Chemistry Reference
In-Depth Information
2.9.2 The Set of Three Coulson's Orthogonal Hybrids
If
v
0
is a hybridization parameter, Coulson's hybrids can be written as:
<
b
0
1
¼
s sin
v
0
þ
z cos
v
0
cos
uþ
y cos
v
0
sin
u
b
0
2
¼
s sin
v
0
þ
z cos
v
0
cos
u
y cos
v
0
sin
u
ð
:
Þ
2
299
:
l
0
¼
s cos
v
0
z sin
v
0
:
Comparison with our set (Equations 2.219) shows that, if the two sets
have to be equivalent, we must have:
1
2
1
2
1
sin
v
0
¼
cos
v
0
cos
u ¼
cos
v
0
sin
u ¼
p
sin
v;
p
cos
v;
p
ð
2
:
300
Þ
so establishing the relation between Coulson's
v
0
and our (
v
) hybridiza-
tion parameters.
We notice, first, that hybrids (Equations 2.299) are correctly normal-
ized, and that:
b
0
1
j
b
0
2
i¼
sin
2
v
0
þ
cos
2
v
0
cos 2
u ¼
S
12
¼h
0
ð
2
:
301
Þ
provided Coulson's relation is satisfied:
2
sin
v
0
cos
v
0
cos 2
u ¼
<
0
ð
2
:
302
Þ
but that the lone pair hybrid l
0
is not orthogonal to either b
0
1
or b
0
2
:
sin
v
0
cos
v
0
ð
S
13
¼
S
23
¼
1
cos
uÞ
=
0
ð
2
:
303
Þ
Schmidt orthogonalization of l
0
to b
1
or b
2
can be done using the explicit
formulae given elsewhere (Magnasco, 2007), so obtaining:
(
b
0
1
;
b
0
2
;
b
1
¼
b
2
¼
ð
2
:
304
Þ
Þ
1
=
2
¼ð
2S
2
ð
l
0
Sb
0
2
Sb
0
1
Þ
l
1
It can be seen that, in doing the orthogonalization process, the coef-
ficient of y in (Equatrions 2.304) becomes identically zero, and we obtain
