Rys quadrature method

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 author s connection to the area covers the implementation of the Rys-Gauss quadrature evaluation of ERIs. However, this article is not intentionally biased towards any method. It is anticipated that after reading this article, readers will be able to form opinions on which method to use in a particular circumstance. 

The Rys-Gauss quadrature developed by Dupuis, Rys, and King in the late 1970s is a method which at first glance seems to have little in common with the other integral schemes. " However, as can be demonstrated the formulae derived for the Rys-Gauss quadrature are connected to the formulae of the incomplete Gamma function based schemes. 

The computation of the incomplete Gamma functions is a vital part of evaluating of the ERIs for all integral methods except the Rys-Gauss quadrature. In the evaluation of Fm(T ) two formulae can be used depending on the value of the argument, T. Firstly, there is the asymptotic formula in which the Laplace formula (see equation 13) is utilized. 

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