Calculation of the one-electron two-center integrals over Slater-type orbitals by means of the ellipsoidal coordinates method

Abstract

A method for the calculation of one-electron two-center integrals is described. Using an ellipsoidal coordinate system, both the overlap, kinetic energy, and nuclear attraction integrals are expressed in terms of the so-called sigma function w introduced by Baba-Ahmed et al. A. Baba-Ahmed and J. Gayoso, Int. J. Quant. Chem. 23, Ž .Ž .x 71 1983 , Eq. 51 and which is easily programmed for a computer. The present investigation offers an advantage in that general formulas are derived which facilitate computation of the overlap and related two-center integrals over all Slater-type orbitals Ž . STOs with eventual nonintegral values of the principal quantum number. The proposed w algorithm permits to avoid the procedure of interpolation A. Baba-Ahmed and J.Ž .x Gayoso, Theor. Chim. Acta, 62, 507 1983 , Eqs. 11᎐14 used to overcome the difficulty introduced by the presence of nonintegral quantum numbers. Finally, numerical aspects of the proposed algorithm are analyzed and comparisons with other approaches are given.

Performance of numerical approximation on the calculation of overlap integrals with noninteger Slater-type orbitals

“…Particularly, the overlap integrals constitute the basic building block of more complicated multicenter integrals. Besides, these integrals arise in Hartree-Fock-Roothaan equations (HFR) both for ab initio and semiempirical methods [4]. Computing such integrals, defined in a nonaligned molecular coordinate system by S nlm,n l m ( p,τ ) = * nlm (ζ,r a ) n l m (ζ ,r b )dV ,…”

“…The NSTOs situation is much more complicated. Outstanding work on obtaining analytical relations for the two-center overlap integrals over NSTOs has been made possible after [13,14] in [4,7,21,22] via the ellipsoidal coordinate system using auxiliary functions and the single-center expansion procedure [23], respectively. It should be noted that the auxiliary functions used in analytical derivation of overlap integrals over NSTOs need to be convergent and numerically stable.…”

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

“…[36], p. 847) proves that the term (see Eq. 11a) depending on p can be bounded above, in absolute value, by a series of the Riemann type (i.e., C x p x1 with x>0), which is obviously convergent.…”

“…Some results on atomic and molecular calculations using STOs with noninteger n can be found in the literature [23±35]. Some recent results on overlap integrals over NISTOs are also available [36]. In the last few years, extensive work due to Koga and coworkers has proven that the extension of the STO principal quantum number from integer to noninteger values considerably improves the quality of the single-zeta [37±39] and the double-zeta [40] wave functions.…”

“…In a previous paper [36], we presented an ecient algorithm for the calculation of some one-electron twocenter integrals over STOs with possible noninteger values of the principal``quantum numbers''. Using a prolate spheroidal coordinate system, explicit formulas for overlap, kinetic energy and nuclear attraction integrals were developed in terms of the so-called``sigma'' function, which depends on the quantum numbers and scaling parameters and on the internuclear distance.…”