Understanding transition state theory

Question

Let's say we have standard chemical reaction given by $$A-B + C \leftrightharpoons [A\cdot\cdot B\cdot\cdot C]^{\dagger} \rightarrow A +B-C$$

I want to estimate the pre-exponential factor of this reaction using Transition State Theory.

From Fundamentals of Chemical Reaction Engineering by Davis and Davis, I see that $$\text{rate} = \nu _i C_{\dagger} = \nu _i \mathcal{K} C_{AB} C_{C}$$ Where $\mathcal{K} = C_{\dagger}/(C_{AB}C_C)$, and $C_i$ is the concentration of species $i$. Then we redefine $\mathcal{K}$ as $$\mathcal{K}_C = \frac{C_{\dagger}}{C_{AB}C_C} = \frac{Q'_{\dagger}}{Q'_{AB}Q'_C}$$ where $Q'_i$ is the partition function per unit volume. We redefine energies to lowest ground state, so $$\text{rate} = \nu _i C_{\dagger} = \nu _i \mathcal{K}e^{-E/k_BT}$$

We know that $Q = Q_tQ_rQ_{\nu}Q_{el}$, where $t$ is for translational, $r$ is for rotational, $\nu$ is for vibrational, $el$ is for electronic partition function. We also know that $Q'_{t} = \frac{(2\pi mk_B T)^{3/2}}{h^3}, Q_{el} \approx 1$.

Everything good so far.

My question lies with the rotational and vibrational partition function. $$Q_r = \frac{8\pi}{h^2 \sigma} (2\pi k_B T)^{3/2} (I_1I_2...)$$ where $\sigma$ is the rotational symmetry number. My question is, what is the rotational symmetry number? How do I find it? How do I find moment of inertia for molecules? More importantly, does a single atom like $C$ even have a rotational partition function? How do I calculate $Q_{v}$ for $A-B$? and $C$? The formula says $$Q_v = \prod_{s=1}^n (1-e^{-hv/k_BT})^{-1}$$

The end result says $$r = \frac{k_B T}{h} \cdot \frac{Q'_{\dagger}}{Q'_{A}Q'_{B}} e^{-E/k_BT}C_{AB}C_C$$

The final question would be, how do I simply something like this? With the $\frac{k_B T}{h}$, what exactly got simplified?

Edit: I figured out I can get moments of inertia from Calculated moments of inertia for molecules are listed in the NIST Computational Chemistry Comparison and Benchmark Database: http://cccbdb.nist.gov/).

Answer 1

So rotational spectroscopy is something one can spend a lot of time talking about so I'll try to keep this brief. Although this isn't a complete answer, it was too long for a comment, but hopefully this sheds some light on the topic.

For rotational transitions, one of the selection rules tells us that in order for EMR to induce a transition in the rotational energy levels, a permanent dipole moment must exist in the molecule (i.e. a single atom won't be observed in rotational spectra, nor would a molecule such as $\ce{H_2}$).

The rotational symmetry number is a variable that is used to account for the amount of rotations one can perform on a molecule that would make it indistinguishable from the starting point (i.e. a 90 degree rotational of $\ce{PCl_5}$ along the principal axis can be performed three times before returning each atom to its original position in space, however, each rotation produces an identical picture of the molecule each time).

Moment of inertia can be calculated from the rotational constant, however, in practice, one would usually calculate the moment of inertia with computational software and then use that to approximate the rotational constants which are employed to predict the energies and allowed transitions for the rotations of a molecule.

This is a link to an appendix with the equations used for calculating the moment of inertia in molecules based on the geometry (e.g. symmetric rotor, linear)

Here is an article that further discusses symmetry numbers and partition functions.

This one talks about rotational partition functions of diatomic gases.

This article talks about transition moment integrals and selection rules.