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 CO_{2}, C_{2}H_{2}
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.
M_{J} 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 M_{J} 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. NH_{3}
and ND_{3}.) 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 infrared portion of the electromagnetic. Hence this
form of spectroscopy is sometimes known as infrared rotational
spectroscopy (to distinguish it from rotational Raman spectroscopy).
