Molecular Vibrations: Stretching and Bending

Introduction

Molecules are never still. Even at very low temperatures, their atoms vibrate back and forth about their average positions. These vibrations are not random: each molecule has a set of specific patterns of motion, called normal modes , each with its own frequency. Infrared spectroscopy detects these vibrations, so understanding the types of motion — stretching and bending — is the key to understanding why IR spectra look the way they do.

Core explanation

Stretching. In a stretching vibration, a bond lengthens and shortens along its axis while the bond angle stays roughly the same. A diatomic molecule such as HCl has only one vibration: a stretch.

Bending. In a bending vibration, the angle between two bonds changes. Bending needs less energy than stretching the same bonds, so bending modes generally appear at lower wavenumbers.

Water (H₂O), a bent triatomic molecule. It has three normal modes:

Mode Description Approximate wavenumber --- --- --- Symmetric stretch Both O–H bonds lengthen and shorten together 3657 cm⁻¹ Asymmetric stretch One O–H lengthens while the other shortens 3756 cm⁻¹ Bend (scissoring) H–O–H angle opens and closes 1595 cm⁻¹

(Values are for isolated gas-phase molecules; liquid water gives broadened bands shifted by hydrogen bonding.)

Carbon dioxide (CO₂), a linear triatomic molecule. It has four normal modes:

Mode Description Approximate wavenumber --- --- --- Symmetric stretch Both C=O bonds stretch together about 1340 cm⁻¹ Asymmetric stretch One C=O lengthens as the other shortens 2349 cm⁻¹ Bend (two equivalent modes) Molecule bends in two perpendicular planes 667 cm⁻¹

Counting modes. A molecule of N atoms has 3N ways to move (three directions for each atom). Three of these are translations of the whole molecule. A non-linear molecule also has three rotations, leaving 3N − 6 vibrations. A linear molecule has only two rotations, leaving 3N − 5 vibrations.

- H₂O: 3(3) − 6 = 3 modes. - CO₂: 3(3) − 5 = 4 modes. - CH₄: 3(5) − 6 = 9 modes.

More bending types. In larger molecules, bending motions of groups such as CH₂ are named scissoring, rocking, wagging and twisting. Many of these contribute to the complex fingerprint region below about 1500 cm⁻¹.

Formulae

Number of vibrational modes: non-linear molecule = 3N − 6; linear molecule = 3N − 5, where N is the number of atoms.

Step-by-step reasoning

To find the number of vibrational modes of a molecule:

1. Count the atoms, N. 2. Decide whether the molecule is linear or non-linear from its shape. 3. Apply 3N − 5 (linear) or 3N − 6 (non-linear). 4. Remember that not every mode need appear in the IR spectrum.

Visual explanation

Draw water as a V shape. For the symmetric stretch, draw arrows on both hydrogens pointing outwards together. For the asymmetric stretch, draw one arrow outward and the other inward. For the bend, draw curved arrows showing the two hydrogens swinging towards each other like the blades of a pair of scissors.

Real-world analogy

Think of two children on a seesaw with springs. They can both bounce up and down in step (symmetric), alternate so one rises as the other falls (asymmetric), or swing sideways changing the angle between them (bending). Each pattern has its own natural rhythm.

Real-world example

The greenhouse effect of carbon dioxide depends largely on its bending mode at 667 cm⁻¹ (about 15 μm), which lies close to the peak of the infrared radiation emitted by the Earth's surface. CO₂ absorbs this radiation and re-emits it, warming the lower atmosphere.

Why?

Why do bending vibrations usually occur at lower wavenumbers than stretching vibrations? Changing a bond angle requires less energy than compressing or extending the bond itself, because the restoring force for bending is weaker than for stretching.

Common misconception

"A molecule has one vibration for each bond." Vibrations involve the whole molecule, and the number of modes is given by 3N − 6 or 3N − 5, not by counting bonds. Water has two bonds but three vibrational modes.

Worked example

Question: How many vibrational modes do (a) ethyne, C₂H₂ (linear), and (b) methanal, H₂CO (non-linear), have?

Reasoning: (a) N = 4, linear: 3(4) − 5 = 7. (b) N = 4, non-linear: 3(4) − 6 = 6.

Answer: (a) 7 modes; (b) 6 modes.

Quick check

1. Which stretching mode of water involves one O–H bond lengthening while the other shortens? Answer: The asymmetric stretch, which occurs at slightly higher wavenumber than the symmetric stretch.

Exam focus

Be able to sketch and name the symmetric stretch, asymmetric stretch and bend of H₂O and CO₂, and to apply 3N − 6 and 3N − 5 correctly. Remember that the linear or non-linear shape must be decided first, often using VSEPR reasoning.

Advanced insight

Real vibrations are not perfectly harmonic. Anharmonicity allows weak "overtone" bands at roughly twice a fundamental wavenumber and "combination" bands where two modes are excited together. For example, a weak overtone of the carbonyl stretch sometimes appears near 3400 cm⁻¹ and can be mistaken for an O–H band.

Summary

Molecular vibrations are stretches (bond length changes) and bends (bond angle changes). Molecules with two or more similar bonds show symmetric and asymmetric stretches. Bends usually occur at lower wavenumbers than stretches. The number of vibrational modes is 3N − 6 for non-linear molecules and 3N − 5 for linear ones: water has three, carbon dioxide four.

Practice questions

1. State the difference between a stretching and a bending vibration. Answer: A stretch changes a bond length; a bend changes a bond angle. 2. Calculate the number of vibrational modes of ammonia, NH₃. Answer: NH₃ is non-linear with N = 4, so 3(4) − 6 = 6 modes. 3. Why does CO₂ have one more vibrational mode than H₂O? Answer: CO₂ is linear, so it has only two rotations, leaving 3N − 5 = 4 modes; bent H₂O has three rotations, leaving 3N − 6 = 3. 4. Which usually requires more energy: stretching or bending a given set of bonds? Answer: Stretching, so stretching vibrations generally appear at higher wavenumbers.