Modern Spectroscopy

106 5 ROTATIONAL SPECTROSCOPY
for the rotational energy levels E r of a diatomic molecule, in the rigid rotor approximation,
was introduced. In this equation, I is the moment of inertia [equal to mr 2 , where the reduced
mass m is equal to m 1m 2=ðm 1 þ m 2ÞŠ and the rotational quantum number J ¼ 0; 1; 2; ....
The same expression applies also to any linear polyatomic molecule but, because I is likely
to be larger than for a diatomic molecule, the energy levels of Figure 1.12 tend to be more
closely spaced.
In practice, what is measured experimentally is not energy but frequency, in the millimetre
wave and microwave regions, or wavenumber, in the far infrared. Therefore we convert the
energy levels of Equation (5.10) to what are known as term values FðJÞ having dimensions
of either frequency, by dividing by h, or wavenumber, by dividing by hc, giving
or
FðJÞ ¼ E r
h
h
¼
8p2 JðJ þ 1Þ ¼BJðJ þ 1Þ ð5:11Þ
I
FðJÞ ¼ Er h
¼
hc 8p2 JðJ þ 1Þ ¼BJðJ þ 1Þ ð5:12Þ
cI
The use of the symbols FðJÞ and B for quantities which may have dimensions of frequency
or wavenumber is unfortunate, but the symbolism is used so commonly that there seems
little prospect of change. In Equations (5.11) and (5.12) the quantity B is known as the
rotational constant. Its determination by spectroscopic means results in determination of
internuclear distances and represents a very powerful structural technique.
Figure 5.2 shows a set of rotational energy levels, or, strictly, term values, for the CO
molecule.
The transition intensity is proportional to the square of the transition moment, which is
given by
ð
Rr ¼ c 0
r*mc 00
r dt ð5:13Þ
analogous to Equation (2.13). The rotational selection rules constitute the conditions for
which the intensity, and therefore R r, is non-zero and are:
1. The molecule must have a permanent dipole moment (m 6¼ 0).
2. DJ ¼ 1.
3. DM J ¼ 0, 1, a rule which is important only if the molecule is in an electric or magnetic
field (see Equation 1.61).
Rule 1 shows that transitions are allowed in heteronuclear diatomic molecules such as CO,
NO, HF and even 1 H 2 H (for which m ¼ 5:9 10 4 D compared with, say, HF, for which
m ¼ 1.82 D), 1 but not in H 2,Cl 2 and N 2. Similarly, ‘unsymmetrical’ linear polyatomic
1 1D’ 3.335 64 6 10 730 Cm.