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.

Randomness

If you wonder why you lost all your money in the financial crisis or why people get it wrong when it comes to statistics, here is what could read: Fooled by Randomness and The Black Swan by Nassim Nicholas Taleb. The first one is about how we are unable to understand chance intuitively and the second one particularly about the rare event that may compensate everything that came before it. And he has a really nice writing style.

According to the "narrative fallacy" you are more likely to believe my point if I tell you a story, alright: Let's say you are playing "Mensch ärgere dich nicht" or some other game with a die. For many rounds now you have not been able to get a 6. Are you more likely to get a 6 in the next round? "Next time it really has to be a 6!" But of course the chance is still 1/6. Or to apply Taleb's street smarts logic: Someone is probably cheating and the chance for a 6 is even lower.

To stay with games, there is also what Taleb calls the "ludic fallacy": People tend to think that the kind of randomness that matters is the one that follows a known statistical distribution. But it is the events that no-one thought about that matter, or the ones that do not happen in "ten billion times the age of the universe according to our model" ... the Black Swans.

So what is the problem: of course evolution. Our rational thinking (or "Reflective System") is fairly new in evolutionary terms and it does not really play a role when it comes to making decisions (at least not as much we would think it does). What really drives us is intuition (or the "Automatic System"). And statistics in the Automatic System come in the sense of heuristics.[1] The problem is that these heuristics only really apply to Stone Age conditions. For example they don't take into account that the person on whose clothes you just spilled your coffee cup is a complete stranger that will be 5000 miles across the ocean within a few hours. So you will apologize and try to help even though this behavior is probably nicer than what would be purely rational. In this case it was good but the problem is that our intuitive heuristics fail in many cases.

Actually this brings me to another nice point which is interesting, at least from a slightly academic point of view. A major question in trading is if it is possible to be on average better than someone who buys purely at random.[2] Let's assume some stock is lower in value than it should be according to some economic considerations. If there is a way to find out, people will buy the stock and the price will go up until the stock is not cheap anymore. Therefore it is very difficult to get rich in the stock market through information unless you are either very smart or very quick. But in our real world stock prices are not ideal but biased by humanness (e.g. because of people that think that they are very smart or very quick, or generally herding phenomena, and so on). So stock prices will reflect the real value with a bias by human (Stone Age) intuition. All you have to do to get rich, is counter-intuitive trades. I like the thought but the problem in economics is always that oversimplified theories lead to elegant results but to the destruction of trillion dollar hedge funds. Anyway in essence my point is similar to what Taleb says only that he gives the examples.

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[1] Some of these things are nicely explained in the first part of "Nudge" by Thaler and Sunstein. In the second part they will probably explain how to nudge people to do the right thing but I am not there yet. "Nudge" is also a nice book but it is a bit too academic to be an easy read ...

[2] Here the point is average. You can (and should) affect the deviation by diversification. In case we are talking about your funds you should keep the deviation low. If it is someone else's money, you should gamble. If it goes up use the hype to get rich. And when it crashes, you should disappear or take the settlement, depending on your contract.

Antisymmetry

Antisymmetry, second quantization and things related to it are something I kind of stayed away so far. And now that I have finally taken a look at it, I notice how nice the math behind it actually is. And the important things that many things work in general and do not require orbitals.

We consider functions of the following form where A is some set and N a natural number[1]:

Let V be the vector space of all such functions. And let's look at operators [2] acting on V. First we could consider the Transposition operator

This is a well defined mapping and the application is straight forward, e.g.

And if you think about it some more you notice that it is linear.

And if I am correct it is both Hermitian and orthogonal with the usual skalar product. This would mean that all the eigenvalues are either 1 or -1. Either way, we are interested in the eigenspaces corresponding to the eigenvalue -1, i.e. functions that are antisymmetric with respect to the transposition. The intersection of all these eigenspaces V_A is the set of functions that are antisymmetric with respect to all the transpositions. It is a vector space since it is formed as a intersection of vector spaces.

All fermion wave functions that comply with the Pauli principle have to be taken out of this space.

Next it helps to be a little bit more general and to introduce the permuation operator (related to a permuation σ out of the symmetric group S_N) as a generalization of the transposition operator

This operator is also linear and I think also Hermitian and orthogonal.

Then you could write the antisymmetric space also like this

since every transposition is a permutation with negative sign and every permutation can be formed out of transpositions and the sign corresponds to how many you take.

Well let's define one more operator, the antisymmetrizer

You can show that this operator is a projection operator. This is equivalent to the condition that applying it twice is the same as applying it once, since the vector is already projected after the first time. The proof is also straight forward. Applying it twice leads to a double sum

But this just means that you sum n! times over all permutations, and it turns out

And next you could prove that the space it projects into, is just the V_A we had before. You have to show that any projected function fullfils the condition for V_A above and that every function which fullfills the condition remains unchanged which is both quite straight forward.

Maybe I should also add why these results are nice. I did have the expression for the Slater determinant before. But by considering that this is nothing but a (normalized) projection of an orbital product into the antisymmetric space it is easier to deal with it. It is no longer some weird expression but nothing but a linear operator. And things like adding Slater determinants are much clearer when you consider this. And that is why it is cool.

And actually as a next step you can consider commutation. These operators commute with the Hamiltonian because they are nothing but relabelling of equivalently treated electrons. Only for that reason you know that eingenfunctions of the Hamiltonian which are also antisymmetric even exist. You restrict the Hamiltonian eigenfunctions to the ones with a -1 eigenvalue of the antisymmetrizer. In the next step you may also do some restrictions according to spatial symmetry - the interacting spaces will transform like irreducible representations of the symmetry group. And finally spin. In the spin case the interacting spaces transform like representations of the unitary group (and I am kind of trying to understand why that is).

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[1] Physically spoken N is the number of particles and A is the set of possible coordinates of the particle.

[2] As I probably said before: An operator is a function that makes a function out of a function. Or if you don't like the word "function", it is a mapping between two vector spaces.

Hubble

For some reason CNN is one of the only channels that I can access from my room here in Prague that is not either in czech or featuring arabic telephone sex commercials. One thing I saw there were some pretty amazing bulls and bears that reminded me of N. N. Taleb. But I also saw some exciting space shuttle features. I like space shuttles but I am wondering what they are actually good for. (I am not an expert on space shuttles but I did once hear an astronomer talk about them.)

As I understand it, a space shuttle is like a Porsche. It's exciting, it's prestigious, but it does not really get you anywhere you could not go without it. (The advantage is that a space shuttle is not as noisy and does not cause traffic jams that I have to carefully bypass on my bicycle without collecting any mirrors.) The question that the people at CNN never asked any of the space shuttle experts is how sending up a 7 austronauts and assembling a telescope in deep space compares to assembling the telescope down here and sending it up by itself. I guess a space shuttle is a nice piece of science fiction, without the need of fiction, but probably not the most cost efficient way of having a telescope in the sky.