Ellipsoidal gaussians

The way forward was proposed by Berne and Pechukas [11] many years later. Their important idea was to consider the overlap between two prolate ellipsoidal gaussian distributions. From the expression for this overlap they evaluated a range parameter which was taken to be the contact distance g and a strength parameter which was set equal to the well depth, e. If the orientations of the two rod-like molecules in the laboratory frame are represented by the unit vectors Ui and Uj and the orientation of the intermolecular vector by the unit vector f then the expression for the angular dependence of the contact distance is... 

Double Zeta + Polarization functions Extended Hartree-Fock Electron Spectroscopy for Chemical Analysis Floating Ellipsoidal Gaussian Orbital Floating Spherical Gaussian Orbital Generalized Atomic Effective Potential Gaussian Type Orbital... 

Work by Frost and co-workers in the mid 1960s abandoned the idea of AO-based functions to arrive at an even more compact basis set. They suggested the use of s-type Gaussians that were not fixed at the atomic centers, but could float in space so as to optimally represent each localized pair of electrons. Because only one function was needed for each pair of electrons, the basis sets used in floating spherical Gaussian (FSGO) scheme were often referred to as subminimal. Extensions of Frost s idea to ellipsoidal Gaussians of the form... 

P. T. van Duijnen and D, B. Cook, Mol. Phys., 21, 475 (1971). Ab Initio Calculations with Small Ellipsoidal Gaussian Basis Sets. 

Keywords Bond functions Ellipsoidal Gaussian functions Polarization functions Unconventional basis functions... 

Bond-centered general ellipsoidal Gaussian functions... 

The ellipsoidal Gaussians that we propose here to be used as basis functions can be written in the... 

Cartesian Gaussians, which reduce to the latter if matrix a is diagonal with all elements equal to a. As such, ellipsoidal Gaussians are expected to be particularly well suited for describing the polarized charge densities in molecular environments. 

Since the purpose of this initial study is to evaluate the potential of ellipsoidal Gaussian basis functions, we did not endeavor to develop a highly efficient integral code. 

A spedlic coordinate system is defined for each ellipsoidal Gaussian. Its origin is the center of the Gaussian, hereafter denoted by E, while its z-axis points in the direction of the bond. In the bond-specific coordinate system, the exponent matrix denoted by a takes the... 

For the ideal, i.e., circular or symmetrically ellipsoidal, bands with the Gaussian analyte concentration profiles, the band centers described by assumptions (1) and (2) are, in fact, identical. 

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). 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

Fig. 4.27 SAXS intensity as a function of wavevector for a PS-P1 diblock (Mw = 60 kg mol-1, 17wt% PS) (points) in dibutyl phthalate with a polymer volume fraction

Recently, in a number of theoretical computational studies the shape of the Gaussian coil has been characterized by the components of radii of gyration Ri, Rj and R3 in the three main directions of the coil (fixed for each conformation). Calculations reveal that the average sh of the coil n be approximated by a three-axial ellipsoid with the ratio of squares of axes... 

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