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.