6.2 Rotational motion of diatomic and polyatomic molecules

Rotational Motion of Molecules

Rotational motion in molecules connects directly to molecular spectra: the patterns of light a molecule absorbs or emits depend on how it tumbles in space. By understanding rotational energy levels, you can extract precise information about bond lengths, molecular geometry, and symmetry from spectroscopic data.

Diatomic molecules are the simplest case, with rotation described by a single moment of inertia. Polyatomic molecules get more involved, requiring classification by their principal moments of inertia (linear, symmetric top, asymmetric top, spherical top). This section covers both.

Rotational Axes and Energy Levels

A diatomic molecule rotates about axes perpendicular to the bond. Since both perpendicular axes give the same moment of inertia, you only need one value of $I$ to describe the rotation. There's no rotation about the bond axis itself because the nuclei lie on that axis and contribute zero moment of inertia around it.

Polyatomic molecules are more complex. Depending on geometry and symmetry, they're classified as:

• Linear molecules (e.g., $CO_2$): Two equal moments of inertia perpendicular to the molecular axis; the moment about the molecular axis is zero. Spectra resemble diatomic molecules.
• Symmetric top molecules (e.g., $NH_3$): Two equal moments of inertia ($I_A = I_B$) and a distinct third ($I_C$). This introduces an additional quantum number $K$.
• Spherical top molecules (e.g., $CH_4$): All three moments of inertia are equal ($I_A = I_B = I_C$). Energy levels are highly degenerate.
• Asymmetric top molecules (e.g., $H_2O$): Three distinct moments of inertia ($I_A \neq I_B \neq I_C$). These produce the most complex rotational spectra.

Because polyatomic molecules can have different moments of inertia around each principal axis, they have a greater number of distinct rotational energy levels and more intricate spectra than diatomics.

Selection Rules and Transitions

Rotational transitions are driven by the interaction of electromagnetic radiation with the molecule's electric dipole moment. The fundamental requirement is:

• A molecule must have a permanent electric dipole moment to show a pure rotational absorption or emission spectrum. Polar molecules like $HCl$ and $H_2O$ are rotationally active; non-polar molecules like $O_2$, $N_2$, and $CO_2$ are not (their rotational transitions are electric-dipole forbidden).

For molecules that do have a permanent dipole:

• Diatomic and linear molecules: $\Delta J = \pm 1$
• Symmetric top molecules: $\Delta J = 0, \pm 1$ and $\Delta K = 0$ (the component of angular momentum along the symmetry axis doesn't change)
• Asymmetric top molecules: Selection rules are more relaxed because the lower symmetry lifts degeneracies and allows additional transitions

The parity of rotational wavefunctions also constrains which transitions occur. For molecules with inversion symmetry, only states of opposite parity are connected by electric dipole transitions ($+ \leftrightarrow -$).

Rotational Energy Levels for Diatomic Molecules

Rigid Rotor Approximation and Hamiltonian

The rigid rotor model treats the diatomic molecule as two point masses separated by a fixed bond length $r$. The molecule rotates freely, and the bond doesn't stretch or compress. This is a good first approximation at low $J$.

The rotational Hamiltonian is:

$\hat{H} = \frac{\hat{J}^2}{2I}$

where $\hat{J}$ is the angular momentum operator and $I$ is the moment of inertia.

Quantized Energy Levels and Wavefunctions

Solving the Schrödinger equation with this Hamiltonian gives quantized energy levels:

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

where $J = 0, 1, 2, \ldots$ is the rotational quantum number and $\hbar$ is the reduced Planck constant.

In spectroscopy, this is often written using the rotational constant $B$:

$E_J = hBJ(J+1), \quad \text{where } B = \frac{\hbar}{4\pi I}$

Each energy level has a degeneracy of $2J + 1$, corresponding to the magnetic quantum number $M = -J, -J+1, \ldots, J$.

The rotational wavefunctions are the spherical harmonics $Y_J^M(\theta, \varphi)$. These describe the probability distribution for the orientation of the molecular axis in space. For $J = 0$, the distribution is isotropic (equal in all directions); for higher $J$, the molecule preferentially orients in certain directions.

Symmetry Effects on Rotational Motion

Degenerate Energy Levels and Symmetry

Molecular symmetry determines which energy levels are degenerate (have the same energy):

• Linear molecules (e.g., $CO_2$): Each level $J$ is $(2J+1)$-fold degenerate due to the $M$ quantum number, the same as a diatomic.
• Symmetric top molecules (e.g., $NH_3$): Levels with $K \neq 0$ are doubly degenerate because $+K$ and $-K$ give the same energy. The total degeneracy for $K \neq 0$ is $2(2J+1)$.
• Spherical top molecules (e.g., $CH_4$): All three moments of inertia are equal, so the energy depends only on $J$. The degeneracy is $(2J+1)^2$.
• Asymmetric top molecules (e.g., $H_2O$): All three moments of inertia differ, so the $K$-degeneracy is lifted. Energy levels are non-degenerate (aside from the $M$-degeneracy), and spectra become considerably more complex.

Electric Dipole Moment and Selection Rules

The connection between symmetry and spectroscopic activity comes down to the dipole moment:

• Polar molecules ($HCl$, $H_2O$, $NH_3$) have allowed pure rotational transitions with $\Delta J = \pm 1$.
• Non-polar molecules ($O_2$, $N_2$, homonuclear diatomics) have no permanent dipole, so pure rotational transitions are electric-dipole forbidden. These molecules can still be studied via Raman spectroscopy, which has the selection rule $\Delta J = 0, \pm 2$.

For molecules with a center of inversion, parity selection rules apply: transitions connect states of opposite parity ($+ \leftrightarrow -$). Asymmetric top molecules, lacking high symmetry, have more relaxed parity constraints and correspondingly denser spectra.

Moments of Inertia for Different Geometries

The moment of inertia quantifies how the mass of a molecule is distributed relative to a rotational axis. Larger $I$ means more closely spaced rotational energy levels (since $E_J \propto 1/I$), so heavier or more extended molecules have denser rotational spectra.

Diatomic and Linear Molecules

For a diatomic molecule, the moment of inertia is:

$I = \mu r^2$

where $\mu = \frac{m_1 m_2}{m_1 + m_2}$ is the reduced mass and $r$ is the equilibrium bond length.

Linear polyatomic molecules (like $CO_2$ or $HCN$) have two equal moments of inertia perpendicular to the molecular axis ($I_B = I_C$) and $I_A = 0$ along the axis. You calculate $I$ by summing $m_i r_i^2$ for each atom, where $r_i$ is the perpendicular distance from the molecular axis (which, for a linear molecule, is the distance from the center of mass).

Symmetric and Asymmetric Top Molecules

Symmetric tops have two equal principal moments of inertia:

• Prolate symmetric top ($I_A < I_B = I_C$): Shaped like a rugby ball. Example: $CH_3Cl$.
• Oblate symmetric top ($I_A = I_B < I_C$): Shaped like a frisbee. Example: $BF_3$.

The rotational energy for a symmetric top is:

$E_{J,K} = BJ(J+1) + (A - B)K^2$

where $A$ and $B$ are rotational constants related to the distinct and degenerate moments of inertia, respectively, and $K = 0, \pm 1, \ldots, \pm J$.

Asymmetric tops (e.g., $H_2O$, $H_2O_2$) have three distinct moments of inertia ($I_A \neq I_B \neq I_C$). There's no closed-form energy expression; energy levels must be found by diagonalizing the rotational Hamiltonian matrix for each $J$.

The principal moments of inertia are calculated from:

$I = \sum_i m_i r_i^2$

where $r_i$ is the perpendicular distance of atom $i$ from the relevant principal axis. Finding the principal axes themselves requires diagonalizing the inertia tensor.

A note on methane ($CH_4$): despite being listed alongside symmetric tops in some texts, methane is actually a spherical top because its tetrahedral symmetry makes all three principal moments of inertia equal.