Representation of Point Groups

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

Table 1:
Point Groups available within Symmetry Code

C₁	C_s	C_i					O
C₂	C_2v	C_2h	D₂	D_2d	D_2h		T
C₃	C_3v	C_3h	D₃	D_3d	D_3h		T_d
C₄	C_4v	C_4h	D₄	D_4d	D_4h	S₄	T_h
C₅	C_5v	C_5h	D₅	D_5d	D_5h		O_h
C₆	C_6v	C_6h	D₆	D_6d	D_6h	S₆	I	I_h
C₇	C_7v	C_7h	D₇	D_7d	D_7h		C	D
C₈	C_8v	C_8h	D₈		D_8h	S₈	R₃

Each point group is represented by a subset of the associated point-group table. For example, the group D_2h 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, I_h, 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 D_2h

Γ	E	C_2y	C_2z	I
A_g
B_1g	1	1	-1	1
B_2g	1	-1	1	1
B_3g	1	-1	-1	1
A_u	1	1	1	-1
B_1u	1	1	-1	-1
B_2u	1	-1	1	-1
B_3u	1	-1	-1	-1

Each point-group is assumed to contain the totally symmetric representation, here A_1g. Operations are represented as 3x3 Euler matrices, thus C_2x, C_2y and C_2z would be represented as in the Figure All operations not given can be generated as products of operations already known, thus C_2x = C_2y C_2z.

Figure:
Representation of Symmetry Operations

C_2x:
1	0	0
0	-1	0
0	0	-1

C_2y:
-1	0	0
0	1	0
0	0	-1

C_2z:
-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., C_2v, C_2h, and D₂ to use the same character table.