Physical Chemistry , 1st ed.

Example 14.14

Which of the following molecules will have pure vibrational spectra?

a.Hydrogen sulfide, H 2 S

b.Oxygen, O 2

c.Ozone, O 3

d.Hydrogen chloride, HCl

e.acetylene, C 2 H 2

Solution

a.Hydrogen sulfide is a bent molecule, like water. It has a permanent dipole

moment varies as the molecule vibrates. It will show a pure vibrational

spectrum.

b.Elemental oxygen, a homonuclear diatomic, will not have a pure vibra-

tional spectrum.

c.Ozone is a bent molecule, which will have a permanent dipole moment. Like

H 2 S, ozone’s dipole moment will change during the course of a vibration, so

we would expect that ozone will have a detectable vibrational spectrum.

d.Hydrogen chloride is diatomic, but it has a dipole moment whose magni-

tude will change as the two atoms change distance. Therefore, it is expected

to have a vibrational spectrum.

e.Acetylene is linear, but like carbon dioxide it is expected that some of its

vibrations will produce a fleeting dipole moment. We therefore expect that

acetylene will have a pure vibrational spectrum.

There is a more specific selection rule that depends on the quantum num-
ber for the vibrational state,v. Since the normal modes of vibration are inde-
pendent of each other (they are orthogonal to each other), this selection rule
is applicable to one normal mode at a time. That is, we will consider the ap-
plication of this new selection rule to each individual normal mode of vibra-
tion. The selection rule does not address simultaneous changes in more than
one normal mode of vibration. As indicated in section 14.2, such specific se-
lection rules are group-theoretical. The transition moment integral for vibra-
tions is

vˆvd

where vand vrepresent the vibrational quantum numbers of the two vibra-
tional states involved, and ˆis the electric dipole operator. In order for this
integral to be nonzero for an allowed transition, the following change in quan-
tum number vis allowed:

v 1 (14.33)

This selection rule is applicable to each normal mode of vibration. In absorp-
tion spectroscopy, the change is 1. This assumes that the normal mode acts
as an ideal harmonic oscillator. Real molecules do not act as ideal harmonic
oscillators, so in some cases it is not uncommon to detect v2,3,...,
transitions. Such observations are part of what is called overtone spectroscopy.
Partly because of the selection rule in equation 14.33, detection of overtones is
difficult because many such absorptions are only weakly represented in a
vibrational spectrum. Lasers, with their high intensities, are frequently utilized
in overtone spectroscopy. However, the majority of vibrational spectroscopy
deals with transitions following equation 14.33.

14.11 Selection Rules for Vibrational Spectroscopy 489