The Rys quadrature revisited: A novel formulation for the efficient computation of electron repulsion integrals over Gaussian functions

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Abstract
A novel formulation of the Rys quadrature algorithm for the calculation of the electron repulsion integrals over Gaussian basis functions is presented. The new algorithm is specifically designed for high contractions. As for the original Rys quadrature algorithm, the new algorithm is very efficient for high angular momentum functions. In addition it is also equally efficient for low angular momentum functions. The new algorithm takes unique advantage of (1) the numerical Rys quadrature methodology in (2) dealing with charge distributions a la McMurchie–Davidson and in (3) scaling integral blocks as a means of transferring angular momentum a la Gill–Head–Gordon–Pople. An analysis of the algorithm suggests very favorable floating-point operation counts.