Energy optimized arrangement in AB5-Molecules

In an AB5-molecule (i.e. an atom consisting of one (say positive charged) center atom and five negative charged ligands) the question of the molecule geometry seems to be very interesting. In the lectures we heard, that a bipyramidal arrangement will appear. I.e. all ligands on the surface of a sphere – the center atom of course in the center – such, that three of them form an equilateral triangle at the equator of the sphere and the other two take place at the poles. Interesting enough, one can calculate, that for this arrangement there is not an equal distribution of the energies between the ligands. That means, that the two ligands at the poles (we call it axial position) have a higher potential energy than the three in equatorial position. Since this isn’t trivial, a closer look might not be wrong.

Looking up a chemical textbook, one can find that for some examples of AB5-molecules the distance of the axial ligands to the center is greater than it is for the equatorial ones. I found this a very uncomfortable fact because it’s an exception in molecule geometry. (Remember, 2 ligands form a (bent) line, 3 form an equilateral triangle, 4 a tetrahedron, 6 a cube, 8 an octahedron, etc. But always, the distance ligand-ligand and ligand-center is constant. Why should it be not constant for coordination number 5?)

A killer argument that will probably satisfy most Chemists is that the ligands need some space and five of them are simply too many to fit all around the center. Instead of accepting this, I preferred to set the volume of the ligands to zero in order to eliminate this bloody argument. Now our problem is to find a way to arrange five point-like negative charges in a three dimensional space so that the potential energy is a minimum. (Keep in mind that the negative charges are also attracted by the positively charged center atom so they don’t scatter all around the universe – otherwise we wouldn’t have any molecules on earth.)

For all other common coordination numbers (4, 6, 8, …) the most favorable arrangement is related to a platonic body. Unfortunately there is no such platonic body with five edges. (Even nature isn’t perfect, it seems.)

To tackle this problem, two strategies seem to be worth a try. Once look for an arrangement that fulfills the condition that all distances ligand-center are constant (1). And second an arrangement that keeps the ligand-ligand distance constant (2). If there is an arrangement that fulfills both requirements, we will find it anyway because both methods should lead to it. If the optimum is a violation of both premises then we have a problem.

An example for an arrangement fulfilling (2) would be a double tetrahedron. But since this is difficult to calculate, we’d better assume, that condition (1) weights heavier. (Come on, we’re Chemists, we needn’t be accurate, just evident!) (1) is of course fulfilled by any arrangement that allows to draw a sphere so that all five ligands take place on it.

So, again, we have reduced our problem to the question of the optimum arrangement of five point-like charges on a sphere. (I admit that we got a little far from our original problem.) Together with some friends of mine, we tried to find a way to solve this problem analytically but it seemed to be too difficult. As I use to do if thinking fails, I started trying, i.e. writing a little script, that tries to find the optimum arrangement.

We take a sphere and place five charges on it, where the coordinates are pure random, and calculate the total potential energy of the system. Now we move the first charge for a little amount. If we have done so, we calculate the energy again and compare. If we’ve made things better, we leave it, if we didn’t we’ll move the charge back to it’s old position. That procedure is repeated with the other four charges and then started again with the first one. To make the calculation faster, we start with a large amount for the movement and scale it down during the iteration. (If we tried too often but can’t find a better position for our charge, we assume that we’ve reached the optimum position with the precision of our iteration and so fore go to a smaller level.)

Interesting enough, it takes only very little iteration to reach a very good precision. Look at the following graphs to see the mechanism work.

Position of a single charge during the iteration.

This graph shows the longitude of a charge during the iteration. You can clearly see that the amount the charge is moved decreases as the iteration goes on.

The total energy of the system during an iteration.

The total energy of the system during the iteration. Note that the function is monotonic decreasing. (Due to the rules the iteration is carried out.) Again, you can identify some nice plateaus.

After I had fired up the script for many times, it always showed the following result: Two charges at the axial position, the remaining three forming an equilateral triangle at the equator. Surprised? Well, sometimes your teacher seems to be true…

Please note: The results of this script do not prove, that this is the best arrangement of all. There might be an even better one. Even tough this isn’t very likely. In other words: If the total potential energy is a function of the coordinates of the five charges, my script might have discovered only a local minimum and there might be a global minimum bellow it. However, it would be very unlikely that the iteration always chooses the way to the same local minimum if there is an other one too.

Here you can download the script to try out yourself:


One Comment

  1. Posted 2009-04-08 at 07:50 | Permalink | Reply

    Quite amazing. By the way, your method for iteratively optimizing the solution by small random perturbations is quite the same idea of (artificial) evolution. However, in evolution-based code you would have started with a whole generation of starting individuals (arrangement of 5 charges) and then “somehow” combined two of this population to generate a new individual. This is done a few times to retrieve a bunch of individuals, which build the second generation. After that you can perturb the individuals in second generation and drop, say, the worst half of the generation. And then you can repeat this procedure to get the third generation. However, I do not expect any improvement for this particular problem.

    Even nature isn’t perfect, it seems.
    This reminds me what is said about Euler: “The physical universe was an occasion for mathematics to Euler, scarcely a thing of much interest in itself; and if the universe failed to fit his analysis it was the universe which was in error.”

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