McMurchie-Davidson method integrals

As demonstrated by the Pople-Hehre method it is possible to achieve considerable reduction in the computational expense of contracted ERIs if large parts of the integral manipulation are performed after the contraction step. The McMurchie-Davidson and the Obara-Saika methods utilization of the transfer equation (17) to minimize the operation count has been shown. This idea can, however, be employed to the extent that all manipulations are performed on fully or partially contracted integrals. Recently a number of methods have been presented along those lines.The method of Gill and Pople will be used as an example of the approach because it is currently one of the most commonly used integral methods. Note the concept of early contraction, however, applies to any of the methods presented in the chapter. 

The PRISM method modifies the McMurchie-Davidson recurrence relations to process contracted rather than primitive integrals. The first step in this procedure is to introduce simultaneous contraction and scaling. Examples of this are... 

The different scaling behaviour of the classical and nonclassical two-electron integrals has important ramifications. Thus, whereas the nonclassical integrals must be evaluated by the standard techniques such the McMurchie-Davidson, Obara-Saika and Rys schemes, the classical integrals may be evaluated, to an accuracy of 10 , more simply by the multipole method developed in Section 9.13. Moreover, to calculate the total Coulomb contribution to the Fock operator or to the energy in large systems, there is no need to evaluate the individual integrals explicitly. Rather, as discussed in Section 9.14, their contribution may be calculated much more efficiently by the fast multipole method, at a cost that scales only linearly with the size of the system. 

Another leading technique for integral evaluation is that of McMurchie and Davidson (1978). According to Saunders (1985), the ultimate efficiency of this method is higher than that of the Rys quadrature method. It has not become as popular as the latter, perhaps because of its slightly more complex logic. Saunders (1983) recommends the combination of the two techniques this method was used in the evaluation of third derivative integrals by Gaw et al. (1984). 

The necessary integrals over cartesian Gaussian functions have been evaluated by Schwerdtfeger and Silberbach [213] using a mixed Gauss-Laplace transform and more recently according to the method of McMurchie and Davidson [155] by Smit [214]. An alternative way to account for core-polarization in ECP calculations has been devised by Jeung et al. [215-219], who based their method on second-order perturbation theory. 

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