Chemistry Reference
In-Depth Information
as reported in the last row of Table 12.5. We can proceed similarly for
D
(I),
Dðs
v
Þ
, and
Dðs
0
v
Þ
. Using the standard techniques of the matrix
eigenvalue problem, it is left as an easy exercise for the reader to verify
that the lower off-diagonal block of the representative matrices
D
(C
2
)
and
Dðs
0
v
Þ
in the original basis set (12.31) are diagonalized in the
symmetry-adapted basis (h
z
h
y
), namely by transformation with the
unitary matrix
0
@
1
A
1
1
2
p
p
U ¼
ð
12
:
36
Þ
1
1
2
p
p
12.3.2 Use of Symmetry in Ground-state NH
3
(
1
A
1
)
The three hydrogen atoms of the NH
3
molecule are chosen to lie in the xy-
plane, numbered anticlockwise with H
1
on the positive x-axis, the origin
of the right-handed coordinate system being taken at their intersection
and the Natomalong the symmetry axis with z positive (Figure 12.3). The
nuclear symmetry now implies three symmetry planes at 120
(
3
),
their intersection determining the threefold symmetry axis C
3
. The
s
1
,
s
2
,
s
z
N
y
C
3
+
H
2
σ
3
H
1
x
0
σ
1
H
3
_
σ
2
C
3
Figure 12.3
Elements of C
3v
symmetry in NH
3
