A novel Rys quadrature algorithm for use in the calculation of electron repulsion integrals

Abstract The half-range Rys polynomials, R n ( x ) , are orthonormal with respect to weight function w ( x ) = e - cx 2 on the interval x ∈ [ 0 , 1 ] and defined with the set of coefficients, α n and β n , in the three term recurrence relation for the polynomials. Full range Rys polynomials, J n ( x ) , are orthonormal with respect to w ( x ) = e - cx 2 on the interval x ∈ [ - 1 , 1 ] . They are defined with the set of β n recurrence coefficients as α n = 0 . The Gauss-Rys quadrature defined with the Rys polynomials are used to evaluate electron repulsion integrals in quantum chemistry computer codes. The present paper proposes a new algorithm for the efficient computation of the Rys quadrature weights and points versus the parameter c in the weight function. The method is based on the full range Rys polynomials and a novel method for the calculation of the positive quadrature points and related weights.