Three-centre two-electron Coulomb and hybrid integrals evaluated using nonlinear D- and (D)over-bar-transformations

Of the two-electron integrals occuring in the molecular context, the three-centre Coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy. The analytical and numerical difficulties arise mainly from the presence of the spherical Bessel function and hypergeometric series in these integrals. The present work accelerates the convergence of these integrals by first manipulating the indices of the hypergeometric function and exploiting relationships to express this function as a finite expansion and exploiting the properties of Bessel functions which satisfy second-order linear differential equations. This is a suitable form of the integrand to apply the nonlinear D (due to D Levin and A Sidi) and (D) over bar (due to A Sidi) transformations. The extensive numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals.