Draft:Archive/2024/Hückel method/Computerising Hückel calculations

Computerising Hückel calculations
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
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:


 * center top}}||Image:Huckel.gif]]||center bottom}}

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

We start with our


 * center top}}||Image:Huckel.gif]]   ethene example ||center bottom}}

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 ||Hückel annotated results for ethene|annotated version]] of the ethene output.

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


 * center top}}||Image:Huckel.gif]]    benzene ||center bottom}}


 * Computational Chemistry|Basic Hückel theory|Odd electron and open shell systems}}