Rys polynomials

In the work of King, Dupuis, and Rys [15,16], the mabix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys polynomials. The potential problem of this method is that to obtain the mabix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to... 

Next, we shall consider the integral basically corresponding to the Rys polynomial problem [15,16]. Letting... 

In the Rys polynomial method, the first step is to use the Gaussian transform of the inverse electronic distance to separate the electron repulsion integral to, x, y and z components (Cook, 1974) using the Gaussian transform... 

RCS version control, 186, 248, 256, 306, 308-309 details, 310-313 repeat statement, 118 repulsion integral processing, 142-147 response matrix, 637 Restricted Hartree-Fock, see RHF Restricted open-shell Hartree-Fock, see ROHF reverse dictionary order, 275, 449 (R)GUHF, see GUHF RHF, 314 Ring, P., 630 ROHF, 407 RPA, 621-628 history of name, 629 Rumer, G., 542n Rys polynomials, 228n, 233... 

H. F. King and M. Dupuis, J. Comput. Phys., 21,144 (1976). Numerical Integration Using Rys Polynomials. M. Dupuis, J. Rys, and H. F. King, J. Chem. Phys., 65, 111 (1976). Evaluation of Molecular Integrals over Gaussian Basis Functions. 

H. B. Schlegel, J. S. Binkley, and J. A. Pople, J. Chem. Phys., 80,1976 (1984). First and Second Derivatives of Two-Electron Integrals over Cartesian Gaussians Using Rys Polynomials. 

In contrast to the incomplete Gamma functions, Fm(T), where the computation is not difficult the evaluation of the roots and weights of the Rys polynomials are more complex. Firstly, there is little work published on the matter the only reports are due to King and Dupuis, and Ishida. Secondly, the tables are not able to be generated as straightforwardly as those for the incomplete Gamma functions. However, parallels can still be made with the procedures presented in the previous subsection. 

Since only the even members of the J-type Rys polynomials are needed, we also introduce the R-type Rys potynomUUs of degree 2n [17]... 

The weight function of the Rys polynomials / (x) and depends on the real parameter... 

O > 0. These polynomials consequently constitute a manifold of polynomials, one for each a. In Figure 9.10, we have plotted the first four Rys polynomials for several values of the weight parameter a. As a increases, the Rys roots shift towards the wigin. At a = 0, the weight fiinctim becomes equal to unity and the Rys polynomials turn into the scaled Legendre polynomials (9.6.2)... 

On the other hand, in the limit as a tends to infinity, we may in (9.11.16) integrate over the fiiU set of real numbers and the Rys polynomials may then be related to the Hermite functions as... 

From these expressions, we conclude that the roots of the Rys polynomials coincide with the roots of the Legendre polynomials for = 0 and with the scaled roots of the Hermite polynomials for large a ... 

Fig. 9.11. The roots and weights of the Rys polynomials K (x). On the left, the roots of f (x) have been plotted as functions or the dotted lines represent the five positive roots of 7/ o(V ) the dots at a = 0 the five positive roots of Ltoix). On the right, the weights of R ix) have been plotted as functions a the dotted lines represent the corresponding five weights of // o(v ) and the dots at a = 0 the five weights of LioW.

These relationships are illustrated in Figure 9.11, where, in the left-hand plot, we have plotted the five roots of the Rys polynomials Rf (jc) as a function of a, superimposed on the positive roots of the Hermite polynomial Hioiy/ax) multiplied hy a (dotted lines) and with the positive toots of the Legendre polynomial Lio( c) indicated with black dots at a = 0. 

Having introduced and discussed the Rys polynomials, we are now in a position to consider the evaluation of Hermite Coulomb integrals by Gaussian quadrature. In this subsection, we consider the one-electron Hermite Coulomb integrals (9.9.9)... 

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