Numerical evaluation of three-center two-electron Coulomb and hybrid integrals over B functions using the HD and H(D)over-bar methods and convergenceproperties

The difficulties of the numerical evaluation of three-center two-electron C oulomb and hybrid integrals over B functions, arise mainly from the presenc e of the hypergeometric series and semi-infinite very oscillatory integrals in their analytical expressions, which are obtained using the Fourier tran sform method. This work presents a general approach for accelerating the convergence of t hese integrals by first demonstrating that the hypergeometric function, inv olved in the analytical expressions of the integrals of interest, can be ex pressed as a finite sum and by applying nonlinear transformations for accel erating the convergence of the semi-infinite oscillatory integrals after re ducing the order of the differential equation satisfied by the integrand. The convergence properties of the new approach are analysed to show that fr om the numerical point of view the H (D) over bar method corresponds to the most ideal situation. The numerical results section illustrates the accuracy and unprecedented ef ficiency of evaluation of these integrals.