Computerising Hückel calculations

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Computerising Hückel calculations[edit]

The presence of the symbols α and β in the equations and the determinant makes it difficult to handle them by computer. However, we can simplify them and in so doing we also get greater insight into what the theory is actually doing.

We take the original equations:

[Equation #1]

We divide through by [beta] and make the substitution:

[Equation #2]

to give:

[Equation #3]

The determinant becomes:

[Equation #4]

This is easily solved, in this case, by hand:

[Equation #5]

It is also easy to solve by computer. The xi terms are the eigenvalues of the matrix:

[Equation #6]

and the C terms are the eigenvectors. The numerical solution for the eigenvalues and eigenvectors of this type of equation is well developed.

This analysis also shows that the solutions of Hückel theory are determined simply by the connectivity matrix - a matrix that has a 1 in the off-diagonal position corresponding to two linked atoms and 0 elsewhere. This has made Hückel theory a branch of graph theory and some of the theorems of graph theory were worked out in the context of Hückel theory by Charles Coulson and others before the graph theorists got around to them.

Running calculations[edit]

You are now in a position to use a simple forms interface to a Hückel theory calculator that allows you to type in the data for any molecule, send the data to the server machine where the calculation is carried out, and then receive back the results. Note that this is going off the wikiversity wiki pages, to a straight HTML web page, which will direct you back here. The image:


is always an indicator of a link out to the calculator. Links have not yet been added.

We start with our

Huckel.gif ethene example

and, to make it even simpler, the data is already loaded into the form.

You will see that the main data is simply to indicate that atoms 1 and 2 are linked. This is all that is needed to construct the connectivity matrix and then perform the calculation.

The results show some features you have not met yet, so we have supplied an annotated version of the ethene output.

Now that you have worked through the ethene example, try studying

Huckel.gif benzene