Diatomic Rotational Energy Levels and Spectroscopy

Interested readers are referred to the chapter by J. C. Tully in Dynamics of Molecular Collisions, Part B, edited by W. H. Miller (Plenum, New York, 1976) for additional information on this subject. 

In the remainder of this chapter, we assume that the Born-Oppenheimer approximation is good, and that Eq. 3.16 holds. In this section we consider the rotational motion of a idealized rigid diatomic rotor, in which R is fixed at Rq. Then U,f R) = U Rq) is a constant that we may set to zero for convenience. Using Eq. 3.15 for V in the Schrodinger equation (3.16), we immediately have 

According to Eq. 3.22, the rotational levels are (2J + l)-fold degenerate. Equation 3.23 gives the rotational levels in joules (J) when h is in J s and I is in kg m. They are more commonly expressed in cm (wave numbers) via 

We now derive the pure rotational selection rules (i.e., in the absence of vibrational or electronic transitions) for the rigid rotor. If a rotor of fixed length Rq has charges + d glued to its ends, it has a permanent dipole moment 

Since the energy changes in cm for transitions from state (J — 1) to state J are equal to 2BJ for a rigid rotor, the observed rotational absorption lines are predicted to be unifomly separ ed by the spacing 2B (Fig. 3.5). This is borne out to within experimental error in the far-infrared spectrum of HCl, Fig. 3.2. If the nuclear masses (and hence the nuclear reduced mass pj ) are independently 

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