Concluding Remarks and Examples

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 

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This section is not intended as a list of all the benefits of interphase formation. They are the subject of this book and the properties of materials are discussed in detail in the individual chapters. It is not appropriate to identify a single property of a material as the most significant. Here are some concluding remarks and examples of other benefits. This will perhaps show that interfacial interactions are not used only for reinforcement. 

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