Efficient and Rapid Evaluation of Three-Center Two Electron Coulomb and Hybrid Integrals Using Nonlinear Transformations

Journal of Computational Physics

Volumn 155, Issue 2, 1999, Pages 331-347

  Safouhi, H ^a   Hoggan, P E ^a  

^a UMR 6504 ^*  (France)

Author keywords

Hybrid integrals; Nonlinear transformations; Rapid evaluation of infinite oscillatory integrals; Three center two electron Coulomb integrals

Indexed keywords

ACCELERATION; BESSEL FUNCTIONS; DIFFERENTIAL EQUATIONS; EXPANSION; FUNCTION EVALUATION; LINEAR SYSTEMS; MATHEMATICAL TRANSFORMATIONS; NONLINEAR EQUATIONS;

% REDUCTIONS; BESSEL'S FUNCTION; HIGH-ACCURACY; HYBRID INTEGRAL; NON-LINEAR TRANSFORMATIONS; OSCILLATORY INTEGRAL; RAPID EVALUATION OF INFINITE OSCILLATORY INTEGRAL; SPHERICAL BESSEL FUNCTIONS; THREE-CENTER TWO ELECTRON COULOMB INTEGRAL; TWO-ELECTRON INTEGRALS;

ELECTRONS;

EID: 0000199191     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.1999.6339     Document Type: Article

References (58)

1. Safouhi H., Pinchon D., Hoggan P. E. Efficient evaluation of integrals for density functional theory: Non-linear D-transformations to evaluate three-center nuclear attraction integrals over B functions. Int. J. Quantum. Chem. 70:1998;181.
2. Safouhi H., Hoggan P. E. Efficient evaluation of Coulomb integrals: The non-linear D- and D -trans-formations. J. Phys. A. 31:1998;8941.
3. Safouhi H., Hoggan P. E. Three-center two electron Coulomb and hybrid integrals evaluated using non-linear D- and D -transformations. J. Phys. A. 32:1999;6203.
4. H. Safouhi, and, P. E. Hoggan, Non-linear transformations for rapid and efficient evaluation of multicenter bielectronic integrals over B functions, J. Math. Chem, in press.
5. I. Shavitt, The Gaussian function in calculation of statistical mechanics and quantum mechanics, in, Methods in Computational Physics. 2. Quantum Mechanics, edited by, B. Alder, S. Fernbach, and M. Rotenberg, Academic Press, New York, 1963, p, 1.
6. Steinborn E. O., Filter E. 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.
7. Filter E., Steinborn E. O. Extremely compact formulas for molecular one-electron integrals and Coulomb integrals over Slater-type orbitals. Phys. Rev. A. 18:1978;1.
8. Filter E. Analytische Methoden zur Auswertung von Mehrzentren-Matrixelementen in der Theorie der Molekülorbitale bei Verwendung exponentialartiger Basissätze. 1978.
9. Weniger E. J. Reduzierte Bessel-Funktionen als LCAO-Basissatz: Analytische und numerische Untersuchungen. 1982.
10. Weniger E. J., Steinborn E. O. The Fourier transforms of some exponential-type functions and their relevance to multicenter problems. J. Chem. Phys. 78:1983;6121.
11. Prosser F. P., Blanchard C. H. On the evaluation of two-center integrals. J. Chem. Phys. 36:1962;1112.
12. Bonham R. A., Peacher J. L., Cox H. L. On the calculation of multicenter two-electron repulsion integrals involving Slater functions. J. Chem. Phys. 40:1964;3083.
13. Bonham R. A. The evaluation of two-center integrals involved in the calculation of the intensity of diffracted electron and X-rays from molecules. J. Phys. Soc. Japan. 20:1965;12.
14. Weniger E. J., Steinborn E. O. Numerical properties of the convolution theorems of B functions. Phys. Rev. A. 28:1983;2026.
15. Weniger E. J., Steinborn E. O. Addition theorems for B functions and other exponentially declining functions. J. Math. Phys. 30:1989;774.
16. 2. Int. J. Quantum. Chem. Symp. 11:1977;509.
17. ++. Int. J. Quantum Chem. Symp. 12:1978;103.
18. Kranz H. H., Steinborn E. O. Implications and improvements of single-center expansions in molecules. Phys. Rev. A. 25:1982;66.
19. E. O. Steinborn, Methods in Computational Molecular Physics, edited by, H. H. Dierckesen and S. Wilson, Reidel, Dordrecht, 1983.
20. Steinborn E. O., Ruedenberg K. Rotation and translation of regular and irregular solid spherical harmonics. Adv. Quantum. Chem. 7:1973;1.
21. Trivedi H. P., Steinborn E. O. 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.
22. Grotendorst J. Berechnung der mehrzentren-molekülintegrale mit exponentialartigen basisfunktionen durch systematische anwendung der Fourier-transformationsmethode. 1985.
23. Grotendorst J., Weniger E. J., Steinborn E. O. 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.
24. Weniger E. J., Grotendorst J., Steinborn E. O. 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.
25. Weniger E. J., Steinborn E. O. Comment on "Molecular overlap integrals with exponential-type orbitals" J. Chem. Phys. 87:1987;3709.
26. Grotendorst J., Steinborn E. O. 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.
27. Weniger E. J., Steinborn E. O. Overlap integrals of B functions: A numerical study of infinite series representations and integrals representations. Theor. Chim. Acta. 73:1988;323.
28. Steinborn E. O., Homeier H. H. H. Möbius-type quadrature of electron repulsion integrals with B functions. Int. J. Quantum. Chem. 24:1990;349.
29. Homeier H. H. H., Steinborn E. O. Numerical integration of a function with a sharp peak at or near one boundary using Mobius transformations. J. Comput. Phys. 87:1990;61.
30. Homeier H. H. H., Steinborn E. O. Improved quadrature methods for three-center nuclear attraction integrals with exponential-type basis functions. Int. J. Quantum. Chem. 39:1991;625.
31. Steinborn E. O., Homeier H. H. H., Weniger E. J. Recent progress on representations for Coulomb integrals of exponential-type orbitals. J. Mol. Struct. 260:1992;207.
32. Homeier H. H. H., Steinborn E. O. Improved quadrature methods for the Fourier transform of a two-center product of exponential-type basis functions. Int. J. Quantum. Chem. 41:1992;399.
33. Homeier H. H. H., Steinborn E. O. On the evaluation of overlap integrals with exponential-type basis functions. Int. J. Quantum Chem. 42:1992;761.
34. Homeier H. H. H., Weniger E. J., Steinborn E. O. Program for the evaluation of overlap integrals with B functions. Comput. Phys. Comm. 72:1992;269.
35. H. H. H. Homeier, Integraltransformationsmethoden und Quadraturverfahren für Molekülintegrale mit B-Functionen, S. Roderer Verlag, Regensburg, 1990; Ph.D thesis, Universität Regensburg, 1990.
36. Levin D., Sidi A. Two new classes of non-linear transformations for accelerating the convergence of infinite integrals and series. Appl. Math. Comput. 9:1981;175.
37. Sidi A. Some properties of a generalization of the Richardson process. J. Inst. Math. Appl. 24:1979;327.
38. Sidi A. Convergence properties of some non-linear sequence transformations. Math. Comput. 33:1979;315.
39. Sidi A. Extrapolation methods for oscillating infinite integrals. J. Inst. Maths. Appl. 26:1980;1.
40. Magnus W., Oberhettinger F., Soni R. P. Formulas and Theorems for the Special Functions of Mathematical Physics. 1966.
41. Arfken G. B., Weber H. J. Mathematical Methods for Physicists. 1995.
42. Watson G. N. A Treatise on the Theory of Bessel Functions. 1944.
43. Condon E. U., Shortley G. H. The Theory of Atomic Spectra. 1970.
44. Nikiforov A. F., Ouvarov V. B. Special Functions of Mathematical and Physics. 1988.
45. Slater J. C. Atomic shielding constants. Phys. Rev. 36:1930;57.
46. Slater J. C. Analytic atomic wave functions. Phys. Rev. 42:1932;33.
47. Gaunt J. A. The triplets of heluim. Phil. Trans. R. Soc. A. 228:1929;151.
48. Homeier H. H. H., Steinborn E. O. Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients. J. Mol. Struct. 368:1996;31.
49. Sébilleau D. On the computation of the integrated product of three spherical harmonics. J. Phys. A. 31:1998;7157.
50. Weniger E. J., Steinborn E. O. Programs for the coupling of spherical harmonics. Comput. Phys. Comm. 25:1982;149.
51. Xu Yu-Lin. Fast evaluation of Gaunt coefficients. Math. Comput. 65:1996;1601.
52. Xu Yu-Lin. Fast evaluation of Gaunt coefficients: recursive approach. J. Comput. Appl. Math. 85:1997;53.
53. Xu Yu-Lin. Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories. J. Comput. Phys. 139:1998;137.
54. Abramowitz M., Stegun I. A. Handbook of Mathematical Functions. 1972.
55. n) transformation. Math. Tables Aids Comput. 10:1956;91.
56. Brezinski C. Algorithmes d'Accélérations de la Convergence. 1978.
57. Levin D. Developement of non-linear transformations for improving convergence of sequences. Int. J. Comput. Math. B. 3:1973;371.
58. Brezinski C., Zaglia M. R. Extrapolation Methods: Theory and Practice. 1991.