Raman spectroscopy also has selection rules. The gross selection rule for a
Raman-active vibration is related to the polarizability of the molecule.
Polarizabilityis a measure of how easily an electric field can induce a dipole
moment on an atom or molecule. Vibrations that are Raman-active have a
changing polarizability during the course of the vibration. Thus, a changing
polarizability is what makes a vibration Raman-active. The quantum-mechan-
ical selection rule, in terms of the change in the vibrational quantum number,
is based on a transition moment that is similar to the form ofMin equation
14.2. For allowed Raman transitions, the transition moment [] is written in
terms of the polarizability of the molecule:
[] *finalinitiald (14.45)
where finaland initialare the final and initial vibrational wavefunctions, re-
spectively. As with vibrations absorbing infrared light, it can be shown that this
integral is exactly zero unless the difference in the quantum numbers offinal
and initialis 1:
v 1 for allowed Raman vibrational transitions (14.46)
IR-active vibrations require a changing dipole moment, which is relatively
easy to visualize by inspection of the normal mode’s atomic vectors. Changes
in polarizability are not as straightforward to visualize. But, as with IR-active
vibrations, we can use the great orthogonality theorem to determine the num-
ber of Raman-active vibrations that a molecule will have. The procedure is the
same as what was done using Table 14.5 in Example 14.19, except for step 9,
in which we use information in the character table to determine which irre-
ducible representations are spectroscopically active. For IR-active vibrations,
we looked for the x,y, and zlabels on the irreducible representations. These la-
bels gave us an indication of the irreducible representation of the dipole mo-
ment operator in those point groups. But according to equation 14.45, the op-
erator for the Raman transition moment is , not . The polarizability has
different irreducible representations in the point groups, and those irreducible
representations are labeled with second-order variables:x^2 ,y^2 ,z^2 ,xy,yz,xz,or
other combinations of second-order functions. Such functions are listed in the
character tables in Appendix 3. Vibrations that have irreducible representations
associated with these labels are Raman-active.
Example 14.22
Use the information in Example 14.19 to determine which vibrations of car-
bon tetrachloride, CCl 4 , are Raman-active.Solution
According to step 8 of Example 14.19, the vibrations of CCl 4 collectively have
the irreducible representations
1 A 1 1 E 2 T 2
If we check the Tdcharacter table in Appendix 3, we find second-order func-
tions listed with A 1 ,E, and T 2 irreducible representations. Therefore, all of the
vibrations of CCl 4 will be Raman-active, and the Raman spectrum will con-
sist of four signals representing one singly degenerate, one doubly degener-
ate, and two triply degenerate vibrations.14.18 Raman Spectroscopy 513