Physical Chemistry , 1st ed.

14.12 Vibrational Spectroscopy of

Diatomic and Linear Molecules

For infrared absorption spectroscopy of diatomic molecules, only heteronu-

cleardiatomic molecules show a vibrational spectrum. Their spectra are rela-

tively simple, since there is only one vibration: the motion of the two atoms

back and forth about their center of mass. This is a good example of a stretch-

ing vibrational mode. Table 14.4 lists, among other data, the stretching vibra-

tions for a series of gaseous diatomic molecules.

To a first approximation, the vibrations of diatomic molecules can be treated

as harmonic, Hooke’s-law-type oscillators. That is, as the molecules are mov-

ing back and forth about their center of mass, the force opposing the motion

is proportional to the distance away from some minimum-energy, equilibrium

distance. Figure 14.27 shows a plot of the potential energy curve, equal to ^12 kx^2 ,

for ideal oscillators. Superimposed on this potential energy curve are the val-

ues of the vibrational energy for the oscillator. For an ideal harmonic oscilla-

tor, the vibrational levels are spaced equally, which is consistent with equation

14.34. The vibrational force constant,k, is a measure of the curvature of the

potential energy plot. It can easily be shown that





2

x

V

 2 k

Therefore, the larger the force constant, the narrower the potential energy

curve.

However, real molecules are not ideal systems. A more accurate potential en-

ergy curve for the vibration of diatomic molecules resembles the curve for a

real molecule in Figure 14.28. The harmonic potential energy curve is super-

imposed for comparison. At low vibrational energies the curve is close to ideal,

but at higher vibrational energies the potential energy curve is much wider

than for the ideal harmonic oscillator. As a result, the vibrational energy lev-

els, shown in the figure, begin to get closer and closer together. This is the trend

observed in Table 14.3 for the energies of the HCl vibration. In reality, our os-

cillator is not harmonic but anharmonic.Also, at some point the molecule has

enough energy that the two atoms move apart—and never move back toward

each other. The molecule has dissociated, and the amount of energy required

to do this is called the dissociation energy.No quantized vibrational levels ex-

ist above the dissociation energy limit. An ideal harmonic oscillator does not

have a dissociation energy until one gets to v.

14.12 Vibrational Spectroscopy of Diatomic and Linear Molecules 491

Table 14.4 Vibrational parameters of various heteronuclear diatomic moleculesa

Molecule  ̃e(cm^1 ) xee(cm^1 ) Internuclear distance, Å

HF 4138.52 90.07 0.9171

HCl 2989.74 52.05 1.275

HBr 2649.67 45.21 1.413

OH 3735.21 82.81 0.9706

OD 2720.9 44.2 0.9699

NO 1904.03 13.97 1.1508

CO 2170.21 13.46 1.1281

LiH 1055.12 13.22 1.5949

Source:G. Herzberg.Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules.Van Nostrand

Reinhold, New York, 1950.a

See also Table 14.2

x  0

V ^12 kx^2

E ^152 h

E ^132 h

E ^112 h

E ^12 h

E ^32 h

E ^52 h

E ^72 h

E ^92 h

Figure 14.27 For an ideal harmonic oscilla-
tor, the potential energy equals ^12 kx^2 and the quan-
tized energy levels are equally spaced.

Internuclear separation

re

Potential energy

Ideal harmonic

potential

Real molecule

Figure 14.28 A more realistic potential energy
surface for the vibration of a molecule is super-
imposed on the ideal harmonic oscillator curve.
Only at low vibrational quantum numbers does
the ideal potential energy curve adequately ap-
proximate the real system. Note how the vibra-
tional energy levels get closer and closer together
as the vibrational quantum number increases.