Physical Chemistry , 1st ed.

implies four molecular orbitals, each with its own characteristic values for c 1 ,

c 2 ,c 3 , and c 4 , and each with its own energies. Recognize that with two electrons

in each molecular orbital, only the two lowest molecular orbitals for buta-

diene will be filled. The other two will be empty (and will be considered ex-

cited states of butadiene). Linear variation theory (see section 12.8) indicates

that the energies can be determined from the following secular determinant:



H 11 ES 11 H 12 ES 12 H 13 ES 13 H 14 ES 14



H 21 ES 21 H 22 ES 22 H 23 ES 23 H 24 ES 24

H 31 ES 31 H 32 ES 32 H 33 ES 33 H 34 ES 34 ^0

H^41 ES^41 H^42 ES^42 H^43 ES^43 H^44 ES^44 

which yields a polynomial having E^4 as the highest power (and hence yields

four roots).Hxyand Sxyare the normally defined energy integrals, respectively,

and overlap integrals between carbon xand carbon y:

Hxy*xHˆyd

Sxy*xyd

as defined in equation 12.28. Since the atomic orbitals used in the expansion

are assumed to be normalized,H 11 H 22 H 33 H 44 , and the value of that

energy integral is usually designated by the Greek letter . Also, the overlap in-

tegrals S 11 ,S 22 ,S 33 , and S 44 are exactly 1. At this point, no other simplification

can be made without approximating a solution.

Hückel put forth some simplifying assumptions. For a Hückel approximation:

1. All other overlap integrals Sxyare zero.


2. All energy integrals Hxybetween nonneighboring atoms are zero.


3. All energy integrals Hxybetween neighboring atoms have the same value.
   This value is usually designated by the Greek letter .
   When these assumptions are made, the above 4 4 determinant takes the
   following form (where the values for H 11 ,H 22 , ...,and S 11 ,S 22 , ...,have
   also been substituted):



 E  0 0



 E  0

0  E 

0 0  E

 0

This is a much simpler determinant to solve (even if it does still lead to a

polynomial having a fourth power ofE). It is called the Hückel determinantfor

the molecular orbitals. The polynomial one gets, when all common terms

are collected, is (    E)^4 3(    E)^2 ^2 

^4 0. Algebraic techniques for

finding solutions to such equations eventually provide the following four pos-

sible values for E:    1.618,    0.618,

0.618, and 

1.618. These

states are illustrated graphically in Figure 15.18. By convention,and are

negative, so the lower-energy states have the sign and the higher-energy

states have the sign. The four electrons in butadiene reside in these mo-

lecular orbitals in Hund’s-rule fashion: two in each orbital, opposite spins. The

highest-energy molecular electronic state that has an electron in it is called the

highest occupied molecular orbital,or HOMO. The lowest-energy molecular

electronic state that has no electron in it (when the molecule is in its overall

ground electronic state) is called the lowest unoccupied molecular orbital,or

LUMO. The lowest-energy electronic transition of a -electron-containing

544 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure

(^1)
(^2)
(^3)
(^4)
Figure 15.17 The orbitals of butadiene. 1
and  2 are occupied in the ground electronic
state (see Figure 15.18).
E 
Energy
 1.618
 0.618
 0.618
 1.618
Figure 15.18 Hückel theory predicts this
arrangement for the four electrons in butadi-
ene. A comparison (see text) suggests that this
molecule is more stable than expected due to the
conjugation of the electrons.