Transition state symmetry factor

The quantity Q tS = Qts(R T) is the pseudo-partition function of the system at the transition state, the transition state location Rt being that value of the reaction coordinate R which, at a given T, minimizes Qts(R T). This partition function has its zero of energy on the reaction path where the potential is Vt = Vrxnpath(R ) The form of Qts will be described in detail below. Qreact is the partition function of the reactants a/at is the ratio of reactant and transition state symmetry factors and ge is the ratio of electronic degeneracy factors for the reactants and transition state. The incorporation of large amplitude transition state motion is through Oxg. 

It follows that the symmetry factor for the equilibrium reaction, A fi, is given by the ratio of the forward and backward reaction degeneracy as crjub, which is correct. We see then that the reaction degeneracy has its roots in the molecular and transition-state symmetry. 

Thus, for a transition between any two vibrational levels of the proton, the fluctuation of the molecular surrounding provides the activation. For each such transition, the motion along the proton coordinate is of quantum (sub-barrier) character. Possible intramolecular activation of the H—O chemical bond is taken into account in the theory by means of the summation of the probabilities of transitions between all the excited vibrational states of the proton with a weighting function corresponding to the thermal distribution.3,36 Incorporation in the theory of the contribution of the excited states enabled us in particular to improve the agreement between the theory and experiment with respect to the independence of the symmetry factor of the potential in a wide region of 8[Pg.135]

The vibrational sum rule (Equation 4.99) applies to transition states even when one of the frequencies is imaginary (and if is negative for that frequency). In that case one finds for ki /k2, with omission of the symmetry number factor, the analogue of Equation 4.105 for the exchange equilibrium constant... 

Compound 89, though also benefiting from the aforementioned stabilizing factors of its symmetry transition state, suffered, however, from the unfavorable... 

The first term in this expansion, when substituted into the integral over the vibrational coordinates, gives )-t i( Re) <%vf I Xvi> > which has the form of the electronic transition dipole multiplied by the "overlap integral" between the initial and final vibrational wavefunctions. The (-t, i(Re) factor was discussed above it is the electronic El transition integral evaluated at the equilibrium geometry of the absorbing state. Symmetry can often be used to determine whether this integral vanishes, as a result of which the El transition will be "forbidden". 

By considering the symmetry of the normal modes of transition states Murrell and Laidler showed that problems encountered when calculating the statistical factors of transition states (which are needed to calculate the partition function in transition state theory) were associated with configurations of too high a symmetry to be transition states (61, 62). 

In order to evaluate the potential dependence of the free energy of activation, without knowledge of the structure of the activated complex, it is assumed that the electrical contribution to the standard free energy of the transition state lies between that to the standard free energy of P and that to the standard free energy of R in the rds. The symmetry factor or transfer coefficient, a, for the rds has the same properties as outlined in Sect. 3.1. 

In connection with transition-state theory, one will also occasionally meet the concept of a statistical factor [13]. This factor is defined as the number of different activated complexes that can be formed if all identical atoms in the reactants are labeled. The statistical factor is used instead of the symmetry numbers that are associated with each rotational partition function (see Appendix A.l) and, properly applied, the... 

Symmetry factors, o, do not appear in eqn. (28) because the numbers of equivalent pathways have been allowed for in the definition of the kinetic isotope effect. F0(d is the critical energy of the decomposition involving the lighter isotope and F0(II) that of the decomposition involving the heavier isotope. The density of states, N(E), of the reactant ion is, of course, common to both decompositions and does not affect the intramolecular kinetic isotope effect. The intramolecular kinetic isotope effect is, therefore, dependent only upon transition state properties. 

---