# Units on rate constants from transition state theory and DFT

I am using density functional theory (DFT) to compute the energy profile and, by extension, kinetic parameters for a catalytic process. I know that the rate constant from transition state theory (TST) can be found via the Eyring equation:

$$k=\frac{k_{B}T}{h} e^{-(\Delta G^\circ)^\ddagger/RT}$$ I believe that the derivation of this equation from TST implies that it strictly holds for reactions of the form $A\leftrightarrow [AB]^\ddagger\rightarrow P$. It's clear by looking at the units that other molecularities won't work with this equation directly (for example, $A \leftrightarrow A^\ddagger \rightarrow P$).

My concern arises because I am using DFT (specifically Gaussian) to compute the thermochemistry (as discussed here). I am able to generate a free energy landscape for my proposed mechanism, but I am concerned about the accuracy of using the Eyring equation for various molecularities.

It's clear to me how the units work out when looking at this from the perspective of partition functions. From TST, we can say that

$$k=\frac{k_{B}T}{h} \frac{Q^\ddagger}{\prod_{i}Q_{i}} e^{-E/RT}$$

for $i$ reactants, where $E$ is the energy computed from DFT, and

$$Q\equiv \frac{q_{t}}{V}q_{r}q_{v}q_{e}$$

such that the translational partition function is effectively written in units of inverse volume. So, changing the number of reactants $i$ thereby changes the units of $k$ appropriately.

Where is the disconnect between this expression for $k$ and the Eyring equation I originally show? Presumably I can use the above expression for $k$ since the partition functions are used by Gaussian to compute the Gibbs free energies, but it would be much easier to use the Eyring equation directly, especially if I am making a Gibbs free energy profile anyway.

I'm familiar with the related question posted on StackExchange in the past. However, it does not address how to actually approach various molecularities.

• The first equation is only applicable to first order reactions $\ce{A \rightarrow B}$ since the rate constant has units $\pu{sec^{-1}}$ , (although it is widely quoted as otherwise). Using the ratio of partition functions Q is the correct way to go for other reactions. – porphyrin May 14 '17 at 9:37