Chemistry Reference
In-Depth Information
9.6.4 The Allyl Radical (N
¼
3)
This case (last row of Figure 9.8) is an interesting example of how to apply
Pauling's rules to an odd-electron system (N
¼
3, S
¼
1
2
). We add a
phantom atom (say d), treat the system as a four-atom system and, at
the end, remove the contribution of the phantom atom from the calcula-
tion. The covalent VB structures and the corresponding superposition
patterns are then the same as those of butadiene. Hence, with reference to
the previous calculation, we can write
<
1
2
K
þ
K
¼
Q
E
þ
1
2
K
H
11
ES
11
¼
Q
E
þ
K
1
2
K
1
2
K
¼
Q
E
þ
1
2
K
H
22
ES
22
¼
Q
E
þ
K
:
1
2
ð
Q
E
þ
K
þ
K
þ
K
Þ¼
1
2
ð
Q
E
Þþ
K
H
12
ES
12
¼
H
21
ES
21
¼
ð
9
:
89
Þ
where we have bolded the contributions which must be removed.
We then obtain the secular equation for the allyl radical:
1
2
x
2
þ
1
x
þ
¼
0
ð
9
:
90
Þ
x
2
þ
1
1
2
x
þ
2
2
1
2
x
2
þ
1
3
3
4
¼
0
)
x
¼
1
4
x
2
x
þ
¼
)
ð
9
:
91
Þ
Taking the positive root, we have the following for the
p
energy of the
radical:
8
<
1
2
K
1 Kekul
e
:
E
¼
Q
þ
ð
9
:
92
Þ
:
1 Kekul
e
þ
1 Dewar
:
E
¼
Q
þ
K
