Chemistry Reference
In-Depth Information
in the sign of the determinant of the corresponding representation matrices in the
(x,y,z)
basis. For proper rotations, the determinant is equal to
+
1. For improper
rotations, it is equal to
1. This minus sign comes from the representation matrix
for the inversion centre, which corresponds to minus the unit matrix:
−
⎛
⎞
−
10 0
0
ı
xyz
=
xyz
⎝
⎠
ˆ
10
00
−
(3.34)
−
1
Since the determinant of a matrix product is the product of the determinants of the
individual matrices, multiplication of proper rotations will yield again a proper ro-
tation, and for this reason, the proper rotations form a rotational group. In contrast,
the product of improper rotations will square out the action of the spatial inversion
and thus yield a proper rotation. For this reason, improper rotations cannot form a
subgroup, only a coset. Since the inversion matrix is proportional to the unit ma-
trix, the result also implies that spatial inversion will commute with all symmetry
elements.
In all the point groups with improper rotations, we shall thus always also have a
rotational subgroup, like
D
n
in
D
nd
or
D
nh
,or
T
in
T
d
and
T
h
, etc. Moreover, this
rotational subgroup is always a halving subgroup, i.e., its order is half the order of
the full group. This can easily be demonstrated. Let
H
rot
be the rotational subgroup
of
G
, and consider an improper symmetry element,
S
i
, as coset generator. The coset
S
i
H
rot
will contain only improper symmetry elements, and its order will be equal to
|
. Now is it possible that the group contains additional improper elements, out-
side this coset? Suppose that we find such an element, say
S
j
. Of course, the product
S
−
i
S
j
is the combination of two improper elements and thus must be a proper rota-
tion, included in the rotational subgroup. Let us denote this element as
R
z
. Hence,
it follows that
H
rot
|
S
i
R
z
=
S
i
S
−
1
S
j
=
S
j
(3.35)
i
This result confirms that
S
j
is included in the coset of
S
i
and thus implies that
there is only one coset of improper rotations, covering half of the set of symmetry
elements.
A group is a
direct product
of two subgroups,
H
1
and
H
2
, if the operations of
H
1
commute with the operations of
H
2
and every operation of the group can be written
uniquely as a product of an operation of
H
1
and an operation of
H
2
. This may be
denoted in general as
G
=
H
1
×
H
2
(3.36)
This is certainly the case when a group is
centrosymmetric
, i.e., when it contains
an inversion centre. Since the inversion operation commutes with all operations,
a centrosymmetric group can be written as the direct product
C
i
×
H
rot
, where
C
i
=
{
E,
ˆ
ı
}
. However, direct product groups are not limited to centrosymmetry. In the
group
D
3
h
, for example, the horizontal symmetry plane forms a separate conjugacy
class, which means that it commutes with all the operations of the group. It thus
