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I've studied Transition State Theory since I asked this a while ago, but so far I have not been able to find a detailed explanation of how tunneling corrections are implemented into reaction rates computations.

I am thinking specifically in reactions involving proton transfer along a hydrogen bond. Electrons don't pose any trouble, because they are treated quantically. But for protons, which are usually beheld as classical particles, semiclassical corrections accounting for tunneling must be made, since a full quantum treatment of protons seems undoable.

First idea: If one performs a molecular dynamics simulation, one could at each step think of the proton "just for a moment" as a quantum particle, find an approximated wavefunction thanks to the WKB method and perform a Monte Carlo simulation to decide "where" will we find the proton at the next step. This seems doable, but it's not what is done, as far as I know (sometimes performing molecular dynamics simulations is not possible, or not useful).

Second idea/ problem: Imagine we have a Free Energy surface (I am concerned with two-dimensional problems, where there are two relevant reaction coordinates to be tracked), which informs us about the precise changes in free energy along infinitely many reaction pathways. For each of those, a classical reaction rate can be computed by means of Eyring equation and then we could take averages over some sets of paths via path integrals or something like that. Even better, one can employ Variational Transition State Theory to derive estimations for the reaction rate, once the Free Energy surface is known. I expect this will not be hard. But I don't have a clue as to how to make the tunneling corrections for the reaction rate in this scenario, although I suspect is in this setting where most tunneling corrections are made.

I've certainly found a lot of bibliography dealing with Quantum Transition State Theory and Tunneling Corrections, but absolutely nowhere do they explain with all the mathematical and physical details the concrete formulas they are employing. Instead, they write a lot (without much equations), give you other references, where you can find still some more references -and so on.

So, in brief, my question is: could anyone provide me with a reference containing a thorough, complete and detailed explanation, not lacking the mathematical details, of how does one compute the tunneling corrections for reaction rate constants?

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Have you checked the original papers by C. Eckart, R. P. Bell, H. Shin, J. O. Hirschfelder and E. Wigner? These contain a fair degree of mathematical and physical treatment.

There is also a nice paper from S. G. Christov called "Some aspects of the theory of proton-transfer processes" published in J. Res. Inst. Catalysis 1968, 16 (1), 169-194 (pdf).

Other possible references:

Johnston, H. S.; Heicklen, J. Tunnelling Corrections for Unsymmetrical Eckart Potential Energy Barriers. J. Phys. Chem. 1962, 66 (3), 532–533. DOI: 10.1021/j100809a040.

Some papers published by D. G. Truhlar together with T. N. Truong, B. C. Garret, R. T. Skodje.

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