Skip to main content
SpringerLink
Log in

Non‐linear transformations for rapid and efficient evaluation of multicenter bielectronic integrals over B functions

  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

This work presents an extremely efficient non‐linear transformation based on a certain Hankel type transform, originally due to A. Sidi. The approach is applied to evaluating Coulomb integrals in the molecular context. These integrals are bielectronic one‐, two‐, three‐ and four‐center terms arising from the interactions of electron distributions over a Slater type orbital basis. They occur in many millions of terms, even for small molecules, and require rapid and accurate evaluation. The present work shows how we can reduce the order of the linear differential equation required to be satisfied by the integrand considerably. Calculation times as short as 10-2 ms were obtained for four‐center terms (the least favorable case) on an IBM RS6000‐340 workstation. This method represents a considerable advance on previous work on Coulomb integrals.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bound on Eigenfunctions of N-Body Schr¨odinger Operators (Princeton University Press, Princeton, NJ, 1982).

    Google Scholar 

  2. S. Agmon, Anal. Math. Appl. 1 (1988) 1.

    Google Scholar 

  3. E.J. Baerends, D.E. Ellis and P. Ros, Chem. Phys. 2 (1973) 41.

    Article  CAS  Google Scholar 

  4. W.W. Bell, Fonctions Spéciales àl'Usage des Chercheurs et Ingénieurs, traduit par G. Fieschi (Dunod, Paris, 1971).

  5. R.A. Bonham, J. Phys. Soc. Japan 20 (1965) 12.

    Google Scholar 

  6. R.A. Bonham, J.L. Peacher and H.L. Cox, J. Chem. Phys. 40 (1964) 10.

    Google Scholar 

  7. R.A. Bonham, J.L. Peacher and H.L. Cox, J. Chem. Phys. 40 (1964) 3083.

    Article  CAS  Google Scholar 

  8. A. Bouferguene, Ph.D. thesis, Université de Nancy I (1992).

  9. A. Bouferguene, M. Fares and D. Rinaldi, J. Chem. Phys. 100 (1994) 8156.

    Article  CAS  Google Scholar 

  10. A. Bouferguene and D. Rinaldi, Int. J. Quantum Chem. 50 (1994) 21.

    Article  CAS  Google Scholar 

  11. S.F. Boys, Proc. Roy. Soc. A 200 (1950) 542.

    Article  CAS  Google Scholar 

  12. C. Brezinski, Algorithmes d'Accélérations de la Convergence (Technip, Paris, 1978).

  13. C. Brezinski and M.R. Zaglia, Extrapolation Methods: Theory and Practice (North-Holland, Amsterdam, 1991).

  14. E. Clementi and G. Coranjia, in: Methods and Techniques in Computational Chemistry, ed. S. Caglian (1997).

  15. E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, Cambridge, 1970).

    Google Scholar 

  16. E.R. Davidson and D. Feller, Chem. Rev. 86 (1986) 681.

    Article  CAS  Google Scholar 

  17. J. Fernández Rico, R. López, A. Aguado, I. Ema and G. Ramírez, J. Comput. Chem. 19 (1998) 1284.

    Article  Google Scholar 

  18. E. Filter, Ph.D. thesis, Universit¨at Regensburg (1978).

  19. E. Filter and E.O. Steinborn, J. Math. Phys. 19 (1978) 79.

    Article  Google Scholar 

  20. E. Filter and E.O. Steinborn, Phys. Rev. A 18 (1978) 1.

    Article  CAS  Google Scholar 

  21. E. Filter and E.O. Steinborn, J. Math. Phys. 21 (1980) 2725.

    Article  Google Scholar 

  22. J. Grotendorst, Ph.D. thesis, Universit¨at Regensburg (1985).

  23. J. Grotendorst and E.O. Steinborn, Phys. Rev. A 38 (1988) 3857.

    Article  CAS  Google Scholar 

  24. J. Grotendorst, E.J. Weniger and E.O. Steinborn, Phys. Rev. A 33 (1986) 3706.

    Article  Google Scholar 

  25. W.H. Hehre, L. Radom, P.R. Schleyer and J. Pople, Ab Initio Molecular Orbital Theory (Wiley, New York, 1986).

  26. H.H.H. Homeier, Integraltransformationsmethoden und Quadraturverfahrer f¨ur Molek¨ulintegrale mit B-Funktionen (S. Roder Verlag, Regensburg, 1990); also: Doctoral dissertation, Universit¨at Regensburg.

    Google Scholar 

  27. H.H.H. Homeier and E.O. Steinborn, J. Comput. Phys. 87 (1990) 625.

    Article  Google Scholar 

  28. H.H.H. Homeier and E.O. Steinborn, Int. J. Quantum Chem. 39 (1991) 625.

    Article  CAS  Google Scholar 

  29. H.H.H. Homeier and E.O. Steinborn, Int. J. Quantum Chem. 41 (1992) 399.

    Article  CAS  Google Scholar 

  30. H.H.H. Homeier and E.O. Steinborn, Int. J. Quantum Chem. 42 (1992) 761.

    Article  CAS  Google Scholar 

  31. H.H.H. Homeier and E.O. Steinborn, Comput. Phys. Comm. 77 (1993) 135.

    Article  CAS  Google Scholar 

  32. H.H.H. Homeier, E.J. Weniger and E.O. Steinborn, Comput. Phys. Comm. 72 (1992) 269.

    Article  CAS  Google Scholar 

  33. International Business Machines, Inc., ESSL, 4th ed. (Engineering and Scientific Subroutine Library, 1990).

  34. H.W. Jones, Phys. Rev. A 38 (1988) 1065.

    Article  CAS  Google Scholar 

  35. H.W. Jones and C.A. Weatherford, J. Mol. Struct. (THEOCHEM) 199 (1988) 233.

    Article  Google Scholar 

  36. T. Kato, Comm. Pure Math. 10 (1957) 151.

    Article  Google Scholar 

  37. W. Kutzelnigg, J. Mol. Struct. (THEOCHEM) 50 (1988) 33.

    Article  CAS  Google Scholar 

  38. D. Levin, Internat. J. Comput. Math. B 3 (1973) 371.

    Article  Google Scholar 

  39. D. Levin, Ph.D. thesis, Tel Aviv University (1975).

  40. D. Levin and A. Sidi, Appl. Math. Comput. 9 (1981) 175.

    Article  Google Scholar 

  41. R. López and G. Ramírez, Int. J. Quantum Chem. 49 (1994) 11.

    Article  Google Scholar 

  42. W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966).

    Google Scholar 

  43. A. Nikiforov and V. Ouvarov, Fonctions Spéciales de la Physique Mathématique (Mir, Moscou, 1978; French translation, 1983).

  44. A.W. Niukkanen, Int. J. Quantum Chem. 25 (1984) 941.

    Article  CAS  Google Scholar 

  45. V.I. Perevozchivok, I.V. Maslov, A.W. Niukkanen, H.H.H. Homeier and E.O. Steinborn, Int. J. Quantum Chem. 44 (1992) 45.

    Article  Google Scholar 

  46. J.A. Pople, D.L. Bevridge and P.A. Dobosh, J. Chem. Phys. 47 (1967) 2026.

    Article  CAS  Google Scholar 

  47. F.P. Prosser and C.H. Blanchard, J. Chem. Phys. 36 (1962) 1112.

    Article  Google Scholar 

  48. J.F. Rico, R. López and G. Ramirez, J. Chem. Phys. 91 1989) 4204.

    Article  Google Scholar 

  49. J.F. Rico, R. López and G. Ramirez, J. Chem. Phys. 91 (1989) 4213.

    Article  Google Scholar 

  50. J.F. Rico, R. López and G. Ramirez, Int. J. Quantum Chem. 37 (1990) 21.

    Article  Google Scholar 

  51. J.L. Rivail, Eléments de la Chimie Quantique (Inter Edition/Editions du CNRS, 1989).

  52. C.C.J. Roothaan, J. Chem. Phys. 19 (1951) 12.

    Article  Google Scholar 

  53. C.C.J. Roothaan, Rev. Modern Phys. 23 (1951) 69.

    Article  CAS  Google Scholar 

  54. H. Safouhi and P.E. Hoggan, J. Phys. A 31 (1998) 8941.

    Article  CAS  Google Scholar 

  55. H. Safouhi and P.E. Hoggan, J. Phys. A (1998), in press.

  56. H. Safouhi, D. Pinchon and P.E. Hoggan, Int. J. Quantum Chem. 70 (1998) 181.

    Article  CAS  Google Scholar 

  57. V.R. Sanders, Methods in Computational Molecular Physics (D. Reidel, Dordrecht, 1983).

  58. D. Shanks, J. Math. Phys. 34 (1955) 1.

    Google Scholar 

  59. I. Shavitt, Methods in Computational Physics, Vol. 2 (Academic Press, New York, 1963).

    Google Scholar 

  60. A. Sidi, J. Inst. Math. Appl. 26 (1980) 1.

    Article  Google Scholar 

  61. A. Sidi, Numer. Math. 38 (1982) 299.

    Article  Google Scholar 

  62. A. Sidi, BIT 30 (1990) 347.

  63. A. Sidi and D. Levin, IMA J. Numer. Anal. 2 (1982) 153.

    Article  Google Scholar 

  64. J.C. Slater, Phys. Rev. 36 (1930) 57.

    Article  CAS  Google Scholar 

  65. E.O. Steinborn and E. Filter, Theor. Chim. Acta 38 (1975) 273.

    Article  CAS  Google Scholar 

  66. E.O. Steinborn and H.H.H. Homeier, Int. J. Quantum Chem. Symp. 24 (1990) 349.

    Article  CAS  Google Scholar 

  67. E.O. Steinborn, H.H.H. Homeier and E.J. Weniger, J. Mol. Struct. (THEOCEM) 260 (1992) 207.

    Article  Google Scholar 

  68. E.O. Steinborn and E.J. Weniger, Int. J. Quantum Chem. Symp. 11 (1977) 509.

    CAS  Google Scholar 

  69. E.O. Steinborn and E.J. Weniger, Int. J. Quantum Chem. Symp. 12 (1978) 103.

    CAS  Google Scholar 

  70. H.P. Trivedi and E.O. Steinborn, Phys. Rev. A 25 (1982) 113.

    Article  CAS  Google Scholar 

  71. H.P. Trivedi and E.O. Steinborn, Phys. Rev. A 27 (1983) 670.

    Article  CAS  Google Scholar 

  72. N. Ullah, Fortschr. Phys. 42(1) (1994) 49.

    Article  CAS  Google Scholar 

  73. E. Van Lenthe, R. Van Leeuwen, E.J. Baerends and J.G. Snijders, in: New Challenges in Computational Quantum Chemistry, ed. Bagus (Groningen, 1994).

  74. G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University Press, Cambridge, 1944).

    Google Scholar 

  75. C.A. Weatherford and H.W. Jones, eds., International Conference on ETO Multicenter Integrals, Tallahasse, 1981 (D. Reidel, Dordrecht, 1982).

    Google Scholar 

  76. E.J. Weniger, J. Grotendorst and E.O. Steinborn, Phys. Rev. A 33 (1986) 3688.

    Article  Google Scholar 

  77. E.J. Weniger and E.O. Steinborn, J. Chem. Phys. 78 (1983) 10.

    Article  Google Scholar 

  78. E.J. Weniger and E.O. Steinborn, J. Chem. Phys. 78 (1083) 6121.

    Article  Google Scholar 

  79. E.J. Weniger and E.O. Steinborn, Phys. Rev. A 28 (1983) 2026.

    Article  CAS  Google Scholar 

  80. E.J. Weniger and E.O. Steinborn, Theor. Chim. Acta 73 (1988) 323.

    Article  Google Scholar 

  81. P. Wynn, Math. Tables Aids Comput. 10 (1956) 91.

    Article  Google Scholar 

  82. P. Wynn, SIAM J. Numer. Anal. 3 (1966) 91.

    Article  Google Scholar 

  83. D. Yarkony, ed., Modern Electronic Structure Theory, Vol. 1 (Plenum, 1996).

Download references

Author information

Authors and Affiliations

  1. SEESIB, UMR 6504, Ensemble Scientifique des Cézeaux, 63177, Aubière Cedex, France

    H. Safouhi & P.E. Hoggan

  2. Laboratoire de Catalyse et Spectrochimie, Université de Caen, UMR 6506, ISMRA, 14050, Caen Cedex, France

    H. Safouhi

Authors
  1. H. Safouhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P.E. Hoggan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Safouhi, H., Hoggan, P. Non‐linear transformations for rapid and efficient evaluation of multicenter bielectronic integrals over B functions. Journal of Mathematical Chemistry 25, 259–280 (1999). https://doi.org/10.1023/A:1019101020660

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019101020660

Keywords

Search

Navigation