The rotational motion of diatomic molecules

The Schrodinger equation for a system of two particles, can be separated [3]. We define the reduced mass, n. 

Then in spherical coordinates, Fig. 6.1, we have (where r, and are the polar coordinates of a nucleus) 

Under the Bom-Oppenheimer approximation Eq. (6.2) separates into three equations of the single variables r, 9 and The solutions for the angular functions ) and are given by  

The total angular wavefiinction is the product of both angular terms, Eq. (6.3) and Eq. (6.4) 

Where the simultaneous wavefiinctions of the angular momentum operators, J and are expressed in the spherical harmonics, Yjm. The total angular momentum due to the rotation of the molecule is 

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The most direct application of particle-on-a-sphere result is to the rotational motion of diatomic molecules in a gas. As with vibrations (see Section 3.2), the real situation looks a little more complicated, but can be solved in a similar way. A molecule actually rotates about its centre of mass the coordinates 8 and can be used to define its direction in space. If we replace the mass in Schrodinger s equation by the reduced mass given by eqn 3.22, and let r be the bond length, then the moment of inertia is... 

Equation (28-71) indicates that the rotational motion of diatomic molecules yields rotation,classical = T/B, where B is a characteristic rotational temperature that is, at most, a few tens of Kelvin (see Table 28-2). The rotational contribution to the internal energy is... 

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