Calculation of two-center Coulomb integrals over STOs in molecular coordinate system using I and A auxiliary functions

In this study, two-center molecular Coulomb integrals over Slater-type atomic orbitals, which are necessary for calculating the electronic structure of molecules whose atoms are not aligned on the same line, were calculated using the I and Q auxiliary functions obtained in the literature using the elliptic coordinate method. These calculations were performed on a personal computer using Fortran programming language with quadratic precision. To obtain analytical expressions to be used in computer programs of these integrals , the STO products on the right and left sides of the Coulomb operator within the integral were linearized using the coupling relations of complex spherical harmonics. To calculate the Gaunt coefficients resulting from this linearization, the analytical expressions of Guseinov et al., which can be calculated separately for each set of quantum numbers, were used. In this case, the resulting two-center Coulomb integrals in the non-lined-up coordinate system are expressed in terms of two-center Coulomb integrals in the lined-up coordinate system, using the rotation coefficients of Guseinov et al. The elliptical coordinate method is used to obtain analytical solutions of these Coulomb integrals. In order to calculate the analytical expression obtained in this way with a computer program, it is sufficient to know only the quantum numbers of the STOs, the screening parameters and the spherical coordinates of the nuclei of the molecule atoms. Using this analytical expression, the results of two-center Coulomb integrals over STOs in both the lined-up and non-lined-up coordinate systems for a wide range of quantum number sets, shielding constants, and different nuclear positions are tabulated. As can be seen from these tables, the Coulomb integral results in the lined-up coordinate system obtained by Bağcı and Hoggan, who performed quadrative precision calculations, are in full agreement with the results obtained in our study, which also performed quadrative precision calculations. Among other results, Jones's results are in full agreement with Bagci and Hogga's results, but Jones's results are low sensitivity results. The sensitivity of the studies led by Guseinov is also quite low, and even with this sensitivity, the results do not fully agree with the literature. For the two-center Coulomb integrals in non-lined-up coordinate systems, which is the main subject of our thesis, only the results obtained in the studies led by Guseinov are available in the literature. In these studies, Guseinov et al. used the method of moving STOs from one center to another to calculate integrals in the lined-up coordinate system. Comparing the results obtained in Guseinov's studies for two-center Coulomb integrals in a non-lined-up coordinate system with those in our study, we obtain agreement between one percent and one hundred trillionth of a percent. A precision of one percent is quite low. However, since there are no other results to compare in the literature, it cannot be said at first glance which study's results have the sensitivity in question. However, since the sensitivity of the results obtained in the calculations of two-center Coulomb integrals over STOs in the lined-up coordinate system pioneered by Guseinov is quite low compared to the literature, and the results obtained in our study are in full agreement with the literature, it is thought that the results obtained in our study are much more sensitive for the two-center Coulomb integrals in the non-lined-up coordinate system. Keywords: Coulomb integrals, electron-repulsion integrals, molecular integrals, Slater-type orbitals