Journal of Mathematical Chemistry

, Volume 54, Issue 3, pp 682–689 | Cite as

On the evaluation of integrals with Coulomb Sturmian radial functions

  • Daniel A. Morales
Brief Communication


It is presented the full evaluation of integrals involving Coulomb Sturmian functions already discussed in this Journal.


Sturmian functions Coulomb Sturmians Slater type orbitals STO molecular integrals Electronic structure theory 

Mathematics Subject Classification

81Q05 81V45 81V55 92EXX 33C05 


  1. 1.
    H. Shull, P.O. Löwdin, Superposition of configurations and natural spin-orbitals. Applications to the He problem. J. Chem. Phys. 30, 617–626 (1959)CrossRefGoogle Scholar
  2. 2.
    M. Rotenberg, Application of Sturmian functions to the Schroedinger three-body problem: Elastic \(\text{ e }^{+}\)-H scattering. Ann. Phys. 19, 262–278 (1962)CrossRefGoogle Scholar
  3. 3.
    M. Rotenberg, Theory and application of Sturmian functions. Adv. Atom. Mol. Phys. 6, 233–268 (1970)CrossRefGoogle Scholar
  4. 4.
    J.E. Avery, New computational methods in the quantum theory of nanostructures. PhD thesis, University of Copenhagen (2011)Google Scholar
  5. 5.
    J.E. Avery, J.S. Avery, Generalized Sturmians and Atomic Spectra (World Scientific, Singapore, 2006)CrossRefGoogle Scholar
  6. 6.
    J.E. Avery, J.S. Avery, The generalized Sturmian method, in Solving the Schrödinger Equation: Has Everything Been Tried? Chap. 6, ed. by P. Popelier (Imperial College Press, 2011), pp. 111–140Google Scholar
  7. 7.
    J.S. Avery, Hyperspherical Harmonics and Generalized Sturmians (Kluwer, Dordrecht, 2000)Google Scholar
  8. 8.
    J. Avery, Many-center Coulomb Sturmians and Shibuya-Wulfman integrals. Int. J. Quantum Chem. 100, 121–130 (2004)CrossRefGoogle Scholar
  9. 9.
    E. Red, C.A. Weatherford, Derivation of a general formula for the Shibuya–Wulfman matrix. Int. J. Quantum Chem. 100, 208–213 (2004)CrossRefGoogle Scholar
  10. 10.
    R. Beals, R. Wong, Special Functions (Cambridge University Press, Cambridge, 2010)CrossRefGoogle Scholar
  11. 11.
    T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics. Comm. Pure Appl. Math. 10, 151–177 (1957)CrossRefGoogle Scholar
  12. 12.
    P. E. Hoggan, Slater-type orbital basis sets: reliable and rapid solution of the Schrödinger equation for accurate molecular properties, in Solving the Schrödinger Equation: Has Everything Been Tried? Chap 7, ed. by P. Popelier (Imperial College Press, Ames, 2011)Google Scholar
  13. 13.
    J. Slater, Analytic atomic wave functions. Phys. Rev. 42, 33–43 (1932)CrossRefGoogle Scholar
  14. 14.
    M.P. Barnett, C.A. Coulson, The evaluation of integrals occurring in the theory of molecular structure. Parts I and II. Phil. Trans. R. Soc. Lond. A 243, 221–249 (1951)CrossRefGoogle Scholar
  15. 15.
    P.O. Löwdin, Quantum theory of cohesive properties of solids. Adv. Phys. 5, 1–172 (1956)CrossRefGoogle Scholar
  16. 16.
    F.E. Harris, H.H. Michels, Multicenter integrals in quantum mechanics. I. Expansion of Slater-type orbitals about a new origin. J. Chem. Phys. 43, S165–S169 (1965)CrossRefGoogle Scholar
  17. 17.
    Y.G. Smeyers, About evaluation of many-center molecular integrals. Theoret. Chim. Acta (Berlin) 4, 452–459 (1966)CrossRefGoogle Scholar
  18. 18.
    I.I. Guseinov, Unified analytical treatment of multicenter multielectron integrals of central and noncentral interaction potentials over Slater orbitals using \(\Psi ^{\alpha }\)-ETOs. J. Chem. Phys. 119, 4614–4619 (2003)CrossRefGoogle Scholar
  19. 19.
    S.F. Boys, G.B. Cook, C.M. Reeves, I. Shavitt, Automatic fundamental calculations of molecular structure. Nature 178, 1207–1209 (1956)CrossRefGoogle Scholar
  20. 20.
    E.J. Weniger, E.O. Steinborn, The Fourier transforms of some exponential-type basis functions and their relevance for multicenter problems. J. Chem. Phys. 78, 6121–6132 (1983)CrossRefGoogle Scholar
  21. 21.
    E.J. Weniger, Weakly convergent expansions of a plane wave and their use in Fourier integrals. J. Math. Phys. 26, 276–291 (1985)CrossRefGoogle Scholar
  22. 22.
    E.J. Weniger, J. Grotendorst, E.O. Steinborn, Unified analytical treatment of overlap, two-center nuclear attraction, and Coulomb integrals of B functions via the Fourier-transform method. Phys. Rev. A 33, 3688–3705 (1986)CrossRefGoogle Scholar
  23. 23.
    D. Antolovic, H.J. Silverstone, On the computation of (2–2) three-center Slater-type orbital integrals of \(1/\text{ r }_{12}\) using Fourier-transform-based analytical formulas. Int. J. Quantum Chem. 100, 146–154 (2004)CrossRefGoogle Scholar
  24. 24.
    I. Shavitt, M. Karplus, Multicenter integrals in molecular quantum mechanics. J. Chem. Phys. 36, 550–551 (1962)CrossRefGoogle Scholar
  25. 25.
    I. Shavitt, M. Karplus, Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43, 398–414 (1965)CrossRefGoogle Scholar
  26. 26.
    J. Fernández-Rico, R. López, I. Ema, G. Ramirez, Reference program for molecular calculations with Slater-type orbitals. J. Comp. Chem. 25, 1987–1994 (2004)CrossRefGoogle Scholar
  27. 27.
    R.S. Mulliken, C.A. Rieke, D. Orloff, H. Orloff, J. Chem. Phys. 17, 1248–1267 (1949)CrossRefGoogle Scholar
  28. 28.
    S. Huzinaga, Molecular integrals. Prog. Theor. Phys. Suppl. 40, 52–77 (1967)CrossRefGoogle Scholar
  29. 29.
    J.C. Browne, Molecular wave functions: calculation and use in atomic and molecular processes. Adv. At. Mol. Phys. 7, 47–95 (1971)CrossRefGoogle Scholar
  30. 30.
    J. Yaisui, A. Saika, J. Chem. Phys. 76, 468–472 (1982)CrossRefGoogle Scholar
  31. 31.
    S. Varganov, A.T.B. Gilbert, E. Deplazes, P.M.W. Gill, Resolutions of the Coulomb operator. J. Chem. Phys 128, 201104-1–201104-4 (2008)Google Scholar
  32. 32.
    P.M.W. Gill, A.T.B. Gilbert, Resolution of the Coulomb operator II. The Laguerre generator. Chem. Phys. 356, 86–92 (2009)CrossRefGoogle Scholar
  33. 33.
    P.E. Hoggan, General two-electron exponential type orbital integrals in polyatomics without orbital translations. Int. J. Quantum Chem. 109, 2926–2932 (2009)CrossRefGoogle Scholar
  34. 34.
    F.E. Harris, H.H. Michels, The evaluation of molecular integrals for Slater-type orbitals. Adv. Chem. Phys. 13, 205–266 (1967)Google Scholar
  35. 35.
    F.E. Harris, Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Int. J. Quantum Chem. 88, 701–734 (2002)CrossRefGoogle Scholar
  36. 36.
    J.E. Avery, J.S. Avery, Molecular integrals for Slater type orbitals using Coulomb Sturmians. J. Math. Chem. 52, 301–312 (2014)CrossRefGoogle Scholar
  37. 37.
    E. Filter, E.O. Steinborn, Extremely compact formulas for molecular two-center one-electron integrals and Coulomb integrals over Slater-type atomic orbitals. Phys. Rev. A 18, 1–11 (1978)CrossRefGoogle Scholar
  38. 38.
    E.J. Weniger, The strange history of B functions or how theoretical chemists and mathematicians do (not) interact. Int. J. Quantum Chem. 109, 1706–1716 (2009)CrossRefGoogle Scholar
  39. 39.
    E.J. Weniger, E.O. Steinborn, Addition theorems for B functions and other exponentially declining functions. J. Math. Phys. 30, 774–784 (1989)CrossRefGoogle Scholar
  40. 40.
    E.J. Weniger, E.O. Steinborn, Numerical properties of the convolution theorems of B functions. Phys. Rev. A 28, 2026–2041 (1983)CrossRefGoogle Scholar
  41. 41.
    J.S. Avery, Hyperspherical Harmonics: Applications in Quantum Theory (Kluwer, Dordrecht, 1989)CrossRefGoogle Scholar
  42. 42.
    I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, seventh edition, ed. by A. Jeffrey and D. Zwillinger (Academic Press, Amsterdam, 2007)Google Scholar
  43. 43.
    R.G. Parr, H.W. Joy, Why not use Slater orbitals of nonintegral principal quantum number? J. Chem. Phys. 26, 424 (1957)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad de Los AndesMéridaVenezuela

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