IR Spectra & Vibrations — IR absorption and molecular...

SECTION 1

Historical Context & Motivation

The story of infrared spectroscopy begins with the realization that light extends beyond the visible spectrum. In 1800, William Herschel placed thermometers in the dispersed colors of sunlight and discovered that the region just beyond visible red produced the greatest heating effect, revealing what he called calorific rays. This invisible radiation—what we now call infrared (IR) radiation—would eventually become one of the most powerful structural characterization tools in chemistry. Over the next two centuries, physicists and chemists connected IR absorption to the quantized vibrational motions of molecules, opening a window into bond strengths, molecular symmetry, and functional group identification that remains indispensable in modern research and industry.

1800

Herschel Discovers Infrared Radiation

William Herschel uses thermometers to detect heating beyond the red end of the visible spectrum, establishing the existence of infrared radiation.

1905

Coblentz Records the First IR Spectra

William W. Coblentz systematically measures the infrared absorption spectra of hundreds of organic and inorganic compounds, laying the experimental foundation for IR spectroscopy as an analytical technique.

1927–1930

Quantum Mechanical Treatment of Vibrations

With the development of quantum mechanics, the harmonic oscillator model is applied to diatomic molecules. The quantization of vibrational energy levels provides a rigorous framework for interpreting IR absorption frequencies.

1950s–1960s

Dispersive IR Spectrometers Become Standard

Prism- and grating-based IR spectrometers become routine analytical instruments in academic and industrial laboratories, enabling rapid functional group identification in organic chemistry.

1969–present

Fourier-Transform IR (FTIR) Revolution

The Cooley–Tukey FFT algorithm and Michelson interferometer are combined to create FTIR spectrometers, dramatically improving speed, sensitivity, and spectral resolution. FTIR is now the dominant platform for IR spectroscopy worldwide.

The central question IR spectroscopy answers is deceptively simple: which bonds in a molecule vibrate at which frequencies, and why? Answering it rigorously requires understanding the quantum mechanics of the harmonic (and anharmonic) oscillator, the selection rules that govern which vibrations absorb IR radiation, and the relationship between molecular symmetry and spectral activity. The sections that follow build this understanding from first principles.

SECTION 2

Core Principles & Definitions

Infrared spectroscopy rests on the interaction between electromagnetic radiation in the IR region (roughly 4000–400 cm⁻¹) and the vibrational modes of molecules. When IR photons of the correct frequency strike a molecule, they can be absorbed if two conditions are met: the photon energy must match the energy gap between vibrational quantum levels, and the vibration must produce a change in the molecular dipole moment. These two requirements form the core selection rules that govern every IR spectrum you will interpret.

Vibrational Modes

A nonlinear molecule with N atoms possesses 3N − 6 normal vibrational modes; a linear molecule has 3N − 5. Each mode corresponds to a specific pattern of atomic displacement—stretching, bending, twisting, or rocking—at a characteristic frequency.

Dipole Moment Change (Selection Rule)

A vibration is IR-active only if it produces a net change in the molecule's electric dipole moment (∂μ/∂Q ≠ 0). Symmetric vibrations in centrosymmetric molecules (e.g., O₂, CO₂ symmetric stretch) are IR-inactive because the dipole moment remains unchanged.

Wavenumber (ν̃) and the IR Region

IR spectra plot transmittance or absorbance versus wavenumber (cm⁻¹), which is directly proportional to energy. The mid-IR range (4000–400 cm⁻¹) captures most fundamental molecular vibrations and is the workhorse region for structural identification.

Harmonic vs. Anharmonic Oscillator

The harmonic oscillator model treats a bond as a perfect spring with equally spaced energy levels. Real molecules behave anharmonically—energy levels converge at higher quantum numbers, allowing overtone and combination bands that enrich the spectrum.

Functional Group & Fingerprint Regions

The region above ~1500 cm⁻¹ contains characteristic absorptions of functional groups (O–H, C=O, N–H). Below ~1500 cm⁻¹ lies the fingerprint region, where complex coupled vibrations produce a pattern unique to each molecule.

✦ KEY TAKEAWAY

Think of each bond in a molecule as a tuning fork with a characteristic pitch. When you sweep IR light across a sample, only those tuning forks whose pitch matches the incoming frequency will resonate and absorb energy—but only if vibrating that fork also shifts the molecule's charge distribution (its dipole moment). An IR spectrum is essentially a record of which tuning forks rang and how loudly.

SECTION 3

Visualizing Molecular Vibrations

To connect macroscopic IR absorption bands with the atomic-level picture, it is essential to visualize the types of normal vibrational modes a molecule can undergo. The diagram below illustrates the six fundamental vibration types for a generic AX₂ group (such as the CH₂ unit). Stretching vibrations change the bond length, while bending vibrations change the bond angle. Each mode absorbs at a different characteristic frequency, and the distinction between these modes is central to spectral interpretation.

The six fundamental vibration types of an AX₂ group. Top row: symmetric stretch, asymmetric stretch, and scissoring (in-plane bending). Bottom row: rocking (in-plane), wagging (out-of-plane, same direction), and twisting (out-of-plane, opposite directions). Symbols ⊙ and ⊗ indicate motion toward and away from the viewer, respectively. Representative CH₂ frequencies are shown for each mode.

Several observations emerge from the diagram. First, stretching vibrations always appear at higher frequencies than the corresponding bending vibrations because changing a bond length requires more energy than changing a bond angle. Second, the asymmetric stretch absorbs at a higher wavenumber than the symmetric stretch because it involves greater displacement of the central atom. Third, out-of-plane bending modes (wagging and twisting) generally require slightly different energies than in-plane bending modes (scissoring and rocking), giving rise to distinct spectral bands that collectively constitute the molecule's vibrational fingerprint.

SECTION 4

Mathematical Framework

The quantitative framework for IR spectroscopy begins with modeling a chemical bond as a quantum harmonic oscillator. While real bonds are anharmonic, the harmonic approximation captures the essential physics and predicts vibrational frequencies with surprisingly good accuracy for fundamental transitions. Starting from Hooke's law and solving the Schrödinger equation for the quadratic potential V(x) = ½kx², one arrives at quantized energy levels and a fundamental vibrational frequency that depends on the force constant and the reduced mass.

HARMONIC OSCILLATOR ENERGY LEVELS

E_v = (v + ½)ℏω = (v + ½)hν

where v = vibrational quantum number (0, 1, 2, …), ℏ = h/2π is the reduced Planck constant, ω = (k/μ)^1/2 is the angular frequency, k = force constant (N m⁻¹), and μ = reduced mass (kg). Note the zero-point energy term ½hν, meaning a bond vibrates even at v = 0.

FUNDAMENTAL VIBRATIONAL WAVENUMBER

ν̃ = (1 / 2πc) × √(k / μ)

where ν̃ is the wavenumber in cm⁻¹, c is the speed of light (cm s⁻¹), k is the force constant, and μ = m₁m₂/(m₁ + m₂) is the reduced mass of the two bonded atoms. This equation reveals two key trends: higher force constants (stronger bonds) increase ν̃, while heavier atoms decrease ν̃.

SELECTION RULE (HARMONIC OSCILLATOR)

Δv = ±1 and (∂μ/∂Q)₀ ≠ 0

The first condition (Δv = ±1) arises from the harmonic oscillator matrix elements and restricts transitions to adjacent energy levels. The second condition requires that the dipole moment μ change with respect to the normal coordinate Q at the equilibrium geometry. Together, these determine which vibrations produce absorption bands and at what energy.

ANHARMONIC (MORSE) ENERGY LEVELS

E_v = hν₀[(v + ½) − x_e(v + ½)²]

where x_e is the anharmonicity constant (typically 0.01–0.05). This correction causes energy levels to converge at higher v, predicts molecular dissociation at sufficiently high v, and allows overtone transitions (Δv = ±2, ±3, …) and combination bands that appear as weak absorptions at roughly 2×, 3×, etc. of the fundamental frequency.

The practical implication of these equations is that a spectroscopist can predict the approximate absorption frequency of any bond by knowing two things: the bond's force constant (related to bond order and bond strength) and the masses of the bonded atoms. For example, C–H stretches appear near 3000 cm⁻¹ because the C–H force constant is moderate (~500 N m⁻¹) and hydrogen's low mass makes μ small, driving ν̃ up. Replacing H with D (deuterium) nearly doubles the reduced mass, shifting the absorption down to around 2200 cm⁻¹—a dramatic isotope effect that confirms the harmonic oscillator model's qualitative predictions.

SECTION 5

Characteristic Group Frequencies

One of the most practically powerful features of IR spectroscopy is that certain functional groups absorb IR radiation in characteristic, predictable wavenumber ranges regardless of the rest of the molecular structure. This allows rapid identification of functional groups from an IR spectrum—a capability that earned IR spectroscopy its central role in organic and analytical chemistry. The table and diagram below summarize the most important group frequency correlations across the mid-IR range.

Common Functional Group IR Absorptions in the Mid-IR Region

Functional Group	Vibration Type	Wavenumber Range (cm⁻¹)	Intensity
O–H (alcohol, free)	Stretch	3580–3650	Strong, sharp
O–H (H-bonded)	Stretch	3200–3550	Strong, broad
N–H (amine)	Stretch	3300–3500	Medium
C–H (sp³)	Stretch	2850–2960	Medium–strong
C–H (sp²)	Stretch	3020–3100	Medium
C–H (sp, ≡C–H)	Stretch	3300	Strong, sharp
C≡N (nitrile)	Stretch	2210–2260	Medium–strong
C≡C (alkyne)	Stretch	2100–2260	Weak–medium
C=O (carbonyl)	Stretch	1650–1800	Strong
C=C (alkene)	Stretch	1620–1680	Variable
C–O (ether, alcohol)	Stretch	1000–1300	Strong
C–F	Stretch	1000–1400	Strong

Overview of the mid-IR spectrum showing the functional group region (4000–1500 cm⁻¹) and the fingerprint region (1500–400 cm⁻¹). Colored boxes indicate approximate absorption ranges for common functional groups. Note that higher wavenumbers correspond to higher energies.

Several systematic trends govern where a particular absorption appears. Bonds to hydrogen (O–H, N–H, C–H) always appear above 2500 cm⁻¹ because of hydrogen's exceptionally low mass. Among bonds of similar reduced mass, the bond order determines the force constant: triple bonds (C≡C, C≡N) absorb near 2100–2260 cm⁻¹, double bonds (C=O, C=C) near 1600–1800 cm⁻¹, and single bonds (C–O, C–C) near 800–1300 cm⁻¹. Hydrogen bonding dramatically broadens and shifts the O–H and N–H stretch bands to lower wavenumbers—a feature that is diagnostically useful when distinguishing between free and associated hydroxyl groups.

💡 Interpreting Band Shape

Band shape carries information beyond position. A broad absorption typically signals hydrogen bonding or a distribution of environments (e.g., the broad O–H stretch of carboxylic acids from 2500–3300 cm⁻¹). A sharp band indicates a well-defined vibrational frequency and a more uniform chemical environment. Pay attention to multiplicity as well: primary amines (–NH₂) show two N–H stretch bands, while secondary amines (–NHR) show only one.

SECTION 6

Worked Example: Calculating and Interpreting a Vibrational Frequency

Consider the following problem: Predict the fundamental stretching frequency (in cm⁻¹) of the C–O bond in carbon monoxide, given that the force constant is k = 1860 N m⁻¹. Then, explain whether this vibration will be IR-active.

Predicting the IR Absorption of Carbon Monoxide

Step 1 — Determine the Reduced MassCarbon monoxide (CO) consists of ¹²C (m₁ = 12.00 u) and ¹⁶O (m₂ = 16.00 u). The reduced mass is μ = m₁m₂/(m₁ + m₂). First convert to kg using 1 u = 1.6605 × 10⁻²⁷ kg: μ = (12.00 × 16.00)/(12.00 + 16.00) u = 192.00/28.00 u = 6.857 u μ = 6.857 × 1.6605 × 10⁻²⁷ kg = 1.139 × 10⁻²⁶ kg

μ = 1.139 × 10⁻²⁶ kg

Step 2 — Apply the Vibrational Frequency EquationUsing ν̃ = (1/2πc)√(k/μ), where c = 2.998 × 10¹⁰ cm s⁻¹: ν̃ = [1/(2π × 2.998 × 10¹⁰)] × √(1860 / 1.139 × 10⁻²⁶) First calculate the term under the square root: k/μ = 1860 / 1.139 × 10⁻²⁶ = 1.633 × 10²⁹ s⁻² √(1.633 × 10²⁹) = 4.041 × 10¹⁴ s⁻¹ Then: ν̃ = 4.041 × 10¹⁴ / (2π × 2.998 × 10¹⁰) ν̃ = 4.041 × 10¹⁴ / 1.884 × 10¹¹ ν̃ ≈ 2145 cm⁻¹

ν̃ ≈ 2145 cm⁻¹

Step 3 — Assess IR ActivityCO is a heteronuclear diatomic molecule. Because C and O have different electronegativities, the C–O bond has a permanent dipole moment. When the bond stretches and compresses, the dipole moment changes (∂μ/∂Q ≠ 0). Therefore, the fundamental stretch of CO satisfies the IR selection rule and is IR-active. In contrast, homonuclear diatomic molecules like O₂ and N₂ have no dipole moment and no dipole change upon vibration, making them IR-inactive.

The C–O stretch at ~2145 cm⁻¹ is IR-active.

Step 4 — Compare with ExperimentThe experimentally observed fundamental absorption of gas-phase CO occurs at 2143 cm⁻¹, in excellent agreement with our harmonic oscillator prediction. The small discrepancy (≈2 cm⁻¹) arises from anharmonicity effects that the harmonic model neglects. This example demonstrates that the simple harmonic oscillator formula provides remarkably accurate frequency predictions for fundamental transitions.

Experimental value: 2143 cm⁻¹ — deviation < 0.1%

SECTION 7

Strengths, Limitations, and Comparison with Raman Spectroscopy

IR spectroscopy is one of several vibrational spectroscopy techniques, and understanding its strengths and limitations is essential for selecting the right analytical method. The most common complementary technique is Raman spectroscopy, which probes the same molecular vibrations but through a different physical mechanism (inelastic scattering of light rather than absorption). The selection rules for the two techniques are complementary: IR spectroscopy requires a dipole moment change, while Raman spectroscopy requires a change in polarizability. For molecules with a center of symmetry, the rule of mutual exclusion states that vibrations active in IR are inactive in Raman and vice versa, making the two techniques truly complementary.

Comparison of IR and Raman Spectroscopy

Feature	IR Spectroscopy	Raman Spectroscopy
Physical process	Absorption of IR photons	Inelastic scattering of visible/UV photons
Selection rule	Change in dipole moment (∂μ/∂Q ≠ 0)	Change in polarizability (∂α/∂Q ≠ 0)
Best for	Polar bonds: O–H, N–H, C=O, C–F	Nonpolar bonds: C=C, S–S, C–C backbone
Water interference	Significant — water absorbs strongly in mid-IR	Minimal — water is a weak Raman scatterer
Sample preparation	Thin films, KBr pellets, ATR; some prep needed	Direct measurement through glass; minimal prep
Spatial resolution	~10 μm (diffraction-limited in mid-IR)	~1 μm (shorter excitation wavelength)
Sensitivity	High for polar functional groups	Intrinsically weaker signal; enhanced by SERS

✦ KEY TAKEAWAY

Think of IR and Raman as two investigators examining the same crime scene but searching for different types of evidence. IR finds vibrations that disturb a molecule's charge separation (dipole moment)—like polar bonds crying out when they stretch. Raman finds vibrations that distort the electron cloud (polarizability)—like nonpolar bonds subtly warping their shape. In a centrosymmetric molecule, the rule of mutual exclusion guarantees that no vibration belongs to both investigators' jurisdiction. Using both techniques together provides a complete vibrational characterization that neither can achieve alone.

SECTION 8

Connection to Group Theory and Advanced Spectral Analysis

While the group frequency approach introduced in Section 5 is powerful for practical identification, a rigorous treatment of IR activity requires group theory and symmetry analysis. By assigning a molecule to its point group and decomposing the reducible representation of all atomic displacements into irreducible representations, one can predict the exact number of IR-active and Raman-active modes without ever recording a spectrum. This formal approach becomes essential when analyzing highly symmetric molecules (e.g., SF₆, benzene) where many vibrations are degenerate or inactive.

Basic vs. Advanced Approaches to IR Spectral Analysis

Concept	Basic Approach (This Lesson)	Advanced Approach (Group Theory)
Predicting IR activity	Does the vibration change the dipole moment? Assess qualitatively.	Decompose Γ₃N into irreducible representations; modes transforming as x, y, or z translations are IR-active.
Number of modes	3N − 6 (nonlinear) or 3N − 5 (linear)	Γ₃N = Γtrans + Γrot + Γvib; character tables give symmetry species of each mode.
Degeneracies	Not explicitly addressed; count total modes	Doubly (E) and triply (T/F) degenerate modes identified from irreducible representation dimensionality.
Frequency prediction	Harmonic oscillator: ν̃ = (1/2πc)√(k/μ)	Normal mode analysis via GF matrix method (Wilson's FG formalism) yields all frequencies simultaneously.
Coupling effects	Acknowledged qualitatively (fingerprint region)	Quantitatively treated through off-diagonal F-matrix elements (interaction force constants) and mass-weighted coordinates.

Beyond group theory, modern research increasingly employs computational quantum chemistry (DFT, MP2, coupled cluster methods) to calculate harmonic and anharmonic vibrational frequencies from first principles. These calculated spectra can be compared directly with experimental IR data to assign ambiguous bands, predict spectra of transient species, and interpret spectra of complex biological macromolecules. Two-dimensional infrared spectroscopy (2D IR) represents another frontier, using ultrafast laser pulses to map vibrational couplings and dynamics on femtosecond timescales—akin to 2D NMR but operating in the vibrational frequency domain.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Molecular nitrogen (N₂) and carbon monoxide (CO) have nearly identical bond lengths and similar force constants, yet N₂ shows no absorption in its IR spectrum while CO shows a strong band near 2143 cm⁻¹. Explain why, referencing the relevant selection rule.

PROBLEM 2 — BASIC CALCULATION

Calculate the expected fundamental stretching frequency (in cm⁻¹) for the H–Cl bond. Use k = 480 N m⁻¹, m(H) = 1.008 u, m(Cl) = 34.97 u, and c = 2.998 × 10¹⁰ cm s⁻¹. Compare your result to the experimental value of 2886 cm⁻¹.

PROBLEM 3 — INTERMEDIATE

When hydrogen (H) in the C–H bond of chloroform (CHCl₃) is replaced with deuterium (D) to form CDCl₃, the C–H stretching absorption shifts from 3020 cm⁻¹ to approximately 2256 cm⁻¹. Using the isotope effect, calculate the expected ratio ν̃(C–D)/ν̃(C–H) and compare with the observed ratio. Assume the force constant is unchanged by isotopic substitution.

PROBLEM 4 — APPLIED

You obtain an IR spectrum of an unknown organic liquid. The spectrum shows: (a) a broad, strong absorption centered at 3350 cm⁻¹; (b) a strong, sharp band at 1710 cm⁻¹; (c) C–H stretches near 2950 cm⁻¹; and (d) a strong band near 1200 cm⁻¹. Identify the functional groups responsible for each absorption and propose a possible compound class.

PROBLEM 5 — CRITICAL THINKING

Carbon dioxide (CO₂) is a linear triatomic molecule with D∞h symmetry. It has 3(3) − 5 = 4 normal modes. (i) Describe each of the four normal modes. (ii) Determine which are IR-active and which are Raman-active. (iii) Explain why the symmetric stretch is IR-inactive despite involving changes in bond length. (iv) Predict the approximate wavenumber range for each IR-active mode based on force constant and mass considerations.

SUMMARY

IR Spectra & Vibrations — Summary

Infrared spectroscopy exploits the interaction between IR radiation and molecular vibrations to reveal structural information. A molecule absorbs IR light when the photon energy matches a vibrational energy gap and the vibration produces a change in dipole moment. The harmonic oscillator model predicts vibrational frequencies through ν̃ = (1/2πc)√(k/μ), linking force constant and reduced mass to absorption frequency. Anharmonicity corrects this model for real bonds, enabling overtone and combination bands.

The mid-IR spectrum divides into the functional group region (4000–1500 cm⁻¹), where characteristic absorptions of O–H, N–H, C–H, C≡N, C=O, and C=C bonds appear, and the fingerprint region (below 1500 cm⁻¹), where complex coupled vibrations create molecule-specific patterns. Stretching vibrations absorb at higher frequencies than bending vibrations, and bonds to hydrogen always appear above 2500 cm⁻¹. IR spectroscopy complements Raman spectroscopy through different selection rules (dipole moment change vs. polarizability change), and the two techniques together provide a complete vibrational characterization. Group theory and computational chemistry extend these concepts to predict spectra from symmetry and first principles.