Below are the molecular orbitals for benzene (left) and 1,3,5-hexatriene (right). For each molecule, the six p-orbitals (one on each carbon) overlap to generate six LCAO molecular orbitals, three bonding and three antibonding.
In a previous topic we saw how benzene is more stable than cyclohexatriene, by about 30 kcal/mol. This stability can be rationalized by looking at the bottom most MO's . When adjacent p orbitals are aligned it is a stabilized bonding interaction. The more stablized bonding interactions the more stable the given MO. Looking at Ψ1 for benzene we see that it has 6 stabilized bonding interactions, while π1 for the hexatriene only has 5. Its the cyclic nature of benzene that gives it one more interaction and explains its "extra" stability relative to that of hexatriene.
It's the shape of the MO diagram above (left) that is the origin of the 4n+2 rule. As we will see, the MO diagram of an aromatic molecule will always starts with a single MO Ψ1 which has an occupancy of two electrons, and then the degenerate MO's, Ψ2 and Ψ3 , which each have two electrons. Therefore a total of 6 electrons (n=1). This is stable since all bonding MO's are full and the antibonding orbitals have no electrons. So there will always be the bottom 2 electrons and then multiples of 4 electrons (ie.e 4n+2).
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