Strategies for evaluation of rys roots and weights

The Rys quadrature method for evaluating molecular integrals requires accurate numerical values of the nodes of a Rys polynomial and associated weight factors. The numerical value of a Rys polynomial for a specified value of its argument can be evaluated by three-term recursion using α and β coefficients. We review existing integration schemes for computing these recurrence parameters, discuss issues related to computational efficiency and numerical precision, and propose a slightly new integration method using Gauss−Rys quadrature. We discuss the advantages and disadvantages of using Golub’s matrix method for the computation of roots and weights.