Centroid methods equilibrium

In the early papers [4,8], the development of the CMD method was guided in part by the effective harmonic analysis and, in part, by physical reasoning. In Paper III, however, a mathematical justification of CMD was provided. In the latter analysis, it was shown that (1) CMD always yields a mathematically well-defined approximation to the quantum Kubo-transformed position or velocity correlation function, and (2) the equilibrium path centroid variable occupies an important role in the time correlation function because of the nature of the preaveraging procedure in CMD. Critical to the analysis of CMD and its justification was the phase-space centroid density formulation of Paper III, so that the momentum could be treated as an independent dynamical variable. The relationship between the centroid correlation function and the Kubo-transformed position correlation function was found to be unique if the centroid is taken as a dynamical variable. The analysis of Paper III will now be reviewed. For notational simplicity, the equations are restricted to a two-dimensional phase space, but they can readily be generalized. 

The steps involved in data reduction are outlined in Table 1. The centroid volume, V, is determined for each concentration by one of the methods described above. These values of V are plotted as a function of concentration with the elution values of the standard proteins used to calibrate the column indicated on the ordinate. If V decreases with increasing concentration, this Indicates that the protein is self-associating under the experimental conditions and that an equilibrium exists. No decrease in V means that the protein is not self-associating and the experiment is over. The next step is to calculate for each value of V using equation (8) and to decide on a plausible self-association model to fit the data. Examination of the plot of V as a function of concentration should aid in this decision. For example, if all of the data points fall between the monomer and dimer elution points (as determined from the column calibration), then the equilibrium is probably a monomer-dimer selfassociation. 

In the polypropylene catalyst shown in Figure 14, an isotactic-atactic block copolymer can be formed by rotation of one ring relative to the Zr-centroid axis. (For descriptions of polymer stereochemistry, see Fig. 9.) The isospecific rac rotomer of the catalyst gives rise to the isotactic block, while the aspecific meso form gives rise to the atactic block (Fig. 14). Using UFF (4) and RFF (85,86) (as weU as ab initio methods and DFT), the workers were able to confirm experimental evidence (89,90) that the indenyl substituent, R in Figure 14, could influence the equilibrium between the rac and meso rotomers. Using RFF the workers were able to successfully predict the relative amount of isotactic and atactic blocks in the polymer and to correlate that with R. 

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