Gaussian-dipole distribution

The field of the gaussian-dipole distribution is calculated from the Green s functions of the two distributions, as a sum in the direct space. (In this calculation the gaussian placed at 0 must be taken into account it plays a... 

There are other parameterizations possible. Thole noted, in his original paper, that it would be more elegant to describe the interaction—between induced dipoles—in terms of two interacting charge distributions instead of his one-particle ansatz. Jensen et al. [35] took up this suggestion in order to arrive at traceless interaction tensors. He started from the interaction between two isotropic Gaussian charge distributions on a distance r ... 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

Fig. 9. The efTect of the dipole-dipole interaction on inhomogeneously broadened infrared absorption peaks. Without and with the interaction turned on for peaks with I, a low frequency tail, II, a symmetric Gaussian distribution and III, a high frequency tail. (Adaptedfrom Persson...

Investigation of the multivariate Gaussian distribution and the dipole moments of perturbed chains expansion factors for perturbed chains. 

In eqn. (2), AH°m is the EPR line width in the limit of such dilution that one can neglect the dipole-dipole interaction and A is a coefficient depending on the EPR line shape and on the character of the spatial distribution of the spin-labelled molecules. In the case of random spatial distribution according to theoretical calculations A = 35 GM 1 for the Gaussian EPR line shape and A = 56GM-1 for the Lorentzian EPR line shape. 

Figure 1.1. Scheme of dipole-moment densities utilized in the method of Ewald (A) Distribution of point dipoles whose field is calculated at the origin 0. (B) distribution of gaussians centered at each point, point 0 included. (C) distribution of dipoles plus gaussians ... 

The introduction of Gaussian function in Equation (3) is based on the prediction by Pake23 for crystals of small molecules, such as 1,2-dichloroethane, where the dipole pairs are isolated from each other. However, in the polymeric materials, not only the dipole interactions between the nearest neighbours but also those between the proton pairs at a longer distance contribute to the FID profile shape. Therefore, the distribution of the dipole interaction may be distorted. This is expressed by the introduction of Weibullian form,24 as follows ... 

We shall now demonstrate how the CTRW in the diffusion limit may be used to justify the fractional diffusion equation. We consider an assembly of permanent dipoles constrained to rotate about a fixed axis (the dipole is specified by the angular coordinate unit circle with fixed angular spacing A. We note that A may not necessarily be fixed for example, if we have a Gaussian distribution of jumps, the standard deviation of A serves as a fixed quantity. A typical dipole may remain in a fixed orientation at a given site for an arbitrary long waiting time. It may then reorient to another discrete orientation site. This is the discrete orientation model. 

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