Rotational Groups and Chiral Molecules

The symmetry operations that we have encountered are either proper or improper. Proper symmetry elements are rotations, also including the unit element. The improper rotations comprise planes of symmetry, rotation-reflection axes, and spatial inversion. All improper elements can be written as the product of spatial inversion and a proper rotation (see, e.g.. Fig. 1.1). The difference between the two kinds of symmetry elements is that proper rotations can be carried out in real space, while improper elements require the inversion of space and thus a mapping of every point onto its antipode. This can only be done in a virtual way by looking at the structure via a mirror. From a mathematical point of view, this difference is manifested 

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. 

This result confirms that Sj is included in the coset of 5, 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. Hi and H2, if the operations of Hi commute with the operations of H2 and every operation of the group can be written uniquely as a product of an operation of Hi and an operation of H2. This may be denoted in general as 

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 x Hrot. where Q = E,i. However, direct product groups are not limited to centrosymmetry. In the group Dsh, 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 

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