Transition state partition function

The minimization of the canonical transition state partition function as in Eq. (2.13) is generally termed canonical variational RRKM theory. This approach provides an upper bound to the more proper E/J resolved minimization, but is still commonly employed since it simplifies both the numerical evaluation and the overall physical description. It typically provides a rate coefficient that is only 10 to 20% greater than the E/J resolved result of Eq. (2.11). 

The expression for the transitional mode contribution to the canonical transition state partition function in flexible RRKM theory is particularly simple [200] ... 

Flexible RRKM theory and the reaction path Hamiltonian approach take two quite different perspectives in their evaluation of the transition state partition functions. In flexible RRKM theory the reaction coordinate is implicitly assumed to be that which is appropriate at infinite separation and one effectively considers perturbations from the energies of the separated fragments. In contrast, the reaction path Hamiltonian approach considers a perspective that is appropriate for the molecular complex. Furthermore, the reaction path Hamiltonian approach with normal mode vibrations emphasizes the local area of the potential along the minimum energy path, whereas flexible RRKM theory requires a global potential for the transitional modes. One might well imagine that each of these perspectives is more or less appropriate under various conditions. 

Related expressions for the microcanonical and E/J resolved transition state partition functions are given by... 

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... 

The iimer integral on the right-hand side is just e so equation (A3.11.185) reduces to the transition state partition function (leaving out relative translation) ... 

Conditions (i) and (ii), plus the usual assumption that bound degrees of freedom have quantized energy levels, lead to the familiar expression (1,29) in which the rate constant k(T) becomes proportional to a transition state partition function, Qt T), which is a sum over quantized states of an activated complex ... 

This result is quite transparent. The consequence of the energy shift Ej in the thermal trace of the CEG expression for the cumulative probability is to factor out the corresponding transition state partition function, expression for r >y (T) that... 

The contribution to the partition function of the activated complex corresponding to motion along the reaction coordinate, in this case translational motion, is factored out of the transition state partition function, <2ahb > to yield ... 

Cases of this kind have been investigated in studies of anodic hydrocarbon oxidation/ While a priori calculations of the values of electrochemical kinetic Isotope effects, particularly with complex organic molecules, cannot be regarded as quantitatively reliable owing to the difficulty of evaluation of the relevant reactant and transition state partition functions for particles in solution, nevertheless sufficient experimental data exist which, coupled with theoretical evidence of trends of values of the isotope effect for different types of reaction pathway, enable useful distinctions in mechanism to be made on the basis of H/D kinetic isotope experiments on electrochemical reactions. 

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... 

Figure 4.17. Groundstate and transition state partition functions for NO and CO on a Cu(l 11) surface computed according to density functional theory. The corresponding pre-exponents of the surface reaction rate constraints are also shown, (van Daelen, Newsam, and van Santen, 1994).

Generalized Transition State Partition Functions in Rectilinear Coordinates... 

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