Rys quadrature

R. Lindh, U. Ryu, and B. Liu, /. Chem. Phys., 95, 5889 (1991). The Reduced Multiplication Scheme of the Rys Quadrature and New Recurrence Relations for Auxiliary Function Based Two-Electron Integral Evaluation. 

Another leading technique for integral evaluation is that of McMurchie and Davidson (1978). According to Saunders (1985), the ultimate efficiency of this method is higher than that of the Rys quadrature method. It has not become as popular as the latter, perhaps because of its slightly more complex logic. Saunders (1983) recommends the combination of the two techniques this method was used in the evaluation of third derivative integrals by Gaw et al. (1984). 

The Rys quadrature method25 is based on the observation that the right-hand side integral in Eq. [11] can be simplified by introducing a new variable t. 

This expression is in the appropriate form for Gauss-Rys quadrature (9.11.19) and may thus be evaluated exactly as... 

To complete the evaluation of the Hermite integrals by Rys quadrature, we must derive a scheme for the evaluation of the modified Hermite polynomials. 

To calculate the one-electron Coulomb integrals by Rys quadrature, we write the integrals in the standard McMurchie-Davidson form (9.9.32)... 

In the Rys-quadrature scheme presented above, the one- and two-dimensional integrals were calculated using the McMurchie-Davidson scheme. These integrals may also be obtained from the Obara-Saika scheme, as we shall now discuss. 

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