McMurchie-Davidson method

N.2 Computational speedup for the direct and reciprocal sums Computational speedups can be obtained for both the direct and reciprocal contributions. In the direct space sum, the issue is the efficient evaluation of the erfc function. One method proposed by Sagui et al. [64] relies on the McMurchie-Davidson [57] recursion to calculate the required erfc and higher derivatives for the multipoles. This same approach has been used by the authors for GEM [15]. This approach has been shown to be applicable not only for the Coulomb operator but to other types of operators such as overlap [15, 62],... 

Alternatively, the McMurchie-Davidson scheme [133] can be used for the computation of the spatial part of the matrix elements of Hpy (see [106]). The central concept of this method is to expand the product of two Gaussians (the so-called overlap distribution) in terms of Hermite functions according to... 

Since the ABSs include Hermites with / > 0, the direct space contributions can be efficiently evaluated by using the McMurchie-Davidson (MD) recursion (McMurchie and Davidson, 1978). This recursion has been used to calculate the required erfc and higher derivatives for multipole interactions (Sagui et al., 2004). This approach was also employed for the Hermite Gaussians (Cisneros et al., 2006b), where it was shown that the MD recursion is applicable to other types of operators besides 1/r. For the reciprocal sums three methods were implemented full Ewald (Ewald, 1921), sPME (Essmann et al., 1995) and FFP (York and Yang, 1994). The latter two methods rely on the use of fast Fourier transforms to approximate the structure factors that arise in the reciprocal term, which results in the efficient evaluation of this term and has been shown to scale as 0[N log N) for sPME (Essmann et al., 1995). 

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. 

On the theoretical hand, calculations have been performed as soon as in the 50ies [56,63] since formaldehyde represents the smallest member of the carbonyl series. References to early works are avalaible in the compilation by Davidson and McMurchie [64] and in references [56-58,63]. Of particular interest for a comprehensive assignment of the experimental transitions are the very fine and accurate calculations by Harding and Goddard using their GVB-CI method [60,65]. 

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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