Physical Chemistry , 1st ed.

where ˆis the electric dipole operator, eis the charge on the particle, and xi,yi,

and ziare the coordinates of the particle. Transitions of this sort are electric

dipole transitions.Transitions between wavefunctions can be prompted by

other interactions (like magnetic dipole or polarization changes), but electric

dipole transitions are the most common.

We can use the conclusions of the previous chapter on symmetry at this

point. In order for the integral in equation 14.2 to have a nonzero numerical

value, the irreducible representations of the three components of the integrand

must contain the totally symmetric irreducible representation of the point

group of the system, usually labeled A 1. That is,

(^) final (^) ˆ (^) initialA 1 (14.3)
The great orthogonality theorem may be needed to determine the irreducible
representations of the product in equation 14.3. If it contains A 1 , then the in-
tegral may be nonzero and the transition between initialand final, caused by
absorption or emission of electromagnetic radiation, is considered allowed.On
the other hand, if the combination of irreducible representations in equation
14.3 does not contain A 1 , then the integral defined in the transition moment
must be identically zero and the transition cannot occur. It is a forbidden
transition.
In reality, some forbidden transitions do occur, since the above definition
does not take into account the nonideality of an atomic or molecular system.
But forbidden transitions almost always have a much lower probability than
do allowed transitions. This means that in spectral measurements, absorptions
or emissions of radiation due to allowed transitions are typically stronger than
for forbidden transitions. This fact is not only useful in understanding spectra
but also reinforces the usefulness of the predictions of symmetry and quantum
mechanics.
Although the above equations imply that a lot of symmetry analyses must
be performed, that is not always the case. Equations 14.2 and 14.3 allow for the
possibility of broad statements about which transitions will and will not be
allowed for particular atomic or molecular systems. Such general statements,
ultimately based on quantum-mechanics and symmetry, are called selection
rules.Selection rules allow us to easily determine which transitions will occur.
When one is faced with a spectrum to interpret, knowledge of the selection
rules is an indispensable tool in deriving physical information from the spec-
trum. Rotational and vibrational spectroscopy, in this chapter, are simplified to
a large extent thanks to selection rules.

14.3 The Electromagnetic Spectrum

Light can be represented by a wave, like the one shown in Figure 14.1. The

wave has a characteristic frequency  and a wavelength , and the two are re-

lated to the speed of light, represented by the letter c, by the equation

c  (14.4)

The speed of light in vacuum is constant at 2.9979 

108 m/s. The speed of light

in vacuum is a universal constant, like Planck’s constant. All light, no matter what

the frequency or wavelength, travels at this speed in vacuum. The wavelength has

units of length, whereas the frequency has units of 1/s, or s^1. Frequency is

thought of as the number of waves of light passing a certain point per second.

Light travels at different speeds in different media (like air or water), but

since gases are so dispersed the speed of light through air is usually treated as

14.3 The Electromagnetic Spectrum 463

Number of waves

passing per second  

Velocity  c



y

x

z

Figure 14.1 Light acts as a wave, with a wave-
length and a frequency . In vacuum, all light
has the same velocity, 2.9979
108 m/s. This
value is given the symbol c.