On the use of shifted Jacobi polynomials in accurate evaluation of roots and weights of Rys polynomials

In this paper it is shown that shifted Jacobi polynomials G(n)(p, q, x) can be used in connection with the Gaussian quadrature modified moment technique to greatly enhance the accuracy of evaluation of Rys roots and weights used in Gaussian integral evaluation in quantum chemistry. A general four-term inhomogeneous recurrence relation is derived for the shifted Jacobi polynomial modified moments over the Rys weight function e(-Tx)/root x. It is shown that for q=1/2 this general four-term inhomogeneous recurrence relation reduces to a three-term p-dependent inhomogeneous recurrence relation. Adjusting p to proper values depending on the Rys exponential parameter T, the method is capable of delivering highly accurate results for large number of roots and weights in the most difficult to treat intermediate T range. Examples are shown, and detailed formulas together with practical suggestions for their efficient implementation are also provided. (C) 2009 American Institute of Physics. [DOI: 10.1063/1.3204437]