Physical Chemistry , 1st ed.

terms of these equilibrium positions. When we realize that (in most cases) the
amount of movement of atoms in a vibration is relatively small, distorting a
molecule very little from its equilibrium symmetry, the definition of a mole-
cule as having a symmetry based on its equilibrium atomic positions is still a
good basis for understanding its behavior.
Consider the vectors that describe the normal vibrations for the H 2 O mol-
ecule in Figure 14.31, and the effects on those vectors by the various symme-
try elements of the point group of the molecule,C2v. The table on the right
side of the figure shows that the group of eigenvalues produced for the vectors
that describe the  1 vibration is the same as the irreducible representation A 1
for the point group. Inspection of the other normal vibrations shows that this
is not a coincidence: the other two normal modes behave like irreducible rep-
resentations of the C2vpoint group also (A 1 and B 1 , to be exact). Although
proof is beyond our scope, the point should be clear: vibrational modes of
molecules can be assigned a label of one of the irreducible representations of
the molecular point group. The powerful mathematical tools of symmetry and

14.13 Symmetry Considerations for Vibrations 497



 

A 1

E

1

H H H H

C 2

1

H H H H

(^) v
1
H H H H
(^) v'
1
H H H H
Figure 14.31 The  1 normal vibration of H 2 O, and the effects of the symmetry elements of
the C2vpoint group on the vibration. In this case, operation of all symmetry elements yields a
motion that is exactly the same as the original motion. Therefore, the eigenvalues of the opera-
tions are all 1, and this vibration can be labeled with the A 1 irreducible representation of the C2v
point group.