Physical Chemistry Third Edition

23.4 Electronic Spectra of Diatomic Molecules 973

whereΛis the quantum number for the projection of the total electronic orbital angular

momentum on the internuclear axis andSis the total electron spin quantum number. The

selection rules for the rotational and vibrational transitions that accompany electronic

transitions are:

∆J± 1 (23.4-1d)

∆v:not restricted (23.4-1e)

whereJis the rotational quantum number (not the quantum number for the total

electronic angular momentum, which does not apply to diatomic molecules) andvis

the vibrational quantum number.

The selection rule of Eq. (23.4-1b) forbids transitions that change the value ofS.

Since our selections rules are approximate, forbidden transitions between triplet (S1)

states and singlet (S0) states do occur, but with low probabilities compared with

allowed transitions.

EXAMPLE23.9

The lowest-lying excited singlet electronic energy level of the CO molecule lies 8.0278 eV

above the ground state. Find the wavelength of the light absorbed in the transition to this

level from the ground state, neglecting changes in rotational and vibrational energy.

Solution

λ

hc

∆E



(6. 6261 × 10 −^34 J s)(2. 9979 × 108 ms−^1 )

(8.0278 eV)(1. 6022 × 10 −^19 JeV−^1 )

 1. 544 × 10 −^7 m 154 .4nm

Exercise 23.7

Find the wavelength of the light absorbed in the transition of the previous example ifJchanges

from 0 to 1 andvchanges from 0 to 2.

Each electronic transition produces a number of bands, with one band for each

vibrational transition and with the lines of each band corresponding to different rota-

tional transitions. Measurement and interpretation of such an electronic band spectrum

can yield not only the energy differences between electronic levels but also between

vibrational and rotational levels. Figure 23.11 depicts an electronic transition for a typ-

ical diatomic molecule. The two curves are the Born–Oppenheimer electronic energies

of two different electronic states. The vibrational energy levels are superimposed on

the graph in the appropriate positions, and a graph of the square of each vibrational

wave function (probability density for internuclear distance) is drawn on the line seg-

ment representing its energy level. The vertical scales of these wave function graphs

are separate from the energy scale. The rotational levels are too closely spaced to be

shown.

We apply theFranck–Condon principle, which states that the nuclei do not move

appreciably during an electronic transition. This principle is closely related to the Born–

Oppenheimer approximation. In Figure 23.11 a line segment is drawn to represent a