An efficient computation of parameters in the Rys quadrature formula

Abstract

We present an efficient procedure for constructing the so-called Gauss-Rys quadrature formulas and the corresponding polynomials orthogonal on (−1, 1) with respect to the even weight function w(t; x) = exp(−xt2), where x a positive parameter. Such GaussRys quadrature formulas were investigated earlier e.g. by M. Dupuis, J. Rys, H.F. King [J. Chem. Phys. 65 (1976), 111 − 116; J. Comput. Chem. 4 (1983), 154 − 157], D.W. Schwenke [Comput. Phys. Comm. 185 (2014), 762 − 763], and B.D. Shizgal [Comput. Theor. Chem. 1074 (2015), 178 − 184], and were used to evaluate electron repulsion integrals in quantum chemistry computer codes. The approach in this paper is based to a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only [(N + 1)/2] nodes and their construction. The method of modified moments is used for getting recurrence coefficients. Numerical experiments are included.