Character tables provide a systematic group-theoretical framework for determining whether molecular vibrational modes are infrared– or Raman-active by examining how those modes transform relative to the dipole moment and polarisability operators.
The first step is to work out the symmetries of the normal modes of vibration of a molecule. For example, H2O lying in the -plane belongs to the
point group. Two of its three normal modes (symmetric stretch and bend), as explained in an earlier article, transform according to the
irreducible representation, while the remaining one (asymmetric stretch) transforms according to
.
The character table (see above table) shows that the linear functions (or Cartesian coordinates)
and
also transform according to
and
respectively. This correspondence means that the collective atomic displacements associated with each of the three H2O normal modes transform in the same way under the symmetry operations of the group as the corresponding Cartesian vector component.
The second step is to recognise that an IR-active vibrational transition requires a change in the molecular dipole moment during the vibration. The dipole moment is a vector quantity whose non-zero components transform as the linear functions ,
and
. If a normal mode transforms according to the same irreducible representation as one of these components, then the corresponding component of the dipole moment changes as the atoms are displaced from their equilibrium positions.
Therefore, the modes that transform according to and
are IR-active.
For vibrations to be Raman-active, we refer to the quadratic functions because the components of the polarisability tensor transform in the same way as these functions. In this case, both the and
modes are Raman-active.
Question
Why are IR and Raman activities mutually exclusive for vibrational modes of centrosymmetric molecules?
Answer
See this article for explanation.