I am not starting my QM tutorial quite yet. Today's post is mostly about graphics to look at. You can see the 6 canonical molecular orbitals with lowest energy.
Actually today's post was just for showing those graphics but I will talk a little bit about them.
First you notice that they look a little bit like the π-MOs. This is because of symmetry. You have two dimensional irreducible representations again.
Adding up orbitals and electrons tells you that there are 30 orbitals, 15 of them populated with two electrons. 6 are seen here, 6 are not shown, and the π-MOs (3 of them populated) are shown in the last post.
This time the situation is not quite as easy as with the π-MOs because every carbon has 3 orbitals in the plane (s, px, py) and every hydrogen one. You can combine them to 24 different σ-MOs. You can see that the first one is mostly carbon's s (the lowest energy orbital in the system). The 6th is mostly made of hydrogen s AOs.
The MOs shown are canonical, meaning they are delocalised irreducible representations of the symmetry group. You could take the 12 populated canonical σ-MOs and rearrange them to two 6 dimensional reducible representations. One of them being only C-C σ-bonds the other one C-H σ-bonds. The electron density, the only physically relevant variable, would not change. It does not matter into which MOs you divide it.
It would not be possible to arrange the three populated π-MOs to localised bonds. This again is something a chemist is familiar with.
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