Numerical evaluation of three-center two-electron Coulomb and hybrid integrals over B functions using the H D and H D̄ methods and convergence properties

Journal of Mathematical Chemistry

Volumn 29, Issue 3, 2001, Pages 213-232

  Safouhi, Hassan ^a  

^a UNIVERSITÉ DU QUÉBEC À MONTRÉAL   (Canada)

Author keywords

Hybrid integrals; Nonlinear transformations for accelerating the convergence of semi infinite integrals; Three center two electron Coulomb integrals

Indexed keywords

EID: 0035536410     PISSN: 02599791     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1010994517423     Document Type: Article

References (55)

1. I. Shavitt, The Gaussian function in calculation of statistical mechanics and quantum mechanics, in: Methods in Computational Physics, Vol. 2, Quantum Mechanics, eds. B. Alder, S. Fernbach and M. Rotenberg (Academic Press, New York, 1963).
2. E.O. Steinborn and E. Filter, Translations of fields represented by spherical-harmonics expansions for molecular calculations. III. Translations of reduced Bessel functions, Slater-type s-orbitals, and other functions, Theor. Chim. Acta 38 (1975) 273.
3. E. Filter and E.O. Steinborn, Extremely compact formulas for molecular one-electron integrals and Coulomb integrals over Slater-type orbitals, Phys. Rev. A 18 (1978) 1.
4. E. Filter, Analytische Methoden zur Auswertung von Mehrzentren-Matrixelementen in der Theorie der Molekülorbitale bei Verwendung exponentialartiger Basissätze, Ph.D. thesis, Universität Regensburg (1978).
5. E.J. Weniger, Reduzierte Bessel-Funktionen als LCAO-Basissatz: Analytische und numerische Untersuchungen, Ph.D. thesis, Universität Regensburg (1982).
6. E.J. Weniger and E.O. Steinborn, The Fourier transforms of some exponential-type functions and their relevance to multicenter problems, J. Chem. Phys. 78 (1983) 6121.
7. F.P. Presser and C.H. Blanchard, On the evaluation of two-center integrals, J. Chem. Phys. 36 (1962) 1112.
8. R.A. Bonham, J.L. Peacher and H.L. Cox, On the calculation of multicenter two-electron repulsion integrals involving Slater functions, J. Chem. Phys. 40 (1964) 3083.
9. 2, Int. J. Quantum. Chem. Symp. 11 (1977) 509.
10. ++, Int. J. Quantum Chem. Symp. 12 (1978) 103.
11. E.O. Steinborn, Methods in Computational Molecular Physics, eds. H.H. Dierckesen and S. Wilson (D. Reidel, Dordrecht, 1983).
12. H.P. Trivedi and E.O. Steinborn, Fourier transform of a two-center product of exponential-type orbitals. Application to one- and two-electron multicenter integrals, Phys. Rev. A 27 (1983) 670.
13. J. Grotendorst, Berechnung der Mehrzentren-Molekülintegrale mit Exponentialartigen Basis Funktionen durch systematische Anwendung der Fourier-Transformationsmethode, Ph.D. thesis, Universität Regensburg (1985).
14. J. Grotendorst, E.J. Weniger and E.O. Steinborn, Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators, Phys. Rev. A 33 (1986) 3706.
15. E.J. Weniger and E.O. Steinborn, Comment on "Molecular overlap integrals with exponential-type orbitals", J. Chem. Phys. 87 (1987) 3709.
16. J. Grotendorst and E.O. Steinborn, Numerical evaluation of molecular one- and two-electron multicenter integrals with exponential-type orbitals via the Fourier-transform method, Phys. Rev. A 38 (1988) 3857.
17. E.J. Weniger and E.O. Steinborn, Overlap integrals of B functions. A numerical study of infinite series representations and integrals representations, Theor. Chim. Acta 73 (1988) 323.
18. E.O. Steinborn, H.H.H. Homeier and E.J. Weniger, Recent progress on representations for Coulomb integrals of exponential-type orbitals, J. Mol. Struct. (THEOCEM) 260 (1992) 207.
19. H.H.H. Homeier and E.O. Steinborn, Improved quadrature methods for the Fourier transform of a two-center product of exponential-type basis functions, Int. J. Quantum Chem. 41 (1992) 399.
20. H.H.H. Homeier and E.O. Steinborn, On the evaluation of overlap integrals with exponential-type basis functions, Int. J. Quantum Chem. 42 (1992) 761.
21. H.H.H. Homeier, E.J. Weniger and E.O. Steinborn, Program for the evaluation of overlap integrals with B functions, Comput. Phys. Commun. 72 (1992) 269.
22. H.H.H. Homeier, Integraltransformationsmethoden und Quadraturverfahren für Molekülintegrale mit B-Functionen (S. Roderer Verlag, Regensburg, 1990); also: Ph.D. thesis, Universität Regensburg (1990).
23. E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Rep. 10 (1989) 189.
24. H. Safouhi and P.E. Hoggan, Three-centre two electron Coulomb and hybrid integrals evaluated using nonlinear D- and D̄-transformations, J. Phys. A: Math. Gen. 32 (1999) 6203.
25. H. Safouhi, Nonlinear transformations for accelerating the convergence of molecular multicenter bielectronic integrals, Ph.D. thesis, Université de Blaise Pascal, Clermont-Ferrand, France (1999).
26. D. Levin and A. Sidi, Two new classes of non-linear transformations for accelerating the convergence of infinite integrals and series, Appl. Math. Comput. 9 (1981) 175.
27. A. Sidi, Extrapolation methods for oscillating infinite integrals, J. Inst. Math. Appl. 26 (1980) 1.
28. H. Safouhi and P.E. Hoggan, Efficient and rapid evaluation of three-center two-electron Coulomb and hybrid integrals using nonlinear transformations, J. Comput. Phys. 155 (1999) 331.
29. E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, Cambridge, UK, 1970).
30. G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists (Academic Press, New York, 1995).
31. J.C. Slater, Atomic shielding constants, Phys. Rev. 36 (1930) 57.
32. J.C. Slater, Analytic atomic wave functions, Phys. Rev. 42 (1932) 33.
33. I.M. Gel'fand and G.E. Shilov, Generalized Functions I. Properties and Operations (Academic Press, New York, 1964).
34. M. Weissbluth, Atoms and Molecules (Academic Press, New York, 1978).
35. J.A. Gaunt, The triplets of heluim, Phil. Trans. Roy. Soc. Ser. A 228 (1929) 151.
36. H.H.H. Homeier and E.O. Steinborn, Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients, J. Mol. Struct. (THEOCEM) 368 (1996) 31.
37. D. Sébilleau, On the computation of the integrated product of three spherical harmonics, J. Phys. A 31 (1998) 7157.
38. E.J. Weniger and E.O. Steinborn, Programs for the coupling of spherical harmonics, Comput. Phys. Comm. 25 (1982) 149.
39. Yu-Lin Xu, Fast evaluation of Gaunt coefficients, Math. Comput. 65 (1996) 1601.
40. Yu-Lin Xu, Fast evaluation of Gaunt coefficients: recursive approach, J. Comput. Appl. Math. 85 (1997) 53.
41. Yu-Lin Xu, Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories, J. Comput. Phys. 139 (1998) 137.
42. R.D. Jenks and R.S. Sutor, Axiom, The Scientific Computation System (Springer, New York, 1992).
43. E.O. Steinborn and H.H.H. Homeier, Möbius-type quadrature of electron repulsion integrals with B functions, Int. J. Quantum. Chem. 24 (1990) 349.
44. H.H.H. Homeier and E.O. Steinborn, Numerical integration of a function with a sharp peak at or near one boundary using Möbius transformations, J. Comput. Phys. 87 (1990) 61.
45. H.H.H. Homeier and E.O. Steinborn, Improved quadrature methods for three-center nuclear attraction integrals with exponential-type basis functions, Int. J. Quantum. Chem. 39 (1991) 625.
46. n) transformation, Math. Tables Aids Comput. 10 (1956) 91.
47. D. Levin, Development of non-linear transformations for improving convergence of sequences, Int. J. Comput. Math. B 3 (1973) 371.
48. E.J. Weniger, J. Grotendorst and E.O. Steinborn, Unified analytical treatement of overlap, two-center nuclear attraction, and Coulomb integrals over B functions via the Fourier-transform method, Phys. Rev. A 33 (1986) 3688.
49. A. Sidi, Some properties of a generalization of the Richardson process, J. Inst. Maths. Appl. 24 (1979) 327.
50. A. Sidi, Convergence properties of some non-linear sequence transformations, Math. Comput. 33 (1979) 315.
51. T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics, Comm. Pure Appl. Math. 10 (1957) 151.
52. S.F. Boys, The integral formulae for the variational solution of the molecular many-electron wave equation in terms of Gaussian functions with direct electronic correlation, Proc. Roy. Soc. Ser. A. 258 (1960) 402.
53. C.A. Weatherford and H.W. Jones, eds., International Conference on ETO Multicenter Integrals, Tallahasse, 1981 (D. Reidel, Dordrecht, Holland, 1982).
54. E.J. Weniger and E.O. Steinborn, Addition theorems for B functions and other exponentially declining functions, J. Math. Phys. 30(4) (1989) 774.
55. E.J. Weniger and E.O. Steinborn, Numerical properties of the convolution theorems of B functions, Phys. Rev. A 28 (1983) 2026.