Physical Chemistry , 1st ed.

Solution

We can use a ratio similar to the one used in Example 14.12, in terms of the

reduced masses of the molecules, and use the wavenumber value directly:



3657

 ̃

c

*

m^1

 





*



Considering the O–H and O–D bonds, the reduced masses (in units of grams

per mole) substituted into the above expression yield



3657

 ̃

c

*

m^1

 ^0

1

.9

.7

4

7

1

8

2

(^) g
g
/

/

m

m

o

o

(^) l
l

3657

 ̃

c

*

m^1

0.7276

 ̃*2661 cm^1

Experimentally in the vibrational spectrum of D 2 O, the symmetric O–D

stretch has a vibrational frequency of 2671 cm^1 , which shows that the di-

atomic approximation applied to parts of molecules can be very good. Such

approximate calculations are useful in understanding vibrational spectra of

molecules.

14.11 Selection Rules for Vibrational Spectroscopy

As with rotational spectroscopy, there are several ways of stating selection rules
for spectral transitions involving vibrational states of molecules. There is a
gross selection rule, which generalizes the appearance of absorptions or emis-
sions involving vibrational energy levels. There is also a more specific, quan-
tum-number-based selection rule for allowed transitions. Finally, there is a
selection rule that can be based on group-theoretical concerns, which were not
considered for rotations.
Recall that light is an oscillating electromagnetic field. It can interact with
other oscillating electromagnetic fields, like the dipole moment of a molecule.
This interaction dictated our gross selection rule for pure rotations: the mole-
cule must have a permanent dipole moment in order to have a pure rotational
spectrum. This is because the rotating dipole acts as an oscillating electric field,
not changing in its magnitude but in its direction. As a vector, the dipole mo-
ment can oscillate by changing either its magnitude, or its direction, or both
in order to be detectable by another oscillating electromagnetic field, the light.
For vibrations, the key to interacting with light is based on the changing
magnitudeof the dipole moment of the molecule during the vibration. The di-
pole momentof a molecule is defined as the charge differential times the dis-
tance between the differential charges. The distance between charges changes
as the atoms of the molecule vibrate. As it changes, an oscillating electric field
is created, which can interact with the electromagnetic field of light.
Suppose there is no dipole moment changing its magnitude due to chang-
ing distance. There may be a fleeting, nonpermanent dipole present as a result
of the symmetry-destroying distortions imposed on a molecule during a vi-
brational motion. This will still be enough to interact with light. If, however,
there is not either a change in dipole magnitude or directions, then there is no
oscillating field to interact with the light, and no light is absorbed or emitted.

14.11 Selection Rules for Vibrational Spectroscopy 487