C_{4h} | E | C_{4}(z) | C_{2} | (C_{4})^{3} | i | (S_{4})^{3} | _{h} | S_{4} | rotations |
functions |
functions |
---|---|---|---|---|---|---|---|---|---|---|---|
A_{g} | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | R_{z} | x^{2}+y^{2}, z^{2} | - |
B_{g} | +1 | -1 | +1 | -1 | +1 | -1 | +1 | -1 | - | x^{2}-y^{2}, xy | - |
E_{g} | +1 +1 |
+i -i |
-1 -1 |
-i +i |
+1 +1 |
+i -i |
-1 -1 |
-i +i |
R_{x}+iR_{y} R_{x}-iR_{y} |
(xz, yz) | - |
A_{u} | +1 | +1 | +1 | +1 | -1 | -1 | -1 | -1 | z | - | z^{3}, z(x^{2}+y^{2}) |
B_{u} | +1 | -1 | +1 | -1 | -1 | +1 | -1 | +1 | - | - | xyz, z(x^{2}-y^{2}) |
E_{u} | +1 +1 |
+i -i |
-1 -1 |
-i +i |
-1 -1 |
-i +i |
+1 +1 |
+i -i |
x+iy x-iy |
- | (xz^{2}, yz^{2}) (xy^{2}, x^{2}y) (x^{3}, y^{3}) |
Number of symmetry elements | h = 8 |
Number of irreducible representations | n = 8 |
Number of real irreducible representations | n = 6 |
Abelian group | yes |
Number of subgroups | 6 |
Subgroups | C_{s} , C_{i} , C_{2} , C_{4} , C_{2h} , S_{4} |
---|---|
Optical Isomerism (Chirality) | no |
Polar | no |
dipole (p) | A_{u}+E_{u} |
---|---|
quadrupole (d) | A_{g}+2B_{g}+E_{g} |
octopole (f) | A_{u}+2B_{u}+2E_{u} |
hexadecapole (g) | 3A_{g}+2B_{g}+2E_{g} |
32-pole (h) | 3A_{u}+2B_{u}+3E_{u} |
64-pole (i) | 3A_{g}+4B_{g}+3E_{g} |
128-pole (j) | 3A_{u}+4B_{u}+4E_{u} |
256-pole(k) | 5A_{g}+4B_{g}+4E_{g} |
512-pole (l) | 5A_{u}+4B_{u}+5E_{u} |