Character Tables for Common Symmetry Groups - The Groups C∞v
and D∞h for Linear Molecules
| C∞v | E | 2C∞Φ | ... | ∞σv | ||
|---|---|---|---|---|---|---|
| A1/Σ+ | 1 | 1 | ... | 1 | z | x2 + y2, z2 |
| A2/Σ- | 1 | 1 | ... | -1 | Rz | |
| E1/Π | 2 | 2cosΦ | ... | 0 | (x, y)(Rx, Ry) | (xz, yz) |
| E2/Δ | 2 | 2cos2Φ | ... | 0 | (x2 - y2, xy) | |
| E3/Φ | 2 | 2cos3Φ | ... | 0 | ||
| ... | ... | ... | ... | ... |
| D∞h | E | 2C∞Φ | ... | ∞σv | i | 2S∞Φ | ... | ∞C2 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| Σg+ | 1 | 1 | ... | 1 | 1 | 1 | ... | 1 | x2 + y2, z2 | |
| Σg- | 1 | 1 | ... | -1 | 1 | 1 | ... | -1 | Rz | |
| Πg | 2 | 2cosΦ | ... | 0 | 2 | -2cosΦ | ... | 0 | (Rx, Ry) | (xz, yz) |
| Δg | 2 | 2cos2Φ | ... | 0 | 2 | 2cosΦ | ... | 0 | (x2 - y2, xy) | |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ||
| Σu+ | 1 | 1 | ... | 1 | -1 | -1 | ... | -1 | z | |
| Σu- | 1 | 1 | ... | -1 | -1 | -1 | ... | 1 | ||
| Πu | 2 | 2cosΦ | ... | 0 | -2 | 2cosΦ | ... | 0 | (x, y) | |
| Δu | 2 | 2cos2Φ | ... | 0 | -2 | -2cosΦ | ... | 0 | ||
| ... | ... | ... | ... | ... | ... | ... | ... | ... |
Reference: F.A. Cotton, "Chemical Applications of Group Theory," 3rd ed., Wiley-Interscience, NY, 1990.