Vibrational generalized transition state

The classical phase-space averages for bound modes in Eq. (11) are replaced by quantum mechanical sums over states. If one assumes separable rotation and uses an independent normal mode approximation, the potential becomes decoupled, and onedimensional energy levels for the bound modes may be conveniently computed. In this case, the quantized partition function is given by the product of partition functions for each mode. Within the harmonic approximation the independent-mode partition functions are given by an analytical expression, and the vibrational generalized transition state partition function reduces to... 

Once the reaction path and the force-constant matrix at points along the reaction path have been found, one can use improved versions of transition-state theory (TST) called generalized transition-state theory to calculate rate constants more accurate than those given by TST [see references cited in D. G. TVuhlar, R. Steckler, and M. S. Gordon, Chem. Rev., 87, 217 (1987)] and can construct a reaction-path Hamiltonian and use it to study such things as vibrational-energy transfer during the reaction [W. H. Miller et al., J. Chem. Phys., 72,99 (1981) W. H. Miller, J. Phys. Chem., SI, 3811 (1983)]. 

F-1 generalized normal modes (F=3N-5 or 3N-6 for N-atom generalized transition states that are linear or nonlinear, respectively). Each term Bj p(s) can be written in terms of the scalar product of the generalized normal mode vector of vibration k and the derivative of the gradient (representing motion along the reaction path) with respect to the reaction coordinate at s. 

In many respects the vibrations of generalized transition states are like those of ordinary molecules, and thus the generalized-... 

This approach has already been shown to provide accurate results for the vibrational partition functions of the bound molecules H2O and S02, and eq. (53) should be equally applicable for generalized transition states. The harmonic partition functions are given bySl... 

We have concentrated on vibrations in this section and have net considered hindered rotations, Coriolis coupling, or related complications. These kinds of complications will be at least as important for generalized transition states as for bound molecules, and these complications will have to be addressed in future work. 

To study the effects of incorporating the anharmonic nature of the generalized normal modes transverse to the MEP on the vibrational partition function factor, Q° (T,s), in the generalized transition state partition function, Q (T,s), in eq. (4), we computed at the saddle point of surface 5SP from sets of either harmonic or anharmonic bound vibrational energy levels E /hc (in wave numbers) [176], where Vj,...,V5 are the vibrational quantum numbers and the energy is measured relative to the saddle point (i.e., from the bottom of the vibrational well). That is, we take... 

The generalized transition state number of states needed for microcano-nical variational theory calculations counts the number of states in the transition state dividing surface at s that are energetically accessible below an energy E. Consistent with approximations used in calculations of the partition functions, we assume that rotations and vibrations are separable to give... 

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... 

For examples of applications of the unified statistical model employing this kind of ad hoc quantization, see B. C. Garrett and D. G. Truhlar, Generalized transition state theory. Quantum effects for collinear reactions of hydrogen molecules, J. Chem. Phys. 83 1079 (1979) 84 682(E) (1980) Application of variational transition state theory and the unified statistical model to H + CI2 -> HCl + Cl, J. Phys. Chem. 84 1749 (1980) B. C. Garrett, D. G. Truhlar, R. S. Grev, and R. B. Walker, Comparison of variational transition state theory and the unified statistical theory with vibrationally adiabatic transmission coefficients to accurate collinear rate constants for T + HD TH + D, J. 

H(v =l) + H reactions would be less than the conventional transition state theory rate constant. Generalized transition state theory would clearly be needed to describe the excited state reaction (of course summed over the final vibrational states). 

This general technique has been applied also to the direct observation of vibrational motion in bound electronic states. Although no transition states as such are involved in vibrational motion there is, again, a transitory change of intemuclear distance. 

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