Angular Momentum in Molecular Rotation—The Rigid Rotor

4-7 Angular Momentum in Molecular Rotation—The Rigid Rotor 

We have seen that the two-particle system of an electron and a nucleus rotating about a center of mass (COM) can be transformed to the one-particle system of a reduced mass rotating about a fixed point. However, this transformation can be made for any two-mass system, and so it applies also to the case of the nuclei of a rotating diatomic molecule. As we now show, the mathematical outcome for the rotating diatomic molecule is strikingly similar to that for the hydrogenlike ion. 

The simplest treatment of molecular rotation ignores vibrational motion by assuming that the distance between the nuclei is fixed. The resulting model is therefore called the rigid-rotor model. Let there be two nuclear masses, mi and m2, separated from the COM by distances r and ri, respectively. Then, because of the way that the COM is defined, we have that 

It is not difficult to show (Problem 4-35) that the same moment of inertia results from a reduced mass /x rotating about a fixed point at a distance r = r - - r2. That is, if 

Therefore, solving the problem of a reduced mass 11 rotating about a fixed point at the fixed distance r = ri -h r2 is equivalent to solving the two-mass rigid-rotor problem. In effect, the rotating-diatomic problem is transformed to a particle-on-the-surface-of-a-sphere problem. 

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