# Tunneling in chemical reactions

We know that quantum tunneling is the reason behind several natural phenomenon like alpha decay and thermonuclear fusion inside the stars. How can it influence chemical reactions by tunnelling a species through the activation energy? If so how does it influence the kinetics and fraction of molecules taking part in the reaction.

The probability of tunnelling at an energy $E$ is given by $p(E)\approx e^{-bA\sqrt{m}}$ where $A$ is proportional to the area of the potential energy barrier above energy $E$, i.e. the top part of the potential barrier, $m$ the mass and $b$ some constants, $\pi, \hbar$. etc. Thus for a given mass and energy if the barrier is narrow, so $A$ is small, tunnelling is more likely than if the barrier is wide. At a given energy for the same barrier if the mass is large tunnelling is small. Thus we tend to see tunnelling only with H and D and not Cl atoms for example. Tunnelling is also important in electron transfer reactions.

As there is not just a single energy in a reaction but a distribution of energies, according to the Boltzmann distribution, it is necessary to modify the expression above to average over the energy but the basic result is the same, which is that the reaction rate constant is reduced by the factor $p(E)$.

[If the potential energy barrier is $V(x)$ then $\displaystyle p(E)=\exp\left(-\frac{2\sqrt{m}}{\hbar}\int_{x_1}^{x_2}\sqrt{V(x)-E}\right)$ where $x_{1,2}$ are the points either side of the barrier with energy $E$.]

Quantum tunneling in chemical reactions will only play a role if the collision energy and the spread in the collision energy of the reaction are very low. Just like in nucleosynthesis, the tunneling probability of a particle through a barrier only gets an appreciable value if the collision energy is very close to a quasi-bound state of the reaction complex (as in the triple alpha process). At this energy, this results in a large increase of the cross section (a scattering resonance), and thus the reaction rate (which is the product of the cross section and the relative velocity of the reaction partners). At high temperature, however, you can no longer speak of a collision of a single energy because different collision energies are mixed (to form a wave packet, if you like). This mixing of collision energies results in a mixing of the cross sections and washes out the sharp resonance of the single reaction channel. Resonances do play a role in astrochemistry where reactions happen at very low temperature and have been observed in the laboratory under controlled conditions, but are in general not important under normal conditions.