Spherical rotors
When all three moments of inertia are equal to some value I, as in CH4 and SF6, the classical expression for the energy is
where J2 = Ja2 + Jb2 + Jc2 is the square of the magnitude of the angular momentum. We can immediately find the quantum expression by making the replacement
J2 →J(J+1) h2 J =0, 1, 2,...
Therefore, the energy of a spherical rotor is confined to the values
The resulting ladder of energy levels is illustrated in Fig. 13.12. The energy is normally expressed in terms of the rotational constant, B, of the molecule, where
The expression for the energy is then
EJ = hcBJ(J + 1) J =0, 1, 2,...
The rotational constant as defined by eqn 13.25 is a wavenumber. The energy of a rotational state is normally reported as the rotational term, F(J), a wavenumber, by division by hc:
F(J) = BJ (J +1)
The separation of adjacent levels is
F(J) − F (J −1) =2BJ
Because the rotational constant decreases as I increases, we see that large molecules have closely spaced rotational energy levels. We can estimate the magnitude of the separation by considering CCl4: from the bond lengths and masses of the atoms we find I=4.85× 10−45 kg m2, and hence B = 0.0577 cm−1.
