An extremely efficient and rapid algorithm for numerical evaluation of three-centre nuclear attraction integrals over Slater-type functions

The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of the three-centre nuclear attraction integrals over Slater-type functions. These integrals are numerous, they occur in many millions of terms, even for small molecules and they require rapid and accurate evaluation. The new algorithm is based on nonlinear transformation methods, on numerical quadrature and on properties of the sine and Bessel functions. The section with numerical results shows the high accuracy and the substantial gain in calculation time realized using the new algorithm. The complete expressions of the three-centre nuclear attraction integrals over B functions and over Slater-type functions are evaluated for different values of the quantum numbers to show the efficiency of the new approach. Numerical results obtained with linear and nonlinear systems and comparisons with numerical results from the literature are listed.

[1]  Avram Sidi,et al.  A user-friendly extrapolation method for oscillatory infinite integrals , 1988 .

[2]  Avram Sidi,et al.  The numerical evaluation of very oscillatory infinite integrals by extrapolation , 1982 .

[3]  J. C. Slater.  Atomic Shielding Constants , 1930 .

[4]  A. Bouferguene,et al.  STOP: A slater‐type orbital package for molecular electronic structure determination , 1996 .

[5]  Shmuel Agmon,et al.  Bounds on exponential decay of eigenfunctions of Schrödinger operators , 1985 .

[6]  Molecular calculations with B functions , 1998, physics/9812049.

[7]  Avram Sidi.  Computation of infinite integrals involving Bessel functions of arbitrary order by the D¯ -transformation , 1997 .

[8]  Hassan Safouhi,et al.  An extremely efficient approach for accurate and rapid evaluation of three-centre two-electron Coulomb and hybrid integrals over B functions , 2001 .

[9]  David Levin,et al.  Two New Classes of Nonlinear Transformations for Accelerating the Convergence of Infinite Integrals and Series , 1981 .

[10]  Hassan Safouhi,et al.  Non‐linear transformations for rapid and efficient evaluation of multicenter bielectronic integrals over B functions , 1999 .

[11]  Hassan Safouhi,et al.  Efficient and rapid numerical evaluation of the two-electron, four-center Coulomb integrals using nonlinear transformations and useful properties of Sine and Bessel functions , 2002 .

[12]  Grotendorst,et al.  Numerical evaluation of molecular one- and two-electron multicenter integrals with exponential-type orbitals via the Fourier-transform method. , 1988, Physical review. A, General physics.

[13]  Hassan Safouhi,et al.  The properties of sine, spherical Bessel and reduced Bessel functions for improving convergence of semi-infinite very oscillatory integrals: the evaluation of three-centre nuclear attraction integrals over B functions , 2001 .

[14]  D. Rinaldi,et al.  Integrals over B functions basis sets. I. Three‐center molecular integrals, a numerical study , 1994 .

[15]  Avram Sidi,et al.  Extrapolation Methods for Oscillatory Infinite Integrals , 1980 .

[16]  E. Filter,et al.  Translations of fields represented by spherical-harmonic expansions for molecular calculations , 1975 .

[17]  J. Freeman,et al.  Measurement of anomalous muon pair production in electron-positron annihilations , 1978 .

[18]  Clemens C. J. Roothaan,et al.  New Developments in Molecular Orbital Theory , 1951 .

[19]  E. J. Weniger,et al.  Addition theorems for B functions and other exponentially declining functions , 1989 .

[20]  D. Rinaldi,et al.  A new single‐center method to compute molecular integrals of quantum chemistry in slater‐type orbital basis of functions , 1994 .

[21]  Ahmed Bouferguene,et al.  Convergence analysis of the addition theorem of Slater orbitals and its application to three-center nuclear attraction integrals , 1998 .

[22]  Hassan Safouhi,et al.  The HD and HD methods for accelerating the convergence of three-center nuclear attraction and four-center two-electron Coulomb integrals over B functions and their convergence properties , 2000 .

[23]  John C. Slater,et al.  Analytic Atomic Wave Functions , 1932 .

[24]  Hassan Safouhi,et al.  Efficient evaluation of integrals for density functional theory: NonlinearD transformations to evaluate three-center nuclear attraction integrals overB functions , 1998 .

[25]  H. Safouhi,et al.  Regular Article: Efficient and Rapid Evaluation of Three-Center Two Electron Coulomb and Hybrid Integrals Using Nonlinear Transformations , 1999 .

[26]  E. J. Weniger,et al.  The Fourier transforms of some exponential‐type basis functions and their relevance to multicenter problems , 1983 .

[27]  H. P. Trivedi,et al.  Fourier transform of a two-center product of exponential-type orbitals. Application to one- and two-electron multicenter integrals , 1983 .

[28]  Avram Sidi,et al.  An algorithm for a generalization of the Richardson extrapolation process , 1987 .

[29]  S. F. Boys,et al.  The integral formulae for the variational solution of the molecular many-electron wave equation in terms of Gaussian functions with direct electronic correlation , 1960, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[30]  P. Hoggan,et al.  Three-centre two-electron Coulomb and hybrid integrals evaluated using nonlinear D- and barD-transformations , 1999 .

[31]  Hassan Safouhi,et al.  Efficient evaluation of Coulomb integrals: the nonlinear D- and -transformations , 1998 .

[32]  Werner Kutzelnigg,et al.  Present and future trends in quantum chemical calculations , 1988 .

[33]  E. Steinborn,et al.  Extremely compact formulas for molecular two-center one-electron integrals and Coulomb integrals over Slater-type atomic orbitals , 1978 .

[34]  Avram Sidi,et al.  Some Properties of a Generalization of the Richardson Extrapolation Process , 1979 .

[35]  Avram Sidi,et al.  An algorithm for a special case of a generalization of the Richardson extrapolation process , 1982 .