Chemistry Reference
In-Depth Information
<
2
6
y
c
ð
sx
cy
Þ
c
ð
sx
cy
Þþ
y
þ
c
ð
sx
þ
cy
Þþ
c
ð
sx
þ
cy
Þ
E
yy
y
¼
P
½
1
3
ð
y
csx
þ
c
2
y
þ
csx
þ
c
2
y
þ
y
csx
þ
c
2
y
þ
csx
þ
c
2
y
Þ¼
y
:
¼
2
6
ð
h
1
ch
3
ch
2
h
1
þ
ch
3
þ
ch
2
Þ¼
0
E
yy
h
1
¼
P
!
8
<
2
6
ð
h
2
ch
1
ch
3
h
3
þ
ch
2
þ
ch
1
Þ¼
1
3
3
2
h
2
3
2
h
3
E
yy
h
2
¼
P
:
1
p
ð
h
2
h
3
Þ
)
G
The reducible representation
of Table 12.7 splits into the following
irreps:
G ¼
4A
1
þ
2E
ð
12
:
40
Þ
in accord with the result of Equation 12.28.
Therefore, the (8
8) secular equation in the original AO basis fac-
torizes into a (4
4) block of symmetry A
1
and two (2
2) blocks of
symmetry E, belonging to the symmetries E
x
and E
y
(Figure 12.4) which
are mutually orthogonal and not interacting.
We end by recalling that MOs resulting from all LCAO methods are
classified according to the irreducible representations to which they
A
1
0
4x4
8x8
=0
E
x
2x2
0
E
y
2x2
Figure 12.4
Factorization under C
3v
of the minimum basis set MO secular equation
for NH
3
