Rotational Spectroscopy

Rotational spectroscopy probes the quantized angular momentum of molecules by observing absorption and emission in the microwave and far-infrared regions of the electromagnetic spectrum, typically at frequencies from 1 GHz to 1 THz (wavenumbers ~0.03–30 cm$^{-1}$).

A molecule can absorb microwave radiation only if it possesses a permanent electric dipole moment. The oscillating electric field of the radiation couples to the rotating dipole, driving transitions between quantized rotational energy levels. This is why homonuclear diatomics such as H$_2$, N$_2$, and O$_2$ are microwave-inactive — they lack a permanent dipole.

From the precisely measured frequencies of rotational transitions, one can extract bond lengths, bond angles, and force constants with extraordinary precision. The technique is among the most accurate methods for determining molecular geometry, with bond lengths routinely measured to $\pm 0.001$ Å. Rotational spectroscopy also plays a pivotal role in radio astronomy, where microwave emission lines identify molecules in the interstellar medium.

Consider a diatomic molecule with atoms of masses $m_1$ and $m_2$separated by a fixed bond length $r_0$. In the center-of-mass frame, the two-body rotation reduces to a single particle of reduced mass:

moving on the surface of a sphere of radius $r_0$. The moment of inertia is:

The Schrödinger Equation for the Rigid Rotor

Since the bond length is fixed, the potential energy is zero and the Hamiltonian contains only kinetic (rotational) energy. In spherical coordinates, the rigid-rotor Hamiltonian is:

where $\hat{L}^2$ is the square of the orbital angular momentum operator. The eigenfunctions of $\hat{L}^2$ are the spherical harmonics $Y_J^{M_J}(\theta, \phi)$, satisfying:

where $J = 0, 1, 2, \ldots$ is the rotational quantum number and$M_J = -J, -J{+}1, \ldots, J{-}1, J$. Substituting into the eigenvalue equation$\hat{H}\psi = E\psi$:

Thus the energy eigenvalues are:

where we define the rotational constant:

Degeneracy

The energy $E_J$ depends only on $J$, not on $M_J$. Since $M_J$ runs from $-J$ to $+J$, there are$2J + 1$ degenerate states for each $J$:

This degeneracy reflects the fact that the energy of a freely rotating molecule is independent of the orientation of its angular momentum in space.

Energy Level Spacing

The energy gap between adjacent levels grows linearly with $J$:

The levels are not equally spaced — the gap increases as $J$increases — but we shall see that the transition frequencies are nearly equally spaced under the selection rule $\Delta J = +1$.

Transition Dipole Matrix Element

A transition between rotational states $|J, M_J\rangle$ and$|J', M_J'\rangle$ is driven by the interaction of the radiation field with the molecular dipole moment $\hat{\boldsymbol{\mu}}$. The transition rate is proportional to the square of the transition dipole matrix element:

For a diatomic molecule with permanent dipole moment $\mu_0$ along the internuclear axis, the dipole operator in the space-fixed frame involves direction cosines. The $z$-component of the dipole is$\hat{\mu}_z = \mu_0 \cos\theta$, and similarly for $x$ and $y$. Using the relation $\cos\theta \propto Y_1^0(\theta,\phi)$, the integral becomes:

This is a standard integral over three spherical harmonics (a Gaunt integral). The properties of the Clebsch–Gordan coefficients dictate that it is nonzeroonly when:

and the molecule must have $\mu_0 \neq 0$ (permanent dipole moment). This is the gross selection rule: a polar molecule is required.

Absorbed Frequencies

For an absorption transition $J \to J + 1$, the photon energy must match the energy gap:

In wavenumber units ($\tilde{\nu} = \nu / c$):

Uniform Spacing of Lines

The first few transitions produce lines at:

The spacing between consecutive absorption lines is constant:

This equal spacing of $2\tilde{B}$ is the hallmark of a pure rotational spectrum in the rigid-rotor approximation.

Why Homonuclear Diatomics Have No Rotational Spectrum

Homonuclear diatomic molecules (H$_2$, N$_2$, O$_2$, etc.) possess a center of symmetry. By symmetry, their permanent electric dipole moment vanishes:$\mu_0 = 0$. Since the transition dipole is proportional to $\mu_0$,every rotational matrix element is zero. These molecules are therefore microwave inactive and do not exhibit a pure rotational absorption spectrum.

Note that homonuclear diatomics can still have rotational Raman spectra, as the Raman selection rule depends on the polarizability rather than the dipole moment.

Real molecules are not perfectly rigid. As the molecule rotates faster (higher $J$), the centrifugal force stretches the bond, increasing the moment of inertia and lowering the rotational energy compared to the rigid-rotor prediction.

Classical Picture: Centrifugal Stretching

For a diatomic rotating with angular momentum $L$, the centrifugal force on the reduced mass is $F_{\text{cent}} = \mu\omega^2 r = L^2/(\mu r^3)$. In equilibrium, this is balanced by the restoring force of the bond, modeled as a spring with force constant $k$:

For small displacements $\delta r = r - r_0 \ll r_0$, we approximate$r \approx r_0$ in the centrifugal term:

Corrected Energy Levels

Treating the centrifugal effect as a perturbation and replacing $L^2$ by its quantum eigenvalue $\hbar^2 J(J+1)$, one obtains the first-order corrected energy:

where $D$ is the centrifugal distortion constant.

Derivation of D

The stretching lowers the rotational energy by an amount proportional to $[J(J+1)]^2$. The detailed calculation yields:

where $\omega$ is the harmonic vibrational frequency of the bond (in the same units as $B$). Since vibrational frequencies are much larger than rotational constants ($\omega \gg B$), we have $D \ll B$, typically$D/B \sim 10^{-4}$ to $10^{-6}$.

Effect on Spectral Lines

The transition frequencies for $J \to J + 1$ now become:

The centrifugal correction causes each line to shift slightly to lower frequency. The spacing between consecutive lines decreases:

This gradual decrease in spacing is clearly observable in high-resolution spectra. By fitting the measured frequencies to the non-rigid rotor formula, one extracts both $\tilde{B}$and $\tilde{D}$, giving information about both the bond length and the force constant.

A symmetric top molecule has two equal principal moments of inertia. For a prolate symmetric top (e.g., CH$_3$Cl),$I_A < I_B = I_C$, while for an oblate top (e.g., benzene),$I_A = I_B < I_C$. We focus on the prolate case, which includes most common examples such as NH$_3$ and CH$_3$F.

Energy Levels

The rotational Hamiltonian for a symmetric top is:

where $\hat{J}^2$ is the total angular momentum squared and$\hat{J}_a$ is its component along the molecular symmetry axis. The eigenvalues of $\hat{J}^2$ are $\hbar^2 J(J+1)$ and those of$\hat{J}_a$ are $\hbar K$ with$K = 0, \pm 1, \pm 2, \ldots, \pm J$. Defining rotational constants$A = \hbar^2/(2I_A)$ and $B = \hbar^2/(2I_B)$:

For a prolate top, $A > B$, so $(A - B) > 0$ and higher$|K|$ increases the energy. For an oblate top, replace $A$ with$C$: $E_{J,K} = BJ(J+1) + (C - B)K^2$ where$C - B < 0$.

Selection Rules

For a symmetric top with a dipole moment along the symmetry axis:

The $\Delta K = 0$ rule means that the component of angular momentum about the symmetry axis does not change. Since the energy depends on $K^2$, and$K$ does not change, the $K$-dependent term cancels in the transition frequency. The result is the same expression as for a linear rotor:

Thus, the pure rotational spectrum of a symmetric top looks like that of a diatomic molecule — equally spaced lines separated by $2\tilde{B}$ — but each line may carry different degeneracy and intensity patterns.

Application: NH$_3$ and CH$_3$Cl

Ammonia (NH$_3$): A prolate symmetric top with $\tilde{B} \approx 9.94$ cm$^{-1}$ and$\tilde{A} \approx 6.30$ cm$^{-1}$. Its famous inversion doubling at 23.87 GHz (the transition used in the first maser) overlays the rotational structure.

Methyl chloride (CH$_3$Cl): With$\tilde{B} \approx 0.443$ cm$^{-1}$, its rotational lines lie in the far infrared. Centrifugal distortion terms $D_J$, $D_{JK}$, and $D_K$ are needed for precise fitting.

At thermal equilibrium, the population of rotational level $J$ is given by the Boltzmann distribution:

where $g_{\text{ns}}$ is the nuclear spin statistical weight (important for molecules with equivalent nuclei), and the rotational partition function is:

In the high-temperature limit ($k_BT \gg B$), this sum can be replaced by an integral, giving $q_{\text{rot}} \approx k_BT/(\sigma B)$ where$\sigma$ is the symmetry number.

Most Populated Level

The population $N_J$ first increases with $J$ (due to the degeneracy factor $2J+1$) then decreases (due to the Boltzmann exponential). To find the maximum, differentiate $f(J) = (2J+1)\exp[-BJ(J+1)/(k_BT)]$ with respect to $J$ and set it to zero. Treating $J$ as continuous:

Simplifying:

Solving for $J$:

This result has a simple physical interpretation: higher temperature populates higher angular momentum states. For HCl at 300 K with $\tilde{B} \approx 10.6$ cm$^{-1}$,$J_{\max} \approx 3$.

Intensity Envelope

The absorption intensity of the $J \to J+1$ transition is proportional to the population of the lower state $N_J$ and to the transition probability, which scales as $(J+1)$ from the squared matrix element. Thus:

The resulting intensity envelope rises steeply at low $J$, peaks near$J_{\max}$, and then decays exponentially. At higher temperatures, the envelope broadens and the peak shifts to higher $J$, distributing intensity over more lines.

The development of microwave spectroscopy was greatly accelerated by radar technology from World War II, which produced high-power microwave sources (klystrons, magnetrons) and sensitive detection systems that were subsequently adapted for laboratory spectroscopy.