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.

The two-center two-electron Coulomb and hybrid integrals arising in relativistic and nonrelativistic ab initio calculations on molecules are evaluated. Compact, arbitrarily accurate expressions are obtained. They are expressed through molecular auxiliary functions and evaluated with the numerical Global-adaptive method for arbitrary values of parameters in the noninteger Slater-type orbitals. Highly accurate benchmark values are presented for these integrals. The convergence properties of new molecular auxiliary functions are investigated. The comparison for two-center two-electron integrals is made with results obtained from single center expansions by translation of the wave function to a single center with integer principal quantum numbers and results obtained from the Cuba numerical integration algorithm, respectively. The procedures discussed in this work are capable of yielding highly accurate two-center two-electron integrals for all ranges of orbital parameters.

This work describes the fully analytical method for the calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper (Bağcı and Hoggan in Phys Rev E 91(2):023303, 2015). The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers.

The Slater-type orbital basis with non-integer principal quantum numbers is a physically and mathematically motivated choice for molecular electronic structure calculations in both non-relativistic and relativistic theory. The non-analyticity of these orbitals at r = 0, however, requires analytical relations for multi-center integrals to be derived. This is nearly insurmountable. Previous papers by present authors eliminated this difficulty. Highly accurate results can be achieved by the procedure described in these papers, which place no restrictions on quantum numbers in all ranges of orbital parameters. The purpose of this work is to investigate computational aspects of the formulae given in the previous paper. It is to present a method which helps to increase computational efficiency. In terms of the processing time, evaluation of integrals over Slater-type orbitals with non-integer principal quantum numbers are competitive with those over Slater-type orbitals with integer principal quantum numbers.Keywords Slater-type orbitals · Multi-center integrals · Auxiliary functions 1 IntroductionIn the first paper [1] of aforementioned series, history and importance of usage the auxiliary function method was summarized. Applications in molecular calculations were briefly