Vibrational generalized transition state partition

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... 

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... 

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... 

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... 

This entropy of activation is determined by the ratio of partition functions, which generally has a slight temperature dependence. The typical value of kc = 10-I-10 5 s 1 corresponds to a drop in AS of 65-85 cal/mol-K. Since only vibrational degrees of freedom are involved in a solid-state reaction, the sole reason for this change may be the increase in their frequencies in the transition state ... 

The rate of molecular desorption may vary over a few orders of magnitude, depending on the rotation of the molecule in the transition state. When molecules go from a rigid adsorption state to a rotating transition state, the pre-exponential factor of the desorption rate constant increases by several orders of magnitude. The partition function of the internal vibration of a molecule remains close to one because the molecular frequencies are generally high compared to kT. 

Fig. 5.19 Logarithm of the Franck-Condon factor ( f o) for transitions from the lowest vibrational level of one electronic state to level n of another state, when the transition is coupled to a single, harmonic vibrational mode with a displacement of 0.1 (squares), 0.2 (circles) or 0.5 (triangles). The vibration frequency (o) is assumed to be the same in the two states. In a transition that is coupled to only one vibrational mode, energy conservation requires that n RJ AEoJhv. More generally, is partitioned among multiple vibrational modes of the molecule and the solvent...

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