4000 cm–^1 (2.5– 40 μm), although some occur between 4000 cm–^1 and the beginning of the visible
region at about 12500 cm–^1 (0.8 μm or 800 nm) in a region known as the near infrared (NIR). A
molecule can absorb energy only if there is a net change in the dipole moment during a particular
vibration, a condition fulfilled by virtually all polyatomic molecules. The absorption spectra of such
molecules are often very complex and the underlying reason for this complexity is best understood by
first considering the spectrum of a heteronuclear diatomic molecule in the gaseous state.
Diatomic Molecules
The classical analogy of a vibrating diatomic molecule is that of two weights connected by a spring.
The potential energy of such a vibrating system is related to the displacement of the weights relative to
each other along the axis of the spring. If the motion is simple harmonic the relation is given by E = 1/2
fx^2 where E denotes potential energy, x the displacement and f the force constant or stiffness of the
spring. The equation describes a parabola, Figure 9.12, and shows that identical changes in potential
energy occur on stretching or compressing the spring. A diatomic molecule behaves rather differently in
that the forces of internuclear repulsion arising when the bond is compressed build up rapidly whilst, on
stretching, the bond weakens and may finally disrupt. The resulting vibrational motion is anharmonic in
nature and the mathematical relation between potential energy and displacement is necessarily more
complex. The corresponding curve is modified in the manner shown in Figure 9.12.
Figure 9.12
Vibrational energy levels of a diatomic molecule.The concept of discrete or quantized energy levels can be superimposed on this diagram by representing
them as a series of horizontal lines, the spacing of which becomes closer with increasing energy due to
the anharmonic nature of the vibration. In quantum mechanical terms, these levels are labelled V =
0,1,2,3... , where V is the vibrational quantum number, and the general mathematical expression for
the potential energy of the system expressed in wavenumbers is given by