Transition state conserved modes

Each reactant state correlates with some state of the products along the potential. Vibrations and rotations that are similar in the reactant and product (conserved modes), remain in the same quantum state throughout the channel, in the sense that their quantum numbers remain the same throughout. Other modes that change between reactants and products (transitional modes), are subject to correlation rules. Channels with the same angular momentum are not permitted to cross, similar to the non-crossing rule in diatomic molecules. 

For the calculation using Variflex, the number of a variational transition q uantum s tates, N ej, w as given b y t he v ariationally d etermined minimum in Nej (R), as a function of the bond length along the reaction coordinate R, which was calculated by the method developed by Wardlaw-Marcus [6, 7] and Klippenstein [8]. The basis of their methods involves a separation of modes into conserved and transitional modes. With this separation, one can evaluate the number of states by Monte Carlo integration for the convolution of the sum of vibrational quantum states for the conserved modes with the classical phase space density of states for the transitional modes. 

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. 

The transitional mode Hamiltonian is given by the last four terms in equation (7.36). In the work of Wardlaw and Marcus, the phase space volume for the transitional modes was calculated versus the center of mass separation R, so that R is assumed to be the reaction coordinate. In recent work, Klippenstein (1990, 1991) has considered a more complex reaction coordinate. The multidimensional phase space volume for the transitional modes can not be determined analytically, but must be evaluated numerically, for example, by a Monte Carlo method of integration (Wardlaw and Marcus, 1984). The density of states is then obtained by dividing the phase space volume by h", where n is the dimensionality of the integral, and differentiating with respect to the energy. The total sum of states of the transition state is obtained by convoluting the density of the transitional modes with the sum of the conserved modes, N(E,J) so that... 

It is most revealing to compare the best RRKM calculation with a fixed transition state with the VTST result. This is shown in figure 7.20. The improvement offered by the variationally located TS is remarkable. To be sure, this is a somewhat extreme example because the transitional modes outnumber the conserved modes and the latter hardly contribute to the density of states since their frequencies are so high. 

A VRRKM/ECC model for product vibrational and rotational distributions was introduced by Wardlaw and Marcus (1988). Subsequently, Marcus (1988) constructed a refined version which successfully describes rotational quantum number distributions of products arising from the decomposition of NCNO (Klippenstein et al., 1988) and CH2CO (Klippenstein and Marcus, 1989). In the latter model, the conserved modes are assumed vibrationally adiabatic (as in SACM) after passage through the transition state and, consequently, the distribution of vibrational quantum numbers for the products is the same as it is at the transition state. The transitional modes are assumed nonadiabatic between the variationally determined TS and a loose TS located at the centrifugal barrier. [These are the same two transition states associated with the TS switching... 

These results are interesting because neither the frontier orbital method nor arguments based on the conservation of orbital symmetry during reactions can account for the difference between the two modes of the Cope rearrangement. According to these theories, rearrangement via boat and chair transition states should be equally allowed. 

For reactions where Ri and R2 are both diatomic and the reaction proceeds through a linear transition state, there will be six vibrations in the transition state of which two will be conserved and are likely to have high frequencies, close to those for the vibrations in Ri and Rj, whereas the other four vibrations are transitional and will be low frequency bending modes. The partition functions (q. ns vibtR Rf and q ibtR will all have values close to unity,... 

Study. The reason is that transition state theory concentrates on the behaviour of the system up to the dividing surface, whereas the product energy disposal is determined by the potential felt by the fragments as they separate. However, Marcus, Wardlaw and Klippenstein have proposed an extension that describes the evolution of the transitional modes along the reaction coordinate and that, at the same time, conserves angular momentum. 

A strictly loose transition state is defined as one in which the conserved vibrational modes are uncoupled to the transition modes and have the same frequencies in the variational transition state as in the associating reagents.(Conserved vibrational modes are modes that occur in both... 

The adiabatic separation between the reaction coordinate and all other F — 1 vibrational degrees of freedom means that quantum states in those modes are conserved through the reaction path. With this approximation, we can label the levels of the generalized transition states in terms of the one-dimensional vibrationally and rotationally adiabatic potentials... 

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