cases. The new methods give significant improvement over previous summation methods. Shifted Borel summation using Fm( delta ) = (summation)jE[sub j][sup (prime)]/ Gamma (j + 1 – m) is considered. The standard deviation of results calculated with different values of the shift parameter m is proposed as a measure of summation accuracy. © 1998 American Institute of Physics.

PII: S0022-2488(98)04210-8
PACS: 31.10.+z, 02.30.Lt      Additional Information


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References

  1. L. D. Mlodinow, ``The large N expansion in quantum mechanics,'' Ph.D. dissertation, University of California, Berkeley, 1981.
  2. L. D. Mlodinow, in Progress in Particle and Nuclear Physics, edited by D. Wilkinson (Pergamon, Oxford, 1982), Vol. 8, pp. 387–399.
  3. P. du T. van der Merwe, J. Chem. Phys. 81, 5976 (1984); [SPIN]
    82, 5293 (1985). [SPIN]
  4. O. Goscinski and V. Mujica, Int. J. Quantum Chem. 29, 897 (1986). [INSPEC]
  5. D. R. Herschbach, J. Chem. Phys. 84, 838 (1986). [SPIN]
  6. J. G. Loeser, J. Chem. Phys. 86, 5635 (1987). [SPIN]
  7. D. Z. Goodson and D. R. Herschbach, J. Chem. Phys. 86, 4997 (1987). [SPIN]
  8. D. R. Herschbach, J. G. Loeser, and D. K. Watson, Z. Phys. D 10, 195 (1988). [INSPEC]
  9. S. Kais and D. R. Herschbach, J. Chem. Phys. 100, 4367 (1994). [SPIN]
  10. D. R. Herschbach, in Dimensional Scaling in Chemical Physics, edited by D. R. Herschbach, J. Avery, and O. Goscinski (Kluwer, Dordrecht, 1993), pp. 7–80, and references therein.
  11. J. G. Loeser and D. R. Herschbach, in New Methods in Quantum Theory, edited by C. A. Tsipis, V. S. Popov, D. R. Herschbach, and J. S. Avery (Kluwer, Dordrecht, 1996), pp. 1–33, and references therein.
  12. P. Serra and S. Kais, Chem. Phys. Lett. 260, 302 (1996); [INSPEC]
    Phys. Rev. A 55, 238 (1997); [SPIN]
    Phys. Rev. Lett. 77, 466 (1996). [SPIN]
  13. V. M. Va(i, breve)nberg, V. D. Mur, V. S. Popov, A. V. Sergeev, and A. V. Shcheblykin, Theor. Math. Phys. 74, 269 (1988) [Teor. Mat. Fiz. 74, 399 (1988)].
  14. M. Dunn, T. C. Germann, D. Z. Goodson, C. A. Traynor, J. D. Morgan III, D. K. Watson, and D. R. Herschbach, J. Chem. Phys. 101, 5987 (1994). [SPIN]
  15. D. Z. Goodson, M. López-Cabrera, D. R. Herschbach, and J. D. Morgan III, J. Chem. Phys. 97, 8481 (1992). [SPIN]
  16. M. O. Elout and D. Z. Goodson (unpublished).
  17. C. M. Bender, L. D. Mlodinow, and N. Papanicolaou, Phys. Rev. A 25, 1305 (1982). [SPIN]
  18. V. M. Va(i, breve)nberg, V. D. Mur, V. S. Popov, and A. V. Sergeev, JETP Lett. 44, 9 (1986) [Pis'ma Zh. Eksp. Teor. Fiz. 44, 9 (1986)]. [SPIN]
  19. V. D. Mur, V. S. Popov, and A. V. Sergeev, Sov. Phys. JETP 70, 16 (1990) [Zh. Eksp. Teor. Fiz. 97, 32 (1990)]; and references therein. [SPIN]
  20. V. M. Va(i, breve)nberg, V. S. Popov, and A. V. Sergeev, Sov. Phys. JETP 71, 470 (1990) [Zh. Eksp. Teor. Fiz. 98, 847 (1990)]. [SPIN]
  21. T. C. Germann, D. R. Herschbach, M. Dunn, and D. K. Watson, Phys. Rev. Lett. 74, 658 (1995). [SPIN]
  22. M. López-Cabrera, D. Z. Goodson, D. R. Herschbach, and J. D. Morgan III, Phys. Rev. Lett. 68, 1992 (1992). [SPIN]
  23. S.-W. Huang, D. Z. Goodson, M. López-Cabrera, and T. C. Germann, Phys. Rev. A 58, 250 (1998). [SPIN]
  24. D. J. Doren and D. R. Herschbach, Phys. Rev. A 34, 2654 (1986); [SPIN]
    J. Chem. Phys. 87, 433 (1987). [SPIN]
  25. D. J. Doren and D. R. Herschbach, Chem. Phys. Lett. 118, 115 (1985). [INSPEC]
  26. M. López-Cabrera, A. L. Tan, and J. G. Loeser, J. Phys. Chem. 97, 2467 (1993). [INSPEC]
  27. L. N. Lipatov, JETP Lett. 25, 79 (1977) [Pis'ma Zh. Eksp. Teor. Fiz. 25, 116 (1977)]. [SPIN]
  28. J. Zinn-Justin, Phys. Rep. 109, 109 (1981).
  29. V. S. Popov and A. V. Sergeev, Phys. Lett. A 193, 165 (1994); [INSPEC]
    JETP Lett. 78, 303 (1994) [Zh. Eksp. Teor. Fiz. 105, 568 (1994)].
  30. D. Z. Goodson and D. R. Herschbach, Phys. Rev. A 46, 5428 (1992). [SPIN]
  31. D. Z. Goodson and D. K. Watson, Phys. Rev. A 48, 2668 (1993). [SPIN]
  32. D. R. Herrick and F. H. Stillinger, Phys. Rev. A 11, 42 (1975). [SPIN]
  33. C. M. Rosenthal, J. Chem. Phys. 55, 2474 (1971). [INSPEC]
  34. M. Dunn and D. K. Watson, Ann. Phys. (N.Y.) 251, 266 (1996); [SPIN]
    251, 319 (1996); [SPIN]
    Few-Body Syst. 21, 187 (1996). [INSPEC]
  35. P. du T. van der Merwe, Phys. Rev. A 36, 3446 (1987). [SPIN]
  36. D. J. Doren and D. R. Herschbach, J. Phys. Chem. 92, 1816 (1988). [INSPEC]
  37. D. D. Frantz and D. R. Herschbach, Chem. Phys. 126, 59 (1988). [INSPEC]
  38. A. L. Tan and J. G. Loeser, in Dimensional Scaling in Chemical Physics, edited by D. R. Herschbach, J. Avery, and O. Goscinski (Kluwer Academic, Dordrecht, 1993), pp. 230–255.
  39. C. A. Traynor and D. Z. Goodson, J. Phys. Chem. 97, 2464 (1993).
  40. J. Killingbeck, J. Phys. A 14, 1005 (1981). [INSPEC]
  41. D. K. Watson and D. Z. Goodson, Phys. Rev. A 51, R5 (1995). [SPIN]
  42. J. D. Baker, D. E. Freund, R. N. Hill, and J. D. Morgan III, Phys. Rev. A 41, 1247 (1990). [SPIN]
  43. B. W. Ninham, J. Math. Phys. 4, 679 (1963).
  44. D. Z. Goodson and M. López-Cabrera, in Dimensional Scaling in Chemical Physics, edited by D. R. Herschbach, J. Avery, and O. Goscinski (Kluwer Academic, Dordrecht, 1993), pp. 275–314.
  45. A. A. Suvernev and D. Z. Goodson, J. Chem. Phys. 107, 4099 (1997). [SPIN]
  46. D. Z. Goodson, Phys. Rev. A 55, 4155 (1997). [SPIN]


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