Why does vibrational frequency influence quantum tunneling of a proton?


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> Furthermore, consideration of the tunneling in the framework of the Wigner theory assumes that the tunneling factor is small. However, the $\ce{O-H}$ frequency is very high ($\gg k\_\mathrm{B}T/h$), and the H atom tunneling under a potential energy barrier when proceeding either from a ground or from other excited vibration levels may be significant.
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**Source:** [German, E. D.; Sheintuch, M. Kinetics of Catalytic OH Dissociation on Metal Surfaces. *J. Phys. Chem. C* **2012,** *116* (9), 5700–5709. DOI:10.1021/jp2106499](http://doi.org/10.1021/jp2106499).


Why does a higher frequency influence the quantum tunneling effect?




It is not the frequency *per se* but the mass that is important,; the probability of tunnelling an energy *E* is given by 


$$p(E)=\exp \left(-\frac{4\pi\sqrt{m}}{h^2}\int\_{x\_a}^{x\_b}\sqrt{V(x)-E}\;dx \right)$$


where $x\_a$ and $x\_b$ are two points either side of the potential barrier through which tunnelling occurs. The integral is in effect a measure of the area of the potential $V$ above position $x\_a$ to $x\_b$ through which tunnelling can occur.


As *m* increases, for a given potential barrier and energy, the tunnelling probability falls. 


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