Elastic Rotator: Centrifugal Distortion

So far, we assumed that rotation does not affect the shape of a molecule (rigid rotor). However, in general there are centrifugal forces that stretch the bonds of a rotating molecule. We will confine our treatment of this problem with the case of linear rotor.

An increased length of the bond corresponds to a higher moment of inertia ($I = \mu r^2$). In turn, the rotational constant $B$ decreases ($B \sim I^{-1}$), i.e. we would expect a rotational energy with a minor slope than in the case of $E_{rot} = B_{rigid} J(J + 1)$.

In this case we have no exact solution of the Schrodinger equation, however, we are can attack the problem using a series expansion and write an expected rotational energy as:

$$F(J) = B\cdot J(J+1) - D\cdot J^2(J+1)^2 + \cdots$$ (47)

Here we assume that the coefficient $D$ is a small correction, in other words, $D$ is much smaller than $B$. This coefficient is called centrifugal distortion constant.

From the treatment above we should expect that the weaker the bond, the more pronounced the deviation from the rigid rotor becomes. A weak bond corresponds to a low vibrational energy: $E_{vib} = \hbar\omega(v + 1/2)$; $\omega = (k/\mu)^{1/2}$, where $k$ is a force constant of the bond. The value of $D$ can be either calculated, or derived from spectral data.

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