where the units on eand Deare the same so that xeis unitless. Definition of
a diatomic molecule as a Morse potential oscillator is useful because it ties in
several important observables about that oscillator: classical frequency, disso-
ciation energy, force constant, anharmonicity. Such relationships are complex
but increase our understanding of the behaviors of such molecules. Table 14.4
also includes information on the anharmonicities of the various diatomic
molecules.Example 14.16
Predict where the v 0 →v6 transition for HCl will occur if it acts as an
ideal Morse oscillator. Use the information in Table 14.4.Solution
Using equation 14.39, we can calculate the following energies for the v 0
and v6 vibrational energy states:
E(v0)he(0 ^12 ) hexe(0 ^12 )^2
(6.626
10 ^34 J s)(2989.74 cm^1 )(2.9979
1010 cm/s)(^12 )
(6.626
10 ^34 J s)(52.05 cm^1 )(2.9979
1010 cm/s)(^12 )^2
E(v0)2.94
10 ^20 J
The term 2.9979
1010 cm/s is the conversion directly from wavenumber
(cm^1 ) to frequency (s^1 ). For E(v6), similarly:
E(v6) 3.42
10 ^19 J
The difference in energy between the two vibrational states is 3.13
10 ^19 J;
using the conversions Eh and c and converting to wavenumber, this
is equal to a transition occurring at 15,753 cm^1. This is consistent with the
trend of overtone absorptions in Table 14.3.Diatomic molecules have a relatively simple vibrational spectrum because
they have only one type of vibrational motion: a stretching motion. For linear
triatomic molecules, the number of vibrations is four [3N 5 3(3) 5
4], which is three more than a diatomic molecule. The descriptions of the nor-
mal modes of vibration start getting a little more complicated. This is because
for a normal mode,the center of mass of the molecule does not move.This means
that all of the atoms in the molecule participate in each normal mode so that
the center of mass stays fixed. Ultimately, this implies a more complicated ex-
act description of the vibrational motion.
Although the exact description may be more complicated, an approximate
description is often utilized in vibrational spectroscopy. Figure 14.30 shows the
normal modes of vibration for linear triatomic molecules that are symmetric
(such as CO 2 ) and asymmetric (such as HCN). Although the sets of normal
modes are labeled similarly, using subscripts on the Greek letter to label the
vibrations, the vibrations themselves are not described similarly. For the sym-
metric triatomic molecule, the vibration labeled 1 has the outside atoms mov-
ing in and out at the same time with respect to the center atom. Thus no over-
all change occurs in the dipole moment during this vibration and it is not
considered IR-active; it is IR-inactive.This motion is called a symmetric stretch-
ing vibration(because both sides move symmetrically). The vibration labeled
3 is also a stretching vibration, but now the two outside atoms are moving one494 CHAPTER 14 Rotational and Vibrational Spectroscopy