976 23 Optical Spectroscopy and Photochemistry
Vibrational Spectra of Polyatomic Molecules
In the harmonic oscillator approximation the vibration of polyatomic molecules is that
of normal modes, each acting like an independent harmonic oscillator. We number
the normal modes with an indexi, ranging from 1 to 3n−5 for linear molecules and
ranging from 1 to 3n−6 for nonlinear molecules. The selection rules for vibrational
transitions are:∆ν0or±1 for one normal mode,∆v0 for all other normal modes (23.5-1a)The classical motion of the normal mode must modulate the molecule’s
dipole moment (23.5-1b)The case in which all∆v’s vanish corresponds to a microwave spectrum. According
to Eq. (23.5-1a),∆v±1 for only one normal mode at a time. Transitions obeying
this selection rule producefundamental bands. There is a fundamental band in the
infrared region for each normal mode that modulates the molecule’s dipole moment.
The vibrational selection rules are less well obeyed than the rotational selection rules.
There areovertone bandsin which∆v2 (and sometimes 3), andcombination bands,
in which two (or more) normal modes change their quantum numbers at once. These
forbidden bands are usually less intense than the fundamental bands.
The selection rule of Eq. (23.5-1b) means that the classically pictured motion must
cause the dipole moment to oscillate in value. It is generally a well-obeyed rule. If a
polyatomic molecule possesses a permanent dipole moment, all of its normal modes
modulate the dipole and give rise to vibrational bands. For example, in nonlinear
triatomic molecules such as H 2 OorSO 2 , all three of the normal modes shown in
Figure 22.6 will produce fundamental bands.EXAMPLE23.10
The infrared spectrum of hydrogen sulfide, H 2 S, shows three strong bands at 1290 cm−^1 ,
2610 .8cm−^1 , and 2684 cm−^1. There are weaker bands at 2422 cm−^1 , 3789 cm−^1 , and
5154 cm−^1. Interpret the spectrum.
Solution
The three strong bands are fundamentals. The lowest frequency generally belongs to the
bend, so that 1290 cm−^1 belongs to the bend, denoted byν 2 .The symmetric stretch is gen-
erally intermediate in frequency, so that the 2610.8cm−^1 frequency is that of the symmetric
stretch, denoted byν 1. The 2684 cm−^1 frequency belongs to the asymmetric stretch,ν 3.
It is customary to number the modes of a triatomic molecule in this way. The weak bands
are identified by trial and error, seeing whether their frequencies approximate a multiple of
a fundamental frequency or a sum of two fundamental frequencies. (The addition is never
exact.) The 2422 cm−^1 frequency is roughly twice that of the bend, so it is the first overtone
of the bend. The 3789 cm−^1 frequency is roughly the sum of 1290 cm−^1 and 2684 cm−^1 ,
and is a combination band of the bend and the asymmetric stretch. The 5154 cm−^1 frequency
is roughly twice as large asν 1 and is also roughly equal toν 1 +ν 3. It has been assigned both
ways but is more likely to be the combination band, by analogy with the H 2 O spectrum.^17(^17) G. Herzberg,Molecular Spectra and Molecular Structure, Vol. II, Infrared and Raman Spectra of Poly-
atomic Molecules, Van Nostrand Reinhold, New York, 1945, p. 283.