Chemistry Reference
In-Depth Information
derived by Lennard-Jones (1937)
1
14
for the orbital energy of the kthMO in
the N-atom linear polyene chain,
the following results are easily
established.
(i) First level (k
¼
1):
p
N
þ
1
«
¼ aþ
2
b
cos
N
!1
«
1
¼ aþ
2
lim
b
ð
7
:
106
Þ
1
Last level (k
¼
N):
p
N
N
þ
1
¼aþ
2
p
«
N
¼aþ
2
b
cos
b
cos
N
!1
«
N
¼a
2
lim
b
1
N
ð
7
:
107
Þ
1
þ
The difference between two successive levels is
"
#
p
N
þ
1
ð
k
þ
1
Þ
cos
p
N
þ
1
k
D«¼«
k
þ
1
«
k
¼
2
b
cos
ð
7
:
108
Þ
sin
2
2k
þ
1
N
þ
1
sin
2
1
N
þ
1
¼
4
b
where use was made of the trigonometric identity
b¼
2sin
aþb
2
sin
ab
2
a
cos
ð
7
:
109
Þ
cos
Hence, for N
!1
,
D«!
0 and we have formation of a continuous
band of molecular levels. The limiting values
aþ
2
b
and
a
2
b
are
reached asymptotically when N
!1
.
(ii) For N
!1
, therefore, the polyene chain becomes the model for
the one-dimensional crystal. We have a bonding band with energy
ranging from
aþ
2
b
to
a
and an antibonding band with energy
ranging from
,whichareseparatedbythesocalledFermi
level, the top of the bonding band occupied by electrons. It is
important to note that using just one
a
to
a
2
b
, equal for single and double
bonds, there is no band gap between bonding and antibonding levels.
b
14
See also McWeeny (1979).
