The elastic rotor; centrifugal distortion

In an elastic rotor R is no longer constant but increases with increasing amount of rotation as a result of centrifugal forces. This effect is known as centrifugal distortion. An estimate of this effect can be obtained from a simple classical picture. As the molecule stretches the centrifugal force F_c is, at some new equilibrium distance R_e', balanced by the elastic binding force F_e, which is harmonic. The centripetal and elastic forces are:

By equating F_c=F_e and by assuming it follows:

The expression for the rotational energy including the centrifugal effect is obtained from:

Now use R_e' for the above equations and expanding the first term of the energy expression it follows:

The quantum mechanical Hamiltonian is obtained by replacing N by the quantum mechanical operator. It is clear that the spherical harmonics Y_NM(Omega) are also solutions of that Hamiltonian. The result for the rotational energy can be expressed as:

where:

is the centrifugal distortion constant. This constant is quite small, e.g. 5.32 x 10^-4 cm^-1 in HCl, but its effect can be quite large for high rotational angular momentum states (N⁴ dependence).
Selection rules for the elastic rotor are the same as for the rigid rotor (see later).

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Last change: 18 February 2001