McMurchie-Davidson recursion

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],... 

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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