Classical mechanics angles, conserved

The differential cross-section refers, as in the classical description, to the scattering angle in the center-of-mass coordinate system. In order to relate to experimentally observed differential cross-sections, one has to transform to the appropriate scattering angle. This transformation takes the same form as discussed previously, essentially, because the expectation value of the center-of-mass velocity V is conserved just as in classical mechanics. 

The 8-dimensional phase space of two 2-dimensional oscillators is reduced by the existence of two conserved actions, Ka and Kb, and by the absence of the conjugate angles, classical mechanical polyad 7feff. The conserved actions appear parametrically in 7feff, thus the phase space accessible at specified values of Ka and Kb is four dimensional. Since energy is conserved, in addition to Ka and Kb, all trajectories lie on the surface of a 3-dimensional energy shell. 

Wardlaw and Marcus (1984, 1985, 1988) have developed a flexible variational TS model for calculating the transition-state sum of states. This method treats the molecule s conserved vibrations in the normal quantized manner, while treating the transitional modes by classical mechanics. Thus for the bent NO2 molecule which dissociates to NO + O, three vibrations are converted into one vibration and two rotations of the NO fragment. The variables that describe the potential energy surface of the transitional modes are two bond distances, N—O and the distance between the center of mass of the NO and the departing O, as well as two angles. 

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