the molecule, the fewer unique vibrational frequencies a molecule has, and the
fewer that have a corresponding change in dipole moment.
There is a strict group-theoretical method for determining exactly the num-
ber of allowed vibrational transitions expected to be observed in a vibrational
spectrum. The method is presented here as a sort of recipe to be followed, and
uses the character tables in Appendix 3. Use of the recipe requires two things.
First, the character tables contain more information than we have used so far.
In particular, note the information in the rightmost column of the character
tables. One or more of the irreducible representations of each character table
has an x,y,or znotation at the right (among other things). These mark the ir-
reducible representation(s) of the components of the electric dipole operator
(see equation 14.2) in that particular symmetry. This information will be nec-
essary in our recipe. Second, we need to differentiate between “proper” and
“improper” rotations. In a broad sense, all symmetry operations can be thought
of as rotations. Proper rotations are Eand Cn, which have angles of rotation of
0° and 360°/n, respectively. Improper rotations are i,Sn, and all planes of sym-
metry; these have angles of rotation of 180°, 360°/n, and 180°, respectively.
Proper and improper rotations are treated slightly differently in two steps of
the recipe.
The recipe for determining the number of IR-active vibrations of a mole-
cule of known symmetry is given in Table 14.5, in a somewhat abbreviated
fashion. Briefly, it rests on finding a set of characters that describe the vibra-
tional degrees of freedom of a molecule, then using the great orthogonality
theorem to reduce that set of characters into a set of irreducible representa-
tions of the molecule’s symmetry group. Then, by finding the x,y, and zlabels
in the character table, we can determine which irreducible representation la-
bels correspond to vibrations that are infrared-active. The following example
goes through the steps in the scheme.500 CHAPTER 14 Rotational and Vibrational Spectroscopy
Table 14.5 Steps for determining the number of infrared-active vibrations of a
polyatomic molecule
Procedure Formula/expression
Construct a blank table with a column for every symmetry class. In successive lines of the
table, do the following:- In the first line, write the number of atoms in the molecule Nstationary
that do not change their position in space under that
operation. - In the next line, determine the angle of the “rotation”
of the symmetry operation.a - Evaluate the expression (1 2 cos ) for each symmetry (1 2 cos )
operation. - Evaluate Nstationary (1 2 cos ), and multiply by 1 for Nstationary (1 2 cos )
proper rotations or 1 for improper rotationsb. This is tot. - Evaluate the character for rotations,r, as (1 2 cos ). r(1 2 cos )
- Evaluate the character for translations,t, as (1 2 cos ) t(1 2 cos )
(depending on whether the operation is proper or improper). - Subtract tand rfrom totto get the character set for vtottr
vibrations,v. - Reduce vinto its irreducible representations using the GOT. v
n - Irreducible representations having x,y,or zlabels in
character table are IR-active.
aFor i, 0°. For S 3 , 60°. For S 4 , 90°. For S 6 , 120°.
bProper rotations are Eand Cn; improper rotations are ,i, and Sn.