Chemistry Reference
In-Depth Information
so that we obtain
1
2
K
<
D
E
¼
E
ð
K
þ
D
Þ
E
ð
K
Þ¼
ð
9
:
93
Þ
0
for the conjugation energy. The conjugation energy in the allyl radical is
hence larger than that of butadiene (nearly twice).
Calculation of the coefficients in the allyl radical proceeds through
solution of the system
<
:
!
x
þ
c
1
þ
1
2
x
2
þ
1
c
2
þ
0
ð
9
:
94
Þ
c
1
þ
c
2
¼
1
1
2
x
c
2
¼
c
1
¼
c
1
for x
¼
1
ð
9
:
95
Þ
2
1
giving
1
c
1
¼
c
2
¼
p
ð
9
:
96
Þ
if we neglect nonorthogonality between the structures.
Therefore, the relative weights of the two structures
c
1
and
c
2
are
50
% c
1
;
50
% c
2
ð
9
:
97
Þ
as it must be for two truly resonant
structures (
c
1
and
c
2
are fully
equivalent).
By writing the complete VB wavefunctions for the two resonant
structures, we can immediately obtain electron and spin density distribu-
tions in the allyl radical, even without doing any effective calculation
of the energy. Assuming orthonormal spin-orbitals, with reference to
Figure 9.8, the complete form of the two structures in terms of their
parents is
1
p ð
abc
Þð
abc
Þ
c
1
¼
ð
9
:
98
Þ
