1292 H The Hückel Method
Theccoefficients in these orbitals can be determined so that the orbitals are normalized.
We omit this step. These orbitals follow the pattern that we have come to expect: The
lowest-energy orbital has no nodes between the carbon atoms, the next lowest has a
single node that passes through the center carbon atom, and the highest-energy orbital
has two nodes that pass between carbon atoms. Since there are three pi electrons, the
lowest-energy orbital is occupied by two electrons and the next orbital is occupied by
one electron.
The results of the Hückel method for benzene are summarized in Chapter 21. The
delocalized pi orbitals in benzene are linear combinations of six unhybridized 2pz
orbitals:φic( 1 i)ψ 1 +c 2 (i)ψ 2 +c 3 (i)ψ 3 +c( 4 i)ψ 4 +c( 5 i)ψ 5 +c( 6 i)ψ 6 (H-17)The treatment is exactly analogous to that of the allyl radical except that we must deal
with six simultaneous equations anda6by6secular equation:
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
α−Wβ 000 β
βα−Wβ 000
0 βα−Wβ 00
00 βα−Wβ 0
000 βα−Wβ
β 000 βα−W∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
0 (H-18)
Note that there is aβin the upper right and lower left corners, corresponding to the
carbons being bonded in a ring. With the same replacement as before, the secular
equation is
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
x 10001
1 x 1000
01 x 100
001 x 10
0001 x 1
10001 x∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
0 (H-19)
We do not go through the solution of this secular equation, but it gives the six values
ofWand the six delocalized orbitals of Figure 21.9.