# Electron hopping among molecules - Marcus equation

I'm running out of professors to talk to, and I need to clarify a couple of things for the sake of making a realistic model of electron travel through a mesh.

This is about calculations of electron hopping using Marcus theory. The Marcus equation results in a frequency of electron jumps (1/s) worked out by electric coupling of a pair of molecules, the Gibbs free energy, reorganization energy of adjacent particles, ambient temperature and the Dirac constant.

This rate of electron hopping is worked out per bond between molecule n and molecule n+1, for instance. So when an electron is 'sitting' on a molecule, there are several paths (bonds) it could follow, to go to other molecules. We're assuming that the probability of following these paths is based on the rate of hopping worked out per bond.

So when I look at one pair of molecules, a given electron hopping rate (a property of the bond) from n to n+1 should give the the percentage chance to move from n to n+1, and a percentage chance to move from n+1 to n. How can I interpret this rate quantity in order to get these probabilities?

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It seems that to get the probabilities from the rates, you'd use an expression like $P(n\rightarrow l) = R_{n\rightarrow l} / \left( \sum_{m\ne n} R_{n\rightarrow m} \right)$, that is, the probability of a particular transition is the rate of a particular transition divided by the total rate of all transitions. – KDN Dec 13 '12 at 1:50
I did that in my previous model where I simply had magnitudes of electron hopping rates around a molecule. I compared all these magnitudes one by one against the total. It turned out that the transfer rate dictated by the bond goes only one way, from n to n+1, while being unable to tell me what a probability for the travel in opposite direction would be. – Adam Dec 13 '12 at 10:41
You have to compute the rate for the transition $n+1\rightarrow n$ separately. The Gibbs free energy change will have the opposite sign from it's value in the $n\rightarrow n+1$ transition. I presume that the rest of the values in the equation will remain unaffected, i.e., the inverse electron hop uses the same reaction coordinate as the forward hop. If this is not the case, the coupling between the $n$ and $n+1$ states would also need to be recomputed. – KDN Dec 16 '12 at 3:37
In case your model is of the kinetic Monte Carlo type (plausible by your description) the only quantities you will need for the time evolution are the hopping rates R. Then the probability of a simulated jump $P(n \rightarrow l)$ is as writen above. You will have to take care that the forward and backward \it{rates} $R_{n \rightarrow l}$ and $R_{l \rightarrow n}$ (prob/s, and not the jump probabilities as such) are defined such that they obey the principle of detailed balance. – Lupercus Dec 16 '12 at 23:11