We can use the soltions for both the particle in a box and particle on a circle to model molecules. For linear molecules the solutions form a set of energy levels that are occupied by electron pairs. The model can also be applied to open shell systems with unpaired electrons. The same basic rules apply regarding transitions and their selection rules. The rule for the linear model is n' = n + 1 for absorption of light.

Particle on a circle model for aromatic molecules

The particle on a circle model is shown above for porphine with 18 electrons. The solutions to the particle on circle have paired levels with positive and negative quantum numbers for all solutions above zero. If we fill a level then the system will automatically follow the 4n + 2 rule for aromatic molecules. For example, for benzene with 6 electrons we would fille m = 0, -1 and +1. To fill the m = +2 and +2 level would require 4 more electrons to make 10 total. The model for porphine has levels filled up to m = -4 and +4. The lowest unoccupied level has m = -5 or +5. The figure above shows one intersting feature of the particle on a circle model. THere are two transitions possible. The one with Delta = + or -1 represents an allowed transition. The transition with Delta m = + or -9 is forbidden. This means that it is a very weak transition. In all aromatic molecules there are intense bands (often known as B bands) and weak bands (known as Q bands). The particle on a circle does predict this behavior.