CESTC 2007

CESTC 2007 is over. It was pretty nice, of course only for scientific reasons ... Anyway what I am going to talk about is only the scientific part. I want to cover a few lectures during the next days.

The first one has the advantage of being mostly mathematical and you don't need to know a lot about theoretical chemistry, but you need linear algebra. I always thought that in theoretical chemistry everything was hermitian or unitary and you didn't need to worry about singular value decomposition. But I guess I don't have an excuse anymore to skip that in my linear algebra script when I will do the exam.

It goes like this: When using configuration interaction singles we have to use a n_occ x n_virt (numbers of occupied and virtual orbitals) matrix C that represents the weights of all possible excitations.




where c_ij represents the weight of the component where excitation goes from the i^th occupied to the j^th virtual orbital.

This is not a very handy information especially when you try to visualise the orbitals.

But we know that for every real matrix orthogonal matrices U and V exist with the following property:


with



We only have to consider n=min(n_occ, n_virt) excitations any more. The orbitals they are taken from and given to, are the columns in U and V respectively. Typically only one singular value λ_i deviates much from zero. And the corresponding MOs are localised on the chromophores.

Give it up for linear algebra!

For more information read I. Mayer's article.

If you want some homework, you can think about how this works for a 4-dimensional CISD excitation tensor.

Still living

What are readers trying to tell me that I get more hits than ever before now that I am not writing? Well, anyway I feel like supporting you with some more important bits of information. These days I am doing some summer research. I can't tell you what I am doing because you might steal the information, publish, and get famous. But I can tell you what it is like. I don't lead the easy life of synthetic chemistry. I don't get to set up a synthesis in the morning and wait around all day until it is done. Life in my field is much rougher [1].

The nice thing about computations is that no work that's older than 10 years is any competition. It's not the fault of the people from 10 years ago, they were alright. They just did not have the computers. I remember our first PC from that time was 20 times slower and 5 times as expensive as mine is now. For the same price computational power increased 100-fold. That's the difference between running five jobs over night or running a PC for a whole month. Thanks to anyone who is funding me through the computer game industry.

Next week I am going to the Central European Symposium for Theoretical Chemistry. That makes me feel pretty smart. I don't know how much I am going to understand. But I can spell the word symposium: symposium.

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[1] A computational chemist sets up a computation in the morning and sits around all day until it is done.

Hückel package (2)

I added a graphics function to the Hückel theory package that I have introduced in my last post. To some people visualising a graphical scheme is more meaningful than just numbers. If you are one of them: here you go.

This is the π energy scheme of benzene. Half of its orbitals are neatly filled with 6 electrons, 4*1 + 2 electrons or an uneven number of electron pairs, however you want to put it. On a second look you notice that the energy scheme is a hexagon, just like the molecule.

import hueckel
h_mol = hueckel.hueckel('***/benzene.mol',\
     calc_everything=True,   print_results=False)
h_mol.show_energy_scheme(can_width=300, can_height=300,\
                bg='white', arrow_length=.2)

If we take a carbon away (but no π-electrons) we have a cyclopentadienyl-anion. With 6 π-electrons, we have again the feeling of a filled subshell (or stability). A second look shows us that the energy scheme is a pentagon.

What does the energy scheme of cyclobutadiene look like? Yes - a square on its corner. But with 4 electrons we have the half-filled subshell which is only stable in transition metals, not here.

In fact it is true for every monocyclic conjugated hydrocarbon that the energy scheme looks like the corresponding regular polygon on its corner. Isn't that pretty cool?

But it kind of loses its magic once you put it in mathematical terms. For a monocyclic system of n C-atoms the energy of the k^th π-orbital is given as:




This corresponds to the polygon on its corner (in the scheme I set α=0 and β=1). Since the polygon is on its corner we need an odd number of electron pairs (4n+2) electrons to fill it properly. That's Hückels rule.

As far as I know there is nothing like Hückels rule for anything else than monocyclic systems. Both benzopyrene and biphenyl are totally happy with an even number of electron pairs.

By the way: all this was an ab initio calculation, I would say. No empirical parameters, just symmtery considerations and plausible simplifications. For more information you can read about the Hückel method at wikipedia or in a quantum chemistry book.

Finally you may ask yourself: Is a cyclopentadienium-cation stable? Let's try it out!

    h_mol = hueckel.hueckel('***/03-c-pentadienyl.mol',\
                  calc_everything=True, print_results=False)
  h_mol.set_total_charge(1)
  h_mol.show_energy_scheme(can_width=300, can_height=300,\
                               bg='white', arrow_length=.2)

Of course not.

The package can be found on my homepage under "Python scripts". When I was little I sometimes used to like writing German programs in Visual Basic. They can be found there, too. But there are two language barriers to overcome: German and Visual Basic.