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

Computing two-center overlap integrals arising in Hartree-Fock-Roothaan equations is considered by using the numerical Global-adaptive method. These integrals are expressed through auxiliary functions in ellipsoidal coordinates. They involve Slater-type basis sets with noninteger principal quantum numbers. A computationally simple, efficient, and reliable program procedure is presented. Comparison is made with the results of numerical three-dimensional adaptive integration procedure presented by Ramanowski, with methods used for analytical solution via auxiliary functions and series expansions by translation to a single center. Highly accurate results can be achieved for overlap integrals by numerical approximations both for integer and noninteger principal quantum numbers also, these extended calculations are efficient with no restriction and over a wide range of orbital parameters.