Distribution Hermite-Gaussian

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). 

The Gaussian mode is a specific case of the more generalized Hermite-Gaussian (HG) modes these are also referred to as transverse electromagnetic (TEM) modes. The TEM modes carry indices / and m, namely TEM/ where / is the number of intensity minima in the direction of the electric field oscillation, and m is the number of minima in the direction of the magnetic field oscillation (basically, the formula describing the TEMqo mode distribution (Equation (3.4)) is modified by multiplication with so-called Hermite polynomials Him x,y, 2 L). 

Since the Hermite Gaussians are to be used as one-centre basis functions for the expansion of Cartesian overlap distributions (over which the integration wiU eventually be carried out), it is important to determine the integrals over these Gaussians. The integral over the x component of the Hermite Gaussians is given by... 

Before we go on to consider the use of Hermite Gaussians as basis functions for overlap distributions, let us establish the relationship between Hermite Gaussians and Hermite polynomials. From Section 6.6.6, we recall that the Hermite polynomials may be generated from the Rodrigues... 

Let us consider the expansion of two-centre Cartesian overlap distributions in Hermite Gaussians. Since the overlap distribution is a polynomial of degree i -p j in xp (9.2.24), it may be expanded exactly in the Hermite polynomials of degree t < i + j. We therefore write... 

From these expressions, the fiill set of Hermite-to-Cartesian expansion coefficients may be generated and the overlap distribution may then be expanded in Hermite Gaussians according to (9.5.1). 

OVERLAP DISTRIBUTIONS FROM HERMITE GAUSSIANS BY RECURSION... 

In Section 9.5.1, the Cartesian overlap distributions were generated from the one-centre Hermite Gaussians by explicit expansion, using the expansion coefficients E /. These distributions may be generated recursively as well. For this purpose, we introduce the auxiliary distributions... 

Using these telations, we may generate the Cartesian overlap distributions recursively from the Hermite Gaussians. 

In Section 9.7, we discussed the evaluation of Coulomb integrals over spherical Gaussians. We now go one step further and consider nonspherical electron distributions as described by Hermite Gaussians. The one-electron Coulomb integral can then be expressed as... 

The definition of Hermite Gaussians in terms of differentiation has thus enabled us to express integrals over nonspherical distributions as derivatives of integrals over spherical distributions. 

This method was subsequently improved by noting that the ao in [ii rj() can be different for each pair ij [15], In this way, the Hermite charge distributions qi p are separated into compact and diffuse sets based on their exponents a . Subsequently, ao is chosen to be infinite for i j pairs where at least one of the two Gaussians is diffuse. This ensures that all pairs involving diffuse Hermites are evaluated in reciprocal space. For all compact ij pairs, ao is chosen so that i,ja(, is constant, that is, given P > 0, a Gaussian distribution qL p, is classified as compact if a, > 2 ) (C set) and diffuse otherwise (D set). Then, for i, j e C, choose a0 so that 1 =... 

For a distribution expanded around the equilibrium position, the first derivative is zero, and may be omitted, while the second derivatives are redundant as they merely modify the harmonic distribution. Since P0(u) is a Gaussian distribution, Eq. (2.28) can be simplified by use of the Tchebycheff-Hermite polynomials, often referred to simply as Hermite polynomials,3. , related to the derivatives of the three-dimensional Gaussian probability distribution by... 

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... 

For chains having fewer than 50 bonds, such as the short chains in a bimodal network for example, the distribution departs markedly from the Gaussian limit. Among various representations of w r) for short chains are the Hermite series (Flory and Yoon (1974, 1974)), the Fixman-Alben distribution, and Monte Carlo simulations (Erman and Mark, 1988). The Fixman-Alben distribution is given by... 

This expansion describes an arbitary distribution, P , for n>0 as a perturbation about a T-distribution, equivalent to the Gram-Charlier series resulting from a perturbation about a Gaussian distribution using Hermite polynomials [46]. 

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