Character Tables for Common Symmetry Groups - The Dnd
Groups
| D2d | E | 2S4 | C2 | 2C2' | 2σd | ||
|---|---|---|---|---|---|---|---|
| A1 | 1 | 1 | 1 | 1 | 1 | x2 + y2, z2 | |
| A2 | 1 | 1 | 1 | -1 | -1 | Rz | |
| B1 | 1 | -1 | 1 | 1 | -1 | x2 - y2 | |
| B2 | 1 | -1 | 1 | -1 | 1 | z | xy |
| E | 2 | 0 | -2 | 0 | 0 | (x, y)(Rx, Ry) | (xy, yz) |
| D3d | E | 2C3 | 3C2 | i | 2S6 | 3σd | ||
|---|---|---|---|---|---|---|---|---|
| A1g | 1 | 1 | 1 | 1 | 1 | 1 | x2 + y2, z2 | |
| A2g | 1 | 1 | -1 | 1 | 1 | -1 | Rz | |
| Eg | 2 | -1 | 0 | 2 | -1 | 0 | (Rx, Ry) | (x2 - y2, xy)(xz, yz) |
| A1u | 1 | 1 | 1 | -1 | -1 | -1 | ||
| A2u | 1 | 1 | -1 | -1 | -1 | 1 | z | |
| Eu | 2 | -1 | 0 | -2 | 1 | 0 | (x, y) |
| D4d | E | 2S8 | 2C4 | 2S83 | C2 | 4C2' | 4σd | ||
|---|---|---|---|---|---|---|---|---|---|
| A1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | x2 + y2, z2 | |
| A2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | Rz | |
| B1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | ||
| B2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | z | |
| E1 | 2 | √2 | 0 | -√2 | -2 | 0 | 0 | (x, y) | |
| E2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | x2 - y2 | |
| E3 | 2 | -√2 | 0 | √2 | -2 | 0 | 0 | (Rx, Ry) | (xz, yz) |
Reference: F.A. Cotton, "Chemical Applications of Group Theory," 3rd ed., Wiley-Interscience, NY, 1990.