Removing Translations and Rotations

Introduction

A 3N displacement representation includes every small Cartesian motion, but spectroscopy of internal vibrations requires a smaller space. Whole-molecule translations and rotations do not change bond lengths or angles in the same way as vibrations. Character tables list how x, y, z and rotational functions transform, allowing these rigid motions to be subtracted algebraically. Water provides a complete calculation, and the linear-molecule exception explains why some molecules use 3N−5 instead of 3N−6.

Core explanation

For a nonlinear molecule, three independent translations move its centre of mass along x, y and z. Three independent rotations change its orientation about x, y and z. These six motions are contained in the 3N space, so Γvib = Γ3N − Γtrans − Γrot. The subtraction refers to representation content: remove one copy of each listed translational or rotational species, keeping nonnegative multiplicities in a correct calculation.

In C₂v with z the C₂ axis and columns E, C₂(z), σ(xz), σ(yz), coordinate functions transform as z:A₁, x:B₁ and y:B₂. Therefore Γtrans = A₁ + B₁ + B₂. Rotational functions are axial vectors; in this convention Rz:A₂, Ry:B₁ and Rx:B₂. Thus Γrot = A₂ + B₁ + B₂. It is important not to assign Rx the same species as x merely because they share an axis. Under reflection, an axial vector transforms differently from an ordinary polar vector.

The previous page found Γ3N for water as (9,−1,1,3). Reducing this row using the C₂v table gives 3A₁ + A₂ + 2B₁ + 3B₂. Subtracting translations removes A₁, B₁ and B₂; subtracting rotations removes A₂, B₁ and B₂. The result is Γvib = 2A₁ + B₂. Its dimension is 2+1=3, equal to 3(3)−6=3. The two A₁ modes correspond to symmetric stretch and bend, and the B₂ mode to antisymmetric stretch in the stated yz-plane convention.

These species label normal-coordinate patterns, not numerical frequencies. Two A₁ modes can have different frequencies because they are distinct vibrational eigenvectors within the same symmetry block. Symmetry permits their coupling in a force-constant calculation and does not by itself decide which frequency is larger. The B₂ mode belongs to a different block and cannot mix with A₁ modes if the equilibrium geometry and harmonic force field preserve C₂v.

For a linear molecule, rotation about its own internuclear axis does not produce an independent nuclear displacement of the equilibrium line. Only two rigid rotations are counted, in addition to three translations, leaving 3N−5 vibrational degrees of freedom. A linear triatomic therefore has 3(3)−5=4 vibrational coordinates. These may appear as fewer distinct frequencies when a bending mode is doubly degenerate, as in ideal linear CO₂. Counting distinct spectral bands is not the same as counting normal-coordinate degrees of freedom.

A free single atom has three translations and no internal vibration. Very small molecules are edge cases where the general nonlinear formula must be applied with care, but for ordinary nonlinear molecules with N≥3 it works directly. In a constrained molecule or solid, boundary conditions change the available rigid motions; the gas-phase point-group formula should not be copied blindly to every environment.

Once Γvib is known, IR and Raman activity are tested by comparing each mode's species with linear coordinates and quadratic functions in the character table. This order matters. The fact that a bond is polar does not alone tell whether a particular collective motion changes the net dipole; symmetry of the entire normal coordinate decides the first-order allowance.

Step-by-step reasoning

Construct and reduce Γ3N. Read x, y and z species from the character table and add them for Γtrans. Read Rx, Ry and Rz separately for Γrot. Subtract their multiplicities irrep by irrep. Check the result's dimension against 3N−6 for nonlinear molecules or 3N−5 for linear molecules, then identify mode patterns and spectral activity.

Visual explanation

Draw three water sketches translated bodily in x, y and z; no internal angle or bond length changes. Draw three whole-molecule rotations. Place six labelled symmetry chips beside them, then remove those chips from a stack representing 3A₁ + A₂ + 2B₁ + 3B₂. The three chips left are 2A₁ + B₂.

Real-world analogy

A camera view of a dancing figure includes movement of the whole figure across the frame, turning of the figure and changes in limb positions. To study the dance's internal pattern, subtract camera-relative translation and rotation. Vibrational analysis similarly isolates internal deformation from rigid movement, but does so with precise vector symmetries.

Real-world example

Water's three internal modes have different spectral frequencies. Group theory predicts two A₁ and one B₂ symmetry types before force constants are known. A calculation that returned four water vibrations from the nine Cartesian displacements would signal that too few rigid motions were removed, whereas a two-mode count would miss one independent internal coordinate.

Why?

Why does a linear molecule subtract only two rotations? Rotating a line around the line itself leaves the nuclear coordinates unchanged to first order and is not an independent rigid-body orientation coordinate in the usual vibrational count. Rotations about perpendicular axes are independent, so a linear molecule has five nonvibrational coordinates rather than six.

Common misconception

Three normal vibrational coordinates do not necessarily mean three distinct spectral bands. Degenerate partner modes can share a frequency. Another error is assigning a rotation about x the same symmetry as x translation; reflections distinguish axial and polar vector transformation rules.

Worked example

For water, start from Γ3N = 3A₁ + A₂ + 2B₁ + 3B₂. Subtract Γtrans = A₁ + B₁ + B₂ to get 2A₁ + A₂ + B₁ + 2B₂. Then subtract Γrot = A₂ + B₁ + B₂ to get 2A₁ + B₂. The dimensions sum to three, matching 3N−6 for N=3. These are the symmetric stretch, bend and antisymmetric stretch species under the chosen axes.

Quick check

1. How many internal vibrational coordinates does a nonlinear five-atom molecule have? Answer: 3(5)−6 = 9. 2. How many does a linear five-atom molecule have? Answer: 3(5)−5 = 10, because it has only two independent rigid rotations.

Exam focus

Specify linear versus nonlinear geometry before applying a count. Obtain rotational species from R functions, not from x, y, z. Subtract representations irrep by irrep and verify the final dimension; this catches most character or coordinate-convention mistakes.

Advanced insight

At a true equilibrium minimum, the mass-weighted Hessian has zero-frequency modes corresponding to rigid translations and rotations in an ideal isolated molecule, apart from numerical and coordinate effects. A negative curvature along an internal coordinate signals an imaginary frequency and a saddle point rather than a stable vibrational mode. Group theory classifies such a coordinate but cannot alone determine curvature sign.

Summary

Internal vibrational species are obtained from Γ3N after removing three translations and three rotations for a nonlinear molecule, or three translations and two rotations for a linear one. In the stated C₂v water convention, Γvib = 2A₁ + B₂. The group labels organise modes, while frequencies and intensities require force and response calculations.

Practice questions

1. Why are Rx and x not necessarily assigned the same irrep in a group with mirror planes? Answer: x is a polar vector component, whereas Rx is an axial vector component. Reflection changes them differently, so their full character patterns can differ. 2. A linear triatomic calculation finds three normal-coordinate dimensions after removing rigid motions. What was likely missed? Answer: It should have 3(3)−5=4 vibrational dimensions; treating it as nonlinear and subtracting three rotations instead of two would remove one too many. 3. Do the two A₁ water vibrations have to have equal frequency? Answer: No. They share a symmetry species but are separate normal modes. Symmetry permits them to mix in the force-constant problem and does not enforce equal eigenvalues.