# Isotopes in Reaction Kinetics by Transition State Theory

Can kinetic isotope effects be handled in density functional theory level calculations? My intuition says that in the context of molecular dynamics simulations that they can be, but can transition state calculations reveal anything about kinetic isotope effects?

For example the relationship between the activation energy and rate is commonly used as follows.

$$k = Ae^{-E_{a}/RT}$$

Isotope effects must be folded into either the prefactor $$A$$ or into the entropy change of $$E_{a}$$. However it is unclear to me right now how this would actually be done, even when considering just trends in the reactivity. I suspect that entropy is not the only thing affecting things, but it is unclear where the correction to $$A$$ would come from.

• I am no expert in reaction kinetics, but isotopes are essentially nuclei of different mass, so any calculation that takes into account the motion of nuclei (phonons, MD) can describe isotope effects. My guess is therefore that, if you include nuclear motion contributions to the transition state calculation, then yes, you should be able to study isotope effects in this phenomenon. If you only consider electronic contributions, then you cannot say anything about different isotopes. – ProfM Oct 22 '20 at 20:05
• +1, but can you also clarify what you mean? QTST (quantum transition state theory) will depend on the isotopes if the isotope masses are taken in to account, and for classical TST do you mean configuration-space TST, or the formally exact phase-space TST, or one of the other classical TSTs. Also, what exactly are you asking: do you want to know if the TST rate constant depends on isotope effects? What is your formula for the TST rate constant (this might clarify precisely what you mean by TST)? – Nike Dattani Oct 23 '20 at 3:21
• I will clarify the question further in a bit, I won't have time till later tonight. If its too vague, we can close it temporarily. – Tristan Maxson Oct 23 '20 at 15:30
• – Tyberius Oct 31 '20 at 5:11
• TST based on DFT calculations can treat kinetic isotope effects. The rate constants depend on the partition functions that change with isotopic substitution. I am on my cell now and can post a complete answer with examples, if someone else does not beat me to it. – Antonio de Oliveira-Filho Oct 31 '20 at 10:48

The canonical transition state theory expression for the thermal rate constant, with tunneling correction, is $$k(T) = \kappa(T) \frac{k_\mathrm{B}T}{hc^\circ}\exp [-\Delta^\ddagger G/RT]$$ which is just the Eyring equation with a tunneling correction $$\kappa(T)$$. The Gibbs free energy of activation, $$\Delta^\dagger G$$, is calculated using the partition functions (translational, vibrational, and rotational) of the transition state and reactants. These depend on the masses of the atoms, so this is how the kinetic isotope effects are included.
There is also tunneling. If we use the simple semiclassical Wigner correction $$\kappa^\mathrm{W}(T) = 1 + \frac{1}{24}|\hbar\omega^\ddagger/k_\mathrm{B}T|^2$$ where $$\omega^\ddagger$$ is the imaginary frequency of the transition state, we can see that an isotopic substitution would change the frequency and, consequently, the rate constant.