Normal Modes of Polyatomic Molecules

Water bends and stretches. It never vibrates in some random tangle of motion. Instead, its atoms move in three very specific coordinated patterns called normal modes, each with its own sharp frequency. Understanding why a molecule vibrates in exactly these patterns, and how many it has, is the key that unlocks infrared and Raman spectroscopy. This idea matters because every vibrational spectrum you will ever interpret is built from normal modes. A carbonyl stretch near 1715 cm⁻¹, a water bend near 1595 cm⁻¹, these are not arbitrary labels. They are the vibrational fingerprints of specific, calculable atomic motions. This article counts vibrational modes using the 3N degrees-of-freedom argument. It classifies them as stretches or bends. It shows how normal coordinates decouple a messy many-body vibration problem into independent harmonic oscillators. Two worked examples, for CO₂ and H₂O, make every number concrete.

Key takeaways

A molecule of $N$ atoms has $3N$ total degrees of freedom, split among translation, rotation, and vibration.
Nonlinear molecules have $3N-6$ vibrational modes; linear molecules have $3N-5$ , because they lack a rotation about the molecular axis.
Each normal mode is an independent, synchronized motion of all atoms at one single frequency, built from stretches and bends.
Normal coordinates diagonalize the vibrational potential and kinetic energy simultaneously, turning coupled oscillators into independent ones.
Symmetric and antisymmetric stretches, plus in-plane and out-of-plane bends, are the common structural vocabulary for describing modes.
Key idea: the true vibrational motion of a polyatomic molecule is a superposition of $3N-6$ (or $3N-5$ ) independent normal modes, each behaving as its own one-dimensional harmonic oscillator with a unique frequency.

The core insight

Although every atom in a molecule can vibrate in a complicated, coupled way, there exists a special set of collective coordinates, the normal coordinates, in which the vibrational Hamiltonian separates exactly into $3N-6$ (or $3N-5$ ) independent harmonic oscillators, each corresponding to all atoms moving together at one single frequency.

The physical picture

Picture a molecule as a set of point masses connected by springs representing chemical bonds and angle-bending forces. If you displace one atom, the restoring forces propagate through the whole structure, and every atom starts moving. Left alone, this motion looks chaotic: a jumble of stretching, bending, and twisting all superposed. The remarkable mechanical fact is that this jumble can always be decomposed into a finite set of special collective motions called normal modes. In a normal mode, every atom moves sinusoidally in phase (or exactly out of phase) with every other atom, all oscillating at the same single frequency. The general vibrational motion of the molecule is simply a linear combination of these normal modes, each with its own amplitude and phase, evolving independently in time. Before counting or building modes, recall the total mechanical freedom in the system. Each of the $N$ atoms can move independently in three Cartesian directions, so the whole molecule has $3N$ degrees of freedom. Some of this freedom is "boring": it moves the whole molecule without deforming it. Three degrees of freedom describe translation of the center of mass along $x, y, z$ . Two or three more describe overall rotation of the rigid molecular frame (two for a linear molecule, three for a nonlinear one, since rotation about a linear axis carries no atomic displacement in the point-mass limit). Whatever freedom is left over is genuine internal deformation: vibration. Vibrations further split conceptually into bond stretches, which change bond lengths, and bends, which change bond or dihedral angles. A stretch stores energy mostly in bond-length distortion, typically at higher frequency because bond-stretching force constants are large. A bend stores energy in angular distortion, typically softer and lower in frequency. Real normal modes are rarely "pure" stretches or "pure" bends; they are frequently mixtures, but for molecules with enough symmetry the mixing is forbidden by symmetry, and modes come out as clean, describable motions such as the symmetric stretch, antisymmetric stretch, and bend of a triatomic molecule.

Figure

Counting the 3N degrees of freedom of a nonlinear triatomic like H2O, N = 3, 3N = 9

Once you know how many modes to expect, the next step is to picture their shapes. For a bent triatomic like H₂O, the three vibrations are visualized as arrows on each atom showing instantaneous velocity direction.

Figure

The three normal modes of a bent XY2 molecule such as H2O

The core relations

\text{degrees of freedom} = 3N

where: $N$ is the number of atoms, each contributing 3 Cartesian coordinates.

n_{\text{vib}} = 3N - 6 \quad \text{(nonlinear molecule)}

where: 3 translations and 3 rotations are removed from $3N$ total coordinates.

n_{\text{vib}} = 3N - 5 \quad \text{(linear molecule)}

where: only 2 independent rotations exist for a linear frame, leaving one extra vibrational mode.

V(\mathbf{q}) = V_0 + \tfrac{1}{2}\sum_{i,j} k_{ij}\, q_i q_j, \qquad T = \tfrac{1}{2}\sum_i \dot{q}_i^2

where: $q_i$ are mass-weighted Cartesian displacement coordinates and $k_{ij}$ is the force-constant (Hessian) matrix.

E_{\text{vib}}(v_1, v_2, \dots, v_{n}) = \sum_{k=1}^{n} \left(v_k + \tfrac{1}{2}\right) h \nu_k

where: each normal mode $k$ contributes independently, with its own quantum number $v_k$ and frequency $\nu_k$ .

Worked examples

Derivation AFrom coupled Cartesian displacements to independent normal modes

Goal. Show how diagonalizing the mass-weighted force-constant matrix decouples $3N$ coupled equations of motion into $3N-6$ (or $3N-5$ ) independent harmonic oscillators, each a normal mode.

Step 1. Write Newton's equations in Cartesian displacements. Let $x_i$ be the small displacement of Cartesian coordinate $i$ (there are $3N$ of them) from equilibrium. Expanding the potential energy to second order (the harmonic approximation, justified because displacements are small compared to bond lengths) gives

m_i \ddot{x}_i = -\sum_j \left(\frac{\partial^2 V}{\partial x_i \partial x_j}\right)_0 x_j .

Each atom's motion is coupled to every other atom's motion through the off-diagonal second derivatives (the force constants).

Step 2. Mass-weight the coordinates. Define

q_i = \sqrt{m_i}\, x_i .

Substituting turns the equations of motion into

\ddot{q}_i = -\sum_j F_{ij}\, q_j, \qquad F_{ij} = \frac{1}{\sqrt{m_i m_j}}\left(\frac{\partial^2 V}{\partial x_i \partial x_j}\right)_0 .

The matrix $F$ is real and symmetric (mixed partials commute), so it can always be diagonalized by an orthogonal transformation.

Step 3. Diagonalize the force-constant matrix. Because $F$ is real symmetric, there exists an orthogonal matrix $L$ such that

L^{T} F L = \Lambda = \text{diag}(\lambda_1, \lambda_2, \dots, \lambda_{3N}),

where each eigenvalue $\lambda_k$ is real. Defining new coordinates $Q_k$ (the normal coordinates) through

q_i = \sum_k L_{ik} Q_k

converts the coupled system into $3N$ independent equations

\ddot{Q}_k = -\lambda_k Q_k .

Step 4. Identify the trivial eigenvalues. Six of the eigenvalues (five for a linear molecule) are exactly zero. These correspond to translation of the whole molecule (3 modes, $\lambda = 0$ , uniform displacement with no restoring force) and rotation of the rigid frame (3 modes for nonlinear, 2 for linear, again zero restoring force since rotating a molecule costs no potential energy). These are not vibrations; they are the trivial "modes" that must be subtracted from the naive count of $3N$ .

Step 5. Interpret the remaining nonzero eigenvalues as vibrations. Each remaining nonzero eigenvalue $\lambda_k > 0$ gives an independent equation

\ddot{Q}_k = -\lambda_k Q_k \quad \Rightarrow \quad Q_k(t) = A_k \cos(2\pi\nu_k t + \phi_k), \qquad \nu_k = \frac{1}{2\pi}\sqrt{\lambda_k}.

This is exactly the equation of a one-dimensional harmonic oscillator (see prerequisite: The Harmonic Oscillator and Vibrational Levels), but now in the collective coordinate $Q_k$ rather than a single bond length. Since translations and rotations remove 6 (or 5) of the $3N$ degrees of freedom, the number of genuine vibrational normal modes is

\boxed{n_{\text{vib}} = 3N - 6 \ \text{(nonlinear)}, \qquad n_{\text{vib}} = 3N - 5 \ \text{(linear)}}

each behaving as an independent harmonic oscillator with its own frequency $\nu_k$ and its own quantized energy ladder $(v_k + \tfrac12)h\nu_k$ .

Worked examples

Example 1: Counting and assigning the modes of water, H₂O (foundational). Water has $N = 3$ atoms (O, H, H), and it is bent (nonlinear), with an equilibrium H–O–H angle near 104.5°. Step 1 — total degrees of freedom:

3N = 3 \times 3 = 9.

Step 2 — subtract translations and rotations for a nonlinear molecule:

n_{\text{vib}} = 3N - 6 = 9 - 6 = 3.

Step 3 — assign the three modes by symmetry and known spectroscopic data. All three are both infrared and Raman active for the $C_{2v}$ water molecule:

$\nu_1$ , symmetric stretch: both O–H bonds lengthen and shorten in phase. Observed at $3657\ \text{cm}^{-1}$ .
$\nu_2$ , bend (scissoring): the H–O–H angle opens and closes. Observed at $1595\ \text{cm}^{-1}$ .
$\nu_3$ , antisymmetric stretch: one O–H bond lengthens while the other shortens. Observed at $3756\ \text{cm}^{-1}$ .

Sanity check: bends are always softer than stretches for the same bond because angle-bending force constants ( $k_{\text{bend}} \sim 50$ – $100\ \text{N m}^{-1}$ ) are typically 5–10 times smaller than bond-stretching force constants ( $k_{\text{stretch}} \sim 500$ – $800\ \text{N m}^{-1}$ ). Indeed $1595\ \text{cm}^{-1} \ll 3657\ \text{cm}^{-1}$ , consistent with this trend. The zero-point vibrational energy of water is

E_{\text{ZPE}} = \tfrac12 h(\nu_1+\nu_2+\nu_3) = \tfrac12 hc\,(3657+1595+3756)\ \text{cm}^{-1} = \tfrac12 hc \times 9008\ \text{cm}^{-1},

using $hc = 1.986\times10^{-23}\ \text{J cm}$ , giving

E_{\text{ZPE}} = \tfrac12 \times 1.986\times10^{-23}\ \text{J cm} \times 9008\ \text{cm}^{-1} = 8.95\times10^{-20}\ \text{J per molecule},

or, multiplying by Avogadro's number $6.022\times10^{23}\ \text{mol}^{-1}$ , about $53.9\ \text{kJ mol}^{-1}$ . This is a physically sensible number, comparable to but smaller than a typical O–H bond dissociation energy of roughly 460 kJ mol⁻¹, as zero-point energy should be.

Example 2: Counting and assigning the modes of carbon dioxide, CO₂ (applied, exam-level). CO₂ has $N = 3$ atoms and is linear (O=C=O, $D_{\infty h}$ symmetry). Step 1 — total degrees of freedom:

3N = 9.

Step 2 — subtract translations (3) and rotations (only 2 for a linear top, since spinning about the O=C=O axis moves no mass):

n_{\text{vib}} = 3N - 5 = 9 - 5 = 4.

Step 3 — assign the four modes. Two of them are degenerate (same frequency, two perpendicular directions), which is why only 3 distinct frequencies are usually quoted:

$\nu_1$ , symmetric stretch: both C=O bonds stretch in phase, center of mass fixed. This mode is infrared inactive (no dipole moment change, by symmetry) but Raman active. Observed at $1388\ \text{cm}^{-1}$ .
$\nu_2$ , bending (doubly degenerate, $\pi_u$ ): two perpendicular bending motions with identical frequency. Infrared active. Observed at $667\ \text{cm}^{-1}$ .
$\nu_3$ , antisymmetric stretch: one C=O bond stretches while the other compresses. Strongly infrared active (this is the mode responsible for CO₂'s greenhouse absorption). Observed at $2349\ \text{cm}^{-1}$ .

Count check: $1 (\nu_1) + 2 (\nu_2, \text{degenerate}) + 1(\nu_3) = 4$ vibrational modes, matching $3N - 5 = 4$ exactly. Sanity check on frequency ordering: the bend (667 cm⁻¹) is softest, as expected since bending force constants are weaker than stretching ones. The antisymmetric stretch (2349 cm⁻¹) lies higher than the symmetric stretch (1388 cm⁻¹), a general pattern because the antisymmetric motion pushes the central atom against a stiffer effective restoring force while the symmetric stretch leaves the central atom's local environment more nearly balanced. This assignment matches the well-known 2349 cm⁻¹ absorption band that makes CO₂ a potent infrared absorber and greenhouse gas.

Limiting cases & common traps

Situation	Correct handling
Linear molecule (e.g. CO₂, HCN, C₂H₂)	Use $3N - 5$ , not $3N - 6$ ; only 2 rotational degrees of freedom exist
Highly symmetric molecule with degenerate modes	Count each degenerate pair/triple according to its dimension (e.g. CO₂'s bend counts as 2 toward $3N-5$ )
Diatomic molecule ( $N=2$ )	$3N - 5 = 1$ : only one vibrational mode exists, the single bond stretch (consistent with prerequisite harmonic-oscillator treatment)
Very floppy or fluxional molecules	Harmonic normal-mode picture becomes an approximation; large-amplitude motions (internal rotation, inversion) may need separate treatment
Overtones and combination bands	Not counted among the $3N-6$ (or $3N-5$ ) fundamentals; they arise from higher quanta or sums of fundamentals, a topic for later spectral analysis

The chemist behind it

The general theory of coupled small oscillations, and its diagonalization into independent normal coordinates, traces back to classical mechanics (Lagrange, and later Rayleigh), but its systematic application to molecular vibrational spectra is closely associated with Gerhard Herzberg, whose monumental treatises on molecular spectra and molecular structure defined how generations of chemists read and assign vibrational spectra of polyatomic molecules, connecting group theory, symmetry, and the normal-mode framework into the practical tool used in every modern IR and Raman laboratory.

Origins · 1 minGerhard Herzberg and how this was discovered

Gerhard Herzberg

1904–1999 · Pioneered systematic vibrational and electronic spectroscopic analysis of polyatomic molecules; awarded the 1971 Nobel Prize in Chemistry

Ask the tutor: Can you show me how symmetry (point groups) predicts which normal modes are infrared active versus Raman active?

Why do we subtract exactly 6 (or 5) from 3N instead of some other number

Three of the $3N$ Cartesian degrees of freedom always describe translation of the whole molecule's center of mass along $x$ , $y$ , and $z$ , since sliding a rigid molecule costs no potential energy. A nonlinear molecule needs three independent angles to specify its orientation in space, so three more degrees of freedom describe rotation, again at zero energy cost for a rigid body. That leaves $3N-6$ true shape-changing (vibrational) coordinates. A linear molecule only needs two angles to fix its orientation, since spinning about its own axis does not move any atoms, so only 5 degrees of freedom are trivial, leaving $3N-5$ vibrations.

Are normal modes the same thing as the bond stretches and bends chemists draw in mechanisms

Not exactly. Chemists often draw idealized "pure" stretches or bends for intuition, and for highly symmetric molecules like CO₂ or H₂O the true normal modes are indeed very close to these idealized pictures because symmetry forbids mixing. In lower-symmetry molecules, however, a real normal mode is usually a blend of several bond stretches and angle bends occurring simultaneously with fixed relative amplitudes and phases, determined by solving the force-constant matrix rather than guessed intuitively.

Does every normal mode show up in an infrared spectrum

No. A vibrational mode is only infrared active if it produces a change in the molecule's dipole moment during the motion. CO₂'s symmetric stretch, for example, leaves the dipole moment at zero throughout the motion (by symmetry) and is infrared-silent, though it is visible in Raman spectroscopy instead. This complementary infrared/Raman activity based on symmetry is explored further in the article on Raman Spectroscopy and the Scattering Mechanism.

How does this normal-mode counting connect to identifying functional groups from a spectrum

Once you know a molecule's normal modes, group frequencies emerge from the empirical observation that certain local bond types (C=O, O–H, C≡N) vibrate at nearly molecule-independent frequencies because their local force constants and reduced masses barely change between molecules. This lets chemists read a spectrum and infer functional groups directly, the subject of the article on Group Frequencies and Functional Group Identification.

What happens to the normal-mode picture if the molecule is very floppy, like a large flexible chain

The harmonic normal-mode approximation still applies locally near an equilibrium geometry, but floppy molecules often have several low-lying conformers, internal rotations, or large-amplitude motions (like ring puckering) that are poorly described by a single harmonic expansion. In such cases, the $3N-6$ counting rule for a single rigid conformer still holds, but the true dynamics can involve anharmonic coupling between formally "independent" modes.

Normal Modes of Polyatomic Molecules

The physical picture

The core relations

Worked examples

Worked examples

Limiting cases & common traps

The chemist behind it

Before this

Where this leads