The next question is to which of the indicated excitations are allowed to happen for an electrical dipole process (such as optical absorption between the electronic levels). Note the vertical scale for the ilustration above does not relate to the energies of the various electronic levels.
For a transition to be allowed, the direct product of the ground state wave function, the excited state wave function and the dipole operator must contain the a1 irredicible representation. For the C2v point group the dipole operator, which transforms as a linear function (x, y and/or z) transforms as a1+b1+b2, the specific irreducible representation(s) depending on the polarization of the electric field.
Let us look in detail at the HOMO-LUMO transtion between levels 4 and 5. The HOMO has an irredicible representation of b1, and the LUMO of a1. The product of functions transforming under these irredicible representations, in turn transforms as b1. If the electric field (incident photon) contains a component polarized along the x, which also transforms as b1, then the product of the molecular orbitals with the electric field transforms as a1. This is therefore a dipole allowed transition.
As a counter example, lets look at the transition between the
fourth orbital (the HOMO) and the sixth (the LUMO+1). The
product of the b1 and b2 wave functions transforms as
a2. Combining any of the dipole
representations with this cannot result in an overall a1
symmetry, so this transition is dipole forbidden. In fact, of the transitions indicated
in the figure, this is the only dipole forbidden transition.
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