An efficient computation of parameters in the RYS quadrature formula

Gradimir V. Milovanović

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 43 (2018) no. 1.

Résumé

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 $w(t;x)=\exp(-x t^2)$, where $x$ a positive parameter. Such Gauss-Rys quadrature formulas were investigated earlier e.g. by M. Dupuis, J. Rys, H.F. King $[$J. Chem. Phys. {\bf65} $(1976)$, $111-116$; J. Comput. Chem. {\bf4} $(1983)$, $154-157$$]$, D.W. Schwenke $[$Comput. Phys. Comm. {\bf185} $(2014)$, $762-763$$]$, and B.D. Shizgal $[$Comput. Theor. Chem.\ {\bf 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.