Physical Chemistry , 1st ed.

By making this ideal harmonic oscillator assumption, we make the wave-
functions and energies for the ideal harmonic oscillator directly applicable
to the diatomic molecule’s vibrations! In particular, since spectroscopy
deals with differences in the energy states, we are particularly interested in
the fact that

Evibh(v ^12 ) v0,1,2,3,... (14.31)

where his Planck’s constant,is the classical frequency as predicted by Hooke’s
law, and vis the vibrational quantum number.† (Be careful not to confuse 
with v.) That is, we expect that the vibrational energy of the diatomic mole-
cule is quantizedand given by equation 14.31 above. If the energies of the
vibrational states are quantized, then the differencesin the energies will have
only certain values.
Diatomic molecules are particularly easy to treat quantum-mechanically be-
cause they are easily described in terms of the classical harmonic oscillator. For
example, the expression



2

1



 



k

 (14.32)

which relates the classical force constant and reduced mass of the oscillator
to its frequency, is a valid mathematical tool. It is relatively easy to extend
some concepts to other vibrational motions of larger molecules: the vibra-
tions act as ideal harmonic oscillators that have certain wavefunctions and
certain quantized energies. However, normal modes are vibrations of all
atoms in a molecule, not just two, so expressions like equation 14.32 aren’t
directly applicable even if the idea of a force constant for a polyatomic mo-
tion is used. On the other hand, many normal modes are largely motions of
only a few connected atoms of a large molecule, so it is not uncommon to
hear of “C–H stretches” or “CH 2 bends” or such localized types of motions
even for large molecules. Technically, such labels are incorrect, but practi-
cally they are useful in qualitativelydescribing the normal mode of the
molecule.

Example 14.12

Assuming that the vibrational frequency of 2886 cm^1 (8.652 

1013 s^1 ) for

hydrogen chloride is^1 H^35 Cl, predict the vibrational frequencies for^1 H^37 Cl

and^2 H^35 Cl. Assume that the molecule is an ideal harmonic oscillator and that

the force constant does not change upon isotopic substitution. (Such as-

sumptions are common in vibrational spectroscopy.)

Solution

If the molecule is acting like an ideal harmonic oscillator and the force con-

stant is not changing, then for the classical frequency of the^1 H^35 Cl oscillator

we have

8.652 

1013 s^1 

2

1



 



k



and

* 

2

1



 



k

*



14.10 Quantum-Mechanical Treatment of Vibrations 485

†In Chapter 11, we used nfor the quantum number for the harmonic oscillator.