hvrotations
functions
functions
/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)/7)| D7h | E | 2C7 | 2(C7)2 | 2(C7)3 | 7C'2 | h |
2S7 | 2(S7)5 | 2(S7)3 | 7v |
rotations |
functions |
functions |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A'1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | - | x2+y2, z2 | - |
| A'2 | +1 | +1 | +1 | +1 | -1 | +1 | +1 | +1 | +1 | -1 | Rz | - | - |
| E'1 | +2 | +2cos(2/7) |
+2cos(4/7) |
+2cos(6/7) |
0 | +2 | +2cos(2/7) |
+2cos(4/7) |
+2cos(6/7) |
0 | (x, y) | - | (xz2, yz2) [x(x2+y2), y(x2+y2)] |
| E'2 | +2 | +2cos(4/7) |
+2cos(6/7) |
+2cos(2/7) |
0 | +2 | +2cos(4/7) |
+2cos(6/7) |
+2cos(2/7) |
0 | - | (x2-y2, xy) | - |
| E'3 | +2 | +2cos(6/7) |
+2cos(2/7) |
+2cos(4/7) |
0 | +2 | +2cos(6/7) |
+2cos(2/7) |
+2cos(4/7) |
0 | - | - | [y(3x2-y2), x(x2-3y2)] |
| A''1 | +1 | +1 | +1 | +1 | +1 | -1 | -1 | -1 | -1 | -1 | - | - | - |
| A''2 | +1 | +1 | +1 | +1 | -1 | -1 | -1 | -1 | -1 | +1 | z | - | z3, z(x2+y2) |
| E''1 | +2 | +2cos(2/7) |
+2cos(4/7) |
+2cos(6/7) |
0 | -2 | -2cos(2/7) |
-2cos(4/7) |
-2cos(6/7) |
0 | (Rx, Ry) | (xz, yz) | - |
| E''2 | +2 | +2cos(4/7) |
+2cos(6/7) |
+2cos(2/7) |
0 | -2 | -2cos(4/7) |
-2cos(6/7) |
-2cos(2/7) |
0 | - | - | [xyz, z(x2-y2)] |
| E''3 | +2 | +2cos(6/7) |
+2cos(2/7) |
+2cos(4/7) |
0 | -2 | -2cos(6/7) |
-2cos(2/7) |
-2cos(4/7) |
0 | - | - | - |
| Number of symmetry elements | h = 28 |
| Number of irreducible representations | n = 10 |
| Abelian group | no |
| Number of subgroups | 8 |
| Number of distinct subgroups | 7 |
| Subgroups
(Number of different orientations) |
Cs (2) , C2 , C7 , D7 , C2v , C7v , C7h |
|---|---|
| Optical Isomerism (Chirality) | no |
| Polar | no |
| Tropylium Cation | Cucurbit[7]uril |
|---|
| dipole (p) | E'1+A''2 |
|---|---|
| quadrupole (d) | A'1+E'2+E''1 |
| octopole (f) | E'1+E'3+A''2+E''2 |
| hexadecapole (g) | A'1+E'2+E'3+E''1+E''3 |
| 32-pole (h) | E'1+E'2+E'3+A''2+E''2+E''3 |
| 64-pole (i) | A'1+E'1+E'2+E'3+E''1+E''2+E''3 |
| 128-pole (j) | A'1+A'2+E'1+E'2+E'3+A''2+E''1+E''2+E''3 |
| 256-pole(k) | A'1+2E'1+E'2+E'3+A''1+A''2+E''1+E''2+E''3 |
| 512-pole (l) | A'1+A'2+E'1+2E'2+E'3+A''2+2E''1+E''2+E''3 |
