Rotation

In classical mechanics, the energy of an object that rotates around some defined axis is given by E = ½ Iω2. Here, ω is the angular velocity and has unit of radians per second. In the equation I = mr2, I is the moment of inertia which is dependent on mass m and distance r from the axis. Having replaced m for I and v for ω, the analogy to kinetic energy E = ½ mv2 is noticeable. An object that rotates freely around three axes (A, B, C), i.e. a principal axes of rotation, therefore has an rotational energy of:

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 moment of inertia are found here. To begin, we assume the shape of the molecule to not be dependent on the rotation, i.e. the length of bonds arenot 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||

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