Chemistry Reference
In-Depth Information
7.4.6 Benzene (N
¼
6)
The H
€
uckel secular equation for the closed chain (ring) with N
¼
6is
x 10001
1
x 1000
01
x 100
001
x 10
0001
x 1
1 0001
x
D
6
¼
¼
0
ð
7
:
93
Þ
where the boldface elements are the two differing from those of the linear
chain (1 and 6 are nowadjacent atoms). By expanding the determinant we
obtain a sixth-degree equation in x that can be easily factorized into the
three quadratic equations:
1
13
D
6
¼
x
6
2
6x
4
þ
9x
2
4
¼ð
x
2
4
Þð
x
2
1
Þ
¼
0
ð
7
:
94
Þ
with the following roots, written in ascending order:
x
¼
2
;
1
;
1
;
1
;
1
;
2
ð
7
:
95
Þ
Because of the high symmetry of the molecule, two levels are now
doubly degenerate. The calculation of the MO coefficients can be done
using the elementary algebraic methods used previously for the allyl
radical and cyclobutadiene. With reference to Figure 7.6, a rather lengthy
.
y
2
.
.
3
1
.
.
x
.
4
6
5
Figure 7.6 Numbering of carbon atoms in benzene and coordinate system
13
The same factorization can be obtained using hexagonal symmetry. In hexatriene the symmetry
is lower and only two cubic equations are obtained, equivalent to using the symmetry plane
bisecting the molecule.
