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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.

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    $\begingroup$ 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. $\endgroup$
    – ProfM
    Oct 22, 2020 at 20:05
  • $\begingroup$ +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)? $\endgroup$
    – Nike Dattani
    Oct 23, 2020 at 3:21
  • $\begingroup$ 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. $\endgroup$
    – Tristan Maxson
    Oct 23, 2020 at 15:30
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    $\begingroup$ Related: en.m.wikipedia.org/wiki/Kinetic_isotope_effect $\endgroup$
    – Tyberius ♦
    Oct 31, 2020 at 5:11
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    $\begingroup$ 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. $\endgroup$
    – Antonio de Oliveira-Filho
    Oct 31, 2020 at 10:48

1 Answer 1

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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.

As your question also mentions Density Functional Theory, I would like to point out that in the Born–Oppenheimer approximation the potential energy surface is independent of the nuclear masses, so all isotopic effects are handled as shown above if one uses transition state theory. Even if the DFT description of the PES is very wrong, I think, that overall trends for kinetic isotope effects would be OK.

For some examples of transition state theory calculations of kinetic isotope effects, I recommend the following papers:

  1. E. F. V. de Carvalho, G. D. Vicentini, T. V. Alves, O. Roberto-Neto, Variational transition state theory rate constants and H/D kinetic isotope effects for CH3 + CH3OCOH reactions. J. Comput. Chem. 41, 231–239 (2020).

  2. L. Simón-Carballido, T. V. Alves, A. Dybala-Defratyka, A. Fernández-Ramos, Kinetic Isotope Effects in Multipath VTST: Application to a Hydrogen Abstraction Reaction. J. Phys. Chem. B. 120, 1911–1918 (2016).

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