Rotational selection rules for molecules determine the probabilities of rotational state transitions observed in spectroscopy.
According to the time-dependent perturbation theory, the transition probability 
between the orthogonal rotational states _{l,m_l})
and _{l',m'_l})
within a given vibrational state of a molecule, as observed by microwave spectroscopy, is proportional to _{l',m'_l}\vert\boldsymbol{\mathit{\hat{\mu}}}\vert\psi(\theta,\phi)_{l,m_l}\rangle\vert^2)
, where 
is the operator for the molecule’s electric dipole moment. In other words, a molecule must possess a permanent electric dipole moment (
) to exhibit a rotational spectrum. Homonuclear diatomic molecules and spherical rotors with no net permanent dipole moment (such as H2 and CH4 in their ground states) are generally rotationally inactive.
Since 
, the components of the dipole moment in polar coordinates are:
Suppose the perturbation on the molecule is caused by a plane-polarised electromagnetic wave with an electric component 
oscillating in the 
-direction. We have _{l',m'_l}\vert\hat{\mu_z}\vert\psi(\theta,\phi)_{l,m_l}\rangle\vert^2)
, with
_{l',m'_l}\mu cos\theta\psi(\theta,\phi)_{l,m_l}sin\theta d\theta d\phi\;\;\;\;\;\;\;\;63)
Substituting the explicit expression of the spherical harmonics wavefunction )
into eq63 gives:
\phi}d\phi\;\;\;\;\;\;\;\;64)
where \frac{(l'-\vert m'_l\vert)!}{(l'+\vert m'_l\vert)!}\biggr\]^{\frac{1}{2}}\biggr\[\biggr\(\frac{2l+1}{2}\biggr\)\frac{(l-\vert m_l\vert)!}{(l+\vert m_l\vert)!}\biggr\]^{\frac{1}{2}})
, and 
and 
are functions of 
.
When 
, the integral with respect to 
is \phi}d\phi =0)
. When 
, it evaluates to \phi}d\phi =2\pi)
. Since eq63 must be non-zero for a transition to be probable in the -direction, or
Substituting back into eq64 and noting that =-sin\theta d\theta)
yields:
\;\;\;\;\;\;\;\;66)
where (2l+1)(l'-\vert m'_l\vert)!(l-\vert m_l\vert)!}{(l'+\vert m'_l\vert)!(l+\vert m_l\vert)!}\biggr\]^{\frac{1}{2}})
.
Substituting the polar coordinate form 
into eq370 (a recurrence relation of the associated Legendre polynomials) results in:
Substituting eq67 into eq66 gives:
+\frac{l-\vert m_l\vert+1}{2l+1}\int_{-1}^{1}P_{l'}^{\vert m'_l\vert}P_l^{\vert m_l\vert}d(cos\theta)\biggr\]\;\;\;\;\;\;\;\;68)
For 
, either integral in eq68 must be non-zero. Each integral is non-zero when the corresponding pair of spherical harmonics is not orthogonal. This occurs if 
and 
for the first integral and 
and for the second integral. In other words, if
or using rotational spectroscopy notations:
For an electric component 
oscillating in the 
-direction, eq63 becomes
_{l',m'_l}\mu sin\theta cos\phi\psi(\theta,\phi)_{l,m_l}sin\theta d\theta d\phi\;\;\;\;\;\;\;\;70)
Substituting the explicit expression of the spherical harmonics wavefunction and 
into eq70 gives:
\phi}+e^{i(m_l-m'_l-1)\phi}] d\phi\;\;\;\;\;\;\;\;71)
When , the integral with respect to equals zero. When 
, it evaluates to 
. Since eq71 must be non-zero for a transition to be probable in the -direction, it must satisfy:
Substituting \phi}+e^{i(m_l-m'_l-1)\phi}] d\phi=2\pi)
back into eq71, and noting that , yields:
\;\;\;\;\;\;\;\;73)
Substituting the polar coordinate form and into eq371 (another recurrence relation of the associated Legendre polynomials) results in:
Substituting eq74 into eq73 gives:
}\biggr\[\int_{-1}^{1}P_{l'}^{\vert m_l\pm 1\vert}P_{l+1}^{\vert m_l\vert+1} d(cos\theta)-\int_{-1}^{1}P_{l'}^{\vert m_l\pm 1\vert}P_{l-1}^{\vert m_l\vert+1} d(cos\theta)\biggr\]\;\;\;\;\;\;\;\;75)
For 
, either integral in eq75 must be non-zero. Each integral is non-zero when the corresponding pair of spherical harmonics is not orthogonal. This occurs if
| Integrals |
Condition 1 |
Cases |
Condition 2 |
Results |
| 1st |
|
 |
 |
 if  |
 |
 if |
 |
 |
if  |
 |
if |
| 2nd |
|
|
|
if |
|
if |
|
|
if |
|
if |
Combining the results, when 
and 
, or in rotational spectroscopy notations:
Repeating the derivation for 
, we arrive at the same selection rules expressed by eq76. Therefore, the rotational selection rules for polar molecules (linear rotors and spherical rotors) subjected to isotropic radiation are:
For symmetric rotors, the electric dipole moment lies along the principal molecular axis (-axis). However, there is a third quantum number 
to consider. The wavefunction )
can be approximated as the product of three functions 
, where 
depends solely on the quantum number . If 
in the matrix element 
is defined with respect to the molecular axis, then the matrix element can be expressed as a product of three integrals, one of which is \hat{\mu}_{z}f_3(K)d\gamma)
. For this integral to be non-zero, the vanishing integral theorem from group theory states that must transform as the totally symmetric irreducible representation of the molecule’s point group (which is the case in 
groups) and 
. In other words, the rotational transition selection rules for symmetric rotors are:
Question
Is the effect of nuclear statistics on rotational states different from rotational selection rules?
Answer
The effect of nuclear statistics on rotational states is a separate, yet related, concept from the general rotational selection rules. While both influence which rotational states are observed in a spectrum, they operate based on different fundamental principles. The key distinction is that rotational selection rules dictate the possible transitions, while nuclear statistics determine the relative populations of the initial states. A transition must be both “allowed” by the selection rules and originate from a “populated” state to be observed. Therefore, nuclear statistics modify the intensity of the allowed transitions without changing the fundamental rule of which transitions are allowed.