Symmetry of Normal Vibrations

Introduction

An N-atom molecule has three Cartesian displacement coordinates per atom, giving a 3N-dimensional space of small motions. Not every motion is a vibration: some move the whole molecule through space or rotate it. Group theory first classifies all 3N displacements, then translations and rotations are removed to expose the normal vibrational species. This page focuses on the first step, including a reliable character-counting method for water.

Core explanation

At a fixed equilibrium geometry, let each atom have small x, y and z displacement components. A symmetry operation permutes equivalent atoms and transforms their displacement vectors. The resulting 3N × 3N matrix is usually too large to write explicitly, but its trace can be found efficiently. A moved atom contributes no diagonal block because its three displacement coordinates are sent to coordinates associated with another atom. An atom that stays at its own site contributes the trace of the operation's 3 × 3 Cartesian transformation matrix. Sum that trace over all fixed atoms.

For identity E, every atom is fixed and the Cartesian matrix is the three-dimensional identity with trace 3. Therefore χ3N(E) = 3N. A 180° rotation about z has coordinate matrix diag(−1,−1,+1) with trace −1. Each atom on the rotation axis and fixed by the operation contributes −1. A reflection in the xz plane has coordinate matrix diag(+1,−1,+1) with trace +1, so each atom lying in that mirror plane contributes +1. A fixed atom does not automatically contribute +3; the vector direction changes matter.

Use bent water in the yz plane with O at the origin and H atoms at (0,±a,b). Its C₂v operations are E, C₂(z), σ(xz) and σ(yz). Identity fixes all three atoms, giving 9. C₂(z) fixes only O, whose vector trace is −1, giving −1. σ(xz) fixes only O and exchanges the H atoms, so its character is +1. σ(yz), the molecular plane, fixes all three atoms but flips every x displacement; each atom contributes +1, giving 3. The full displacement character row is therefore Γ3N = (9,−1,1,3).

The character row can be reduced into irreducible representations. With the stated C₂v table, Γ3N = 3A₁ + A₂ + 2B₁ + 3B₂. All species here are one-dimensional, and the dimensions add 3+1+2+3=9, as required. This decomposition includes translations and rotations, so it must not be read as nine vibrational modes. A nonlinear three-atom molecule has only 3N−6 = 3 normal vibrational modes after six rigid-body degrees of freedom are removed.

At a harmonic equilibrium geometry, the mass-weighted Hessian or force-constant matrix can be diagonalised into normal modes. When the force field respects molecular symmetry, modes can be chosen with definite symmetry species. A multidimensional irrep can correspond to degenerate partner modes under exact symmetry. Group theory predicts the types and possible degeneracies before frequencies are calculated, but it does not give the numerical force constants or mode frequencies.

The fixed-atom rule needs careful geometry. An atom lying on a rotation axis remains at its position; an atom in a mirror plane remains at its position; identical atoms exchanged by an operation do not count as fixed even though the whole structure is unchanged. Isotopic substitution changes masses and can affect normal-mode patterns, while the formal symmetry used should match the actual distinguishable nuclear arrangement in the problem.

Step-by-step reasoning

List the N atoms and choose x, y and z axes. For each operation, mark which atoms remain at their own sites. Write the 3 × 3 transformation of a displacement vector and calculate its trace. Multiply by the number of fixed atoms and sum any distinct contributions. Verify χ(E)=3N, then reduce the complete row before removing rigid motions on the next step.

Visual explanation

Draw bent water with tiny x, y and z arrows attached to each atom. Under C₂(z), O stays but the H atoms exchange; show O's x and y arrows reversing and z staying, for trace −1. Under σ(yz), all atoms stay but every x arrow reverses while y and z remain, giving trace +1 per atom. Write the character totals beside the sketches.

Real-world analogy

A choreography can move dancers to other positions while also changing the direction each dancer faces. To count a transformation's trace, only dancers remaining in their own spots contribute, and their orientation changes still matter. This resembles fixed-atom displacement counting, though molecular normal modes are continuous vector coordinates rather than literal people.

Real-world example

Before assigning water's infrared bands, a spectroscopist needs the symmetry of its three vibrations. Building Γ3N provides the raw material. Removing translation and rotation later gives two A₁ and one B₂ vibrational mode in the stated axes, allowing their dipole-derivative activity to be checked against the C₂v table.

Why?

Why not count only atoms left fixed? A displacement is a vector. A fixed atom can have some displacement components reversed by rotation or reflection, contributing negative or positive diagonal values. The Cartesian trace captures these directional changes, while a plain fixed-atom count would give incorrect characters and therefore wrong mode symmetries.

Common misconception

Γ3N is not the vibrational representation. It includes three translations and three rotations for a nonlinear molecule. Another error is to assign +3 for every fixed atom under any operation; only identity has Cartesian trace 3. A fixed atom under C₂ can contribute −1.

Worked example

For water under C₂(z), O is fixed and both H atoms exchange. The vector transformation at O is (δx,δy,δz) → (−δx,−δy,+δz), so its trace is −1−1+1=−1. Exchanged H coordinates contribute zero diagonal trace. Hence χ3N(C₂)=−1. Combined with χ(E)=9, χ(σxz)=1 and χ(σyz)=3, the row is (9,−1,1,3), ready for irreducible reduction.

Quick check

1. What is χ3N(E) for a five-atom molecule? Answer: Fifteen, because identity leaves all 3N Cartesian coordinates unchanged. 2. Does an atom exchanged with an equivalent atom contribute to the 3N character? Answer: No. Its displacement coordinates map to another atom's coordinates and give no diagonal contribution in the atom-centred basis.

Exam focus

Write the atom positions and Cartesian transformation before counting. State the operation order, calculate a fixed atom's vector trace and verify χ(E)=3N. Keep the full Γ3N result separate from Γvib until translations and rotations have been subtracted.

Advanced insight

Mass weighting changes the numerical normal-mode eigenvectors and frequencies but does not change the basic point-group transformation of an ideal equilibrium displacement space when symmetry-related atoms have equal masses. If isotope substitution makes related sites distinguishable, the applicable molecular symmetry is lower and the representation should be rebuilt for that isotopologue.

Summary

The 3N displacement representation describes all Cartesian motions of an N-atom molecule. Its character for an operation comes from fixed atoms multiplied by the trace of the vector transformation. For C₂v water in the stated axes, Γ3N = (9,−1,1,3), containing both rigid motions and vibrations. Removing the rigid motions is the next required step.

Practice questions

1. A C₂ rotation fixes two atoms on its axis in a molecule. What is their total contribution to Γ3N's character? Answer: Each fixed atom contributes the C₂ Cartesian trace −1, so together they contribute −2. Atoms exchanged by the rotation contribute zero. 2. Why does a reflection of water in its molecular yz plane have Γ3N character 3 rather than 9? Answer: All three atoms stay fixed, but each displacement vector has trace −1+1+1=1 because its x component reverses. Three atoms contribute 3(1)=3. 3. If a nonlinear four-atom molecule has Γ3N dimension twelve, how many of those degrees of freedom are vibrations? Answer: Six are vibrations because 3N−6 = 12−6 = 6; the other six are three translations and three rotations.