Physical Chemistry , 1st ed.

86,500 GHz. Therefore, it absorbs only the light that has the same frequency:

86,500 GHz. Light of this frequency has a wavelength of about 3.46 microme-

ters, or 3.46 microns, which is in the infrared region of the spectrum. In units

of wavenumbers, this light has a frequency of 2886 cm^1.

Consider the possible vibrations of the three atoms in the H 2 O molecule,

however. This molecule has 3(3)  6 3 vibrational degrees of freedom. What

are they? One can imagine that the three atoms in water move about in a com-

plicated dance that might be difficult to understand. However, the vibrational

spectrum of water shows three and only three distinct absorptions: at 3756,

3657, and 1595 cm^1. These must correspond to the frequencies of the atoms

in the three normal modes of water. The motions of the atoms in these nor-

mal modes (which can be determined mathematically) are illustrated in Figure

14.24. Notice that the vectors illustrated have different lengths so as to keep the

center of mass in the same position. Again, only part of the motion is illus-

trated. The atoms also move in the reverse direction in the course of a single

vibration. The normal vibrational modes of H 2 O are generally referred to as

(in decreasing order of wavenumber) the asymmetric stretching mode, the

symmetric stretching mode, and the bending mode.

No matter how complicated the molecule, the motions of the atoms with

respect to each other can be treated solely as if those atoms were moving as

shown by the normal modes. This allows us to consider onlythe normal modes.

More importantly, because of symmetry, some of the normal modes of large

molecules are exactly the same as others. Consider again the C–H stretches of

benzene, C 6 H 6. Because of the sixfold symmetry of benzene, we might expect

that they can be described equivalently, and to a certain extent this is true. As

such, these normal modes have the same vibrational frequency. The total num-

ber ofuniquenormal modes therefore depends on two things: the number of

atoms in the molecule (as indicated by the 3N5 or 3N6 number of nor-

mal modes) and the symmetry of the molecule. The higher the symmetry, the

fewer the number of independent normal modes.

14.10 Quantum-Mechanical Treatment of Vibrations

When one considers a diatomic molecule, it can be compared to the ideal har-

monic oscillator as defined by Hooke’s law:

Fkx (14.28)

where the force Facting against a displacement xis proportional to x.(We are

ignoring the vector aspects ofFand xhere.) The proportionality constant,k,

is called the force constant.It has units of force/distance, like N/m or mdyn/Å.

An ideal harmonic oscillator is also defined as having a potential energy of

VF dx^12 kx^2 (14.29)

A plot of a potential energy versus displacement is shown in Figure 14.25. In

the first approximation, diatomic molecules can be considered in terms of a

quantum-mechanical harmonic oscillator having reduced mass , which, re-

call, is related to the two masses of the atoms m 1 and m 2 :



m

m

1

1

m

m

2

2



or, equivalently, 



1



m

1

1

 

m

1

2



(14.30)

484 CHAPTER 14 Rotational and Vibrational Spectroscopy

H

O

H

H

O

H

H

O

H

Figure 14.24 The three normal modes of
H 2 O. The lengths of the vectors indicate the rela-
tive distances that each atom moves. The actual
distance that each atom moves is very small, less
than 0.1 Å.

x  0

V ^12 kx^2

Figure 14.25 Potential energy diagram for an
ideal harmonic oscillator. Usually, this diagram is
applicable only for low-energy (that is, low quan-
tum number) vibrations.