18.3 Vibrational spectroscopy

Vibrational Spectroscopy Principles

Vibrational spectroscopy probes how molecular bonds stretch, bend, and twist when they absorb infrared radiation. Because every molecule has a unique set of vibrational frequencies, the resulting spectrum acts as a molecular fingerprint. For physical chemistry, this technique connects quantum-mechanical models of bonding directly to observable spectral data.

Fundamentals of Vibrational Spectroscopy

Molecules absorb IR radiation at frequencies that match the vibrational energies of their chemical bonds, producing characteristic absorption spectra. Three factors determine these vibrational frequencies:

• The masses of the atoms involved in the vibration
• The strength (force constant) of the chemical bonds
• The molecular geometry, which dictates how many modes exist and how they couple

Two main techniques fall under vibrational spectroscopy:

• Infrared (IR) spectroscopy measures direct absorption of IR photons by molecular vibrations
• Raman spectroscopy measures inelastic scattering of monochromatic light, detecting vibrational transitions through shifts in the scattered photon's frequency

These two methods are complementary because they obey different selection rules, so together they can reveal vibrational modes that either technique alone would miss.

Applications in Physical Chemistry

• Functional group identification: Characteristic absorption frequencies for groups like C=O, O-H, and N-H allow rapid identification in unknown samples.
• Conformational analysis: Different molecular conformations (e.g., gauche vs. trans in n-butane) produce distinct spectral signatures because the geometry change alters vibrational coupling.
• Intermolecular interactions: Hydrogen bonding broadens and shifts O-H and N-H stretching bands, making IR spectroscopy a direct probe of these interactions.
• Reaction mechanisms: Vibrational spectroscopy can characterize reaction intermediates and transition states, especially when combined with time-resolved methods.
• Quantitative analysis: FTIR spectroscopy, combined with Beer-Lambert law, enables concentration measurements in complex mixtures.

Selection Rules for Vibrational Transitions

Selection rules determine which vibrational transitions produce observable spectral bands. They arise from the quantum-mechanical requirements for a transition to have a nonzero probability.

Electric Dipole and Polarizability Selection Rules

IR activity requires a change in the molecule's dipole moment during the vibration. If the dipole moment stays constant throughout the vibrational motion, no IR photon can be absorbed.

Raman activity requires a change in the molecule's polarizability during the vibration. Polarizability describes how easily the electron cloud deforms in an external electric field.

A key consequence for centrosymmetric molecules (those with an inversion center): the rule of mutual exclusion applies. Vibrations that are IR active are Raman inactive, and vice versa. This makes comparing IR and Raman spectra a powerful way to determine whether a molecule has a center of symmetry.

Consequences for Spectral Interpretation

• IR and Raman spectra of the same molecule generally show different bands, providing complementary structural information.
• Overtone bands (transitions of $\Delta v = 2, 3, \ldots$) and combination bands (simultaneous excitation of two different modes) can appear due to anharmonicity, though they are typically much weaker than fundamental transitions.
• The absence of expected vibrational bands in IR or Raman spectra is just as informative as their presence. Missing bands constrain the molecule's symmetry and point group assignment.
• Selection rules guide the assignment of observed bands to specific vibrational modes, which is essential for extracting structural information from spectra.

Interpreting Vibrational Spectra

Spectral Presentation and Band Characteristics

IR spectra are typically plotted as transmittance (or absorbance) vs. wavenumber ($\tilde{\nu}$, in $\text{cm}^{-1}$). Raman spectra plot scattering intensity vs. Raman shift (also in $\text{cm}^{-1}$).

The position of a vibrational band depends on the force constant $k$ and the reduced mass $\mu$ of the vibrating atoms, following the harmonic oscillator relationship:

$\tilde{\nu} = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}$

where $\mu = \frac{m_1 m_2}{m_1 + m_2}$ for a diatomic oscillator. Stronger bonds (larger $k$) vibrate at higher frequencies, and heavier atoms (larger $\mu$) vibrate at lower frequencies.

Band intensity reflects the magnitude of the dipole moment change (IR) or polarizability change (Raman) during the vibration. Band width and shape carry information about the molecular environment: hydrogen bonding broadens O-H stretches dramatically, while crystalline samples often show sharper bands than amorphous or solution-phase samples.

Spectral Analysis Techniques

Characteristic group frequencies are the backbone of spectral interpretation. Some commonly referenced regions:

• O-H stretch: ~3200–3600 $\text{cm}^{-1}$ (broad when hydrogen-bonded)
• N-H stretch: ~3300–3500 $\text{cm}^{-1}$
• C-H stretch: ~2850–3300 $\text{cm}^{-1}$
• C=O stretch: ~1650–1800 $\text{cm}^{-1}$
• C=C stretch: ~1600–1680 $\text{cm}^{-1}$

Isotopic substitution is a powerful assignment tool. Replacing H with D (deuterium) reduces the reduced mass ratio by roughly $\sqrt{2}$, shifting the stretching frequency of that bond downward by a factor of about 1.4. This confirms which bands involve hydrogen motion.

Fermi resonance occurs when an overtone or combination band has nearly the same frequency and symmetry as a fundamental. The two modes couple, causing the bands to split apart and share intensity. Recognizing Fermi resonance prevents misassignment of "extra" bands in a spectrum.

Vibrational Spectroscopy for Molecular Characterization

Determining Molecular Structure and Symmetry

The total number of vibrational modes for a molecule with $N$ atoms is:

• Nonlinear molecules: $3N - 6$ vibrational modes
• Linear molecules: $3N - 5$ vibrational modes

Not all of these modes will necessarily appear in both IR and Raman spectra. By counting the observed bands in each spectrum and comparing with predictions from group theory for various point groups, you can determine (or confirm) the molecule's symmetry.

Band positions and intensities also reveal bonding details. A lower-than-expected C=O stretching frequency, for instance, suggests conjugation or electron delocalization that weakens the bond. Comparing force constants extracted from spectra with theoretical predictions connects experimental data to electronic structure models.

Studying Molecular Interactions and Dynamics

Conformational analysis: Different rotational isomers have different vibrational spectra because the spatial arrangement of atoms changes the coupling between modes. Temperature-dependent IR studies can reveal the relative energies of conformers through changes in band intensities.

Hydrogen bonding: IR spectroscopy is particularly sensitive to hydrogen bonds. The O-H stretching band shifts to lower frequency and broadens significantly as hydrogen bonding strengthens. The magnitude of the frequency shift correlates roughly with the hydrogen bond strength.

Time-resolved techniques extend vibrational spectroscopy into the ultrafast domain:

• Pump-probe IR spectroscopy tracks vibrational population changes on femtosecond to picosecond timescales, revealing energy transfer pathways and relaxation dynamics.
• Coherent anti-Stokes Raman spectroscopy (CARS) provides enhanced Raman signals with high spatial and temporal resolution, useful for studying reaction kinetics and molecular dynamics in real time.

Vibrational Spectroscopy Methods: Advantages vs. Limitations

Infrared (IR) and Raman Spectroscopy

These two techniques are most powerful when used together, since their complementary selection rules ensure more complete vibrational mode coverage.

Advanced Vibrational Spectroscopy Techniques

Fourier-transform IR (FTIR) spectroscopy uses an interferometer rather than a dispersive monochromator. This provides three key advantages over older dispersive instruments:

1. Multiplex (Fellgett) advantage: All frequencies are measured simultaneously, giving faster data acquisition.
2. Throughput (Jacquinot) advantage: No slits are needed, so more light reaches the detector.
3. Improved signal-to-noise ratio: Multiple scans can be co-added rapidly.

FTIR is now the standard for most IR measurements, especially for quantitative work and complex mixtures.

Surface-enhanced Raman spectroscopy (SERS) exploits the electromagnetic enhancement from nanostructured metal surfaces (typically gold or silver) to amplify Raman signals by factors of $10^6$ or more. This enables detection of trace analytes and the study of molecules adsorbed at surfaces.

Choosing the right vibrational spectroscopy method depends on the molecular system, the information you need, and practical constraints like sample state and available instrumentation. Combining multiple techniques almost always yields a more complete picture of molecular structure and dynamics.