Physical Chemistry , 1st ed.

two examples of a high-probability and a low-probability transition using

Franck-Condon principles. The transition marked A has a low probability, be-

cause the vibrational wavefunctions do not overlap well, and a high-probability

region is in the same nuclear position as a low-probability region. The transi-

tion marked B has a higher probability because high probabilities overlap.

Mathematically, the transition moment of a vibronic transition depends on

an overlap integral in terms of the electronic and vibrationalwavefunctions.

The form of the transition moment is

Mel,uppervib,upperˆel,lowervib,lowerd (15.23)

where the “el” refers to the electronic wavefunction and “vib” refers to the vi-

brational wavefunction. The operator ˆis the electric dipole operator. Since

the change in the molecule involves one of its electrons, to a good approxima-

tion the dipole moment operator affects the electronic wavefunction and not

the vibrational wavefunction. The above integral can therefore be separated as

Mel,upperˆel,lowerdvib,uppervib,lowerd (15.24)

The first integral represents a “normal” transition moment. The value of the

second integral is not determined by orthonormality, since it represents dif-

ferent vibrational wavefunctions ofdifferentelectronic states. This second in-

tegral is known as the Franck-Condon overlap integral,and it is a measure of

the amount of overlap between two different vibrational wavefunctions. The

larger the overlap (see Figure 15.14), the larger the transition probability.

The Franck-Condon principle is applicable to polyatomic molecules also.

However, as might be expected, the potential energy diagrams get more com-

plicated, in part because there are now 3N 6 vibrational degrees of freedom

and therefore 3N 6 potential energy diagrams to consider for eachelectronic

state. Many electronic spectra are actually vibronic spectra. In some electronic

spectra, the vibrational structure is visible, in others it is not resolved. Figure

15.15 shows an example of an electronic spectrum at low resolution, so no vi-

brational structure is seen. Compare this with Figure 15.13, which is a much

higher resolution spectrum. See the difference?

15.8 Electronic Spectra of Polyatomic Molecules

Since most chemical species are polyatomic molecules, a discussion of the elec-

tronic spectra of molecules covers most matter. However, the subject is so large

(the saying “books are written about it” is especially true here) that we can

cover only a few specific topics.

The electronic states of polyatomic molecules can be labeled using the irre-

ducible representations of the symmetry point group of the molecule. (This is

another example of how symmetry is important in the understanding of spec-

tra.) As such, the same rule involving the direct product of the irreducible rep-

resentations applies:

*upperoperatorlowerA 1 (15.25)

or whatever label is the totally symmetric irreducible representation in that

symmetry point group. Here,*upperis the (complex-conjugated) irreducible

representation of the upper electronic state,loweris the lower electronic

state, and operatoris the irreducible representation of the appropriate dipole

moment operator. The irreducible representation labels of the dipole moment

15.8 Electronic Spectra of Polyatomic Molecules 541

200

Wavelength (nm)

800

Relative intensity

700600500400300

C 60

20

Figure 15.15 Many electronic transitions have
vibrational structure, which shows up only
under a high resolving power. Compare the low-
resolution electronic spectrum of C 60 with the
higher-resolution spectrum of methylaniline in
Figure 15.13.Source:H. Ajie et al.,J. Phys. Chem.,
1990, 94: 8633.