Statistical computer-aided calculation of molecular integrals

An improved Monte Carlo procedure is described for the evaluation of molecular integrals, which is particularly suitable for multicenter and/or two-electron calculations. The method is almost independent of the complexity of the atomic orbitals involved, and the results can be obtained with an uncertainty which is fairly adequate for most applications and with very moderate waste of computer resources. In the version presented here there is only a restriction concerned with the positive value of some functions involved, as described in the text, but this possibility does not arise in much of the practical work. An example with a two-dimensional exchange integral is worked out in detail.     

Recurrence formulas for molecular integrals over Laguerre Gaussian-type functions

Recurrence formulas for overlap, nuclear attraction, and electron-repulsion integrals over Laguerre Gaussian-type functions are presented. They have been derived using compact recurrence relations for homogeneous solid spherical harmonic operators but are rather lengthy as compared to those over Cartesian Gaussian-type functions. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 273–279, 1998     

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.     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals I. Single‐center expansion method

The multicenter charge‐density expansion coefficients [I. I. Guseinov, J Mol Struct (Theochem) 417 , 117 (1997)] appearing in the molecular integrals with an arbitrary multielectron operator were calculated for extremely large quantum numbers of Slater‐type orbitals (STOs). As an example, using computer programs written for these coefficients, with the help of single‐center expansion method, some of two‐electron two‐center Coulomb and four‐center exchange electron repulsion integrals of Hartree–Fock–Roothaan (HFR) equations for molecules were also calculated. Accuracy of the results is quite high for the quantum numbers, screening constants, and location of STOs. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 146–152, 2000     

Evaluation of one-electron integrals for arbitrary operators V(r) over Cartesian Gaussians: Application to inverse-square distance and Yukawa operators

A general procedure is presented for generating one-electron integrals over any arbitrary potential operator that is a function of radial distance only. The procedure outlines that for a nucleus centered at point C integrals over Cartesian Gaussians can be written as linear combinations of 1-D integrals. These Cartesian Gaussian functions are expressed in a compact form involving easily computed auxiliary functions. It is well known that integrals over the Coulomb operator can be expressed in terms of F_n(T) integrals, where By means of a substitution for F_n(T) by other simple functions, algorithms that form integrals over an arbitrary function can be generated. Formation of such integrals is accomplished with minor editing of existing code based on the McMurchie–Davidson formalism. Further, the method is applied using the inverse-square distance and Yukawa potential operators V(r) over Cartesian Gaussian functions. Thus, the proposed methodology covers a large class of one-electron integrals necessary for theoretical studies of molecular systems by ab initio calculations. Finally, by virtue of the procedure's recursive nature it provides us with an efficient scheme of computing the proposed class of one-electron integrals. © 1993 John Wiley & Sons, Inc.     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. II. Two‐center expansion method

Using expansion formulas for the charge‐density over Slater‐type orbitals (STOs) obtained by the one of authors [I. I. Guseinov, J Mol Struct (Theochem) 1997, 417, 117] the multicenter molecular integrals with an arbitrary multielectron operator are expressed in terms of the overlap integrals with the same screening parameters of STOs and the basic multielectron two‐center Coulomb or hybrid integrals with the same operator. In the special case of two‐electron electron‐repulsion operator appearing in the Hartree–Fock–Roothaan (HFR) equations for molecules the new auxiliary functions are introduced by means of which basic two‐center Coulomb and hybrid integrals are expressed. Using recurrence relations for auxiliary functions the multicenter electron‐repulsion integrals are calculated for extremely large quantum numbers. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 117–125, 2001     

Recursion formulae for calculation of overlap integrals

Multi-ζ Slater-type orbitals are frequently used in molecular orbital calculations. Master formulae and numerical tables are available in literature for overlap integrals between s, p, and d atomic orbitals up to principal quantum number (n) = 3 and for some other selected quantum numbers. However, no master formula or numerical table is available for quantum numbers n = 5 and above and involving ? orbitals. In this article recursion formulae have been presented for the calculation of the overlap integral between any two s, p, d, and ? atomic orbitals formed by a linear combination of Slater-type orbitals. These formulae, when expanded, would give rise to all the master formulae reported in the literature as well as formulae hitherto unreported.     

Optimal use of the recurrence relations for the evaluation of molecular integrals over Cartesian Gaussian basis functions

We consider the tree search problem for the recurrence relation that appears in the evaluation of molecular integrals over Cartesian Gaussian basis functions. A systematic way of performing tree search is shown. By applying the result of tree searching to the LRL2 method of Lindh, Ryu, and Liu (LRL) (J. Chem. Phys., 95 , 5889 1991), which is an auxiliary function-based method, we obtain significant reductions of the floating point operations (FLOPS) counts in the K⁴ region. The resulting FLOPS counts in the K⁴ region are comparable up to [dd|dd] angular momentum cases to the LRL1 method of LRL, currently the method requiring least FLOPS for [dd|dd] and higher angular momentum basis functions. For [ff|ff], [gg|gg], [hh|hh], and [ii|ii] cases, the required FLOPS are 24, 40, 51, and 59%, respectively, less than the LRL1 method in the K⁴ region. These are the best FLOPS counts available in the literature for high angular momentum cases. Also, there will be no overhead in either the K² or K⁰ region in implementing the present scheme. This should lead to more efficient codes of integral evaluations for higher angular momentum cases than any other existing codes. © 1993 John Wiley & Sons, Inc.     

Modified rotation transformation method for the calculation of repulsion integrals

A modified rotation transformation method for two electron repulsion integrals is introduced in this paper. The percentage of nonzero matrix needed to perform the calculation is less than 30 in this method. This revised version was published online in July 2006 with corrections to the Cover Date.     

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     

Modified rotation transformation method for the calculation of repulsion integrals

A modified rotation transformation method for two electron repulsion integrals is introduced in this paper. The percentage of nonzero matrix needed to perform the calculation is less than 30 in this method.     

Gaussian expansion method for molecular integrals of molecular properties

The finite Gaussian Expansion method for molecular integrals proposed by Taketa, O-ohata and Huzinaga has been extended to the integrals of molecular properties. The integral formulas of so-called moment, field and field gradient integrals have been derived. It has been numerically shown that in order to evaluate the field and the field gradient integrals based on Slater type orbitals, eight- or ten-term Gaussian expansions are sufficient but this method fails to attain sufficient effective numbers for the moment integrals.