New translation method for STOs and its application to calculation of two-center two-electron integrals

The new translation method for Slater-type orbitals (STOs) previously tested in the case of the overlap integral is extended to the calculation of two-center two-electron molecular integrals. The method is based on the exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. Fair uniform convergence and stability under wide changes in molecular parameters are obtained for all studied two-center hybrid, Coulomb, and exchange repulsion integrals. Ten-digit accuracy in the final numerical results is achieved through multiple precision arithmetic calculation of common angular coefficients and Gaussian numerical integration of some of the analytical formulas resulting for the radial integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 91–100, 2000     

Spherically symmetrical properties of two-center overlap integrals over arbitrary atomic orbitals and translation coefficients for Slater-type orbitals

The orthogonality relations are derived for the rotation coefficients of two-center overlap integrals over arbitrary atomic orbitals (AAOs) and expansion coefficients for translation of Slater-type orbitals (STOs). Using these formulas, a very interesting theorem regarding the angular dependence is established. If we add the products of all the overlap integrals or all the translation coefficients with the same n and l values, but different m values, the result is independent of orientation. The final results are of a simple structure and are, therefore, especially useful for machine computations of multielectron multicenter molecular integrals by expanding one- and two-center electron charge density over STOs in terms of STOs about a new center.     

Computer-generated formulas for two-center coulomb integrals over slater‒ type orbitals

Formulas can be automatically generated for all two-center Coulomb integrals over Slater-type orbitals by means of the “C-matrix” single-center expansion method with use of “computer algebra.” The formula coefficients may be stored in two-dimensional arrays.     

Unified analytical treatment of one-electron two-center integrals with noninteger n Slater-type orbitals

A unified treatment of one-electron two-center integrals over noninteger n Slater-type orbitals is described. Using an appropriate prolate spheroidal coordinate system with the two atomic centers as foci, all the molecular integrals are expressed by a single analytical formula which can be readily and compactly programmed. The analysis of the numerical performance of the computational algorithm is also presented. Received: 1 April 1999 / Accepted: 2 July 1999 / Published online: 2 November 1999     

Unified treatment of two-center Coulomb,hybrid, and exchange integrals over Slater-type orbitals

Using Neumann expansion for 1/r₁₂ in elliptical coordinates a combined formula has been obtained for two-center Coulomb, hybrid, and exchange integrals with Slater-type orbitals. © 1995 John Wiley & Sons, Inc.     

Calculation of three-center electric and magnetic multipole moment integrals using translation formulas for Slater-type orbitals

 By the use of translation formulas for the expansion of Slater-type orbitals (STOs) in terms of STOs at a new origin, three-center electric and magnetic multipole moment integrals are expressed in terms of two-center multipole moment integrals for the evaluation of which closed analytical formulas are used. The convergence of the series is tested by calculating concrete cases. Computer results with an accuracy of 10⁻⁷ are obtained for 2^ν– pole electric and magnetic multipole moment integrals for 1≤ν≤5 and for arbitrary values of screening constants of atomic orbitals and internuclear distances. Received: 28 October 1999 / Accepted: 15 February 2000 / Published online: 5 June 2000     

Slater-orbital molecular integrals by numerical Fourier-transform methods. II. Four-center integrals over is orbitals

The four-center nonplanar electron repulsion integrals over 1s Slater-type atomic orbitals are considered by a numerical Fourier-transform method. It is shown that the highly oscillating integrand appearing in the Fourier inversion formula could be successfully treated by using Tchebyscheff quadrature. The resulting formulas are thoroughly discussed with particular emphasis on their numerical features and convergence properties. It follows that the aforementioned integrals may be calculated with a good accuracy with a moderate amount of computing time.     

Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals

Multicenter integrals over noninteger n Slater type orbitals with integer and noninteger values of indices u of screened Coulomb type potentials, f(u)(eta,r)=r(u-1)e(-etar), and their first and second derivatives with respect to Cartesian coordinates of the nuclei of a molecule are described. Using complete orthonormal sets of Psi(alpha) exponential type orbitals and rotation transformation of two-center overlap integrals, these integrals are expressed through the noncentral potential functions depending on the molecular auxiliary functions A(k) and B(k). The series expansion formulas derived for molecular integrals of screened Coulomb potentials and their derivatives are especially useful for the computation of multicenter electronic attraction, electric field, and electric field gradient integrals. The convergence of series is tested for arbitrary values of parameters of potentials and orbitals.     

Application of modified Neumann expansion for analytical evaluation of two-center,two-electron integrals over slater-type orbitals

A modified form of the Neumann expansion in terms of products of orthogonal polynomials for the inverse interelectronic distance r¹₁₂ is proposed. This expansion has been applied in order to derive a unified analytical formula for two-center and two-electron integrals over Slater-type orbitals. The results are equivalent to those given recently by Yasui and Saika, but the expansion itself can be used for building up a realistic algorithm for evaluation of three- and four-electron integrals determined by using correlated variational wave functions.     

Translation of STO charge distributions

Barnett and Coulson's zeta-function method (M. P. Barnett and C. A. Coulson, Philos. Trans. R. Soc., Lond. A 1951, 243, 221) is one of the main sources of algorithms for the solution of multicenter integrals with Slater-type orbitals. This method is extended here from single functions to two-center charge distributions, which are expanded at a third center in terms of spherical harmonics times analytical radial factors. For s-s distributions, the radial factors are given by a series of factors corresponding to the translation of s-type orbitals. For distributions with higher quantum numbers, they are obtained from those of the s-s distributions by recurrence. After analyzing the convergence of the series, a computational algorithm is proposed and its practical efficiency is tested in three-center (AB/CC) repulsion integrals. In cases of large basis sets, the procedure yields about 12 correct significant figures with a computational cost of a few microseconds per integral.     

STOP: A slater-type orbital package for molecular electronic structure determination

A general ab initio package using Slater-type atomic orbitals is presented. This package, called STOP, uses the one-center two-range expansion method to evaluate the multicenter electronic integrals. Thoroughly optimized numerical techniques, in particular, convergence accelerators and suitable Gauss quadratures, are used in the algorithms which provide accurate numerical values for all these integrals. STOP thus provides wavefunctions for general molecular structures at the self-consistent field level for the first time over a Slater-type orbital basis. © 1996 John Wiley & Sons, Inc.     

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.     

Comment on “Analytical evaluation for two-center nuclear attraction integrals over Slater type orbitals by using Fourier transform method” by S. ?zcan and E. ?ztekin (J. Math. Chem. DOI 10.1007/s10910-008-9398-z)

S. ?zcan and E. ?ztekin, (J. Math. Chem. doi:) published formulas for evaluating the two-center nuclear attraction integrals over Slater type orbitals. It is shown that the analytical relations for these integrals through the expansion coefficients of the electron charge density for the one-center case and the overlap integrals presented in Sect. 3 of this work can easily be derived by means of a simple algebra from the formulas published in our papers (I.I. Guseinov, J Mol Struct (Theochem) 417:117, 1997; J Math Chem 42:415, 2007 and B.A. Mamedov, Chin J Chem 22:545, 2004). It should be noted that the formulas of overlap integrals presented by E. ?ztekin et al., in previous paper (E. ?ztekin, M. Yavuz, Ş. Atalay, J Mol Struct (Theochem) 544:69, 2001) for the calculation of two-center nuclear attraction integrals also are obtained from our papers (see Comment: I.I. Guseinov, J Mol Struct (Theochem) 638:235, 2003).     

Evaluation of two-center overlap and coulomb integrals derived from slater-type orbitals

The strategy for the evaluation of two-center overlap and Coulomb integrals is illustrated by using computer-generated formulas produced from 1s orbitals. There is no loss in generality as these formulas have the same structure for higher quantum numbers. For small parameter values, singularities may be removed from the formulas by expanding exponential functions and collecting coefficients of like powers by machine. Suitable terminated expansions, along with exact formulas, permit high accuracy throughout the entire possible range of parameter values.     

Strategies for an efficient implementation of the Gauss-Bessel quadrature for the evaluation of multicenter integral over STFs

In a previous work, a new Gauss quadrature was introduced with a view to evaluate multicenter integrals over Slater-type functions efficiently. The complexity analysis of the new approach, carried out using the three-center nuclear integral as a case study, has shown that for low-order polynomials its efficiency is comparable to the SD. The latter was developed in connection with multi-center integrals evaluated by means of the Fourier transform of B functions. In this work we investigate the numerical properties of the Gauss-Bessel quadrature and devise strategies for an efficient implementation of the numerical algorithms for the evaluation of multi-center integrals in the framework of the Gaussian transform/Gauss-Bessel approach. The success of these strategies are essential to elaborate a fast and reliable algorithm for the evaluation of multi-center integrals over STFs.     

Poisson equation for molecular exchange,hybrid and coulomb electron repulsion integrals

The various multicenter exchange, hybrid and Coulomb electron repulsion integrals that occur in molecular quantum mechanics are shown to satisfy a Poisson equation in which an overlap integral plays the role of a source distribution function. Two-, three-and four-center exchange integrals arise from four-center source functions; two- and three-center hybrid integrals arise from three-center distributions; and one- and two-center Coulomb integrals have two-center sources.     

Beyond the MNDO model: Methodical considerations and numerical results

It is suggested to improve the MNDO model by the explicit inclusion of valence-shell orthogonalization corrections, penetration integrals, and effective core potentials (ECPs) in the one-center part of the core Hamiltonian matrix. Guided by analytic formulas and numerical ab initio results, the orthogonalization corrections are expressed in terms of the resonance integrals that are represented by a new empirical parametric function. All two-center Coulomb interactions and ECP integrals are evaluated analytically in a Gaussian basis followed by a uniform Klopman–Ohno scaling. One particular implementation of the proposed NDDO SCF approach is described and parameterized for the elements H, C, N, O, and F. In a statistical evaluation of ground-state properties, this implementation shows slight but consistent improvements over MNDO, AM1, and PM3. Significant improvements are found for excited states, transition states, and strong hydrogen bonds. Possible further enhancements of the current implementation are discussed. © 1993 John Wiley & Sons, Inc.     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

Molecular one-electron integrals over slater-type atomic orbitals and irregular solid spherical harmonics

The one-electron integral over Slater-type atomic orbitals centered at A and B and irregular solid spherical harmonics as operator centered at C is evaluated analytically by using elliptical coordinates and translation of the solid spherical harmonics from center C to either focus A or B. A special case is the three-center nuclear attraction integral which is also evaluated by means of the Neumann expansion and expressed without associate Legendre functions of the second kind. The strict observation of the charge distribution concept leads to compact expressions for the integral. It has the further advantage that the charge density distributions which have been developed for the two-center cases and used in calculations for diatomic molecules can be utilized.     

On bound states in two-body systems

The application of the Σ-separation method to the calculation of multicenter two-electron molecular integrals with Slater-type basis functions is reported. The approach is based on the approximation of a scalar component of the two-center atomic density by a two-center expansion over Slater-type functions. A least-squares fit was used to determine the coefficients of the expansion. The angular multipliers of the atomic density were treated exactly. It is shown that this approach can serve as a sufficiently accurate and fast algorithm for the calculation of multicenter two-electron molecular integrals with Slater-type basis functions. © 1995 John Wiley & Sons, Inc.