17.3 Rigid rotor

The Rigid Rotor Model

Concept and Application

The rigid rotor is a model for molecular rotation that treats a molecule as a rigid body with a fixed bond length and a constant moment of inertia. By assuming the molecule doesn't vibrate and that bond lengths stay constant during rotation, you can cleanly separate rotational motion from vibrational motion. This separation is what makes the math tractable.

The model applies best to small, rigid molecules: diatomics like $H_2$ and $CO$, and linear polyatomics like $CO_2$ and $HCN$. In these systems, rotational energy levels are well-separated from vibrational energy levels, so the rigid approximation holds up. The primary use of the model is predicting and interpreting rotational spectra, which arise from transitions between quantized rotational energy levels.

Limitations and Advanced Models

The rigid rotor ignores several real-world effects:

• Molecular vibrations: Real bonds stretch and compress, but the model treats them as fixed.
• Centrifugal distortion: As a molecule rotates faster, centrifugal force stretches the bonds outward, shifting energy levels away from the rigid rotor prediction. This effect grows more pronounced for molecules with longer bonds or at higher rotational quantum numbers.
• Rotation-vibration coupling: In reality, rotational and vibrational motions influence each other, producing complex rovibrational spectra.

For non-rigid or highly flexible molecules like $H_2O$ and $NH_3$, you need more advanced treatments (the non-rigid rotor or vibrating rotor models) that incorporate these effects.

Energy Levels and Wavefunctions of a Rigid Rotor

Schrödinger Equation and Solutions

The Hamiltonian for the rigid rotor contains only rotational kinetic energy (no potential energy term), expressed through the angular momentum operators and the moment of inertia $I$:

$\hat{H}_{rot} = \frac{\hat{J}^2}{2I} = \frac{1}{2I}\left(\hat{J}_x^2 + \hat{J}_y^2 + \hat{J}_z^2\right)$

The wavefunctions depend on the angular coordinates $\theta$ and $\phi$, and the solutions turn out to be the spherical harmonics $Y_J^m(\theta, \phi)$. These are simultaneous eigenfunctions of both $\hat{J}^2$ and $\hat{J}_z$:

$\hat{J}^2 Y_J^m(\theta, \phi) = \hbar^2 J(J+1)\, Y_J^m(\theta, \phi)$

$\hat{J}_z Y_J^m(\theta, \phi) = \hbar\, m\, Y_J^m(\theta, \phi)$

Note the use of $J$ (rather than $l$) as the quantum number here, which is the standard convention for molecular rotation.

Energy Levels and Degeneracy

The quantized energy levels depend only on the rotational quantum number $J$:

$E_J = \frac{\hbar^2}{2I}\, J(J+1), \qquad J = 0, 1, 2, \ldots$

Each level has a degeneracy of $2J+1$, corresponding to the $2J+1$ allowed values of the magnetic quantum number $m_J = -J, -J+1, \ldots, J-1, J$. This degeneracy exists because the energy doesn't depend on the orientation of the angular momentum vector in space. In the absence of an external field, all $2J+1$ orientations have the same energy.

Notice that the energy spacing between adjacent levels increases with $J$. The gap between $J$ and $J+1$ is $\frac{\hbar^2}{2I}\left[2(J+1)\right]$, so higher levels are progressively farther apart.

Selection Rules for Rotational Transitions

Allowed Transitions and Selection Rule

Not every transition between rotational levels is allowed. The selection rule for a rigid rotor is:

$\Delta J = \pm 1$

Only transitions between adjacent levels occur (e.g., $J = 0 \to 1$, $J = 1 \to 2$). This rule comes from evaluating the transition dipole moment:

$\mu_{fi} = \langle \psi_f | \hat{\mu} | \psi_i \rangle$

This integral is non-zero only when $\Delta J = \pm 1$. There's an additional prerequisite that's easy to overlook: the molecule must possess a permanent electric dipole moment for pure rotational transitions to occur. Homonuclear diatomics like $H_2$ and $N_2$ have no permanent dipole and therefore show no pure rotational spectrum.

Rotational Spectrum and Line Intensities

For absorption ($\Delta J = +1$), the transition energy from level $J$ to $J+1$ is:

$\Delta E = 2B(J+1)$

where $B = \frac{\hbar^2}{2I}$ is the rotational constant. This means the spectrum consists of a series of lines spaced by $2B$. The equal spacing is a hallmark of the rigid rotor.

The intensity of each line depends on the population of the initial level, governed by the Boltzmann distribution:

$N_J \propto (2J+1)\, e^{-E_J / k_B T}$

The $(2J+1)$ factor (degeneracy) causes the population to initially increase with $J$, while the exponential factor causes it to eventually decrease. The result is that line intensities rise to a maximum at some intermediate $J$ value and then fall off. The most intense line in the spectrum corresponds roughly to the most populated rotational level at that temperature.

Rotational Constants and Moment of Inertia

Relationship between Rotational Constant and Moment of Inertia

The rotational constant $B$ directly encodes structural information about the molecule:

$B = \frac{\hbar^2}{2I}$

Larger, more extended molecules have larger moments of inertia and therefore smaller rotational constants (and more closely spaced spectral lines). For a diatomic molecule, the moment of inertia is:

$I = \mu r^2$

where $\mu$ is the reduced mass and $r$ is the bond length. This means that if you measure $B$ from a spectrum, you can solve directly for the bond length, which is one of the most powerful applications of rotational spectroscopy.

Principal Moments of Inertia in Polyatomic Molecules

Polyatomic molecules have three principal moments of inertia ($I_a$, $I_b$, $I_c$) about three mutually perpendicular axes, with corresponding rotational constants:

$A = \frac{\hbar^2}{2I_a}, \qquad B = \frac{\hbar^2}{2I_b}, \qquad C = \frac{\hbar^2}{2I_c}$

The relative magnitudes depend on molecular geometry:

• Linear molecules (e.g., $CO_2$): $I_a \approx 0$ and $I_b = I_c$, so you effectively have one rotational constant $B$.
• Symmetric tops (e.g., $NH_3$, $CH_3Cl$): Two moments are equal ($I_b = I_c$), giving two distinct constants $A$ and $B$.
• Spherical tops (e.g., $CH_4$, $SF_6$): All three moments are equal ($I_a = I_b = I_c$).
• Asymmetric tops (e.g., $H_2O$): All three moments differ, making the spectrum considerably more complex.

Experimental Determination of Rotational Constants

Rotational constants are extracted by fitting observed transition frequencies to the rigid rotor energy expression. For a linear molecule, the transition frequencies are:

$\nu = 2B(J+1)$

High-resolution techniques used for these measurements include microwave spectroscopy (for pure rotational transitions) and Fourier-transform infrared spectroscopy (FTIR) (for rovibrational spectra). The precision of these methods is remarkable, often yielding bond lengths accurate to $\pm 0.001$ Å.

Applications and Limitations of the Rigid Rotor Model

Applicability to Diatomic and Linear Polyatomic Molecules

For diatomic molecules like $HCl$ and $CO$, the rigid rotor model accurately predicts equally spaced rotational lines. Measuring the line spacing gives $2B$, from which you extract the moment of inertia and then the bond length.

Linear polyatomic molecules like $CO_2$ and $HCN$ produce similar equally spaced spectra for rotation about the axis with the largest moment of inertia. The analysis follows the same logic as for diatomics.

Limitations and Deviations from Ideal Rigid Rotor Behavior

Real spectra deviate from the rigid rotor prediction in several ways:

• Centrifugal distortion causes the line spacing to decrease slightly at higher $J$. The corrected energy expression adds a distortion term: $E_J = BJ(J+1) - DJ^2(J+1)^2$, where $D$ is the centrifugal distortion constant ($D \ll B$).
• Rotation-vibration coupling means the effective rotational constant $B$ changes depending on which vibrational state the molecule occupies. This produces rovibrational spectra with P-branch and R-branch structure flanking a vibrational transition.
• Non-rigid structures: Molecules like $H_2O$ and $NH_3$ undergo large-amplitude motions (inversion in $NH_3$, for example) that the rigid rotor simply cannot capture.

Despite these limitations, the rigid rotor remains the starting point for rotational spectroscopy. You learn the ideal model first, then layer on corrections (centrifugal distortion, vibration-rotation interaction) as needed to match experimental data.