Posted by: Mark Foreman | February 14, 2012

Orbitials of benzene

Dear Reader,

Benzene is a molecule which is rather like one of my favourite aerials, I ♥ large loops for HF radio. My favourite loop is 80 meters of wire which is stretched between trees in a Swedish forest. This loop is known as the “German quad” as it was designed by a German man many years ago. While another favourite aerial of mine is the triangular loop in my garden which has three sides which are ten meters long.

When a loop is used at a very low frequency (where it is small compared with the wavelength) then the current is constant throughout the whole loop. For example if I use my 10 meter delta loop to tune in to BBC radio 4 (198 kHz, λ = 1500 meters) then the current distribution in the loop will be constant. Here is a diagram of the loop (dark blue) with the polar plot of the current (purple).

Current distribution in a small loop

Here is a different way to look at the data for the small loop, now the current is plotted against the angle around the loop. Here is the graph.

Current distribution in the small loop as a function of the angle around the loop

Now when the loop is used as a higher frequency, it now has a circumference equal to one wavelength, now the loop is a resonant loop. Here is the new graph of the current distribution in the loop. For those of you who are more interested in organic chemistry than radio aerials, please note that the graph of current distribution now looks like the HOMO of benzene. It has two nodal points, also the sign of the line changes from one part of the graph to another.

The loop is now resonant at its lowest frequency, look how the current in the loop looks like the wavefunction for the HOMO of benzene.

Now if we move to a higher frequency for the loop (twice as high as the lowest resonant frequency) the graph of the current against angle now looks like the LUMO of benzene. Here it is below.

Loop now at twice the fundamental, it now looks like the wavefunction of the LUMO of benzene.

Now if we move to the third harmonic for the loop, we will now see a current distribution which looks rather like the wavefunction for the highest pi orbital for benzene. Here we are.

Loop now at the third harmonic, see how it looks like the wavefunction for the highest pi orbital of benzene


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