Statistical, displacement

It has been known for 40 years, since the initial PGSE experiment of Stejskal and Tanner [20], that NMR is an excellent tool for measuring the statistical displacements of molecules with time, usually called diffusion. The exponential attenuation a of the PGSE signal as a function of time between the PGSE pulses is given by [21]... 

This equation is interesting in so far as it shows that the maximum value of pi corresponds to = 0, as is the case for completely free rotation. Thus it appears that, in some cases at least, restricted rotation leads to a decrease in the mean statistical dipole, whilst it almost always causes an increase in the mean statistical displacement. This is due to the fact that trans configurations are energetically more probable than cis configurations. 

In conclusion to this section, we note that the statistical mean dipole moment of the molecule, as well as its mean displacement, should be proportional to the square root of the degree of polymerization. Hindrance to free rotation tends to decrease the mean statistical dipole moment and to increase the mean statistical displacement. Since hindrance to rotation must disappear at sufficiently high temperatures, the mean dipole moment of the isolated molecule should increase with temperature whilst the mean displacement should be a decreasing function of temperature. 

Ohigaidii (99) has suggested that stress applied parallel to the long axb of the polymer molecules might reduce the sterk hindrance that causes the statistical displacement... 

There is an intimate connection at the molecular level between diffusion and random flight statistics. The diffusing particle, after all, is displaced by random collisions with the surrounding solvent molecules, travels a short distance, experiences another collision which changes its direction, and so on. Such a zigzagged path is called Brownian motion when observed microscopically, describes diffusion when considered in terms of net displacement, and defines a three-dimensional random walk in statistical language. Accordingly, we propose to describe the net displacement of the solute in, say, the x direction as the result of a r -step random walk, in which the number of steps is directly proportional to time ... 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

The squared deviations between the calculated and actual responses are shown in Table 12.6c (see column labeled SSq). The AIC values are calculated according to Equation 11.30. The values are shown in Table 12.6c. It can be seen that the fit to the curves with a mean Emax and slope gives a lower AIC value. Therefore, this model is statistically preferable. It is also the most unambiguous model for simple competitive antagonism since it fulfills the criteria of parallel dextral displacement of dose-response curves with no diminution of maxima. The calculated curves are shown in Figure 12.7b. 

In order to discuss how we can use statistical methods to estimate the errors in measuring the displacement we use the notation h u a). This is the condi-... 

Several brevetoxins have been examined for their respective abilities to competitively displace tritiated brevetoxin PbTx-3 from its specific site of action in brain synaptosomes. Analysis of IC q values revealed no marked differences in the displacing abilities between any of the type-1 toxins, and similarly there was no apparent difference between displacing abilities of PbTx-1 or -7, both type-2 toxins. Although some specific details require correlation, a gross comparison indicates that sodium channels in brain are similar in the systems examined. In the system studied most extensively, the rat brain synaptosome, t-test analysis revealed no significant differences between PbTx-2 and PbTx-3 IC q, or between PbTx-1 and PbTx-7 IC q, but statistically significant differences were found between the two classes (P<0.01) (5). If the Cheng-Prusoff equation (15) is applied ... 

Kuhn has shown how a real polymer chain may be approximated by an equivalent freely jointed chain. Instead of taking the individual bonds as statistical elements, one may for this purpose choose sequences of m bonds each. In Fig. 79, arbitrarily chosen statistical elements consisting of five bonds are indicated, the displacement vectors for these elements being shown by the dashed lines. The direction assumed by a statistical element will be nearly independent of the direction of the preceding element, provided the number m of bonds per... 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

From the statistics of this random displacement, a Gaussian short-time transition probability follows ... 

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f . The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of J (the root-mean-square displacement) on either side of it, is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2f is equal to the total area under the curve minus the area under the curve up to 2f. This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

As mentioned earlier, a random and statistical cyclization with two different and regio-defined Ti-compounds would produce a synthetically unattractive mixture of ten different zirconacycles. In reality, however, there are a few factors that can be exploited to produce a single desired zirconacycle. A systematic investigation has revealed that there are several discrete types offive-membered zirconacycle formation, as shown in Scheme 1.53 [88,89] (Generalization 20). In the Type I reaction, the cross-selective cyclization is kinetically favored. Presumably, little ethylene is displaced during the reaction. Type I reactions cannot be readily observed with ZrCp2 complexes with 1-butene. In contrast, Type II cyclization must be thermodynamically controlled, as 1-butene is readily displaced by a number of better Ti-ligands. It is predicted, however, that the cross-combination of the two Ti-com-... 

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