Reciprocal Space Methods for Integral Evaluation

The use of molecular densities results in the need to compute a large number of two center integrals for the intermolecular interaction. A significant computational speedup can be achieved by using reciprocal space methods based on Ewald sums. In this way, the integrals are calculated in direct or reciprocal space depending on the exponent of the Gaussian Hermites. 

This was later improved by the realization that the Ewald exponent, may be different for each pair ij (Darden, 2007). Thus, is chosen to be infinite for ij pairs where at least one of the Gaussians is diffuse. In this way, all pairs that involve diffuse 

the structure factors 5 /(m involve derivatives of the Fourier exponential with respect to the Hermite centers, Esei/iPi) is the correction due to the self energy of each Hermite interacting with 

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). 

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