Reaction path degeneracy

Reaction path degeneracy if any is grouped into the Zit/Zi factor the transmission coefficient is here set equal to unity. 

The derivation of RRKM theory given here is also deficient because it has assumed that there is a single unique reaction path between the reactants and the products. In fact there may be several such paths related by symmetry, each with its own transition state. For example in the dissociation of CH4 to CH3 + H there are obviously four equivalent paths. The correct way to allow for this reaction path degeneracy has been the subject of much controversy [25,27,29-34]. However this was finally resolved by Pechukas [35], and the resolution of the problem is relatively simple. 

The main effect is already taken into account if symmetry numbers are included in the densities of states. The symmetry number is a correction to the density of states that allows for the fact that indistinguishable atoms occupy symmetry-related positions and these atoms have to obey the constraints of the Pauli principle (i.e. the wave function must have a definite symmetry with respect to any permutation), whereas the classical density of states contains no such constraint. The density of states is reduced by a factor that is equal to the dimension of the rotational subgroup of the molecule. When a molecule is distorted, its symmetry is reduced, and so its symmetry number changes by a proportion that is equivalent to the number of indistinguishable ways in which the distortion may be produced. For example, the rotational subgroup of the methane molecule is T, whose dimension is 12, whereas the rotational subgroup of a distorted molecule in which one bond is stretched is C3, whose dimension is 3. The ratio of these symmetry numbers, 4, is the number of ways in which the distortion can occur, i.e. the reaction path degeneracy. 

Preference for the Hofmann product olefin (i.e., the olefin with the least number of alkyl substituents and, therefore, the least stable thermodynamically) in reactions where more than one olefin may be formed (i.e., secondary and tertiary esters) was noted by DePuy and King in their early review of ester elimination reac-tions . This preference is apparent from the data of Scheer et al. (Table 3). The observed reaction path ratios (Column 2) may be compared with the simple reaction path degeneracy ratios. Column 3, i.e., statistical ratio = ratio of the number of reactive j -H atoms for each path) and with the ratios of the estimated... 

Hi) Divide the weighted symmetry number of the reactant by that of the transition state to get the reaction path degeneracy for the forward direction. 

A transition state for concerted 4-center elimination from the molecule is depicted to the right in Scheme 1, " and we consider here the reaction path degeneracy for each conformation as it passes through this transition state. Internal rotation about the phenyl-oxygen bond is considered explicitly as a... 

The Eigen model requires two different hydrogen-bonded intermediates eorresponding to the two alkene produets that are formed. Eaeh intermediate leads to a different transition state, structures 2 and 3 drawn in Scheme 2. The former, 2, leads ultimately to 1-methylcyclopentene as the final product. This transition state has no overall molecular symmetry, but retains the methyl rotor as well as free rotation about the carbon-nitrogen single bond for a net symmetry number a = 9. Because transition state 2 is chiral, the reaction path degeneracy is equal to (18/ ) = 4. 

Let us compare the ratio of reaction path degeneracies in Scheme 2 (4/3) to the ratio expected for net proton transfer via the Lewis model, which proceeds via unimolecular decomposition of a single intermediate, the quarternary ammonium ion 4 in Scheme 3. The reactive intermediate has Cs-symmetry and two internal 3-fold rotors, for a = 9. Ion 4 can dissociate via two competing transition states, 5 and 6, both of which... 

Values in parentheses are obtained by making proper corrections for symmetry (reaction path degeneracy) and subtracting from S°t, 4.0 gibbs/mole for every rotor involved. 

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). 

Other molecular parameters are obtained from CBS-QB3 ab initio calculations. Since the ab initio data do not yield the observed -factor, we adjusted the reaction path degeneracy in those applications that calculate the TST rates directly from molecular data (ChemRate, Unimol). Programs such as CARRA and MultiWell accept high-pressure rate constants in Arrhenius form as input and no adjustments were needed in these cases. The programs CARRA MSC and CARRA ME calculate the density of states from three characteristic frequencies instead of taking the complete set of frequencies, which we obtained with a separate fitting program. 

As was done in paper I for H+CH3, the free rotor and harmonic oscillator versions of Vtr have been constructed and the resulting rates for D+CH3 are included in the figure under the labels FR and HO, respectively. In paper I, these rates were shown to be very similar in value to those from "traditional" VTST calculations. The HO rate in the figure was constructed with a reaction path degeneracy of two and a quadratic form of Vtr accurate as 0 approaches the 0 = 0° line in Fig. 1. There is of course another limiting quadratic form of Vtr that arises as 0 approaches the 0 = 180° line in Fig. 1. This is not included in the Vtr used in Eq. (10) for F(R,T) but approximately accounted for by the reaction path degeneracy of two. Implicitly, all previous studies of this reaction have used this approach. However as discussed in paper I, for H+CH3 up to 10% overestimations of the rate can occur. 

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