An optical (electric dipole) transition can be forbidden by symmetry. Let us restrict the analysis to cases where a there is no change in overall symmetry between the ground and excited states. Whether an electronic transition is forbidden or not can be ascertained by examining the product of the symmetries of the ground and excited states of the possible transition. If the product contains the irreducible representation that corresponds to the dipole (x, y, and/or z), then the transition is allowed in principle.
Equivalently, if the product irreducible
representations of the ground, excited and dipole-operator contains
the totally symmetric irreducible representation, then the transition
is allowed. This selection rule can be expressed as:
|GS||is the irreducible representation of the electronic ground state|
|p||is the irreducible representation of the dipole operator|
|XS||is the irreducible representation of the electronic excited state, and|
|Sym||is the totally symmetric irreducible representation of the point group (usually denoted by A, A1, A1g, Ag, or A')|
For specific point groups, the direct products if irreducible representations with those transforming as x, y, and/or z have been tabulated. (Follow links labelled product tables or by the symbol.)
In a similar way, vibrational replicas of dipole transitions follow the selection rule:
GS p XS LVM Sym
where the symbols are as before, and LVM is the irreducible representation of the vibrational mode. Clearly a totally symmetric mode of vibration is always a possible replica. To determine whether modes transforming as other irreducible representations can couple to the electronic transition requires application of the selection rule.
Fortunately this is straight forward, and the direct product tables already contain the relevant information.
Furthermore, in general the product of an irreducible representation with itself always contains the totally symmetric irreducible representation. Moreover, the products of non-degenerate irreducible representations with each other (but not with themselves) do not. These results have the consequence that the product GS p XS directly represents all group theoretically possible modes that can couple with the electronic excitation.
Note, dipole forbidden transitions follow the same selection rule, and consequently totally symmetric modes cannot be phonon replicas of forbidden transitions.