# Is there a simple way to predict the molecular pi orbitals in conjugated pi systems?

Particularly for longer alkyl chains. Or is predicting these molecular orbitals from quantum mechanics too complex to be able to be translated easily as a geometric pattern? If Frost Circles exist, I just figured something could for this as well. The only pattern I've been able to identify is that each molecular orbital is symmetrical, starts with no nodes and goes up by one with each higher energy orbital but for systems with many carbon atoms, this isn't a specific enough set of rules.

• How many carbon atoms are you intending to look at? From looking at this figure there does seem to be a pretty clear pattern in the same vein as a frost circle (at least for the even numbered case). @Elmer
– Tyberius
Commented Apr 18, 2018 at 18:34
• It all boils down to eigenvalues of adjacency matrices. Commented Apr 18, 2018 at 18:42
• @IvanNeretin That isn't really useful. Even if the questioner knows what those words are, they wouldn't know that they should look for the Huckel method. Commented Apr 18, 2018 at 18:48
• @pentavalentcarbon That's why this was a comment and not an answer. Anyway, thank you for making it more useful. Commented Apr 18, 2018 at 18:53
• @Tyberius Let's say we're looking at 10 carbons. And what I'm looking for are the positions of the node(s) at each orbital. All I know now is that they go up by one each time starting at zero and ending between each pi orbital and that the positions of the nodes are arranged symmetrically, but this doesn't help me predict the orbitals for molecules beyond a certain length. Commented Apr 18, 2018 at 20:00

This should work for any even number of conjugated carbons (and with slight modification, for cations/radicals/anions of uneven numbers of carbons). You can follow a simple procedure to generate all the $\pi$ orbitals.