Physical Chemistry , 1st ed.

group theory are therefore applicable to the study of molecular vibrations. We

will use such tools to some extent in the material to come. Group theory is also

applicable to other forms of spectroscopy, as we will find.

Example 14.18

From the representation of the normal modes of a symmetric linear mole-

cule shown in Figure 14.30, draw the changes in the vectors upon operation

of each symmetry element and assign irreducible representation labels to the

normal modes of CO 2. You will have to use the Dhcharacter table in

Appendix 3.

Solution

The drawing is left to the student. If the drawings are done properly, it can

be seen that the symmetric stretching vibration can be assigned a label ofg^ ;

the asymmetric stretching vibration is assigned to u^. The doubly degener-

ate bending motion is u.

The degeneracy of a vibration is related to its character of the identity ele-

ment of its irreducible representation label. Doubly degenerate vibrations al-

ways have an irreducible representation label having E2. Triply degenerate

vibrations always have an irreducible representation label having E3. There

are no higher degeneracies for vibrations.

14.14 Vibrational Spectroscopy of Nonlinear Molecules

Moving on to nonlinear molecules, there are few truly new concepts. The num-

ber of vibrational degrees of freedom is now 3N6, and the list of descrip-

tions for the vibrations increases somewhat. Perhaps the biggest difference in

considering nonlinear molecules is how the symmetry of the molecule affects

the number of independent vibrations of the molecule.

Figure 14.32 shows the normal vibrations for the ammonia molecule. They

are numbered  1 , 2 , 3 , and  4. The numbering of the vibrations follows a

498 CHAPTER 14 Rotational and Vibrational Spectroscopy

 1

H

N

H

H

 2

H

N

H

H

 3

H

N

H

H

 4

H

N

H

H

Figure 14.32 The normal modes of vibration for ammonia, NH 3. All are IR-active.