Rotational Selection Rules
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A selection rule is a statement about which transitions are allowed (and thus which lines may be observed in a spectrum). The classical idea is that for a molecule to interact with the electromagnetic field and absorb or emit a photon of frequency ν, it must possess, even if only momentarily, a dipole oscillating at that frequency.

A gross selection rule is one which makes some statement about the general features a molecule must have if it is to produce a spectrum.

In the case of rotation, the gross selection rule is that the molecule must have a permanent electric dipole moment. i.e. only polar molecules will give a rotational spectrum.

Symmetrical linear molecules, such as CO2, C2H2 and all homonuclear diatomic molecules, are thus said to be rotationally inactive, as they have no rotational spectrum. Spherical rotors must also be rotationally inactive, unless their geometry is sufficiently distorted by rotation that they can possess a permanent dipole while they are rotating.

The basis of this selection rule lies in classical mechanics. A rotating molecule with a permanent electric dipole appears, to a stationary observer, to possess a fluctuating dipole. Classically, this fluctuating dipole can be regarded as inducing oscillations in the surrounding electromagnetic field, and this interaction allows absorption of a photon.

Specific rotational selection rules may be obtained by a detailed quantum mechanical treatment of the situation, and for a linear molecule, the selection rules prove to be:

The permitted change in the quantum number J reflects the fact that a photon has an intrinsic angular momentum of one unit. Thus by the conservation of momentum,  the possible change in J is restricted to 1 unit.

MJ is a quantum number which, like K, measures the component of the angular momentum about an axis. However, in this case, the axis is an externally defined one.

(K specifies the component about the principle axis of the molecule, which remains in the same position relative to the molecule whatever orientation the molecule is in. This is not an externally defined axis. An example of an externally defined axis would be one vertically upwards in a laboratory. This axis remains fixed in space regardless of the orientation of the molecule.)

The permitted changes in MJ are also a consequence of the application of conservation of angular momentum to the situation, taking into account the direction which the photon enters or leaves the molecule.

For symmetric rotors, a selection rule is needed for K. This rule is:

We may apply these selection rules to the expressions for the energy terms of a rigid rotor to obtain the wavenumbers of the allowed transitions. For a J + 1 J transition:

When we include the centrifugal distortion constant to compensate for the effect of centrifugal distortion, the expression we obtain is:

The second term is typically very much smaller than the first, so the appearance of the spectrum is often in close accord with that predicted by the first equation. The form of the spectrum predicted by the first equation is demonstrated by this diagram:

Note that the wavenumbers of the spectral lines will lie at 2B, 4B, 6B, so the lines are spaced equally with a distance 2B between them.
Measurement of the spacing thus allows calculation of B, and from this the moment of inertia perpendicular to the principle axis of the molecule.
For diatomic molecules, this allows calculation of the bond length, since the atomic masses are known.
For polyatomic molecules, it is not possible to carry out this calculation, as there are various different bond lengths and angles to be considered. One way round this is to obtain rotational spectra of the same molecule but using different isotopes of the atoms. (e.g. NH3 and ND3.) If we then make the assumption that the bond lengths and angles are the same for both isotopes we can obtain values for them.

Note that the gaps between rotational energy levels are such that the frequencies corresponding to transitions typically lie in the infra-red portion of the electromagnetic. Hence this form of spectroscopy is sometimes known as infra-red rotational spectroscopy (to distinguish it from rotational Raman spectroscopy).

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