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 $times$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.