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 a_{1} irredicible representation. For the C_{2v} point group the dipole operator, which transforms as a linear function (x, y and/or z) transforms as a_{1}+b_{1}+b_{2}, the specific irreducible representation(s) depending on the polarization of the electric field.
Let us look in detail at the HOMOLUMO transtion between levels 4 and 5. The HOMO has an irredicible representation of b_{1}, and the LUMO of a_{1}. The product of functions transforming under these irredicible representations, in turn transforms as b_{1}. If the electric field (incident photon) contains a component polarized along the x, which also transforms as b_{1}, then the product of the molecular orbitals with the electric field transforms as a_{1}. 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 b_{1} and b_{2} wave functions transforms as
a_{2}. Combining any of the dipole
representations with this cannot result in an overall a_{1}
symmetry, so this transition is dipole forbidden. In fact, of the transitions indicated
in the figure, this is the only dipole forbidden transition.
