Chemistry Reference
In-Depth Information
A root of the quartic equation (2.208) is given in this case by the simple
trigonometric relation:
cot
u ¼
cos
v
ð
2
:
214
Þ
45
. This shows that for
v
=
provided
u >
0 the interbond angle resulting
from energy optimization opens beyond 90
o
.
In fact, in this case, we have
ð
x
¼
cos
uÞ
:
2
p
x
2
2x
2
1
x
2
2x
4
cos
2
v ¼
sin
2
v ¼
x
2
;
;
sin2
v ¼
ð
2
:
215
Þ
1
1
x
2
1
x
2
and the coefficients become:
<
:
4x
2
A
¼
2
ð
1
x
2
Þ
p
x
2
2x
4
B
¼
8
2
ð
1
x
2
Þ
2x
2
x
4
4
1
C
¼
ð
2
:
216
Þ
2
ð
1
x
2
Þ
p
8
x
2
x
2
2x
4
D
¼
2
ð
1
x
2
Þ
4
x
2
2x
4
E
¼
2
ð
1
x
2
Þ
giving, upon substitution in Equation (2.208):
8
<
:
Ax
4
Bx
3
Cx
2
P
4
ð
x
Þ¼
þ
þ
þ
Dx
þ
E
8x
3
p
8x
3
p
4x
6
4x
2
8x
4
4x
6
4x
2
8x
4
þ
x
2
2x
4
þ
x
2
2x
4
þ
¼
¼
0
2
ð
1
x
2
Þ
ð
2
:
217
Þ
so that cot
u¼
cos
v
is a solution satisfying the quartic equation.
We now construct two sp
2
hybrids b
1
and b
2
directed towards the H
atoms simply by doing the further orthogonal transformation of the
functions b (belonging toA
1
symmetry) and y (belonging toB
2
symmetry):
