Results of the Normal-Coordinates

In the second case (i j), either i or j can be zero The internal motion is decoupled from the center-of-mass motion. Equation 3.145 indicates that the autocorrelation of q,(0 is lost exponentially with a relaxation time of t,. 

We can obtain the mean square amplitude of q,(0 by setting t to 0 in the above equation  

The higher-order mode has a smaller ampUmde, as we will see in the Rouse and other models. 

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Fig. 1 summarizes the results for highly syndiotactic polyvinyl chloride. It gives the infrared spectra with the dichroism of each band and the result of the normal coordinate treatment and the frequency calculations [2]. 

Symmetiy Relations. Each normal coordinate and every wavefunction involving products of the normal coordinates, must transform under the symmetry operations of the molecule as one of the symmetry species of the molecular point group. The ground-state function in Eq. (3 a) is a Gaussian exponential function that is quadratic in Q, and examination shows that this is of Xg symmetry for each normal coordinate, since it is unchanged by any of the symmetry operations. From group theory the symmetry of a product of two functions is deduced from the symmetry species for each function by a systematic procedure discussed in detail in Refs. 4, 5,7, and 9. The results for the D i, point group apphcable to acetylene can be summarized as follows ... 

In the case of 1,2,4-trioxolane (2), where both the normal species and various isotopic modifications (3,3-D2, D4,1803) were also studied (76JPC1238), all the IR frequencies could be assigned (Table 5) as a result of a normal coordinate analysis. Starting from the molecular geometry determined by microwave spectroscopy (cf. Section 4.33.2.2.2), this analysis led to an excellent agreement between observed and calculated IR frequencies, thus confirming the O—O half-chair conformation of the 1,2,4-trioxolane ring. For a series of substituted... 

Ilie time correlation functions of the normal coordinates can be calculated immediately from eqn (4.19) by using the results of Section... 

Substituting u ) from (2.47) into the Hamiltonian H of (2.3), we can express H in terms of the normal coordinates Q(j). A certain amount of algebra is required to obtain the result which is derived in Appendix C. One obtains... 

Here we do not aim at presenting the standard methods of solving the Newton vibrational equations. It should be emphasized that essential results of these calculations are the transfonnation coefficients Lij that define the relative contribution of each internal coordinate to the respective normal vibrations in the molecule. As underlined, the availability of accurate vibrational form coefficients are needed in intensity analysis. This is detennined simply by the fact that vibrational intensities in the infrared spectra of molecules in the gas-phase (at low pressure so that no considerable intermolecular interaction is present) are governed by two principal factors (1) the intramolecular charge rearrangements accompanying vibrational distortions and (2) the form of the normal vibrations as expressed in the coefficients of the normal coordinate transfonnation matrix L. The elements of L are determined by solving systems of linear equations of the type [4,6]... 

We present results of such calculations to illustrate some of the effects of normalmode mixing. There are obviously a large number of possible examples and a few of them will be included here. The calculations are performed for nti = m2 = ttij = 0, that is, for the transition from the lowest vibrational level of the optically excited initial s state into the vibrational manifold of the lower electronic state (the zero-temperature limit). The results that we shall display are the final-state vibrational distribution (as given by the multidimensional surface of 13) and the relative nomadiative decay probability, both represented as functions of the normal coordinate rotation. 

To measure a residence-time distribution, a pulse of tagged feed is inserted into a continuous mill and the effluent is sampled on a schedule. If it is a dry miU, a soluble tracer such as salt or dye may be used and the samples analyzed conductimetricaUy or colorimetricaUy. If it is a wet mill, the tracer must be a solid of similar density to the ore. Materials hke copper concentrate, chrome brick, or barites have been used as tracers and analyzed by X-ray fluorescence. To plot results in log-normal coordinates, the concentration data must first be normalized from the form of Fig. 20-15 to the form of cumulative percent discharged, as in Fig. 20-16. For this, one must either know the total amount of pulse fed or determine it by a simple numerical integration... 

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