The 57 groups recognized in MOPAC are given in Table 1.
Each point group is represented by a subset of the associated point-group table. For example, the group D2h is represented by the subset shown in Table 2. The operations selected for the subgroup are the identity, E, and that minimum set of operations which is sufficient to allow all the operations to be generated as products of these operations. Thus, for the highest finite point group, Ih, the generating operations are: E, I, C3, and C5. Although it is not obvious, all 120 operations of the group can be generated as products of these four operations.
Each point-group is assumed to contain the totally symmetric representation, here A1g. Operations are represented as 3x3 Euler matrices, thus C2x, C2y and C2z would be represented as in the Figure All operations not given can be generated as products of operations already known, thus C2x = C2y C2z.
Figure:
Representation of Symmetry OperationsC2x: | ||
1 | 0 | 0 |
0 | -1 | 0 |
0 | 0 | -1 |
C2y: | ||
-1 | 0 | 0 |
0 | 1 | 0 |
0 | 0 | -1 |
C2z: | ||
-1 | 0 | 0 |
0 | -1 | 0 |
0 | 0 | 1 |
In order to minimize storage, the characters in character tables are stored separately from the point groups. This allows, e.g., C2v, C2h, and D2 to use the same character table.