E = ½ IAωA2 +½ IBωB2+ ½ ICωC2
with Ji = Ii ⋅ ωi (J = angular momentum) we obtain:
E = JA2/(2IA) + JB2/(2IB) + JC2/(2IC)
Some formulas to calculate a moment of inertia are found here. To begin with, we assume the shape of the molecule not to be dependent on the rotation, i.e. the length of bonds not to be affected by centrifugal forces. The molecule is thus regarded as rigid rotor in contrast to an elastic rotor. We have to deal with the following cases
Spherical top | IA = IB = IC = I | e.g.
CH4, CCl4, SF6 |
Symmetric top | IA = IB = I⊥
and IC = I||
I||>I⊥ oblate I||<I⊥ prolate |
e.g.
Benzene CH3I |
Linear rotor (a special case of symmetric top) |
IA = 0, IB = IC | all diatomic molecules, e.g.
NO, C2N2, CO2 |
Asymmetric top | IA ≠ IB ≠
IC, IA ≠
IC
IA < IB < IC |
e.g. H2O |