# Why does vibrational frequency influence quantum tunneling of a proton?

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.

Why does a higher frequency influence the quantum tunneling effect?

$$p(E)=\exp \left(-\frac{4\pi\sqrt{m}}{h^2}\int_{x_a}^{x_b}\sqrt{V(x)-E}\;dx \right)$$
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.
• @wangge Don't know other than $k_BT/h$ has units of $s^{-1}$ but in this case is not a frequency but units of a first order rate constant. – porphyrin Jun 25 '17 at 20:34
• I believe the comparison to $k_BT/h$ is more commonly talked about as the dimensionless ratio $\hbar\omega/k_bT$. I have seen this comparison made quite a few times when discussing nuclear quantum effects, as it essentially measures whether or not we are in the classical regime where thermal effects will be most important. For something like $\ce{O-H}$, this ratio is quite large even at room temperature. Also note that this shows up in the tunneling rate given above by means of $E$ which will be quite close to the vibrational energy of the $\ce{O-H}$. Hence talk of the vibrational frequency. – jheindel Dec 12 '17 at 22:00