Reduction formula for point group C6v
Characters for molecular motions
| Motion |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3 v |
3d |
| Cartesian 3N |
36 |
0 |
0 |
0 |
0 |
4 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
| Vibration |
30 |
-4 |
0 |
2 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Cartesian 3N |
4 |
2 |
2 |
4 |
6 |
6 |
24 |
| Translation (x,y,z) |
1 |
0 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Vibration |
3 |
1 |
2 |
4 |
4 |
6 |
20 |
Molecule Parameter
| Number of Atoms (N) |
12 |
| Number of internal coordinates |
30 |
| Number of independant internal coordinates |
3 |
| Number of vibrational modes |
20 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Linear (IR) |
3 |
1 |
2 |
4 |
4 |
6 |
7 / 13 |
| Quadratic (Raman) |
3 |
1 |
2 |
4 |
4 |
6 |
13 / 7 |
| IR + Raman |
3 |
1 |
2 |
4 |
4 |
- |
7 / 7 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3v |
3d |
| linear |
1 |
30 |
-4 |
0 |
2 |
0 |
4 |
| quadratic |
2 |
465 |
8 |
0 |
17 |
15 |
23 |
| cubic |
3 |
4.960 |
-10 |
10 |
32 |
0 |
72 |
| quartic |
4 |
40.920 |
8 |
0 |
152 |
120 |
256 |
| quintic |
5 |
278.256 |
-4 |
0 |
272 |
0 |
680 |
| sextic |
6 |
1.623.160 |
7 |
55 |
952 |
680 |
1.904 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
| linear |
1 |
3 |
1 |
2 |
4 |
4 |
6 |
| quadratic |
2 |
51 |
32 |
34 |
38 |
76 |
79 |
| cubic |
3 |
434 |
398 |
396 |
432 |
818 |
832 |
| quartic |
4 |
3.518 |
3.330 |
3.362 |
3.430 |
6.796 |
6.844 |
| quintic |
5 |
23.380 |
23.040 |
22.996 |
23.336 |
46.330 |
46.422 |
| sextic |
6 |
135.999 |
134.707 |
134.886 |
135.498 |
270.360 |
270.675 |
Literature
- J.K.G. Watson, J. Mol. Spec. 41 229 (1972)
The Numbers of Structural Parameters and Potential Constants of Molecules
- X.F. Zhou, P. Pulay. J. Comp. Chem. 10 No. 7, 935-938 (1989)
Characters for Symmetric and Antisymmetric Higher Powers of Representations:
Application to the Number of Anharmonic Force Constants in Symmetrical Molecules
Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement