Physical Chemistry Third Edition

23.5 Spectra of Polyatomic Molecules 977

In molecules without a permanent dipole, some of the normal modes can produce

a fluctuating dipole that oscillates about zero magnitude, and thus produce infrared

absorption. For example, the CO 2 molecule is linear and has no permanent dipole

moment, although each CO bond is polar. The normal modes of CO 2 were shown

in Figure 22.6. The two bending modes, which have the same frequency, produce

temporary dipoles that are perpendicular to the molecule axis and fluctuate about zero

magnitude. They produce a vibrational band that is called aperpendicular band. The

asymmetric stretch produces an oscillating dipole parallel to the molecule axis, because

it stretches one bond as it compresses the other. The spectral band that it produces is

called aparallel band. The symmetric stretch increases and then decreases both bond

dipoles simultaneously, not modulating the dipole moment and not giving rise to a

spectral line. The infrared spectrum of carbon dioxide contains only two fundamental

bands, the parallel band at 1340 cm−^1 and the perpendicular band at 667 cm−^1.

The perpendicular band of a linear molecule like carbon dioxide exhibits aQbranch

corresponding to∆J0 in addition toPandRbranches. The two bending modes

together can produce a motion in which the center atom moves around in a circle

perpendicular to the axis of the molecule. This motion is similar to a rotation of a

bent molecule, which turns out to permit∆J0 as well as∆J±1.^18 Figure 23.13

shows the carbon dioxide perpendicular band at 667 cm−^1 , containingP,Q, andR

branches. The line widths are such that the lines are not completely resolved from each

other, even with a high-resolution instrument.

618 648 667 727

Q branch

P branch

R branch

Reciprocal wavelength/cm^21

Absorbance/arbitrary unit

Figure 23.13 The Perpendicular

Band of Carbon Dioxide.From G.

Herzberg, Molecular Spectra and

MolecularStructure, Vol. II, Infrared

and RamanSpectra of Polyatomic

Molecules, Van Nostrand Reinhold,

New York, 1945, p. 273.

The situation with nonlinear triatomic molecules is similar. Vibrational motions in

which the dipole oscillates perpendicular to a single symmetry axis produce a band

with aQbranch as well asPandRbranches. For example, the asymmetric stretch in

a nonlinear triatomic molecule such as H 2 OorH 2 S produces a band with aQbranch.

In order for a given normal mode of a polyatomic molecule to give rise to a vibra-

tional band (be “infrared active”), the transition dipole moment integral for the two

vibrational wave functions of the normal modes must be nonzero. This integral can be

studied by group theory.^19 However, it is often possible by inspection of the normal

modes to identify those that modulate the dipole moment of the molecule.

EXAMPLE23.11

The normal modes of cyanogen, C 2 N 2 , are shown in Figure 23.14. (The last two diagrams

represent two modes each, of equal frequency.) Determine which modes are infrared active.

Solution

The molecule is symmetrical, and has a zero dipole moment in its equilibrium configuration.

The motions of modes 1 and 2 are symmetrical about the center of the molecule so that any

changes in the bond dipole moments cancel. These modes are not infrared active. Mode 3

corresponds to one end of the molecule moving in as the other end moves out. It produces

an oscillating dipole and is infrared active. Mode 4 corresponds to rotation of the polar C–N

bonds such that they remain parallel while the nonpolar C–C bond rotates. All changes in the

dipole moment cancel and the mode is not infrared active. Mode 5 corresponds to the relatively

negative nitrogen atoms moving in one direction while the relatively positive carbon atoms

move in the other direction, producing an oscillating dipole moment. It is infrared active.

NNCC

ν 1

ν 2

ν 3

ν 4

ν 5

Σ^1 g

Σ^1 g

Σ^1 u

Πg

Πu

Figure 23.14 The Vibrational Nor-

mal Modes of Cyanogen, C 2 N 2 .The

arrows show the directions of motion

of the nuclei during half of the period of

vibration, and the lengths are propor-

tional to the amplitudes of the nuclear

oscillations. From G. Herzberg,Molec-

ularSpectra andMolecularStructure,

Vol. II,InfraredandRamanSpectra of

Polyatomic Molecules, Van Nostrand

Reinhold Co., New York, 1945, p. 181.

(^18) I. N. Levine,op. cit., p. 255ff (note 7).
(^19) P. W. Atkins,Molecular Quantum Mechanics, 2nd ed., Oxford University Press, Oxford, 1983,
p. 303ff; N. C. Craig and N. N. Lacuesta,J. Chem. Educ., 81 , 1199 (2004).