D_{4h} | E | 2C_{4} (z) | C_{2} | 2C'_{2} | 2C''_{2} | i | 2S_{4} | _{h} | 2_{v} | 2_{d} | rotations |
functions |
functions |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{1g} | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | - | x^{2}+y^{2}, z^{2} | - |
A_{2g} | +1 | +1 | +1 | -1 | -1 | +1 | +1 | +1 | -1 | -1 | R_{z} | - | - |
B_{1g} | +1 | -1 | +1 | +1 | -1 | +1 | -1 | +1 | +1 | -1 | - | x^{2}-y^{2} | - |
B_{2g} | +1 | -1 | +1 | -1 | +1 | +1 | -1 | +1 | -1 | +1 | - | xy | - |
E_{g} | +2 | 0 | -2 | 0 | 0 | +2 | 0 | -2 | 0 | 0 | (R_{x}, R_{y}) | (xz, yz) | - |
A_{1u} | +1 | +1 | +1 | +1 | +1 | -1 | -1 | -1 | -1 | -1 | - | - | - |
A_{2u} | +1 | +1 | +1 | -1 | -1 | -1 | -1 | -1 | +1 | +1 | z | - | z^{3}, z(x^{2}+y^{2}) |
B_{1u} | +1 | -1 | +1 | +1 | -1 | -1 | +1 | -1 | -1 | +1 | - | - | xyz |
B_{2u} | +1 | -1 | +1 | -1 | +1 | -1 | +1 | -1 | +1 | -1 | - | - | z(x^{2}-y^{2}) |
E_{u} | +2 | 0 | -2 | 0 | 0 | -2 | 0 | +2 | 0 | 0 | (x, y) | - | (xz^{2}, yz^{2}) (xy^{2}, x^{2}y), (x^{3}, y^{3}) |
Number of symmetry elements | h = 16 |
Number of irreducible representations | n = 10 |
Abelian group | no |
Number of subgroups | 25 |
Number of distinct subgroups | 13 |
Subgroups
(Number of different orientations) |
C_{s} (3) , C_{i} , C_{2} (3) , C_{4} , D_{2} (2) , D_{4} , C_{2v} (4) , C_{4v} , C_{2h} (3) , C_{4h} , D_{2h} (2) , D_{2d} (2) , S_{4} |
---|---|
Optical Isomerism (Chirality) | no |
Polar | no |
[Al_{4}]^{2-}-Ion | Xenontetrafluoride | Squaric Acid Dianion |
dipole (p) | A_{2u}+E_{u} |
---|---|
quadrupole (d) | A_{1g}+B_{1g}+B_{2g}+E_{g} |
octopole (f) | A_{2u}+B_{1u}+B_{2u}+2E_{u} |
hexadecapole (g) | 2A_{1g}+A_{2g}+B_{1g}+B_{2g}+2E_{g} |
32-pole (h) | A_{1u}+2A_{2u}+B_{1u}+B_{2u}+3E_{u} |
64-pole (i) | 2A_{1g}+A_{2g}+2B_{1g}+2B_{2g}+3E_{g} |
128-pole (j) | A_{1u}+2A_{2u}+2B_{1u}+2B_{2u}+4E_{u} |
256-pole(k) | 3A_{1g}+2A_{2g}+2B_{1g}+2B_{2g}+4E_{g} |
512-pole (l) | 2A_{1u}+3A_{2u}+2B_{1u}+2B_{2u}+5E_{u} |