## Representation of Point Groups

The 57 groups recognized in MOPAC are given in Table 1.

Table 1:

Point Groups available within Symmetry Code
 C1 Cs Ci O C2 C2v C2h D2 D2d D2h T C3 C3v C3h D3 D3d D3h Td C4 C4v C4h D4 D4d D4h S4 Th C5 C5v C5h D5 D5d D5h Oh C6 C6v C6h D6 D6d D6h S6 I Ih C7 C7v C7h D7 D7d D7h C D C8 C8v C8h D8 D8h S8 R3

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.

Table 2:

Subset of Group D2h
 Γ E C2y C2z I Ag B1g 1 1 -1 1 B2g 1 -1 1 1 B3g 1 -1 -1 1 Au 1 1 1 -1 B1u 1 1 -1 -1 B2u 1 -1 1 -1 B3u 1 -1 -1 -1

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 Operations
 C2x: 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.