¶ … Symmetry in Inorganic Chemistry
Molecules are often classified in terms of their symmetry. All molecules that have the same basic shape share a number of common properties. Classification of molecules by symmetry is based upon the presence of symmetry "elements," such as mirror planes and rotational axes, and corresponding symmetry "operations" such as reflection through such a plane and rotation about such an axis, shared by all molecules in a given "point" group. A symmetry operation is a real or imagined change that can be made to the molecule that leaves it in an indistinguishable position from the initial position. For example, if a benzene molecule were rotated by 60, 120, 180, 240 or 300 degrees about it's principle axis of rotation, the molecule will be left in an identical configuration with respect to the initial one. Any molecule said to have C2v symmetry (or to "belong" to the C2v point group) will have these and only these same symmetry elements present, and will share a number of properties that can be predicted using the principles of symmetry and group theory (Kitaev et al.).
There are five symmetry operations and elements. These are identity, rotations and rotation axes, reflections and reflection planes, improper rotations and improper rotation axes, and inversion and center of inversion (Mann).
Identity is considered to be both an operation and an element. A symmetry operation is a way in which a molecule or its parts can be moved without changing the relative locations of similar atoms. Identity indicates that all the atoms in the molecule end up exactly where they started. It is the same as not moving them at all, and it's also equivalent to rotating the molecule through 360 degrees in any direction. The motion has no effect, as long as the atoms all end up in exactly the...
Every molecule has the identity symmetry element.
Rotation refers to moving a molecule about an axis. The rotation itself, the act of moving the atoms around, is the symmetry operation. The axis about which the rotation occurs is the symmetry element. The axis always goes through the center of the molecule. Some molecules have more than one symmetry operation associated with a single symmetry axis.
Reflection is a movement of the atoms through a plane. The motion of the atoms as they pass through the plane is the reflection operation, while the plane itself is the symmetry element associated with reflection. Planes of reflection are denoted as lowercase Greek letter sigma and are usually subscripted with the letters of the axes that are contained in the plane.
Improper rotation is a combination of rotation followed by reflection in the plane perpendicular to the axis of rotation. As with the other symmetry operations, the motion of the atoms is the operation. The symmetry element associated with improper rotation is an axis, which is the same axis as that used for the rotation part of the improper rotation. While there is a plane of reflection involved, it does not need to be specified for the improper rotation axis because it is always perpendicular to the rotation axis. Improper rotations are designated with the letter S. Reflection across the xy plane, which is perpendicular to the z axis doesn't move any of the atoms for this planar molecule.
The inversion symmetry operation is the motion of the atoms through the center of the molecule in a straight line. Each atom moves along the line through the center until the distance between the atom and the center is the same as it was originally. The symmetry element associated with the inversion operation is a point, or the…
