Reaction coordinate degeneracy

Motion along the reaction coordinate was limited to classical mechanics, whereas the sum and density (or, to be precise, the degeneracy) of states should be evaluated according to quantum mechanics. The integral in Eq. (7.49) should really be replaced by a sum N (E) is not a continuous function of the energy, but due to the quantization of energy, it is only defined at the allowed quantum levels of the activated complex. That is, the sum of states G (E ) should be calculated exactly by a direct count of the number of states ... 

It can be seen that, as a result of the allyl nature of the 1-2-3 array, there is no degeneracy along the reaction coordinate. With the excited state of reactant affording the lowest states of the two free radical primary products, the reaction is allowed. 

Fig. 11. Reaction coordinate diagram for ECL system involving rubrene (A) and 9,10-diphenylan-thracene (B). Potential energy curves are presented in the zero-order approximation, without removing the degeneracy at the crossing points of the potential energy curves. Broken lines represent the vibronically excited triplet state.

This problem is related to the question of appropriate electronic degeneracy factors in chemical kinetics. Whereas the general belief is that, at very low gas pressures, only the electronic ground state participates in atom recombination and that, in the liquid phase, at least most of the accessible states are coupled somewhere far out on the reaction coordinate, the transition between these two limits as a function of solvent density is by no means understood. Direct evidence for the participation of different electronic states in iodine geminate recombination in the liquid phase comes from picosecond time-resolved transient absorption experiments in solution [40, 44] that demonstrate the participation of the low-lying, weakly bound iodine A and A states, which is also taken into account in recent mixed classical-quantum molecular d5mamics simulations [42. 43]. 

Thus, reaction rate coefficients can be estimated from the thermochemistry of the transition states, whose molecular properties can be calculated with quantum chemical programs. In calculating reaction rate coefficients, the only negative second derivative of energy with respect to atomic coordinates (called imaginary vibrational frequency ) from the transition state is ignored, so that there are only 37/-7 molecular vibrations in the transition structure (37/ — 6 if linear) and all internal and external symmetry numbers have to be included in the rotational partition functions (then any reaction path degeneracy is usually included automatically). 

The reaction has a factor of 12 degeneracy of the reaction coordinate. The high preexponential factor of 8 X 10 s is due to the high degeneracy rather than to a high, abnormal entropy of activation. The calculated A5 was found to be only 3 cal/ (K mol). An additional two-step pathway via propylidene (H3C—CH2—HC ) formation as an intermediate that was proposed by Hettinger et al. has 2 kcal/mol higher barrier (of 67 kcal/mol) [37]. 

The various methods of the quantum chemical calculations and the evaluation of rate parameters of the various elementary steps on the potential energy surface are discussed in detail. The evaluation required the knowledge of the energetics of the transition states and the intermediates, and their entropy, the degeneracy of their reaction coordinates,... 

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. 

Both H and HI2 have electronic degeneracies of 2 since each has a single unpaired electron being a symmetric diatomic molecule, has a symmetry number of 2. The diatomic molecule I2 has two rotational and one vibrational degree of freedom. The linear complex HI2 has two rotational and four vibrational modes because one possible vibrational mode corresponds to the reaction coordinate, three are true vibrational degrees of freedom. From (9.14) the preexponential factor B(T) is... 

E = total energy h = Planck constant kg = Boltzmeinn constant k(B) = unimolecular micro canonical rate constant k(7) = canonical rate con stant Nq = initial number of ions formed M E - E< = number of states of the transition state up to - Elj above the critical energy E P E) = distribution of internal energies R(E,tj = rate of dissociation T = temperature Af = activation enthalpy A5 = activation entropy A5j = microcanonical entropy of activation Vr = reaction coordinate frequency p E) = density of states of the parent ion at internal energy E o= degeneracy of reaction path. 

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