Marcus-Levich-Hush equation

Here is an interesting piece of math:

Or in other words show that in the Marcus-Levich-Hush theory, electron transfer stops when you approach zero temperature. Mathematica tells me that this is the case but I could not really show it. The problem has the following type:

It is not an apparent l'Hospital application and I do not really know what else to do. Anyway, I believe Mathematica and it is also the thing you'd expect physically.

In the full quantum picture you actually have a non-zero transfer rate even when approaching zero temperature that comes from nuclear tunneling (as was derived here and is reviewed here).

Interestingely it is quite easy to find derivations of this quantum formula. But I did not find much about the semi-classical formula, that is shown above. Only that it is some kind of application of Fermi's golden rule which apparently gives this prefactor for the rate equation.

Excited state intramolecular proton transfer

Actually there is a second article with my name on it that just came out. So I'll use the chance to advertise for it a little bit as well. It is a review (at the time of submission also partially a preview) of excited state intramolecular proton transfer cases. The context is that it is part of a special birthday issue. In fact it is kind of interesting how professors have symposia instead of birthday parties and give each other journal issues for presents ... but that is a different question.

This figure, which is similar to the graphical abstract, kind of shows what's going on: You have a molecule with an intramolecular hydrogen bond (the red atom is typically oxygen, the green one oxygen or nitrogen). You excite with a high frequency UV-photon. Then in an ultrafast transfer process with no or almost no barrier the proton moves to the other side. The new tautomer may either emit a photon with strongly Stokes shifted fluorescence (as for example bipyridyl-diol from the last post) or it may exhibit internal conversion (as e.g. 2-(2'-hydroxyphenyl)-benzothiazole in gas phase).

Bipyridyldiol

Nice, I just noticed that my Master's thesis article is already available online [1]. Well, you can take a look at it if you are interested in excited state proton transfer or if you want to know what I did for my Master's thesis. I'll put some additional things here that did not make it there. This is basically what happens:


Bipyridyl-diol has two intramolecular hydrogen bonds. You excite it with UV light wait a few femtoseconds and the protons get transferred. It was understood that the double-transfer product DK is finally formed. The main question was wether there were sequential and/or concerted transfers. The general idea was that there would be a branched reaction path: An ultrafast (100 fs) first step that was either a single or double proton transfer and a "very fast" (10 ps) step from MK to DK. According to us it looks more like there is no branched reaction but rather a dynamical equilibrium between MK and DK that cools toward DK. Well I hope some experimental groups are still interested enough in this system to test for this hypothesis.

This is one trajectory, a simulation of the molecule for 300 fs after UV excitation.[2] You can see a very quick initial transfer and then some more transfers.



Actually I wanted also to show the development of the normal modes in the video. To compare them with the results of Stock et al.'s experiment. But this does not seem to work out here because the videos need to have a fixed 4:3 format. So I'll just show a figure. The important thing is that there is strong participation of the totally symmetric modes (blue, red) even if the process does not conserve the symmetry. Another very interesting thing is that activation of the non-totally symmetric (black) mode is a violation of the Franck-Condon rules. A way to explain this is that the Franck-Condon rules work only under ideal assumptions and not with a strongly anharmonic reactive potential.

Here is another trajectory for comparison. In this case the second proton transfer occured only a little bit later.




Actually another nice figure would be this one. What I am doing is projecting the trajectories onto a normal mode. And then I can average for every time step over the 36 trajectories that we ran. This time-dependent average should represent the coherent motions. Here I am showing 17a_g, an aromatic breathing vibration, which is the classical case for a coherent Franck-Condon excitation (in the context of proton transfer the lower frequency skeletal modes were of more interest). In the harmonic vibrational analysis that we did at the DK equilibrium geometry, the mode has a wavenumber of 682/cm. This corresponds to a period of about 49 fs. Well and there really is a coherent oscillation with just that frequency. So we see that the harmonic vibrational analysis at the minimum and the dynamics nicely work together. If I compute the standard deviation over time of this time-dependent average then I get only one number per normal mode. These numbers are what we are showing in Fig. 10. And by the way: The tools to do this are in the new Newton-X version (aside from many other nice things ...).

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[1] And interestingly there is a direct link to facebook which of course I had to click.

[2] One of these 300 fs RI-CC2/SVP-SV trajectories takes about a month on one processor.

Excited state H transfer

I liked the introduction of this article by Röthlisberger because it nicely explains the processes in excited state proton or hydrogen atom transfer. Most of it is well explained in this Figure.[1]



In the ground state the n and π orbitals (shown on the left-top and right-bottom) are each doubly occupied. Excitations into the two virtual orbitals shown (left-bottom and right-top) lead to three states of interest: ππ*, nπ*, πσ*.

In the ππ* and nπ* states it can be seen that electron density is shifted from the O to the N. This increases both the acidity of the O and the basicity of the N. In the cluster shown this induces proton transfer through the ammonia molecules. Actually there is a very nice movie showing this transfer in their supporting information.

The situation is completely different in the πσ* state. If the anti-bonding σ* orbital is populated, the bond is no longer stable. The molecule stabilizes by dissociation of a hydrogen radical (i.e. hydrogen atom). If the hydrogen atom takes part in a hydrogen bond, you can have excited state hydrogen atom transfer. The orbital corresponding to this is shown in the left bottom. It is very diffuse and has probably also some Rydberg character.

When I decided to write the post, I had only read the introduction which is very nice and helped me finally understand the difference between excited state proton and hydrogen atom transfer. The part that seems kind of strange is that they only computed one trajectory. And for this one trajectory they had 1024 processors on a Blue Gene/L. With atom centered basis sets in Turbomole you could almost do it in real time if you had 1024 CPUs (or pretty fast anyway).[2] So it may be a problem with the plane waves. Or I got something wrong. Or I am just jealous because I never had 1024 CPUs at my service.

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[1] Call it advertisement ...

[2] Not quite real time as one CPU cycle is already about 3 orders of magnitude longer than the process observed.