Physical Chemistry , 1st ed.

benzene. This generality is called the 4n 2 rule,where nis any nonnegative

integer and the expression 4n 2 yields the number ofelectrons in the sys-

tem. (For example,n1 predicts 6 electrons, which is what is found for

benzene.) The rule is of limited value, because in large molecules the deviation

from planarity is large. However, it is useful for predicting whether or not

heterocyclic compounds (that is, cyclic compounds that have atoms other than

carbon) or ions composed of cyclic carbon rings will be unusually stable.

The ultraviolet spectrum of benzene (and other aromatic compounds) is

dominated by transitions of the electrons from the lower orbitals to the

higher-lying, normally unoccupied orbitals. A strong absorption occurring

at 1800 Å marks the beginning of such transitions. (The electronic spectrum

of benzene has absorptions at lower energies, corresponding to light having

wavelengths of2600 Å. Such absorptions were historically very well known

and are one of the earliest recognized examples of transitions involving an

electronic transition that is formally forbidden but made allowed by the vi-

brations of the molecule.

The extended Hückel methodfor molecular orbitals includes a treatment of

all valence electrons (and ), not just the electrons. Atomic orbitals from

atoms are used to determine molecular orbital energies by defining the inte-

grals Hxyand Sxyin a fashion similar to that just presented for the electrons.

Although similar in principle, it requires larger matrices because all valence

electrons are treated. Other concerns preclude a detailed discussion here, but

other references (like J. P. Lowe,Quantum Chemistry,2nd ed., Academic Press,

Boston, 1993) can be consulted for details.

15.11 Fluorescence and Phosphorescence

In a perfect molecule, electronic transitions would go like this: absorption of a

photon excites a molecule from initial (usually ground) state to excited state;

excited state emits a photon having the same energy/frequency/wavelength and

molecule goes from excited state to previous initial ground state. The first

process,excitation,would be followed by the exact opposite process, called de-

excitationor decay.Such processes would follow quantum-mechanical selec-

tion rules strictly.

In reality, electronic transitions stray somewhat from the ideal selection

rules. In particular, when an excited electronic state decays to a lower electronic

state, a photon having the same energy as the excitation photon might not be

emitted. Instead, the molecule may de-excite by transferring the extra energy

into various vibrational, rotational, or solid-state vibrational (called “phonon”)

modes of the sample. Ultimately, this excess energy is converted into heat en-

ergy. Such processes are called radiationless transitions.

There are other mechanisms for energy loss. The initial excited electronic

state of a molecule is best thought of as a manifold of vibrational and rota-

tional states superimposed on the electronic potential energy curve. (Such a

view has been discussed previously.) In many cases, this manifold overlaps the

energy range of another manifold of rotational and vibrational energies of a

different electronic state (usually having lower electronic energy) that has the

same spin multiplicity. This is an important requirement, because allowed

transitions have the selection rule of S0. Such a system is illustrated in

Figure 15.22. In some cases, the molecule will spontaneously change its state

from the initial electronic state to the lower-energy electronic state of the same

multiplicitywithout the emission of a photon. In doing so, any excess energy

548 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure

Singlet manifold

Triplet manifold

Radiationless

transition

Vibrational

relaxation

(a type of

internal

conversion)

T 1

T 2

S 0

S 1

Excitation

Figure 15.22 Many molecules have overlap-

ping singlet (that is,S0, so 2S 

1 1) and

triplet (S1, so 2S 

1 3) electronic states.

Each of these electronic states has its own vibra-

tional state manifold. In some cases, absorbed

electronic energy is simply dissipated by being re-

distributed to the vibrations of the molecules, as

shown. Normally, the singlet manifold of elec-

tronic states does not interact with the triplet

manifold of electronic states via allowed elec-

tronic transitions. The numerical labels in the S

and T states are used to differentiate one singlet

(or triplet) state from another.