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

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.

[1]  Shmuel Agmon,et al.  Lectures on exponential decay of solutions of second order elliptic equations : bounds on eigenfunctions of N-body Schrödinger operators , 1983 .

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

[3]  Warren J. Hehre,et al.  AB INITIO Molecular Orbital Theory , 1986 .

[4]  Herbert H. H. Homeier,et al.  Recent progress on representations for Coulomb integrals of exponential-type orbitals , 1992 .

[5]  Herbert H. H. Homeier,et al.  Improved quadrature methods for the Fourier transform of a two-center product of exponential-type basis functions , 1992 .

[6]  R. López,et al.  Calculation of integrals with slater basis from the one‐range expansion of the 0s function , 1990 .

[7]  H. L. Cox,et al.  On the Calculation of Multicenter Two‐Electron Repulsion Integrals Involving Slater Functions , 1964 .

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

[9]  P. Wynn,et al.  On a Device for Computing the e m (S n ) Transformation , 1956 .

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

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

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

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

[14]  Herbert H. H. Homeier,et al.  Möbius-Type Quadrature of Electron Repulsion Integrals with B Functions , 1990 .

[15]  Guillermo Ramírez,et al.  Molecular integrals with Slater basis. I. General approach , 1989 .

[16]  A. W. Niukkanen.  Fourier transforms of atomic orbitals. I. Reduction to four-dimensional harmonics and quadratic transformations , 1984 .

[17]  E. Steinborn,et al.  A matrix representation of the translation operator with respect to a basis set of exponentially declining functions , 1980 .

[18]  Herbert H. H. Homeier,et al.  Programs for the evaluation of nuclear attraction integrals with B functions , 1993 .

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

[20]  J. Pople,et al.  Approximate Self‐Consistent Molecular‐Orbital Theory. V. Intermediate Neglect of Differential Overlap , 1967 .

[21]  Jones,et al.  Analytical evaluation of multicenter molecular integrals over Slater-type orbitals using expanded Löwdin alpha functions. , 1988, Physical review. A, General physics.

[22]  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.

[23]  Herbert H. H. Homeier,et al.  On the combination of two methods for the calculation of multicenter integrals using exponential–type orbitals , 1992 .

[24]  G. Diercksen,et al.  Methods in Computational Molecular Physics , 1983 .

[25]  David Feller,et al.  Basis Set Selection for Molecular Calculations , 1986 .

[26]  Claude Brezinski,et al.  Extrapolation methods - theory and practice , 1993, Studies in computational mathematics.

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

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

[29]  David Levin,et al.  Development of non-linear transformations for improving convergence of sequences , 1972 .

[30]  E. Steinborn,et al.  The three‐dimensional convolution of reduced Bessel functions and other functions of physical interest , 1978 .

[31]  G. Watson.  Bessel Functions. (Scientific Books: A Treatise on the Theory of Bessel Functions) , 1923 .

[32]  C. C. J. Roothaan,et al.  A Study of Two‐Center Integrals Useful in Calculations on Molecular Structure. I , 1951 .

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

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

[35]  Herbert H. H. Homeier,et al.  On the Evaluation of Overlap Integrals with Exponential-type Basis Functions , 1992 .

[36]  Charles A. Weatherford,et al.  The löwdin α function and its application to the multi-center molecular integral problem over slater-type orbitals , 1989 .

[37]  C. H. Blanchard,et al.  On the Evaluation of Two‐Center Integrals , 1962 .

[38]  J. Fernández Rico,et al.  Reference program for molecular calculations with Slater‐type orbitals , 1998 .

[39]  N. Ullah.  Non‐Linear Transformations and Their Applications in Many‐Body Physics , 1994 .

[40]  Guillermo Ramírez,et al.  Molecular integrals with Slater basis. II. Fast computational algorithms , 1989 .

[41]  Grotendorst,et al.  Unified analytical treatment of overlap, two-center nuclear attraction, and Coulomb integrals of B functions via the Fourier-transform method. , 1986, Physical review. A, General physics.

[42]  Herbert H. H. Homeier,et al.  Programs for the evaluation of overlap integrals with B functions , 1992 .

[43]  Herbert H. H. Homeier,et al.  Numerical integration of functions with a sharp peak at or near one boundary using Mo¨bius transformations , 1990 .

[44]  R. P. Soni,et al.  Formulas and Theorems for the Special Functions of Mathematical Physics , 1967 .

[45]  Herbert H. H. Homeier,et al.  Improved quadrature methods for three‐center nuclear attraction integrals with exponential‐type basis functions , 1991 .

[46]  Evert Jan Baerends,et al.  Self-consistent molecular Hartree—Fock—Slater calculations I. The computational procedure , 1973 .

[47]  David Levin,et al.  Rational Approximations from the d-Transformation , 1982 .

[48]  G. Shortley,et al.  The Theory of Atomic Spectra , 1935 .

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

[50]  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 .

[51]  Grotendorst,et al.  Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators. , 1986, Physical review. A, General physics.

[52]  P. Wynn,et al.  On the Convergence and Stability of the Epsilon Algorithm , 1966 .

[53]  H. P. Trivedi,et al.  Numerical properties of a new translation formula for exponential-type functions and its application to one-electron multicenter integrals , 1982 .

[54]  E. J. Weniger,et al.  Numerical properties of the convolution theorems of B functions , 1983 .

[55]  E. J. Weniger,et al.  Overlap integrals of B functions , 1988 .

[56]  S. F. Boys.  Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

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