Gaussians Hermite

In this section we describe the methods to extend Ewald sum methodologies to accelerate the calculation of the intermolecular interactions using PBC. For simplicity, we begin with a generalization of Ewald sums to interacting spherical Hermite Gaussians (e.g. GEM-0 [14]). This is followed by the extension to arbitrary angular momentum. Finally, we describe the implementation of methods to speed up both the direct an reciprocal terms in the Ewald sum [62],... 

Cisneros GA, Elking D, Piquemal J-P, Darden TA (2007) Numerical fitting of molecular properties to Hermite Gaussians. J Phys Chem A 111 12049... 

Novotny, L., Sanchez, E. J., and Xie, X. S. 1998. Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams. Ultramicrosc. 71 21-29. 

A. Koster, Hermite gaussian auxiliary functions for the variational fitting of the Coulomb potential in density functional methods. J. Chem. Phys. 118, 9943-9951 (2003)... 

The computer codes of Sambe and Felton (SF) and Dunlap et al. (DCS) are based on the choice of a Hermite-Gaussian expansion set. Applying the variational theorem with the trial function of Eq. (37) and the LSD Hamiltonian of Eq. (36) leads to the usual matrix pseudo-eigenvalue problem ... 

The overlap and Hamiltonian matrix elements over Hermite Gaussians must then be evaluated. The one-electron operators present no special problems and need not be discussed further. The Coulomb integrals are identical to those in Hartree-Fock theory, i.e. 

Integrals involving the exchange-correlation potential r,c or the exchange-correlation energy density cannot be evaluated analytically so that further sets of auxiliary functions are introduced. (In practice and 6,<, behave similarly so that a common set is used to fit both functions.) The exchange-correlation basis (XCB) also consists of Hermite Gaussians... 

We shall also require the Hermite Gaussian functions, defined by... 

The zeroth-order ( s-type ) Hermite gaussians have non-zero norm (they are... 

Coulombic lattice sums occur in the charge fitting, Eq. (40), in the matrix elements of the KS Hamiltonian with respect to the orbital basis, and in the total energy. The charge neutral generalization of Eq. (7) is an example of the form when written in Hermite gaussian orbital and chaige-fit basis sets ... 

For GEM the ABSs consist of Hermite Gaussians, Atuv(r). The expansion coefficients for the approximate density p may be obtained by minimizing Eq. 8.2 using some metric 0 (Dunlap et al., 1979 Eichkorn etal., 1995 Roster, 1996 Roster etal., 2002). 

Briefly, we have expanded on the work by Challacombe et al., who have shown that Hermite Gaussians have a simple relation to elements of the Cartesian multipole tensor (Challacombe et al., 1996). Once the Hermite coefficients have been determined, they may be employed to calculate point multipoles centered at the expansion sites. Thus, if hctuv represents the coefficient of a Hermite Gaussian of order Atuv, then if this Hermite is normalized we have... 

A further advantage of this approach to distributed multipoles is that, unlike some conventional multipole expansions (Stone, 2005 Popelier et al., 2001b), the (spherical) multipole expansion obtained from Hermite Gaussians in this way is intrinsically finite of order t+ + V (i.e., the highest angular momentum in the ABS) as shown in (Cisneros et al., 2006a), similar to the multipoles obtained by Volkov and Coppens (Volkov and Coppens, 2004). 

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