Gaussian radial density function

A hydrogen atom is described by the Gaussian radial density function... 

W(Xy yy z)y W r) Density and radial distribution functions for the end-to-end coordinates of a polymer chain (usually Gaussian functions). 

Radial basis function networks (RBF) are a variant of three-layer feed forward networks (see Fig 44.18). They contain a pass-through input layer, a hidden layer and an output layer. A different approach for modelling the data is used. The transfer function in the hidden layer of RBF networks is called the kernel or basis function. For a detailed description the reader is referred to references [62,63]. Each node in the hidden unit contains thus such a kernel function. The main difference between the transfer function in MLF and the kernel function in RBF is that the latter (usually a Gaussian function) defines an ellipsoid in the input space. Whereas basically the MLF network divides the input space into regions via hyperplanes (see e.g. Figs. 44.12c and d), RBF networks divide the input space into hyperspheres by means of the kernel function with specified widths and centres. This can be compared with the density or potential methods in pattern recognition (see Section 33.2.5). 

Figure 15. Radial distribution function T r) of the low-density a-C (p = 2.20 g/cm ) obtained from TBMD simulation (solid curve) compared with the neutron scattering data of Ref. 64 (dotted curve). The simulation result has been broadened by A Gaussian function with a width of 0.085 A. (From Ref. 62.)...

Fig. 9.7. The distance dependence of the nomalized segment density distribution function for 1, an exponential function, 2, a radial Gaussian function and 3, a constant segment density function (after Smitham and Napper, 1979).

In a volume-oriented density function such as that used by ROCS, Gaussian functions are atom-centered. In the surface-oriented formulation, the M. of Equation 2.2.1 are Gaussians with peaks at the atomic surface (set by the atomic radii, denoted fi). By itself, the sum over the M produces internal molecular surfaces in addition to external ones. The E[ of Equation 2.2.2 defines Gaussians on local radial co-ordinates around each observer point from set P, with peaks at the molecular surface (set by the minimum distances from the observers to the molecule, denoted d,). When yis chosen carefully, the integral of the product of two molecules surface-density functions R (defined in Eq. 2.2.3) is very closely approximated by the morphological similarity function used by Surflex-Sim [31]. 

Radial basis functions networks are good function approximation and classification as backpropagation networks but require much less time to train and don t have as critical local minima or connection weight freezing (sometimes called network paralysis) problems. Radial basis fimction CNNs are also known to be universal approximators and provide a convenient measure of the reliability and confidence of its output (based on the density of training data). In addition, the functional equivalence of these networks with fuzzy inference systems have shown that the membership functions within a rule are equivalent to Gaussian functions with the same variance (o ) and the munber of receptive field nodes is equivalent to the number of fuzzy if-then rules. 

The mathematical description of the model is out of the scope of this paper. Briefly, in this model, each reactant beam density is fitted to gaussian radial and temporal distribution functions, the spread in relative translational energy is neglected and the densities are assumed to be constant within the probed volume, which is smaller than the reaction zone. These assumptions result in a simple analytic expression of the overlap integral. Calculations are carried out for each rovibrational state of the outcoming molecule and for extreme velocity vector orientations, i.e, forwards and backwards. An example of the correction function, F, obtained for the A1 + O2 reaction at = 0.49 eV is displayed on Fig. 1, together with the... 

Figure 6. Predicted interchain radial distribution function for a hard-core polyethylene melt described by three single-chain models atomistic RIS at 430 K, overlapping (lid = 0.5) SFC model with appropriately chosen aspect ratio and site number density (see text), and the Gaussian thread model (shifted horizontally to align the hard core diameter with the value of rld = l).

The first question to be asked is why the Brownian diffusion model of Kirkwood should give reasonable results for the unlike-ion friction constants, as mentioned in Section 3.4, when the Coulomb potential is ignored and the experimental radial distribution function used. The assumptions in the Brownian diffusion model are difficult to evaluate but Douglass et have shown it to be a factor of njl greater than their result using a Gaussian autocorrelation function. Now from molecular dynamics Alder et have shown for hard spheres at high densities that the autocorrelation... 

In order to render the expression for d AFa) in a usable form, it remains to evaluate pk and pi. We have already pointed out that the average segment density of a molecule will be greatest at the center of gravity and that it will decrease smoothly as the distance 5 (Fig. 114,a) from the center is increased. While the distribution will not be exactly a Gaussian function of s, it may be so represented without introducing an appreciable error in our final result, which can be shown to be insensitive to the exact form assumed for the radial dependence of the segment density. Hence we may let... 

Additionally, from Fig. 29 one sees that, if, as proposed by Frost 42), a spherical gaussian function is a fair representation of the distribution of charge within an electride ion, there should he, as found by Slater 97>, a very good correlation, and in many cases practically an equality, between the atomic radii. . . and the calculated radius of maximum radial charge density in the outermost shell of the atom". 

Gaussians centered on the same atom A. By using the spherical form of the Gaussian function, rather than the Cartesian one, we can take advantage of the separation into the radial and the angular contributions. For example, the hard density ha given in (16) becomes... 

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