Chemistry Reference
In-Depth Information
Figure 2.22
Geometry of the pyramidal NH
3
molecule of C
3v
symmetry. Reprinted
from Chemical Physics Letters, 477, Magnasco, V., H ¨ ckel transformation theory of
the hybridization problem inNH
3
. 392-396, Copyright (2009), with permission from
Elsevier
2
3
p
sin
g ¼
sin
u
ð
2
:
221
Þ
Using the usual self-explanatory notation for the seven valence AOs, we
call s, x, y, z the 2s and 2p orbitals on N, and h
1
, h
2
, h
3
the 1s orbitals on
the H atoms (H
1
on the positive x axis).
Leaving out s for the moment, the AOs belonging to C
3v
symmetry are
(Magnasco, 2009a):
<
1
A
1
:
z
;
h
z
¼
p ð
h
1
þ
h
2
þ
h
3
Þ
1
p
ð
E
x
:
x
;
h
x
¼
2h
1
h
2
h
3
Þ
ð
2
:
222
Þ
:
1
E
y
:
y
;
h
y
¼
p ð
h
2
h
3
Þ
so that the 6
6H
€
uckel secular equation can be factorized into the three
2
2 equations:
¼
«
H
zh
z
H
zh
z
H
yy
«
H
yh
y
H
yh
y
H
zz
H
xx
«
H
xh
x
H
xh
x
0
H
h
z
h
z
«
¼
0
H
h
x
h
x
«
¼
0
H
h
y
h
y
«
SymmetryE
y
ð
SymmetryA
1
SymmetryE
x
:
Þ
2
223
