molecular axis. Figure 14.22 shows these rotational degrees of freedomfor
linear and nonlinear molecules. For rotations, as for translations, the atoms of
the molecule are moving together in such a way as to move the molecule.
Unlike translations, in rotations the center of mass of the molecule is not mov-
ing in space. However, like the translational degrees of freedom, again the
atoms are not moving with respect to each other.
This leaves either 3N5 (for linear molecules) or 3N6 (for nonlinear
molecules) combinations of the motions of the atoms of the molecule. In these
combinations, the atoms are moving with respect to each other, but the center
of mass of the entire molecule does not change. These internal atomic motions
are the vibrational degrees of freedom,or more simply the vibrations,of the
molecule.
There are many ways to describe the possible motions of the atoms in a
molecule. However, a mathematical treatment of vibrations shows that there
will always be a way to assign the changes in coordinates such that all of the
possible motions of the atoms can be broken down into 3N5 (for linear
molecules) or 3N6 (for nonlinear molecules) independent motions where
for each motion the frequency of every atom’s vibration is exactly the same.
Such coordinate changes are called normal modes of vibration,or just the nor-
mal modes.
In vibrational spectroscopy, the frequencies of the vibrating atoms in the
molecule are probed. The changes in the energies of the vibrations are such
that the radiation typically used is in the infrared region of the spectrum.
Hence, IR spectroscopy is usually synonymous with vibrational spectroscopy.
Very low frequency vibrations will be detected in the microwave region of the
spectrum, whereas high-frequency changes impinge on the visible spectrum.
As with rotational spectroscopy, the wavenumber (cm^1 ) unit is common, but
so is the unit that describes the wavelength of the light involved, usually ex-
pressed as micrometers or microns.
Why do linear molecules have a different term (3N5 instead of 3N6)?
Recall that linear molecules have no defined rotation about their internuclear
axis, so they lack a rotational degree of freedom. This lack is made up for by hav-
ing an extra vibrational degree of freedom. We will find that linear molecules
have at least one vibration that moves perpendicular to another vibration of the
same frequency (and therefore they have the same energy: they are degenerate).482 CHAPTER 14 Rotational and Vibrational Spectroscopy
DCBADCBADCBA(b)(a)Figure 14.22 The rotations in the 3Ndegrees of freedom are either (a) two for linear, or (b)
three for nonlinear.Figure 14.21 Three of the 3Ndegrees of free-
dom correspond to translations of the molecule
as a whole.