Group Theory
Symmetry Operations
1. . The identity operation. This operator leaves the object unchanged.
2. . The proper rotation operator. This operator performs a counterclockwise rotation about an axis through the center of mass by an angle of . For example, rotates by 120; by 240; and by 120. Note the operator equalities and . The operator rotates by 90 and by 180. Therefore, . If the object has several rotation axes, the axis with the highest order (largest n) is the principal axis. Usually, the center of mass of the molecule is located at the origin of the Cartesian coordinate system, and the principal axis is aligned with the z axis. When several axes have the same order, the z axis is taken along the one passing through the greatest number of atoms.
3. . The inversion operation. This operator projects the object through the center of symmetry to an equal distance on the other side of the center.
4. . The horizontal mirror plane reflection. The mirror plane is perpendicular to a principal axis. This operator reflects the object across the plane of symmetry.
5. . The vertical mirror plane reflection. The mirror plane contains a principal axis. This operator reflects the object across the plane of symmetry.
6. . The dihedral (or diagonal) mirror plane reflection. The mirror plane contains a principal axis and bisects the angle between two horizontal axes. (These horizontal axes are perpendicular to the principal axis.) The operator reflects the object across the plane of symmetry.
7. . The improper rotation operator. This operation performs a counterclockwise rotation about an axis by an angle of followed by a reflection across a mirror plane perpendicular to that axis. In other words, this is a rotation followed by reflection across a mirror plane.
If an object has and as symmetry elements, then it must also have as symmetry element. However, it is possible for to be a symmetry element for an object when and are not symmetry elements.
Point Groups
If two symmetry operations are symmetry elements of an object, then their product is also a symmetry element. Any set of operations for which any product of members of that set is a member of the set is called a group. Group theory is the study of the mathematical properties of such a collection of symmetry elements. Groups containing only symmetry elements that leave one point in the molecule unchanged are called point groups. In addition, crystals also have symmetries arising from translation through space and are called space groups. However, here we consider only point groups. Each distinct group of symmetry elements has a name.
We now present the Schoenflies naming system for point groups:
Groups with no proper rotation axis.
. Contains the symmetry element only.
. Contains symmetry elements and .
. Contains symmetry elements and .
. Contains symmetry elements and . Note that and . Therefore, these groups are and respectively.
Groups with one proper rotation axis.
. Contains the symmetry elements and only.
. Contains the symmetry elements , , and n vertical mirror planes .
. Contains the symmetry elements , , and (perpendicular to ).
Dihedral Groups
. Contains the symmetry elements , , and n axes perpendicular to . Note that for n = 2, the group has three mutually perpendicular axes.
. Contains the symmetry elements , , n axes, and (mirror plane is perpendicular to the principal axis). This group also has n vertical mirror planes and if n is an even number.
. Contains the symmetry elements , , n axes, and n vertical mirror planes . This group also has an improper axis of double order parallel to and if n is an odd number.
Linear Groups
These groups are special cases of the above groups for (i.e. operation about the axis by any angle is symmetry element).
. This group is linear unsymmetrical. For example, HCl and HCN.
. This group is linear symmetrical. For example, .
Cubic groups
. This group has the symmetry of a regular tetrahedron. For example, .
. This group has the symmetry of a regular octahedron.
. This group has the symmetry of a regular icosahedron. For example, .
The groups and are often difficult to distinguish. Both have , n, and n mirror planes. has a horizontal mirror plane as well as vertical mirror planes (, each plane contains a axis). But for the groups, the vertical mirror planes lie diagonally between the axes and are therefore .
In determining Schoenflies point groups, it is useful to examine the molecule systematically. Mortimer provides a step-by-step chart for classifying molecules according to their point groups. The following is a different sequence of steps for finding point groups. Please note that it is unnecessary to use both strategies; you only need to use the one that you find easiest to understand.
1. Determine if a molecule has or symmetry. In both of these cubic point groups, there are four axes along the body diagonals of the cube. If the molecule does not belong to either of these point groups, but contains more than one threefold axis (i.e. more than one axis of symmetry greater than a twofold), it is of a point group not discussed here.
2. Determine if the molecule has any axes. If so, then go to step 3. If not, then look for a symmetry plane, in which case it belongs to the point group , or a center of symmetry, in which case it belongs to . If no symmetry elements are found, then the molecule belongs to the point group , which has only the identity element.
3. If at least one axis is present, find the principal axis (i.e. the with the largest n). [Note that there may not be a unique principal axis. For example, the point group has three axes perpendicular to each other.] Next determine if there is an improper axis collinear with the principal axis. If there is an axis but no other elements of symmetry (except i), then the molecules belongs to one of the (n even) point groups. Otherwise, if there are no axes or if there is an axis along with other symmetry elements, go to step 4.
4. Determine if the point group is dihedral () by observing if n axes (equally spaced) lie in a plane perpendicular to the principal axis. If so, go to step 5. If not, go to step 6.
5. To distinguish between the point groups , determine if a horizontal mirror plane () is perpendicular to the axis. If there is a , then the point group is . If not, determine if there are vertical mirror planes () containing the axis. [The planes must bisect the angle formed by two horizontal axes that are perpendicular to the principal axis.] If planes are present, then the point group is . If the molecule contains no mirror planes of any kind, then the point group is .
6. If a molecule does not have n horizontal axes perpendicular to the axis, then it belongs to either the , , or groups, which contain only one proper rotation axis. If the axis contains a horizontal mirror plane (), then it belongs to the group. If the axis has n vertical mirror planes intersecting the axis, then the molecule belongs to the group. If the molecule contains neither horizontal nor vertical mirror planes, then the point group is .
Further discussions of these topics are found in Mortimer pages 651-655, 682-689, 707.