Gaussian Multipole Expansions

One deduces the form of a Gaussian multipole upon considering Eq. (1.58) and the orthogonality of the spherical harmonics [19] 

A point multipole 5 (r — Ro) is best described as a Gaussian multipole [Eq. (1.60)] in the limit of infinite exponent [19,47] 

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The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

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). 

This applies to explicit charge-charge interactions, while the error in multipoles remains controlled by the order at which the multipole expansions are truncated. The explicit pairwise interactions between Gaussian charges in the leaf boxes are still computed explicitly using the erf(f,yr,y)/r,y kernel. 

Very fast multipole methods have been developed in order to calculate these electron repulsion integralsl . The near field is determined by analytical Gaussian calculations. The far field is calculated usiug multipole expansions to treat the distant charges and their interactions. The scahng for this approach has been reduced to Fast quadrature... 

In general, the elementary charge distribution (49) is reduced to the sum of a finite number of Gaussian functions, from which it is immediate to derive an exact multipole expansion composed by a finite number of terms the upper term is the sum of the angular momentum quantum numbers of the two functions. There is still, of course, a penetration term, related to the exponential decay of the functions. Hall has shown, with numerical examples, that this penetration term is reasonably small [75]. 

The elimination of multipole expansion for the elementary distributions regarding Gaussian functions with a low orbital exponent, a, introduced in Ref [98], pays attention to a problem we have already mentioned. The exact potential V may be divided into two components (Eq. 46) in which one component, which in the expansion method is neglected, may be considered to be the error intrinsic in the multipole expansion. A large contribution to is given by the exponential tails of elementary distributions with low a s, which are quite spread. An exact representation of these components of E (r) reduces the extent of the neglected contributions to the MEP. 

A key concept is the idea of the sphere of divergence. This is defined as the sphere centered on the expansion center (typically the center-of-mass) and just enclosing all charges. For a molecule this poses a problem as the electronic density formally extends to infinity. But it has been shown (Stone and Alderton 1985) that for charge densities expressed as the sum of Gaussian functions, it is sufficient that the sphere enclose all nuclear sites. The multipole expansion is valid only if the spheres of divergence of all interacting molecules do not overlap. 

A closely related approach, using the same kind of molecular cavity has been developed by Tomasi et al. (1981). In this model, the multipole expansion is replaced by a numerical computation of the electrostatic potential inside the cavity, due to the polarization of the boundary by the solute. This model also has been improved along the years. It has various versions all known under the acronym of PCM (Canc s et al. 1997) which are implemented in the Gaussian suites of programs, and a closely related model is available in the Jaguar package (Marten et al. 1996 Tannor et al. 1994). 

For example, to calculate an [ss ss] integral with exponents C/i = Cb = 1 to n accuracy of 10 using the multipole expansion, centers A and B have to be separated by Ra + Rb = 4.0A. Similar expressions can be derived for higher angular momenta, but the expression for s Gaussians is usually sufficiently accurate. Together with judicious convergence criteria, the multipole expansion for ERIs produces results that are numerically exact for all practical purposes. 

So far we have only derived the multipole expansion for primitive Gaussian distributions. As pointed out in the introductory example, one of the main strengths of the multipole expansion is that it can be used to treat the interactions of several primitive charge distributions simultaneously by combining them into one single, albeit more complicated, distribution. It will turn out to be useful to translate the centers of multipole expansions to different points in space e.g., if (A) is an expansion about A, we must find a way to convert it to a series about A — t, where t is the translation vector. 

From the analytic expressions for the two-electron integral (ffalffe) over s Gaussians, we pointed out that the error caused by employing the multipole expansion for calculating the ERI is on the order of e, if the bra-ket distance R is chosen such that the following equation holds ... 

We must keep track of the spatial extents of Gaussian distributions in order to know when the multipole expansion is applicable and when it is not in other words, we must know the size of the NF for each Gaussian. To that end, a well-separatedness criterion (WS) or, synonymously, an NF width parameter is introduced, which stores the number of boxes by which a pair of equal Gaussian distributions have to be separated to be treated as an FF pair ... 

At this stage, when the nature of the basis is known, we return to the question, how the MME needed to evaluate (i > I FPerm. cj) and (i > I Fp0i. cj) is done in practice. We use a method that takes advantage of basic properties of Gaussian functions, which is also very similar to the distributed multipole analysis of Stone [123,124,125], For an arbitrary pair of basis functions we use the orbital expansion... 

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