C_{5h} | E | C_{5} | (C_{5})^{2} | (C_{5})^{3} | (C_{5})^{4} | _{h} | S_{5} | (S_{5})^{7} | (S_{5})^{3} | (S_{5})^{9} | rotations |
functions |
functions |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A' | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | R_{z} | x^{2}+y^{2}, z^{2} | - |
E'_{1} | +1 +1 |
+ +^{*} |
+^{2} +^{2*} |
+^{2*} +^{2} |
+^{*} + |
+1 +1 |
+ +^{*} | +^{2} +^{2*} | +^{2*} +^{2} | +^{*} + | x+iy x-iy |
- | (xz^{2}, yz^{2}) [x(x^{2}+y^{2}), y(x^{2}+y^{2})] |
E'_{2} | +1 +1 |
+^{2} +^{2*} |
+^{*} + |
+ +^{*} |
+^{2*} +^{2} |
+1 +1 |
+^{2} +^{2*} | +^{*} + | + +^{*} | +^{2*} +^{2} | - | (x^{2}-y^{2}, xy) | [y(3x^{2}-y^{2}), x(x^{2}-3y^{2})] |
A'' | +1 | +1 | +1 | +1 | +1 | -1 | -1 | -1 | -1 | -1 | z | - | z^{3}, z(x^{2}+y^{2}) |
E''_{1} | +1 +1 |
+ +^{*} |
+^{2} +^{2*} |
+^{2*} +^{2} |
+^{*} + |
-1 -1 |
- -^{*} | -^{2} -^{2*} | -^{2*} -^{2} | -^{*} - | R_{x}+iR_{y} R_{x}-iR_{y} |
(xz, yz) | - |
E''_{2} | +1 +1 |
+^{2} +^{2*} |
+^{*} + |
+ +^{*} |
+^{2*} +^{2} |
-1 -1 |
-^{2} -^{2*} | -^{*} - | - -^{*} | -^{2*} -^{2} | - | - | [xyz, z(x^{2}-y^{2})] |
Number of symmetry elements | h = 10 |
Number of irreducible representations | n = 10 |
Number of real irreducible representations | n = 6 |
Abelian group | yes |
Subgroups | C_{s} , C_{5} |
---|---|
Optical Isomerism (Chirality) | no |
dipole (p) | E'_{1}+A'' |
---|---|
quadrupole (d) | A'+E'_{2}+E''_{1} |
octopole (f) | E'_{1}+E'_{2}+A''+E''_{2} |
hexadecapole (g) | A'+E'_{1}+E'_{2}+E''_{1}+E''_{2} |
32-pole (h) | 2A'+E'_{1}+E'_{2}+A''+E''_{1}+E''_{2} |
64-pole (i) | A'+2E'_{1}+E'_{2}+2A''+E''_{1}+E''_{2} |
128-pole (j) | 2A'+E'_{1}+2E'_{2}+A''+2E''_{1}+E''_{2} |
256-pole(k) | A'+2E'_{1}+2E'_{2}+2A''+E''_{1}+2E''_{2} |
512-pole (l) | 2A'+2E'_{1}+2E'_{2}+A''+2E''_{1}+2E''_{2} |