3.8 Rotational Groups and Chiral Molecules
45
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:
ˆ
ı
xyz
=
xyz
⎛
⎝
−10
0
0 −10
00 −1
⎞
⎠
(3.34)
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 rotation, 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 matrix, 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 ,orT 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, ˆ
S i , as coset generator. The coset
ˆ
S i H rot will contain only improper symmetry elements, and its order will be equal to
|H rot |. Now is it possible that the group contains additional improper elements, outside this coset? Suppose that we find such an element, say ˆ
S j . Of course, the product
ˆ
S
−1
i
ˆ
S j is the combination of two improper elements and thus must be a proper rotation, included in the rotational subgroup. Let us denote this element as ˆ
R z . Hence,
it follows that
ˆ
S i ˆ
R z = ˆ
S i ˆ
S
−1
i
ˆ
S j = ˆ
S j
(3.35)
This result confirms that S j is included in the coset of ˆ
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 =
{ ˆ
E, ˆ
ı}. However, direct product groups are not limited to centrosymmetry. In the
group D 3h , 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
45
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:
ˆ
ı
xyz
=
xyz
⎛
⎝
−10
0
0 −10
00 −1
⎞
⎠
(3.34)
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 rotation, 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 matrix, 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 ,orT 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, ˆ
S i , as coset generator. The coset
ˆ
S i H rot will contain only improper symmetry elements, and its order will be equal to
|H rot |. Now is it possible that the group contains additional improper elements, outside this coset? Suppose that we find such an element, say ˆ
S j . Of course, the product
ˆ
S
−1
i
ˆ
S j is the combination of two improper elements and thus must be a proper rotation, included in the rotational subgroup. Let us denote this element as ˆ
R z . Hence,
it follows that
ˆ
S i ˆ
R z = ˆ
S i ˆ
S
−1
i
ˆ
S j = ˆ
S j
(3.35)
This result confirms that S j is included in the coset of ˆ
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 =
{ ˆ
E, ˆ
ı}. However, direct product groups are not limited to centrosymmetry. In the
group D 3h , 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