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 I_Aω_A² +½ I_Bω_B²+ ½ I_Cω_C²

with J_i = I_i ⋅ ω_i (J = angular momentum) we obtain:

E = J_A²/(2I_A) + J_B²/(2I_B) + J_C²/(2I_C)

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	I_A = I_B = I_C = I	e.g. CH₄, CCl₄, SF₆
Symmetric top	I_A = I_B = I_⊥ and I_C = I_\|\| I_\|\|>I_⊥ oblate I_\|\|<I_⊥ prolate	e.g. Benzene CH₃I
Linear rotor (a special case of symmetric top)	I_A = 0, I_B = I_C	all diatomic molecules, e.g. NO, C₂N₂, CO₂
Asymmetric top	I_A ≠ I_B ≠ I_C, I_A ≠ I_C I_A < I_B < I_C	e.g. H₂O

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