Burnside's lemma

Right at the merge between chemistry and mathematics lies Burnside's lemma, group theory at its best.

Alright, Ambrose Burnside did not invent sideburns and Burnside's lemma. In fact not even William Burnside that it is named after, came up with it. Cauchy and Frobenius knew about it before. But I don't think they are making a fuss about it. Maybe because they are famous anyway [1].

Burnside's lemma can be used for determining isomeric structures. It is a way to systematically find the number of isomeric structures or possible different substitutions. It is a rather laborious way of determining isomeres but it helps you make sure you are not missing anything.

Today I am trying to show what's happening. But you don't really need to know that to apply Burnside's lemma to a problem. Later I am thinking about showing two examples: dichlorobenzenes and dichloro-n-hexanes.

First you need a molecule (let's say benzene) and a substition rule (dichlorobenzenes). Now we look at the molecule fixed in space and make all possible substitutions. This set we call X. It has 6c2 = 15 elements. .

We are not actually interested in this set X. What we would like to know is how many "different" structures there are, meaning structures that cannot be transformed into each other.

To define "different" we need a group G of operations g (bijective functions that act on X). This group G is like the symmetry group, but it makes sense to modify it. First you shouldn't include reflections in order to keep chiral structures. Second you have to include rotations over σ-bonds (unless your are interested in conformeres). [2]

No we can say that two structures in X are different if there is no operation in G that transforms one into the other. X is divided into disjoint subsets of structures that can be transformed into each other by operations in G but only inside the subset. These subsets are called "orbits" [3]. Each one of these orbits corresponds to one isomeric structure (in the example we have ortho, meta, and para orbits each consisting of 5 fixed stuctures).

In other words: We have a set X and a group G acting on it. We are interested in the number of orbits |X/G|. Burnside's lemma states that this number can be found using "fixed sets" X^g. A set X^g consists of all elements in X that stay the same when g is applied (e.g. all the para structures would be the fixed set of the 180° rotation). The proof is not obvious [4] but apparently the following is true:


What we have to do is look at all operations g in G and count the number of elements of X that stay the same when g is applied. From this we know the number of isomeric structures.

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[1] Those days were cool when one single person could come up with hundreds of different new things.

[2] For picky readers: Originally you define the operations as something that is being done to the unsubstituted molecule. The operations in G are X->X (between the fixed structures) but that's just a group isomorphism I think.

[3] Readers that want to keep their chemical maths up can think about how orbits and irreducible representations are related. I think that every irreducible representation is also an orbit.

[4] In other words: I don't know. Or I am just scared of weird terms and symbols (which people are too often I think). Anyway, as long as I can put my problem into mathematical terms I can just ask any mathematician to solve it for me.

Protein purification program

In case you are interested in protein purification, you can check out this computer program. We are doing this in our biochemistry lab because actual protein purification would take too much time. You can see that my school isn't really focused on biochemistry. But we are growing actual molds tonight.

Shaping copper

I thought I'd hate inorganic technology. But it is one of the coolest labs I've ever done. If you know anything about inorganic technology, you can skip this post. But if you think of metals as one big homogeneous phase (like I did) then you may want to take a closer look at metal microstructure.

Normally I am kind of sad that I spend so much time on lab reports that only one person will ever read. So I am happy that I can talk about it a little bit here.

What we did is compress copper and look at its grains. The interesting thing is that all the properties change with rolling. Especially the hardness increases. If you heat it up you can undo parts of those changes depending on temperature and time.

This is the copper as we started out with it. You can see a few large grains. The black spots are Vickers hardness imprints.


If you compress the thing by 30% you notice that the grains are looking in the same direction.


60% compression and everything looks the same direction.
85% and everything looks pretty flat.

You can't roll grains that are flat like this. So what do you do if you want the plate even thinner? Heat it up to 673 K, leave it there for 0.9 ks [1]. And you can pretty much start over again.
To get images like these you have to cut your sample with a diamond saw. Then you have to grind and polish it with SiC and diamond. After that we etched the surface with FeCl₃ and took pictures with a microscope that has a camera attached to it.

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[1] It's important to always use SI units at all costs.

CO2 MO-scheme

If you talk about MO schemes you don't get to talk about exciting molecules. But CO₂ is interesting because a simple correlation consideration shows you things that the Lewis structure doesn't.

The bonding energy of CO₂ (1 816 kJ/mol) is 176 kJ/mol higher than twice the C=O bonding energy (820 kJ/mol) known from ketones. The reason is just another kind of delocalisation [1]. Let's see where it comes from.

What does the MO scheme look like? We have 3 sets of 4 valence orbitals and 16 valence electrons. That means (with a minimal LCAO base) there will be 12 orbitals, 8 of them populated .

CO₂ is of the D_∞h point group. But I don't think you need to know that. It basically means that you have Σ and Π symmetry races, in other words there are σ- and π-orbitals.

The first three orbitals can be considered the bonding, non-bonding and anti-bonding combination of the s-orbitals. p_z-orbitals are mixing in wherever it fits, mainly at the C in the second one and at the O's in the third one.

The next one is a (twice degenerate) bonding π-MO. Then comes a σ-MO which is mainly made of p_z-orbitals and a little bit of s. Again the bonding pσ-MO is higher in energy than the bonding pπ-MO. This is because of the s mixing in just as we had it in diatomic molecules. The HOMO is a twice degenerate non-bonding π-orbital. You can also see the unoccupied MOs with more nodal planes.

The interesting thing about CO₂ is its delocalised π-system. It has to be described as a 4-electron-3-center-MO. In both the xz- and the yz-plane there is a bonding and a non-bonding occupied π-MO which is delocalised over the whole molecule. This is totally different to allene even though it has the same Lewis structure. Allene has two localised two-center π-MOs perpendicular two each other. CO₂ has the same formal bond order as allene but the bonding MOs are much lower in energy because they are 3-centered MOs. This causes the high bonding energy in CO₂ and it is just a special case of delocalisation.

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This is the electrostatic potential. You can clearly see that the carbon atom attracts a negative charge. Besides that the LUMO is big at the C, so a nucleophilic attack will also work as far as orbitals are concerned (if we assume that this simple approach is good enough).

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[1] The reason why delocalisation causes lowering of the energy is not as simple as it seems. The first explanation would be that delocalisation means lower kinetic energy. Explain it with Heisenberg (higher position uncertainty means lower momentum uncertainty and therefore lower momentum in a stationary state) or with Schrödinger (delocalisation means less curvature). But that's only part of the truth because the virial theorem has to hold. I mentioned that a while ago.

jmol

As much as I like jmol, I have to admit that there is a problem with it if you use it in blogspot. It seems that it makes firefox shut itself down. The same thing happened at chemblog. So I guess it is not all about blogspot. Maybe the problem only occurs when I have two jmol applets open and is because I am initialsing twice. Anyway, I am sorry if I made your browser freeze.

Add a new dimension to molecular graphics

If you like 3D molecular structures you could buy a plastic model set. Then you can use pretty much all your senses for understanding the structure. But what do you do when you run out of atoms? Here's something that's almost as real as a model set and has no atom limit: a 3D-viewer.

You look at double images like this one (created with PyMol's stereo function). The two images look slightly different just like the images your eyes see if you look at one 3-dimensional object are slightly different. The mirrors inside the viewer enable you to look at only the left picture with your left eye and the right picture with you right eye. In your head it seems like you are looking at one 3-dimensional object.


The structures look pretty nice. And the 3D-viewer probably helps you for understanding a structure faster.

If you look at yourself in the mirror it looks kind of creepy because your eye appears in the wrong place.


Actually at our institute for cristallography they had more sophisticated viewers that block out the things outside of the screen better. But my professor said they would not produce those anymore. This one is pretty cool, too. Maybe I should make the eye holes smaller though.