Numerical integration using rys polynomials

We define and discuss the properties of manifolds of polynomials J_n(t, x) and R_n(t, x), called Rys polynomials, which are orthonormal with respect to the weighting factor exp(-xt²) on a finite interval of t. Numerical quadrature based on Rys polynomials provides an alternative approach to the computation of integrals commonly encountered in molecular quantum mechanics. This gives rise to a curve fitting problem for the roots and quadrature weights as a function of the x parameter. We have used Chebyshev approximation for small x and an asymptotic expansion for large x. A modified Christoffel-Darboux equation applicable to Rys polynomials is derived and used to obtain alternative formulas for Rys quadrature weight factors.