Physical Chemistry , 1st ed.

As coupling of the angular momenta increases with atomic number, more and

more “forbidden” transitions are observed in the electronic spectra.

Example 15.11

Given that the ground state of O 2 is^3 g^ , list the term symbols of the elec-

tronic states that can be accessed by allowed electronic transitions. Do not

consider the changes in .

Solution

A state means that its value is 0, so possible excited states can have

0 or 1, which will be or states. A multiplicity of 3 means that

S1, and since S0 the excited states will also have an Sof 1 and a mul-

tiplicity of 3. Since the ground state is gerade, the excited states must be

ungerade. Therefore, possible allowed excited states are^3 u^ and^3 u.Note

the sign on the state.

15.7 Vibrational Structure and the Franck-Condon Principle

Recall that, generally, electronic states are separated by more energy than are

vibrational states (which are in turn separated by more energy than are rota-

tional states). It is common to consider that every electronic state of a mole-

cule has its own collection, or manifold,of vibrational states. The following dis-

cussion is easiest if one assumes a diatomic molecule (although the ideas are

applicable to all molecules).

When a molecule absorbs a photon that excites an electron to a higher-

energy state, the state of the molecule is described by a different wavefunction.

For the ground-state wavefunction, a diatomic molecule has a certain equilib-

rium bond distance. Even though it is probably vibrating in its lowest vibra-

tional quantum state (recall the existence of zero-point energy for quantized

vibrations), it is presumably vibrating about an average bond distance known

as the equilibrium bond distance.It is usually labeled Reor re.

An excited-state electronic wavefunction is similar. It too has its own low-

est vibrational quantum state and equilibrium bond distance. However, there

is no guarantee that the equilibrium bond distances will be the same. Normally,

equilibrium bond distances change with electronic state. This is illustrated in

Figure 15.12, which shows two electronic states, their respective vibrational

state manifold, and an energy minimum that occurs at different internuclear

distances.

If these two states are involved in an allowed transition, there are several

considerations. First, a Born-Oppenheimer type of approximation is applica-

ble in that an electronic transition occurs so fast (on the order of 10^15 s†) that

the nuclei do not have time to move: that is, translations and vibrational and

rotational motions do not occur on the timescale of electronic transitions. On

a diagram such as Figure 15.12, a system in its ground electronic state would

move to an excited state by moving straight up in the figure.This means that

the internuclear distance does not change. This idea is called the Franck-

Condon principle.(It is named after the German physicist James Franck and

15.7 Vibrational Structure and the Franck-Condon Principle 539

†Compare this to a single vibration of an H 2 molecule, which lasts  8  10    15 s, or 8 fs.

Internuclear separation

Excited

state

Ground

state

Potential energy

Re Re*

Figure 15.12 Different electronic states have
different minimum-energy internuclear distances
as well as different vibrational energy manifolds
within each. This complicates the electronic spec-
tra of even the simplest, diatomic molecules.