Physical Chemistry Third Edition

962 23 Optical Spectroscopy and Photochemistry

All∆Jvalues forbidden for a molecule with zero permanent dipole moment

(23.3-2b)

whereJis the quantum number for the rotational angular momentum. The rotational

selection rules are well obeyed by diatomic molecules with^1 Σelectronic states.

The selection rule of Eq. (23.3-2b) can be understood classically. In order for a

molecule to interact with the electric field of the radiation, it must exhibit a periodically

varying electric dipole moment of the correct frequency. A rotating diatomic molecule

with a permanent dipole moment does present a periodically varying dipole to the

radiation, so it should absorb or emit radiation if it is rotating with the correct frequency.

If a molecule has no permanent dipole moment it does not exhibit any periodic variation

in the dipole moment.

Rotational transitions correspond to photon wavelengths in the microwave region.

The radiation sources in microwave spectrometers are klystron tubes, which were

originally developed for radar apparatuses in World War II. Hollow metal wave guides

carry the radiation to the sample cell, which is a hollow metal cavity, and the resonant

radiation in the cavity is sampled to detect absorption. Microwave spectroscopy has

played an important role in identifying molecules in interstellar space, but it is not a

common tool in many chemical laboratories.

From the selection rule, the photon energy for an allowed transition is

Ephotonhν

hc

λ

Eν,J+ 1 −EνJ (23.3-3)

whereJis the value of the rotational quantum number for the lower-energy state and the

vibrational quantum numbervhas the same value for both states. Since the vibrational

quantum numbervdoes not change, the reciprocal of the wavelength is

̃ν

1

λ



1

hc

(Eν,J+ 1 −EνJ) (23.3-4)

The reciprocal wavelength is usually expressed in cm−^1 , sometimes called “wave

numbers.”

Equation (22.2-40) gives the energy levels for rotation in the rigid rotor approxima-

tion. The reciprocal wavelength corresponding to absorption is

̃ν

1

λ

 ̃Be[(J+1)(J+2)−J(J+1)]

 ̃Be[J^2 + 3 J+ 2 −J^2 −J] 2 ̃B(J+1) (23.3-5)

whereJ is the quantum number for the initial state. SinceJ can take on values

0, 1, 2,..., this corresponds to a set of equally spaced spectral lines with recipro-

cal wavelengths equal to 2 ̃Be,4 ̃Be, and so on. Figure 23.7a shows the energy levels

with the allowed transitions and Figure 23.7b shows a simulated spectrum for carbon

monoxide. The intensities of the lines are related to the populations of the rotational

levels.

EXAMPLE23.4

The splitting between the spectral lines in the CO spectrum is 3.8626 cm−^1. Find the value

ofre, the equilibrium internuclear distance.