Physical Chemistry Third Edition

988 23 Optical Spectroscopy and Photochemistry

mode in a polyatomic molecule will be Raman active if it produces a periodic change

in the shape or the orientation of the ellipsoid of polarizability. The normal modes

of carbon dioxide are shown in Figure 22.6. The asymmetric stretch, which is seen

in the infrared, is not seen in the Raman spectrum. The stretching of one bond is

accompanied by the compression of the other bond so that the effects cancel, and the

ellipsoid of polarizability is not changed. The symmetric stretch alternately compresses

and stretches the ellipsoid of polarizability since both bonds stretch and compress

simultaneously. It is Raman active. The bending modes, which are seen in the infrared,

are not seen in the Raman spectrum, because the bonds do not stretch appreciably as

the bond angle bends and because the variations in the bond directions cancel.

There is arule of mutual exclusion, which states: In a molecule with a center of

symmetry, a normal mode that is seen in the infrared spectrum will not be seen in the

Raman spectrum, and vice versa.^24 The normal modes of carbon dioxide illustrate this

rule. In molecules with more than three atoms, it is sometimes possible to determine

whether a normal mode will be Raman active or IR active by inspection of the normal

mode motions. Group theory is often used to simplify the analysis.^25

EXAMPLE23.13

Identify the normal modes of cyanogen shown in Figure 23.14 that will give rise to Raman

lines.

Solution

In modes 1 and 2, the C–C bond oscillates in length and the two C–N bonds oscillate in

unison, so that the polarizability is modulated. Both of these modes will be Raman active. In

mode 3, the C–C bond length does not oscillate, and the C–N bonds oscillate out of phase,

so this mode is not seen. Mode 5 is similar to the bend in carbon dioxide and this mode will

not be seen in the Raman spectrum. Mode 4 must be Raman active because there is a rocking

motion of the molecule. The ellipsoid of polarizability follows this rocking motion. Note that

these conclusions could all have been reached by use of the rule of mutual exclusion.

Exercise 23.11

Figure 23.18 shows sketches representing some of the vibrational normal modes of ethylene. The

direction of motion of each atom is shown for one-half of the period. For motions perpendicular

to the plane, a positive sign (+) indicates upward motion and a negative sign (−) indicates

downward motion. Tell which modes are infrared active and which are Raman active. There is

a center of symmetry, so the rule of mutual exclusion applies.

It is possible to deduce the same kinds of structural information from Raman spec-

tra as from infrared and microwave spectra. From the selection rule for rotation,

Eq. (23.7-3a), the Raman shift of the Stokes rotational lines of a diatomic molecule is

given in the rigid-rotor approximation by

̃ν− ̃ν′(EJ+ 2 −EJ)/hc ̃Be(4J+6) (23.7-8)

where terms inαandDhave been neglected.

(^24) I. N. Levine,op. cit., p. 268 (note 7).
(^25) N. B. Colthup, L. H. Daly, and S. E. Wiberley,Introduction to Infrared and Raman Spectroscopy,
3rd ed., Academic Press, San Diego, 1990, p. 109ff.