Gaussian overlap distributions derivatives

As the SIBFA approach relies on the use of distributed multipoles and on approximation derived form localized MOs, it is possible to generalize the philosophy to a direct use of electron density. That way, the Gaussian electrostatic model (GEM) [2, 14-16] relies on ab initio-derived fragment electron densities to compute the components of the total interaction energy. It offers the possibility of a continuous electrostatic model going from distributed multipoles to densities and allows a direct inclusion of short-range quantum effects such as overlap and penetration effects in the molecular mechanics energies. 

The Obara-Saika [50] recursion relations for the three-center overlap matrix elements can be used to derive the recursion relations for the ion-wall source interaction. Specific formulae will be listed below after we consider the interaction with the soft Gaussian part of the charge distribution. 

The first application of differentiation to spectroscopy was, of course, the deconvolution of superposed peaks [41-43], which are frequently found in spectral investigations. From then on, many scientists analyzed peak overlapping by summation of synthetic signals, mostly of the Gaussian type [12,14,15, 44-48]. Derivatives, zero crossing, and extrema of different distribution functions such as Gauss, Lorentz, Student, T3, and others, are not difficult to estimate unless there are superpositions of two or more bands. For pure Gaussian functions, see Table 2-3. 

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