σe denoting the equilibrium vibration frequency and χe the anharmonicity constant. This expression may
be used to predict the appearance of the absorption spectrum of a diatomic molecule. At room
temperature most molecules are in the ground vibrational state (V = 0) so that only transitions from this
level need be considered. The spectroscopic selection rule allows any change in the value of V, so for
transitions originating in the ground state,
where n may be 1,2,3....
Substitution of appropriate values of n into the equation leads to a set of expressions giving the
wavenumber of each transition, e.g. the first three and their designations are
where σspec is the wavenumber of the spectroscopic absorption.
The transition probability falls off rapidly with increasing n, and the spectrum therefore consists of two
or three lines of diminishing intensity, each one being separated from the next by rather less than the
value of σe. Although basically a simple spectrum simultaneous rotational transitions produce a pattern
of fine structure lines on either side of the vibrational transition. Rotational fine structure is not usually
resolved for samples run in the liquid state or in solution. This is due to collisional broadening and the
resulting spectrum would have the appearance shown in Figure 9.13. For a given molecule, and using
the classical analogy considered earlier, the positions of the fundamental and hence the overtones, are
determined by the atomic masses m 1 and m 2 , and the stiffness of the bond. These are related by the
equation
Figure 9.13
Infrared spectrum of a diatomic
molecule in solution.