Hermite coefficients

In this contribution, we present the theory behind the GEM method and recent advances and results on the application of two hybrid GEM potentials. In Section 8.2, we provide a brief review of the analytical and numerical density fitting methods and its implementation, including the methods employed to control numerical instabilities. This is followed by a review of the procedure to obtain distributed site multipoles from the fitted Hermite coefficients in Section 8.3. Section 8.4 describes the extension of reciprocal space methods for continuous densities. Section 8.5 describes the complete form for GEM and a novel hybrid force field, GEM, which combines term from GEM and AMOEBA for MD simulations. Finally, Section 8.6 describes the implementation and initial applications of a multi-scale program that combines GEM and SIBFA. 

In this subsection, we present the methodology to obtain Cartesian point multipoles from the Hermite coefficients obtained in the fitting procedure. In all our work we have purposefully employed ABSs with a maximum angular momentum of 2, which results in... 

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... 

The Hermite polynomials Hn ) form an orthogonal set over the range —oo oo with a weighting factor e . If we equate coefficients of stY on each side of equation (D.12), we obtain... 

The variable s is a dummy variable in the sense that it does not enter die final result. Thus, if the exponential function in Eq. (94) is expanded in a power series in s, the coefficients of successive powers of s are just the Hermite polynomials divided by u . It is not too difficult to show that Eqs. (93) and (94) are equivalent definitions of the Hermite polynomials. 

As this relation is correct for all values of s, the coefficients in brackets must vanish The result yields the important recursion formula for the Hermite polynomials,... 

Other commonly employed and related sets of approximating polynomials are Hermite polynomials and B splines. Particularly in the latter case, the functions possess the desired properties of smoothness across patch boundary intersections, strong locality leading to simplification of the A coefficient matrix, and efficiency of computation. In the following discussion the B functions may be viewed, up to specific values, as any of the aforementioned types. 

The coefficients c in this probability distribution are referred to as the quasimoments of the distribution. Because of the orthogonality of Hermite polynomials, the quasimoments of a function are obtained by integration of the product of the function and the related Hermite polynomial over all space. For the one-dimensional case,... 

A typical application is given by Debets et al. A quality criterion for the characterization of separation in a chromatogram is modified by using Hermite polynomial coefficients in order to enhance the performance. The quality criterion can be used in... 

The non-vanishing A(C) coefficients are not all independent. From the Hermiticity of the dipole operator, p, and the symmetry property,... 

In quantum mechanics and other branches of mathematical physics, we repeatedly encounter what are called special functions. These are often solutions of second-order differential equations with variable coefficients. The most famous examples are Bessel functions, which we wiU not need in this book. Our first encounter with special functions are the Hermite polynomials, contained in solutions of the Schrodinger equation. In subsequent chapters we will introduce Legendre and Laguerre functions. Sometime in 2004, theU.S. National Institute of Standards and Tec hnology (NIST) will publish an online Digital Library of Mathematical Functions, http / /dlmf. nist. gov, including graphics and cross-references. 

The coefficient of the Hermite polynomial can be found using the orthogonality property of the polynomial. Thus... 

In the relation (7.63) the coefficients W exhibit symmetry properties which follow from the hermiticity of the operator HU1. In particular... 

For the present purpose, instead of developing the theory of the polynomials // ( ), called the Hermite polynomials, from the relation between successive coefficients given in Equation 11-7, it is more convenient to introduce them by means of another definition ... 

This identity in the auxiliary variable s means that the function hjyg property that, if it is expanded in a power series in 8, the coefficients of successive powers of s are just the Hermite polynomials H ( ), multiplied by 1/n . To show the equivalence of the two definitions 11-13 and 11-14, we differentiate S n times with respect to s and then let s tend to zero, using first one and then the other expression for S the terms with v < n vanish on differentiation, and those with v > n vanish for s —> 0, leaving only the term with v = n ... 

Since this equation is true for all values of a, the coefficients of individual powers of s must vanish, giving as the recursion formula for the Hermite polynomials the expression... 

By this application of the recursion formula for ( ) we have expressed %2Hn( ) in terms of Hermite polynomials with constant coefficients. By squaring this we obtain an expression for 4 (f), which enables us to express the integral in Equation 23-26 as a sum of integrals of the form... 

Note that odd powers of W do not occur in this expansion because of the anti-hermiticity of W. With the requirement that different powers of W be linearly independent, we arrive at the following unitarity conditions for the coefficients ... 

The fitting coefficients qs are determined by minimizing the spurious Coulomb energy associated with the difference between the fitted and exact densities. (Notice that the set Q can and often does contain several different types of Hermite GTOs, e.g., those that correspond to s-, d- and pz type atomic symmetries.)... 

What was believed to be the most reasonable way of weighting each of the ratios in forming the final function, herein called the non-Hermiticity factor, was chosen according to the importance of the component Hjj in (j> // ). This is given by the eigenvector coefficient product CjCj. 

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