Calculation of overlap integrals over Slater-type orbitals using translational and rotational transformations for spherical harmonics

Using translation and rotation formulas for spherical harmonics the finite sums through the basic overlap integrals and spherical harmonics are derived for the arbitrary overlap integrals over Slater-type orbitals (STOs). The recurrence relations for the evaluation of basic overlap integrals have been established recently [Guseinov II, Mamedov BA (1999) J Mol Struct (THEOCHEM) 465:1]. By the use of the derived expressions the overlap integrals can be calculated most efficiently and accurately, especially for large quantum numbers of STOs. Received: 2 May 2000 / Accepted: 31 May 2000 / Published online: 11 September 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.     

An expansion for four-center integrals over Slater-type orbitals

A new expression is given for the electron repulsion integral over Slater-type orbitals on four different centers. It is based on the asymptotic expansion derived from the bipolar expansion of a previous paper. The expression has the form where q_p = {n_p, l_p, m_p}. Both F and σ are closed expressions. The quantity F is a combination of incomplete gamma functions, Laguerre polynomials and spherical harmonics. It depends upon the relative coordinates of a point P on the AB axis and a point Q on the CD axis. The functions σ_nlm(A, B) depend on the charge distribution (χ_Aχ_B); they have the character of overlap integrals and are of the form      

Exact formulas for overlap integrals of Slater-type orbitals with equal screening constants

Exact formulas for 147 overlap integrals between Slater-type orbitals with equal screening constants are presented in the most simplified form. This represents all combinations of orbitals with quantum numbers: 1 ≤ N ≤ 5, 0 ≤ L ≤ 3, and M ≤ L. The formulas are automatically generated by computer using the “C-matrix” single-center expansion method. There are no limitations to the applicability of this method to orbitals of higher quantum numbers.     

Fast and stable algorithm for the analytical computation of two-center Coulomb and overlap integrals over Slater-type orbitals

Proceeding from analytical expressions for two-center kernel functions that we derived recently, we present new analytical formulas for the two-center Coulomb and overlap integrals over Slater-type orbitals. These formulas are of an exceptionally simple analytical structure and high numerical efficiency. An especially important point is that for the most frequently needed ranges of discrete quantum numbers, the formulas are completely stable in the cases of nearly equal scaling parameters or vanishing interatomic distances, except for one particular case of the Coulomb integral. No special asymptotic formulas are needed any more to compute the two-center integrals over Slater-type orbitals in these case. Furthermore, a largely recursive formulation makes the integral evaluation very economical and fast. In particular, we assess the numerical performance of a new kind of angular momentum recurrences that we have proposed in a previous article [W. Hierse and P.M. Oppeneer, J. Chem. Phys. 99 , 1278 (1993)]. © 1994 John Wiley & Sons, Inc.     

On the evaluation of overlap integrals with the same screening parameters of Slater-type orbitals using binomial coefficients

A general formula has been established for the overlap integrals with the same screening parameters of Slater-type orbitals in terms of bionomial coefficients. The final results are especially useful for the calculation of these integrals for large quantum numbers, which occur in the multicenter integrals. © 1996 John Wiley & Sons, Inc.     

Atomic three-electron integrals for correlated Slater-type s orbitals

A compact series expansion method is described for evaluation atomic three-electron integrals which involve odd powers of the three interelectronic distances and Slater-type s orbitals. Only one dimensional integrals appear in the final expression, and these are readily amenable to machine computation. Convergence of the series is discussed.     

Analytical evaluation of molecular electric and magnetic multipole moment integrals over Slater-type orbitals

The analytical expressions are derived for the magnetic multipole moment integrals in terms of electric multipole moment integrals for which the closed formulas through the overlap integrals are obtained. By the use of the derived expressions in terms of overlap integrals, the electric and magnetic multipole moment integrals, the electric and magnetic properties of molecules can be evaluated most efficiently and accurately. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 145–150, 1998     

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.     

Unified treatment of complex and real rotation-angular functions for two-center overlap integrals over arbitrary atomic orbitals

The new combined formulas have been established for the complex and real rotation-angular functions arising in the evaluation of two-center overlap integrals over arbitrary atomic orbitals in molecular coordinate system. These formulas can be useful in the study of different quantum mechanical problems in both the theory and practice of calculations dealing with atoms, molecules, nuclei and solids when the integer and noninteger n complex and real atomic orbitals basis sets are emploed. This work presented the development of our previous paper (I.I. Guseinov in Phys Rev A 32:1864, 1985).     

Evaluation of the auxiliary function $${G_{-ns}^q}$$ using basic overlap integrals over Slater type orbitals

This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs) for the evaluation of auxiliary function which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function is completely general for p _a ≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function is compared with other method.     

Evaluation of multicenter integrals over Slater-type atomic orbitals by expansion in terms of complete sets

Multicenter integrals over Slater-type orbitals can be expanded in series with known coefficients if they are considered as elements of appropriate Hilbert spaces. Some test calculations for the three-center nuclear attraction integrals are given.     

The scaling approach to multicenter molecular integrals with Slater-type orbitals

A scaling approach to multicenter molecular integrals with Slater-type orbitals (STOs) is presented. The result is significant in that it shows (1) the existence of a simple relationship between multicenter integrals and (2) an implied computational savings. Operation count estimates indicate that the significant savings would occur for a system having large numbers of STOs on each atom.     

Evaluation of two-center,two- and three-electron integrals involving correlation factors over Slater-type orbitals. I. Basic integrals

The modified and extended version of the Neumann expansion of the interrelectronic distance function r for u = ?1, 0, 1, 2, using the set of orthogonal polynomials normalized to unity, is presented. This expansion has been utilized to obtain analytical expressions for evaluating two-center two- and three-electron integrals in the Slater orbital basis occurring if variational correlated functions are used.     

Formulas and numerical table for the radial part of overlap integrals with the same screening parameters of Slater-type orbitals

 The numerical properties of the radial part of overlap integrals with the same screening parameters in the form of polynomials in p = ξR over Slater-type orbitals have been studied and obtained by using three different methods. For that purpose, the characteristics of auxiliary functions were used first, then Fourier transform convolution theorem, and recurrence relations for the basic coefficients of A ^s _{n l λ, n ′ l ′λ} were used. The calculations of the radial part of overlap integrals with the same screening parameters were made in the range 1 ≤ n ≤ 75, 1 ≤ n′ ≤ 75, and 10⁻⁶ ≤ p. Received: 18 January 2001 / Accepted: 5 April 2001 / Published online: 27 June 2001     

Evaluation of two-center overlap integrals using slater type orbitals in terms of bessel type orbitals

Using the definition of STOs in terms of BTOs, we have presented analytical formula for two-center overlap integrals. The obtained formula contains generalized binomial coefficients and Mulliken integrals A_k and B_k. Taking into account the recent advances on the efficient calculation of Mulliken integrals (Harris, Int. J. Quantum Chem., 100 (2004) 142), we have obtained many more satisfactory results for two-center overlap integrals, for arbitrary quantum numbers, scaling parameters, and location of atomic orbitals.PACS No: 31.15.+qAMS Subject Classification: 81V55, 81–08