Overlap integrals analytical evaluation

In our present PS-DFT module, the numerical integration is performed using the procedures proposed by Becke which are based upon classical numerical quadratures. Other approaches include fitting F to an expansion of Gaussian functions and evaluating the resulting three center overlap integrals analytically. However, PS methods can profitably be applied to this term as well, and one can expect substantial accelerations in DFT calculations via this technique. 

This ultrasimple classical theory is, of course, too crude for practical applications, especially for highly excited states of the parent molecule. Its usefulness gradually diminishes as the degree of vibrational excitation increases, i.e., as the initial wavefunction becomes more and more oscillatory. If both wavefunctions oscillate rapidly, they can be approximated by semiclassical WKB wavefunctions and the radial overlap integral of the bound and the continuum wavefunctions can subsequently be evaluated by the method of steepest descent. This leads to analytical expressions for the spectrum (Child 1980, 1991 ch.5 Tellinghuisen 1985, 1987). In particular, relation (13.2), which relates the coordinate R to the energy E, is replaced by... 

The hydrocarbon ("oil") fraction of a coal pyrolysis tar prepared by open column liquid chromatography (LC) was separated into 16 subfractions by a second LC procedure. Low voltage mass spectrometry (MS), infrared spectroscopy (IR), and proton (PMR) as well as carbon-13 nuclear magnetic resonance spectrometry (CMR) were performed on the first 13 subfractions. Computerized multivariate analysis procedures such as factor analysis followed by canonical correlation techniques were used to extract the overlapping information from the analytical data. Subsequent evaluation of the integrated analytical data revealed chemical information which could not have been obtained readily from the individual spectroscopic techniques. The approach described is generally applicable to multisource analytical data on pyrolysis oils and other complex mixtures. 

The vibrational overlap integrals written in the form (m,M21diI 2) be evaluated analytically for harmonic oscillators. The coefficients Af are given by... 

Boundary effects on the electrophoretic migration of a particle with ion cloud of arbitrary thickness were also investigated by Zydney [46] for the case of a spherical particle of radius a in a concentric spherical cavity of radius d. Based on Henry s [19] method, a semi-analytic solution has been developed for the particle mobility, which is valid for all double layer thicknesses and all particle/pore sizes. Two integrals in the mobility expression must be evaluated numerically to obtain the particle velocity except for the case of infinite Ka. The first-order correction to the electrophoretic mobility is 0(A3) for thin double layer, whereas it becomes 0(A) for thick double layer. Here the parameter A is the ratio of the particle-to-cavity radii. The boundary effect becomes more significant because the fluid velocity decays as r l when the double layer spans the entire cavity. The stronger A dependence of the first order correction for thick double layer than that obtained by Ennis and Andersion [45] results from the fact that the double layers overlap in... 

Cartesian Gaussians and products of two such Gaussians (i.e. the Cartesian overlap distributions) play an important role in the evaluation of molecular integrals. In the present section, we prepare ourselves for the study of integration techniques by examining the analytical properties of single Gaussians and their overlap distributions. 

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