Parallel-mode approximation

The extension of the formulas to degenerate accepting modes which occur when a Jahn-Teller effect in the excited state is present is relatively easy. In this case the products of distribution functions can be rewritten by convolution into fundamental distributions which does not change the overall expression of Eq. (29) [38, 66]. Also, it is possible to consider an intermixing of modes in the excited state by virtue of the bi-linear term in Eq. (1) (Duschinsky effect [67]). Since it is difficult to decide from most of the spectra if this effect is really observed in the case of the present complex compounds, we will not consider it here and refer to the literature [68,69]. This is justified as long as we can explain the experimental spectra satisfactorily applying the parallel mode approximation leading to the line shape function of Eq. (29) as it has been described in the method above. 

A first and simple approximation is to neglect the mode mixing and consider a one-to-one relation between the modes of the initial and final states, with the Duschinsky transformation matrix J equal to the identity matrix (notice that VG and AS models belong by definition to this approximation). The interest of this approximation, called parallel-mode approximation, is that the multidimensional FC integrals can be calculated as products of one-dimensional integrals using the relation... 

Figure4.5 The multidimensional distribution / (0,n a,b) (in the parallel-mode approximation) for moderately strongly coupled states a = 2, b = —0.2 and different orders of degeneracy i.

The explicit separation of vibrations in the multidimensional GF of Theorem 4.4 allows an effective reduction of the number of degrees of freedom and facilitates the calculation of transition rates. In this regard, it cannot be too strongly emphasized that the calculation of transition rates in the parallel-mode approximation has a fundamental inadequacy that is evident from the derivation above. The defect emerges, if we return to the case of N-vibrational modes some of which are not parallel to each other. This situation occurs if the latter have the same symmetry, especially if they are totally symmetric in the molecular group. In this case, the complexity introduced by the reciprocal (4.69a) and interactive displacement parameters (Equations 4.69b and 4.69c) is considerable and this fact cannot be overlooked in any parallel-mode estimate of the transition probability. Even with considerable effort, it is impossible to factor the exact GF into a product of one-dimensional GF. 

Note that the interactive displacement parameters are not quantities with physical significance. They merely represent the error incurred in making a crude approximation, assuming that all modes are parallel. Another remarkable feature of mode mixing is the dependence of the transition rate on the cross-frequency parameters (see Equation 4.71). (Note that in the parallel-mode approximation the transition rate depends solely upon and 3j, = per mode). Both of these findings are... 

The summation is over all the spanwise modes. One can use the above ansatz in three-dimensional Navier-Stokes equation and linearize the resultant equations after making a parallel flow approximation to get the following Orr-Sommerfeld equation for the Fourier- Laplace transform f of v ... 

Structural information of LB films has also been obtained from FTIR studies. In the carboxylate form of the fatty acid, the relative intensities of the vs(C02-) and va(C02-) signals are dependent on the orientation of the chain axis. The dipole moments of the vs(C02-) and va(C02 ) stretches are parallel to and perpendicular to the chain axis, respectively. In transmission mode the electric vector of the IR radiation interacts strongly with dipole moments parallel to the substrate. This means that in transmission mode the vs(C02-) will be most intense, and the va(C02-) the weakest, for films with the chain axis perpendicular to the substrate. The opposite is true for the FTIR-RA mode. There is general consensus that in M-FA films the chain axis is approximately perpendicular to the substrate while the protonated form of the acid after exposure to H2S has a tilt relative to the substrate. Further discussion of FTIR as an investigative tool into the reaction of M2+-FA films with dihydrogen chalcogenides is given in later sections. 

Since smokeless powder burns by parallel layers the shape of the powder grains can have a decisive influence on the mode of burning neutral (with an approximately constant burning surface), progressive (increasing surface), degressive (decreasing surface) (Fig. 187). 

The low-temperature VEEL spectra of the species on Ru(0001) at 170 K and on Pt(lll) at 90 K are similar in contour and show broad and strong soft-mode absorptions at ca. 2610 and 2690 cm 1, respectively. The strengths of these features imply that the median plane of the flexible C5 skeleton is approximately parallel to the metal surface so that multiple C-H - M contacts are once again possible. The RAIR investigation at ca. 140 K of cyclopentane on Pt(lll) gives high resolution and shows the soft-mode absorption to be very broad. The positions of the nonagostic r>CH absorptions and of the VEELS features at wavenumbers less than 1500 cm 1 are as expected for the intact, nondissociatively adsorbed species (257, 259). 

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