Efficient and Rapid Numerical Evaluation of the Two-Electron, Four-Center Coulomb Integrals Using Nonlinear Transformations and Useful Properties of Sine and Bessel Functions

Two-electron, four-center Coulomb integrals are undoubtedly the most difficult type involved in ab initio and density functional theory molecular structure calculations. Millions of such integrals are required for molecules of interest; therefore rapidity is the primordial criterion when the precision has been reached. This work presents an extremely efficient approach for improving convergence of semi-infinite very oscillatory integrals, based on the nonlinear D̄-transformation and some useful properties of spherical Bessel, reduced Bessel, and sine functions. The new method is now shown to be applicable to evaluating the two-electron, four-center Coulomb integrals over B functions. The section with numerical results illustrates the unprecedented efficiency of the new approach in evaluating the integrals of interest.