8.1 Microwave spectroscopy and rotational spectra

Microwave spectroscopy uses microwave radiation to probe the rotational energy levels of molecules in the gas phase. By analyzing how molecules absorb this radiation, you can extract precise information about bond lengths, bond angles, dipole moments, and overall molecular geometry.

This topic covers the principles behind rotational spectroscopy, how to read and interpret rotational spectra, the selection rules that govern allowed transitions, and how molecular symmetry shapes the spectra you observe.

Microwave Spectroscopy Principles

Fundamentals of Microwave Spectroscopy

Microwave spectroscopy measures the absorption of microwave radiation (typically 1–100 GHz) as molecules transition between quantized rotational energy levels. Because angular momentum is quantized, molecules can only rotate at specific discrete energies rather than at any arbitrary rate.

The spacing between these energy levels depends on the molecule's moment of inertia ($I$), which reflects how mass is distributed relative to the rotation axis.

• Lighter molecules with mass concentrated near the center of mass have larger rotational constants and more widely spaced energy levels.
• Heavier molecules or those with mass spread far from the center have smaller rotational constants and more closely spaced levels.

The rotational constant $B$ captures this relationship quantitatively (more on this below).

Interaction of Microwave Radiation with Molecules

A molecule must have a permanent electric dipole moment to show a pure rotational spectrum. The oscillating electric field of the microwave radiation couples to this dipole, driving transitions between rotational states. Homonuclear diatomics like $N_2$ and $O_2$ have no dipole moment and are therefore microwave-inactive.

Microwave spectroscopy is especially powerful for small, gas-phase molecules (e.g., $HCl$, $H_2O$, $NH_3$), where rotational lines are sharp and well-resolved. Beyond static structure, rotational spectroscopy can also reveal dynamic phenomena such as internal rotation (e.g., methyl group torsion), quantum tunneling motions, and the effects of molecular collisions on rotational energy transfer.

Rotational Spectra Analysis

Interpretation of Rotational Spectra

For a rigid linear molecule, the rotational spectrum consists of a series of equally spaced absorption lines. The spacing between adjacent lines is $2B$, where $B$ is the rotational constant defined as:

$B = \frac{h}{8\pi^2 I}$

Here $h$ is Planck's constant and $I$ is the moment of inertia. A larger $B$ means wider line spacing; a smaller $B$ means the lines are packed more closely together.

The intensity of each rotational line depends on two factors:

• Population of the initial state: Rotational state populations follow a Boltzmann distribution. Lower-energy states are more populated, but the degeneracy of each level ($2J+1$) increases with $J$. The interplay of these two effects means that line intensity rises with $J$, reaches a maximum, and then falls off.
• Transition dipole moment: This determines the intrinsic probability of the transition.

The overall appearance of the spectrum depends on molecular geometry. Linear molecules produce simple, equally spaced patterns. Nonlinear molecules ($H_2O$, $NH_3$) have multiple distinct moments of inertia, leading to more complex spectra with additional quantum numbers and selection rules.

Extracting Molecular Structure and Properties

For a diatomic molecule, you can extract the bond length directly from the rotational constant. The relationship is:

$r = \sqrt{\frac{h}{8\pi^2 \mu B}}$

where $\mu$ is the reduced mass of the two atoms ($\mu = \frac{m_1 m_2}{m_1 + m_2}$).

Step-by-step: Finding a bond length from a rotational spectrum

1. Measure the spacing between adjacent spectral lines. This spacing equals $2B$.
2. Divide by 2 to get $B$ (in frequency units).
3. Calculate the moment of inertia: $I = \frac{h}{8\pi^2 B}$.
4. Compute the reduced mass $\mu$ from the atomic masses.
5. Solve for the bond length: $r = \sqrt{I / \mu}$.

At higher $J$ values, you'll notice the line spacing gradually decreases. This is centrifugal distortion: as the molecule spins faster, centrifugal force stretches the bond slightly, increasing $I$ and reducing the effective $B$. The centrifugal distortion constant $D$ quantifies this effect and connects rotational spectra to the molecule's vibrational force constant.

Isotopic substitution is another powerful technique. Replacing an atom with a heavier isotope (e.g., $H \rightarrow D$) changes $\mu$ and therefore $B$, without altering the electronic structure or bond length. By comparing rotational constants of isotopologues, you can pinpoint the position of the substituted atom within a polyatomic molecule.

Selection Rules for Rotational Transitions

Angular Momentum Conservation and Selection Rules

Selection rules determine which transitions are allowed and therefore observable. For pure rotational transitions, the key rule is:

$\Delta J = \pm 1$

This arises from angular momentum conservation. A microwave photon carries one unit of angular momentum ($\hbar$), so absorbing or emitting a photon must change the molecule's rotational quantum number by exactly one.

• $\Delta J = +1$ ($J \rightarrow J+1$): absorption of a photon
• $\Delta J = -1$ ($J \rightarrow J-1$): emission of a photon

In a thermal sample, absorption lines dominate because lower-$J$ states are more populated.

Selection Rules for Different Molecular Geometries

Linear molecules have the simplest case. The $\Delta J = \pm 1$ rule produces equally spaced lines with transition frequencies:

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

where $J$ is the quantum number of the lower state. So the first few lines appear at $2B$, $4B$, $6B$, and so on.

Symmetric top molecules (e.g., $NH_3$, $CH_3Cl$) require an additional quantum number $K$, which is the projection of angular momentum onto the molecular symmetry axis. The selection rules are $\Delta J = \pm 1$ and $\Delta K = 0$. Since $K$ doesn't change, you get a series of equally spaced lines for each value of $K$, with spacings determined by the rotational constants $A$ and $B$.

Asymmetric top molecules (e.g., $H_2O$, $SO_2$) have three distinct moments of inertia and are labeled by quantum numbers $J$, $K_a$, and $K_c$. The selection rules become $\Delta J = 0, \pm 1$ with $\Delta K_a$ and $\Delta K_c$ each equal to $0$ or $\pm 1$, subject to symmetry constraints. Their spectra are considerably more complex and typically require computational methods to assign.

Molecular Symmetry and Rotational Spectra

Classification of Molecules Based on Symmetry

Molecular symmetry determines how many distinct moments of inertia a molecule has, which in turn controls the complexity of its rotational spectrum. There are four categories:

Linear molecules (e.g., $HCN$, $CO_2$) have one unique moment of inertia ($I_B = I_C$, with $I_A = 0$ along the bond axis). Their energy levels follow:

$E(J) = BJ(J+1)$

This gives the simplest rotational spectra.

Spherical top molecules (e.g., $CH_4$, $SF_6$) have three equal moments of inertia ($I_A = I_B = I_C$). Because of their high symmetry, they have no permanent dipole moment and therefore produce no pure rotational spectrum.

Symmetric top molecules (e.g., $NH_3$, $CH_3Cl$) have two equal moments of inertia and one distinct moment. They come in two flavors:

• Prolate (cigar-shaped): $I_A < I_B = I_C$ (e.g., $CH_3Cl$)
• Oblate (disk-shaped): $I_A = I_B < I_C$ (e.g., $BCl_3$)

Their energy levels depend on both $J$ and $K$:

$E(J,K) = BJ(J+1) + (A - B)K^2$

Asymmetric top molecules (e.g., $H_2O$, $SO_2$) have three distinct moments of inertia ($I_A \neq I_B \neq I_C$). Their energy levels require three rotational constants ($A$, $B$, $C$) and cannot be written in a simple closed-form expression. Numerical diagonalization of the rotational Hamiltonian is typically needed.

Effects of Molecular Symmetry on Rotational Spectra

Symmetry can cause certain transitions to be forbidden, leading to missing lines in the spectrum. A classic example is $NH_3$: its umbrella-like inversion motion splits each rotational level into a symmetric and an antisymmetric component. Transitions between states of the same symmetry type are forbidden, so some expected lines are absent.

Nuclear spin statistics also affect line intensities. When a molecule contains identical nuclei (e.g., the two $H$ atoms in $H_2$, or the three $H$ atoms in $NH_3$), the total wavefunction must obey the Pauli exclusion principle. Different nuclear spin arrangements have different statistical weights, which means rotational levels with certain $J$ values are more heavily populated than others. This produces an alternating intensity pattern in the spectrum. For $CO_2$, which has two identical $^{16}O$ nuclei (spin-0 bosons), every other rotational level is missing entirely.