Efficiency of the algorithms for the calculation of Slater molecular integrals in polyatomic molecules

The performances of the algorithms employed in a previously reported program for the calculation of integrals with Slater‐type orbitals are examined. The integrals are classified in types and the efficiency (in terms of the ratio accuracy/cost) of the algorithm selected for each type is analyzed. These algorithms yield all the one‐ and two‐center integrals (both one‐ and two‐electron) with an accuracy of at least 12 decimal places and an average computational time of very few microseconds per integral. The algorithms for three‐ and four‐center electron repulsion integrals, based on the discrete Gauss transform, have a computational cost that depends on the local symmetry of the molecule and the accuracy of the integrals, standard efficiency being in the range of eight decimal places in hundreds of microseconds. © 2004 Wiley Periodicals, Inc. J Comput Chem 25: 1987–1994, 2004

[1]  T. Özdoǧan.  Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater-type orbitals with noninteger principal quantum numbers , 2006 .

[2]  Hassan Safouhi,et al.  An extremely efficient and rapid algorithm for numerical evaluation of three-centre nuclear attraction integrals over Slater-type functions , 2003 .

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

[4]  Martin Karplus,et al.  Gaussian‐Transform Method for Molecular Integrals. I. Formulation for Energy Integrals , 1965 .

[5]  D. Andrae.  Numerical self-consistent field method for polyatomic molecules , 2001 .

[6]  P. Hoggan,et al.  Three‐center nuclear attraction, three‐center two‐electron Coulomb and hybrid integrals over B functions evaluated using the nonlinear S\documentclass{article}\pagestyle{empty}\begin{document}$\overline{D}$\end{document} transformation , 2002 .

[7]  I. Guseinov.  Computation of molecular integrals over Slater-type orbitals. IX. Calculation of multicenter multielectron molecular integrals with integer and noninteger n Slater orbitals using complete orthonormal sets of exponential functions , 2002 .

[8]  M. Orbay,et al.  Evaluation of two-center overlap and nuclear attraction integrals over slater-type orbitals with integer and noninteger principal quantum numbers , 2002 .

[9]  B. Mamedov,et al.  Evaluation of overlap integrals with integer and noninteger n Slater-type orbitals using auxiliary functions , 2002, Journal of molecular modeling.

[10]  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.

[11]  P. Knowles,et al.  An efficient method for the evaluation of coupling coefficients in configuration interaction calculations , 1988 .

[12]  J. Budzinski.  Evaluation of two‐center, three‐ and four‐electron integrals over Slater‐type orbitals in elliptical coordinates , 2004 .

[13]  F. Harris.  Comment on “Computation of Two-Center Coulomb Integrals over Slater-Type Orbitals Using Elliptical Coordinates” , 2003 .

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

[15]  E. J. Weniger,et al.  New representations for the spherical tensor gradient and the spherical delta function , 1983 .

[16]  J. F. Rico,et al.  Simplified expansion of Slater orbitals about displaced centers , 1988 .

[17]  Michael P. Barnett,et al.  Digital erosion in the evaluation of molecular integrals , 2002 .

[18]  B. Mamedov,et al.  Algorithm for the storage of expansion coefficients for the product of associated Legendre functions in elliptical coordinates useful for the calculations of molecular integrals , 2004 .

[19]  V. Aquilanti,et al.  Sturmian approach to one-electron many-center systems: integrals and iteration schemes , 2002 .

[20]  J. Fernández,et al.  Four-center integrals for Gaussian and exponential functions , 2001 .

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

[22]  Martin Karplus,et al.  Multicenter Integrals in Molecular Quantum Mechanics , 1962 .

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

[24]  B. Mamedov,et al.  Use of addition theorems in evaluation of multicenter nuclear-attraction and electron-repulsion integrals with integer and noninteger n Slater-type orbitals , 2002 .

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

[26]  G. Arrighini,et al.  Computational quantum chemistry in terms of multicenter Slater‐type orbitals: Entirely numerical procedure for the accurate evaluation of the basic integrals , 2003 .

[27]  Rafael López,et al.  Polarized basis sets of Slater‐type orbitals: H to Ne atoms , 2003, J. Comput. Chem..

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

[29]  R. López,et al.  Molecular integrals with Slater basis. III. Three‐center nuclear attraction integrals , 1991 .

[30]  M. Paniagua,et al.  Fock potentials. II. Atomic potentials and intra-atomic electronic interactions , 1984 .

[31]  C. Tablero,et al.  Molecular integrals with Slater basis. V. Recurrence algorithm for the exchange integrals , 1994 .

[32]  R. López,et al.  Auxiliary functions for Slater molecular integrals , 1993 .

[33]  P. Hoggan,et al.  New methods for accelerating the convergence of molecular electronic integrals over exponential type orbitals , 2003 .

[34]  B. Mamedov,et al.  On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. IV. Use of recurrence relations for basic two‐center overlap and hybrid integrals , 2002 .

[35]  H. Safouhi.  Convergence properties of the SD̄ transformation and a fast and accurate numerical evaluation of molecular integrals , 2002 .

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

[37]  M. Orbay,et al.  Calculation of nuclear-attraction and modified overlap integrals using Gegenbauer coefficients* , 2002 .

[38]  B. Mamedov,et al.  Calculation of molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals and basic one-center Coulomb integrals , 2002, Journal of molecular modeling.

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

[40]  I. Guseinov.  Unified analytical treatment of multicentre electron attraction, electric field and electric field gradient integrals over Slater orbitals , 2004 .

[41]  I. Guseinov.  New complete orthonormal sets of exponential‐type orbitals and their application to translation of Slater orbitals , 2002 .

[42]  R. López,et al.  Recurrence relations for the expansion of Slater‐type orbitals about displaced centers , 1986 .

[43]  J. Fernández Rico.  Long‐Range multicenter integrals with slater functions: Gauss transform‐based methods , 1993, J. Comput. Chem..

[44]  Frank E. Harris,et al.  Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion , 2002 .

[45]  A. Aguado,et al.  New program for molecular calculations with Slater‐type orbitals , 2001 .

[46]  R. López,et al.  Calculation of the one‐electron two‐center integrals with STOS using recurrence‐based algorithms , 1988 .

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

[48]  I. Guseinov.  Addition theorems for Slater-type orbitals and their application to multicenter multielectron integrals of central and noncentral interaction potentials , 2003, Journal of molecular modeling.

[49]  Hassan Safouhi,et al.  Multicentre two-electron Coulomb and exchange integrals over Slater functions evaluated using a generalized algorithm based on nonlinear transformations , 2004 .

[50]  I. Guseinov.  Comment on “Evaluation of Two‐Center Overlap and Nuclear‐Attraction Integrals over Slater‐Type Orbitals with Integer and Noninteger Principal Quantum Numbers” , 2003 .

[51]  E. J. Weniger,et al.  Nuclear attraction and electron interaction integrals of exponentially decaying functions and the Poisson equation , 1992 .

[52]  B. Mamedov,et al.  Computation of molecular integrals over Slater-type orbitals. X. Calculation of overlap integrals with integer and noninteger n Slater orbitals using complete orthonormal sets of exponential functions , 2002 .

[53]  Martin Karplus,et al.  Gaussian‐Transform Method for Molecular Integrals. II. Evaluation of Molecular Properties , 1965 .

[54]  J. Rico,et al.  Molecular integrals with Slater basis. IV: Ellipsoidal coordinate methods for three-center nuclear attraction integrals , 1992 .

[55]  Besselian: calculation of multicenter matrix elements , 2003 .

[56]  Hassan Safouhi,et al.  A new algorithm for accurate and fast numerical evaluation of hybrid and three-centre two-electron Coulomb integrals over Slater-type functions , 2003 .

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

[58]  B. Mamedov,et al.  On the calculation of arbitrary multielectron molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals. III. Auxiliary functions and , 2002 .

[59]  Rafael López,et al.  Correspondence between GTO and STO molecular basis sets , 2001, J. Comput. Chem..

[60]  Jaime Fernández Rico,et al.  Rotation of real spherical harmonics , 1989 .

[61]  R. López,et al.  Improved algorithm for the calculation of one‐electron two‐center integrals with STOs , 1989 .

[62]  B. Mamedov,et al.  Calculation of molecular electric and magnetic multipole moment integrals of integer and noninteger n Slater orbitals using overlap integrals , 2003 .

[63]  P. Knowles,et al.  An efficient internally contracted multiconfiguration–reference configuration interaction method , 1988 .

[64]  R. López,et al.  Fock potentials. I. Separation of short- and long-range electronic interactions , 1984 .

[65]  Michael P. Barnett,et al.  Molecular integrals and information processing , 2003 .

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

[67]  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 .

[68]  I. Ema,et al.  Master formulas for two‐ and three‐center one‐electron integrals involving Cartesian GTO, STO, and BTO , 2000 .

[69]  Guillermo Ramírez,et al.  Molecular integrals for Gaussian and exponential‐type functions: Shift operators , 2000 .

[70]  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.

[71]  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 .

[72]  Rogério Custodio,et al.  Exact Gaussian expansions of Slater‐type atomic orbitals , 2002, J. Comput. Chem..

[73]  A. Özmen,et al.  Computation of two-center Coulomb integrals over Slater-type orbitals using elliptical coordinates , 2003 .

[74]  I. Guseinov.  Unified analytical treatment of one‐electron multicenter integrals of central and noncentral potentials over Slater orbitals , 2002 .

[75]  V. R. Saunders,et al.  An Introduction to Molecular Integral Evaluation , 1975 .

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

[77]  J. Pople,et al.  Self‐consistent molecular orbital methods. XX. A basis set for correlated wave functions , 1980 .

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

[79]  P. Knowles,et al.  A second order multiconfiguration SCF procedure with optimum convergence , 1985 .

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

[81]  Guillermo Ramírez,et al.  Large gaussian expansions of sto's for the calculation of many-center molecular integrals with slater basis , 1987 .

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

[83]  P. Knowles,et al.  An efficient second-order MC SCF method for long configuration expansions , 1985 .