E 8 C 3 3 C 2 6 S 4 6 d
Nstationary 52 1 1 3
0° 120° 180° 90° 180°
1 2 cos 3 0 1 1 1
Nstationary(1 2 cos ) 15 0 1 1 3
r 3 0 1 1 1
t 3 0 1 1 1- Subtract rand tfrom totto get v:
E 8 C 3 3 C 2 6 S 4 6 d
Nstationary 5 2 1 1 3
0° 120° 180° 90° 180°
1 2 cos 3 0 1 1 1
Nstationary(1 2 cos ) 15 0 1 1 3
r 3 0 1 1 1
t 3 0 1 1 1
v 9 0 1 1 3- Use the great orthogonality theorem to reduce vinto its irreducible
representations for the Tdpoint group. Upon doing so (and the details
are left to you), one finds that
1 A 1 1 E 2 T 2- Check the Tdcharacter table in Appendix 3. In the rightmost column,
the T 2 irreducible representation has the x,y, and zlabels. Therefore,
only the T 2 -labeled vibrations will be IR-active, and the conclusion is
that CCl 4 will have only two IR-active vibrational modes. These modes,
being triply degenerate, represent six of the nine normal vibrations
of CCl 4.
The above example illustrates two things. First, the vibrational spectrum of
a molecule can be enormously simplified because of symmetry. The nine pos-
sible (and presumably different) normal vibrations of CCl 4 are reduced to only
twoIR-active, triply degenerate vibrations. When you measure a vibrational
spectrum of CCl 4 , you will expect only two absorptions to appear in the spec-
trum instead of nine. Second, the use of group theory and character tables is
indispensable in quantifying this. Group theory and symmetry are crucial to
the understanding of vibrational spectroscopy in particular and all of spec-
troscopy in general. Although this is the first time we utilize the power of sym-
metry for a specific spectroscopic purpose, it should be understood that sym-
metry considerations and group-theoretical arguments are a fundamental
aspect of spectroscopy.
The majority of molecules have relatively low symmetry, so they can be ex-
pected to have a large number of IR-active vibrations. However, this section
has introduced a powerful concept that can be very useful in spectroscopy, es-
pecially in understanding how the structure of a molecule is related to its vi-
brational spectrum.502 CHAPTER 14 Rotational and Vibrational Spectroscopy