Chemistry Reference
In-Depth Information
1
4
x
2
4
x
1
4
x
r
b
¼ f
2
1
¼
2
1
þ
2
2
þ
2
3
ð
7
:
66
Þ
This gives for the electron density
P
ð
r
Þ¼r
a
þr
b
¼ x
2
2
2
3
1
þx
2
þx
ð
7
:
67
Þ
so that the charge distribution of the
electrons is uniform (one electron
onto each carbon atom), as expected for an alternant hydrocarbon,
9
and
for the spin density we have
p
1
2
x
1
2
x
Q
ð
r
Þ¼r
a
r
b
¼
2
2
3
1
þ
ð
7
:
68
Þ
unpaired electron
is
2
on atom 1 and
2
on atom 3, being zero at the central atom. This is
contrary to the ESR experimental observation that some
According to (7.68), in the H
uckel spin density the
€
a
a
spin at the end
atoms induces some
spin at the middle atom. This wrong result is due to
the lack of any electron correlation in the wavefunction, which belongs to
the class of IPM functions. Both (7.67) and (7.68) satisfy the appropriate
conservation relations:
10
b
trP
¼
3 (the total number of
p
electrons) and
trQ
¼
1 (the unpaired
p
electron of
a
spin).
bond energy in allyl is 2
2
p
The
p
¼
2
:
828 (units of
b
), comparedwith 2
p
for the
bond energy of the ethylenic double bond (the prototype of the
double bond). The difference
D
E
p
ð
allyl
ÞD
E
p
ð
ethylene
Þ¼
2
:
828
2
¼
0
:
828
ð
7
:
69
Þ
is called the delocalization energy of the double bond in the allyl
radical. It corresponds to a stabilization of the conjugated
p
system in
the radical.
9
Alternant hydrocarbons are conjugatedmolecules inwhich the carbon atoms can be divided into
two sets, crossed and circled, such that no two members of the same set are bonded together. All
molecules considered here are alternant hydrocarbons, inwhich energy levels occur in pairs, with
a
bond energy
x, and coefficients of the pairedMOs which are either the same or change sign
(Murrell et al., 1985).
10
Equations 6.31 and 6.32.
p
