+Table of Contents
This is an old revision of the document!
Orientation XYZ
Symmetry Operations
In the Oh Point Group, with orientation XYZ there are the following symmetry operations
Operator | Orientation |
---|---|
E | {0,0,0} , |
C3 | {1,1,1} , {1,1,−1} , {1,−1,1} , {−1,1,1} , {−1,−1,1} , {−1,1,−1} , {1,−1,−1} , {−1,−1,−1} , |
C2 | {1,1,0} , {1,−1,0} , {1,0,−1} , {1,0,1} , {0,1,1} , {0,1,−1} , |
C4 | {0,0,1} , {0,1,0} , {1,0,0} , {0,0,−1} , {0,−1,0} , {−1,0,0} , |
C2 | {0,0,1} , {0,1,0} , {1,0,0} , |
i | {0,0,0} , |
S4 | {0,0,1} , {0,1,0} , {1,0,0} , {0,0,−1} , {0,−1,0} , {−1,0,0} , |
S6 | {1,1,1} , {1,1,−1} , {1,−1,1} , {−1,1,1} , {−1,−1,1} , {−1,1,−1} , {1,−1,−1} , {−1,−1,−1} , |
σh | {1,0,0} , {0,1,0} , {0,0,1} , |
σd | {1,1,0} , {1,−1,0} , {1,0,−1} , {1,0,1} , {0,1,1} , {0,1,−1} , |
Character Table
E(1) | C3(8) | C2(6) | C4(6) | C2(3) | i(1) | S4(6) | S6(8) | σh(3) | σd(6) | |
---|---|---|---|---|---|---|---|---|---|---|
A1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
A2g | 1 | 1 | −1 | −1 | 1 | 1 | −1 | 1 | 1 | −1 |
Eg | 2 | −1 | 0 | 0 | 2 | 2 | 0 | −1 | 2 | 0 |
T1g | 3 | 0 | −1 | 1 | −1 | 3 | 1 | 0 | −1 | −1 |
T2g | 3 | 0 | 1 | −1 | −1 | 3 | −1 | 0 | −1 | 1 |
A1u | 1 | 1 | 1 | 1 | 1 | −1 | −1 | −1 | −1 | −1 |
A2u | 1 | 1 | −1 | −1 | 1 | −1 | 1 | −1 | −1 | 1 |
Eu | 2 | −1 | 0 | 0 | 2 | −2 | 0 | 1 | −2 | 0 |
T1u | 3 | 0 | −1 | 1 | −1 | −3 | −1 | 0 | 1 | 1 |
T2u | 3 | 0 | 1 | −1 | −1 | −3 | 1 | 0 | 1 | −1 |
Product Table
A1g | A2g | Eg | T1g | T2g | A1u | A2u | Eu | T1u | T2u | |
---|---|---|---|---|---|---|---|---|---|---|
A1g | A1g | A2g | Eg | T1g | T2g | A1u | A2u | Eu | T1u | T2u |
A2g | A2g | A1g | Eg | T2g | T1g | A2u | A1u | Eu | T2u | T1u |
Eg | Eg | Eg | A1g+A2g+Eg | T1g+T2g | T1g+T2g | Eu | Eu | A1u+A2u+Eu | T1u+T2u | T1u+T2u |
T1g | T1g | T2g | T1g+T2g | A1g+Eg+T1g+T2g | A2g+Eg+T1g+T2g | T1u | T2u | T1u+T2u | A1u+Eu+T1u+T2u | A2u+Eu+T1u+T2u |
T2g | T2g | T1g | T1g+T2g | A2g+Eg+T1g+T2g | A1g+Eg+T1g+T2g | T2u | T1u | T1u+T2u | A2u+Eu+T1u+T2u | A1u+Eu+T1u+T2u |
A1u | A1u | A2u | Eu | T1u | T2u | A1g | A2g | Eg | T1g | T2g |
A2u | A2u | A1u | Eu | T2u | T1u | A2g | A1g | Eg | T2g | T1g |
Eu | Eu | Eu | A1u+A2u+Eu | T1u+T2u | T1u+T2u | Eg | Eg | A1g+A2g+Eg | T1g+T2g | T1g+T2g |
T1u | T1u | T2u | T1u+T2u | A1u+Eu+T1u+T2u | A2u+Eu+T1u+T2u | T1g | T2g | T1g+T2g | A1g+Eg+T1g+T2g | A2g+Eg+T1g+T2g |
T2u | T2u | T1u | T1u+T2u | A2u+Eu+T1u+T2u | A1u+Eu+T1u+T2u | T2g | T1g | T1g+T2g | A2g+Eg+T1g+T2g | A1g+Eg+T1g+T2g |