Overlapping distribution method

The overlap distribution method is more reliable than the direct averaging or single-direction FEP/NEW calculation. For example, it is capable of producing a good estimate of the chemical potential of dense fluids where the latter methods fail... 

It should be noted that Eq. (6.33) is identical in its form to the overlapping distribution method [22]. For later references, we denote the right hand side of Eq. (6.33) as Since d/i is independent of the energy coordinate R(tj) must be constant with respect to the variation of tj in principle. Hence, the constancy of R(t]) is a good measure of the statistical convergence. For the numerical robustness we multiply Eq. (6.33) by a weight function W(r]) and integrate with respect to rj, which reads,... 

Equation 4.44 is solved using Simpson s Rule to integrate the area of overlap. The method is easily transferred to eomputer eode for high aeeuraey. See Appendix XII for a diseussion of Simpson s Rule used for the numerieal integration of a funetion. Equation 4.44 permits the ealeulation of reliability for any eombinations of distributions for stress and strength provided the partial areas of G and H ean be found. 

These experiments use the product state distribution technique to allow a qualitative characterization of the competition between multiple electronic states. In contrast to the pathway competition in the molecular channel of formaldehyde (Section V.D), where the correlated product state distributions delineate the two channels quite cleanly, it will likely more often be the case that the product state distribution method allows only qualitative separation, due to overlapping distributions. Nevertheless, such experiments provide critical insight into pathway competition. 

Zimmerman, H. E. Alabugin, I. V. Energy Distribution and Redistribution and Chemical Reactivity. The Generalized Delta Overlap-Density Method for Ground State and Electron Transfer Reactions A New Quantitative Counterpart of Electron-Pushing. J. Am. Chem. Soc. 2001, 123, 2265-2270. 

Alternatively, the McMurchie-Davidson scheme [133] can be used for the computation of the spatial part of the matrix elements of Hpy (see [106]). The central concept of this method is to expand the product of two Gaussians (the so-called overlap distribution) in terms of Hermite functions according to... 

This approach may yield a result for that is superior to the simple test particle result if the overlap in the two distribution functions is significant. Contrarily, if the distributions do not overlap significantly, it is difficult to extract a reliable result from either the test particle or overlapping distribution formulae, and an alternative approach like thermodynamic integration or the staged insertion method is favored. 

Whereas at the lower end of its range mercury porosimetry overlaps with the gas adsorption method, at its upper end it overlaps with photomicrography. An instructive example is provided by the work of Dullien and his associates on samples of sandstone. By stereological measurements they were able to arrive at a curve of pore size distribution, which was extremely broad and extended to very coarse macropores the size distribution from mercury porosimetry on the other hand was quite narrow and showed a sharp peak at a much lower figure, 10nm (Fig. 3.31). The apparent contradiction is readily explained in terms of wide cavities which are revealed by photomicrography, and are entered through narrower constrictions which are shown up by mercury porosimetry. 

The limitations of one-dimensional (ID) chromatography in the analysis of complex mixtures are even more evident if a statistical method of overlap (SMO) is applied. The work of Davis and Giddings (26), and of Guiochon and co-workers (27), recently summarized by Jorgenson and co-workers (28) and Bertsch (29), showed how peak capacity is only the maximum number of mixture constituents which a chromatographic system may resolve. Because the peaks will be randomly rather than evenly distributed, it is inevitable that some will overlap. In fact, an SMO approach can be used to show how the number of resolved simple peaks (5) is related to n and the actual number of components in the mixture (m) by the following ... 

The log CLD method can sometimes provide better quality data than the conventional Mayo method. It is less sensitive to experimental noise and has application in measuring the transfer constant to polymeric species where the distributions of the transfer agent and the polymer product partially overlap.24... 

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