Selection rules specify the possible transitions among quantum levels due to absorption or emission of electromagnetic radiation. Incident electromagnetic radiation presents an oscillating electric field \(E_0\cos(\omega t)\) that interacts with a transition dipole. The dipole operator is \(\mu = e \cdot r\) where \(r\) is a vector pointing in a direction of space.
A transition dipole moment is a transient dipolar polarization created by an interaction of electromagnetic radiation with a molecule
In an experiment we present an electric field along the z axis (in the laboratory frame) and we may consider specifically the interaction between the transition dipole along the x, y, or z axis of the molecule with this radiation. If \(\mu_z\) is zero then a transition is forbidden. The selection rule is a statement of when \(\mu_z\) is non-zero.
Rotational transitions
We can use the definition of the transition moment and the spherical harmonics to derive selection rules for a rigid rotator. Once again we assume that radiation is along the z axis.
\[(\mu_z)_{J,M,{J}',{M}'}=\int_{0}^{2\pi } \int_{0}^{\pi }Y_{J'}^{M'}(\theta,\phi )\mu_zY_{J}^{M}(\theta,\phi)\sin\theta\,d\phi,d\theta \nonumber \]
Notice that m must be non-zero in order for the transition moment to be non-zero. This proves that a molecule must have a permanent dipole moment in order to have a rotational spectrum. The spherical harmonics can be written as
\[Y_{J}^{M}(\theta,\phi)=N_{\,JM}P_{J}^{|M|}(\cos\theta)e^{iM\phi} \nonumber \]
where \(N_{JM}\) is a normalization constant. Using the standard substitution of \(x = \cos q\) we can express the rotational transition moment as
\[(\mu_z)_{J,M,{J}',{M}'}=\mu\,N_{\,JM}N_{\,J'M'}\int_{0}^{2 \pi }e^{I(M-M')\phi}\,d\phi\int_{-1}^{1}P_{J'}^{|M'|}(x)P_{J}^{|M|}(x)dx \nonumber \]
The integral over f is zero unless M = M' so \(\Delta M = \) 0 is part of the rigid rotator selection rule. Integration over \(\phi\) for \(M = M'\) gives \(2\pi \) so we have
\[(\mu_z)_{J,M,{J}',{M}'}=2\pi \mu\,N_{\,JM}N_{\,J'M'}\int_{-1}^{1}P_{J'}^{|M'|}(x)P_{J}^{|M|}(x)dx \nonumber \]
We can evaluate this integral using the identity
\[(2J+1)x\,P_{J}^{|M]}(x)=(J-|M|+1)P_{J+1}^{|M|}(x)+(J-|M|)P_{J-1}^{|M|}(x) \nonumber \]
Substituting into the integral one obtains an integral which will vanish unless \(J' = J + 1\) or \(J' = J - 1\).
\[\int_{-1}^{1}P_{J'}^{|M'|}(x)\Biggr(\frac{(J-|M|+1)}{(2J+1)}P_{J+1}^{|M|}(x)+\frac{(J-|M|)}{(2J+1)}P_{J-1}^{|M|}(x)\Biggr)dx \nonumber \]
This leads to the selection rule \(\Delta J = \pm 1\) for absorptive rotational transitions. Keep in mind the physical interpretation of the quantum numbers \(J\) and \(M\) as the total angular momentum and z-component of angular momentum, respectively. As stated above in the section on electronic transitions, these selection rules also apply to the orbital angular momentum (\(\Delta{l} = \pm 1\), \(\Delta{m} = 0\)).