Physical Chemistry , 1st ed.

molecule is the HOMO →LUMO transition, which may or may not be an

allowed transition. (It usually is, so the HOMO and LUMO are important in

the electronic spectra of such molecules.)

Example 15.13

Perform a Hückel approximation treatment of ethylene, CH 2 CH 2.

Solution

This treatment is much simpler than butadiene, because only two carbon

atoms are involved. It should be easy to apply Hückel’s approximations to get

the following 2 2 Hückel determinant:



 E 

 E^0

Upon multiplying out the terms in the determinant, one gets

(  E)^2    ^2  0

(  E)^2 ^2

(  E)

E

for the two electronic orbitals in ethylene. They are illustrated in Figure

15.19, with the lower of the two orbitals having energy 

. This is the

HOMO of ethylene. The LUMO of ethylene has an energy of    .The

electronic spectrum of ethylene has an absorption at about 2000 Å that has

been assigned to the transition between the HOMO and LUMO.

If we compare the answers for ethylene and for butadiene, there is a slight

difference from what we might expect. (Ethylene has the simplest electron

system, so comparisons to its energy levels are common.) If butadiene were

just two ethylenic systems, then the energies of the four electrons should be

simply 4(

)  4 

 4 . However, as seen above, the total energy of the

four butadiene electrons, which occupy the two lowest-energy electronic states,

is 2(

1.618) 2(

0.618)  4 

4.472 , or 0.472lower in en-

ergy than expected. (Recall that itself is negative.) This lower total energy is

due to the fact that the electrons in butadiene are not confined to a single

double bond (a situation termed “localized”) but have some probability of

being found along the entire length of the conjugated system (they are

“delocalized”). This extra energy stability of the four  electrons of

butadiene, 0.472, is called the delocalization energyof the electron system.

Values ofand are measured spectroscopically, and the electronic spec-

troscopy of many electron systems shows that the Hückel approximation

works fairly well. Many transitions between electronic states occur in the

visible or ultraviolet region of the spectrum. These transitions are the cause

of color in conjugated electron systems. In the Hückel approximation, all

of the molecular orbitals end up with a value of energy having the form

E

K, where the value for Kdepends on the system. Therefore, only the

values ofKand determine the molecule’s energy level pattern, which is

what is probed in an experimental spectrum. However, because of how it is de-

fined,has a similar value for most systems: about 75 kJ/mol. The value

15.9 Electronic Spectra of Electron Systems: Hückel Approximations 545

E 

Energy





Figure 15.19 Hückel theory predicts the above
arrangement for the two electrons in ethylene.