A knowledge of the symmetry species of the normal modes of vibration of a molecule may be used to rationalise the molecule's vibrational spectra.

The intensity with which any given transition is observed is proportional to an integral. Without specifying its form explicitly, we may note that the integrand contains the excited state of the molecule reached by the transition, the ground state (usually totally symmetric) and a term which varies according to the type of spectrum being recorded.

As previously noted, the integrand must be totally symmetric or it will vanish (the transition will be forbidden). Since the ground state is totally symmetric, we only require that the direct product of the symmetry species of the excited state and the other term contains the totally symmetric symmetry species. i.e. we require that the excited state and the other term have at least one symmetry species in common.

For infra-red vibrational spectra, the "other term" is the electric dipole moment transition operator. This transforms as x, y or z.

Thus for a normal mode to be infra-red active, it must transform as (i.e. have the same symmetry species as) x, y or z.

Any mode which does so will be infra-red active (though the transition may be too weak to be observed for other reasons).

For Raman spectra, the "other term" is the electric polarisability. This transforms as the quadratic functions, x², y², z², xy, xz and yz.

Thus for a normal mode to be Raman active, it must transform as x², y², z², xy, xz or yz.

Any mode which does so will be Raman active.

Note one very general result which comes from these statements is formalised in the mutual exclusion rule.

This rule, which applies only to molecules with a centre of inversion, states that in such a molecule no mode may be both infra-red and Raman active.

This is very simple to justify: The electric dipole moment operator changes sign under inversion, so is labeled u. Any infra-red active mode must thus also be of u symmetry. The electric polarisability does not change sign under inversion, so is labeled g. Any Raman active mode must thus be g. Since no mode can be of both u and g symmetry (no mode can both change sign and remain unchanged under inversion), no mode in a centrosymmetric molecule can be active in both forms of spectroscopy.