Physical Chemistry , 1st ed.

b.HClO has, roughly, an H–Cl stretch, a Cl–O stretch, and a doubly degen-

erate bending motion.

c.BeF 2 has a symmetric Be–F stretch, an asymmetric Be–F stretch, and a

doubly degenerate bending motion.

d.HCC would be expected to have an H–C stretch, a doubly degenerate

H–C–C bend, and a C–C stretching vibration.

Keep in mind that these descriptions are rough, not exact. Each atom in the

molecule moves. This point gets lost when one begins to consider larger mol-

ecules, but it is no less true for simple triatomic molecules. Notice that we have

introduced another type of vibrational motion: the bending motion. Such a

motion does not fit the classical definition of a Hooke’s-law system, which as-

sumes that two masses are moving back and forth with respect to each other.

However, even for bending motions, the case is made that the atoms are mov-

ing back and forth about some presumed equilibrium position. As such, we

can assume some sort of force constant such that the farther away the atoms

are from that equilibrium position, the stronger the restoring force. Therefore,

bending force constantscan be defined. However, in a bending motion, the con-

cept of reduced mass of the oscillator is much more complicated. Bending mo-

tions in molecules therefore do not follow such simple mathematical relation-

ships as do stretching motions, which were illustrated in Examples 14.12 and

14.13. (They do follow mathematical relationships, but they are a little more

complicated. See G. Herzberg,Molecular Spectra and Molecular Structure. II.

Infrared and Raman Spectra of Polyatomic Molecules,Van Nostrand Reinhold,

New York, 1945.) For example, whereas the O–H and O–D stretches in H 2 O

are predicted in Example 14.13 to have a frequency ratio of about 0.73, which

corresponds to the square root of the reduced mass ratio of the O–H and O–D

bonds, the C–O–H bending motions of CH 3 OH and CH 3 OD have a frequency

ratio of about 0.64, which is substantially less than the O–H/O–D reduced

mass ratio.

Other linear molecules (acetylene, C 2 H 2 , for example) have similarly de-

scribed vibrational spectra: either stretching vibrations or bending vibrations.

It is only when a molecule becomes nonlinear that additional complexities

arise. Unfortunately, most molecules are nonlinear. Fortunately, similar rough

descriptions of the vibrations can be applied. Also fortunately, symmetry con-

siderations combine with the change-in-dipole-moment selection rule to limit

the number of IR-active vibrational motions of large, symmetric molecules.

The next few sections will illustrate some of the procedures used to simplify

our understanding of molecular vibrations.

14.13 Symmetry Considerations for Vibrations

A brief aside into symmetry is useful here. Recall that all molecules can be as-

signed a point group describing the symmetric arrangement (if any) of their

atoms. This assumes, however, that the atoms are fixed in space. The very

thought of vibrations suggests that the atoms are not fixed, and that specific

symmetry designations are useless because the atoms are constantly moving

around. Does this mean that vibrations of molecules destroy the symmetry of

a molecule and that symmetry is not as applicable to molecules as we thought?

No, it doesn’t mean that. All normal modes of a molecule oscillate about an

equilibrium position, and the averagegeometry of a molecule is defined in

496 CHAPTER 14 Rotational and Vibrational Spectroscopy