Chemistry Reference
In-Depth Information
C
2
ℵ
C
2
C
2
ℵ
C
2
C
2
ℵ
C
2
Ta b l e 7 . 4
Character table for
the double group
D
2
E
ℵ
A
1
1
1
1
1
B
1
1
1
1
−
1
−
1
B
2
1
1
−
1
1
−
1
B
3
1
1
−
1
−
1
1
E
1
/
2
2
−
2
0
0
0
Fig. 7.4
D
3
trischelate
complex: orientation of the
twofold axes on the positive
hemicircle in the
(x
,y
)
plane
table; as an example,
D
(C
2
x
)
×D
(C
2
y
)
=D
(
ℵ
C
2
z
)
(7.42)
From the multiplication table one can obtain the conjugacy classes. The double
group,
D
2
, has five classes; hence, it will contain one extra irrep, as compared
with the parent group. The character table is shown in Table
7.4
. The irreps are
of two different kinds. The
orbital
irreps are not changed under
ℵ
and thus retain
the characters of the single group. The other kind is the
spin
irreps, which are anti-
symmetric under
. Direct product tables may also be constructed. Note that the
totally symmetric component in this case belongs to the anti-symmetrized part of
the direct square of the spinor irreps. Relevant tabular information concerning the
double groups and spinor irreps is gathered in Appendix
G
.
As a further example, in the
D
3
symmetry group, the three
C
2
operators, which
bisect the chelating ligand (cf. Fig.
6.5
), all lie in the
(x
,y
)
plane. In order to obey
the conventions, we have to take the poles of these axes in the positive semicircle
with
x
>
0(seeFig.
7.4
). The
(α,n
x
,n
y
,n
z
)
labeling in the primed coordinate
ℵ
