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Applied vibrational spectroscopy 329

Polyatomic normal modes

A diatomic molecule possesses only one mode of vibration, the stretching and
compression of the bond between the two atoms. The number of distinct modes of
vibration in a non-linear polyatomic molecule containing N>2 atoms is 3N−6. (To specify
the displacement of each of N atoms in three dimensions requires a total of 3N
coordinates. Three of these coordinates specify the position of the center of mass of the
molecule, and therefore correspond to the translational modes of the molecule, and three
coordinates specify the orientation in space of the molecule, and therefore correspond to
the rotational modes of the molecule.) A linear molecule of N atoms possesses 3N−5
vibrational modes since only two angles are required to specify the orientation in space of
a linear molecule.
The number of vibrational modes increases rapidly with the size of the molecule. For
example, H 2O is a non-linear triatomic molecule and has three modes of vibration, CO 2 is
a linear triatomic molecule and has four modes of vibration, whereas benzene, C 6H 6, has
30 modes of vibration.
It is easier to visualize the vibrational modes of a polyatomic molecule when particular
combinations of bond stretches or bends are considered together. These collective
vibrational displacements, in which the atoms all move in phase and with the same
frequency, are called normal modes. Exactly 3N−6, or 3N−5, independent normal modes
of molecular vibration can be derived for non-linear, or linear, polyatomic molecules,
respectively. Each normal mode behaves like a harmonic oscillator with a reduced mass,
µ, and force constant, k, that depend on which atoms and bonds contribute to the
vibration (Table 2).
Figure 1 illustrates the normal modes of vibration of H 2O and CO 2. The bending
vibration of CO 2 is doubly degenerate since the bending motion can also be drawn
perpendicular to the plane of the paper in Fig. 1b. The degeneracy accounts for the
required additional vibrational mode of linear