Reduction formula for point group C6v
Characters for molecular motions
| Motion |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3 v |
3d |
| Cartesian 3N |
9 |
6 |
0 |
-3 |
3 |
3 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
1 |
1 |
| Rotation (Rx,Ry)* |
2 |
1 |
-1 |
-2 |
0 |
0 |
| Vibration |
4 |
3 |
1 |
0 |
2 |
2 |
* Linear molecule
Decomposition into Irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Cartesian 3N |
3 |
0 |
0 |
0 |
3 |
0 |
6 |
| Translation (x,y,z) |
1 |
0 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry)* |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
2 |
0 |
0 |
0 |
1 |
0 |
3 |
* Linear molecule
Molecule Parameter
| Number of Atoms (N) |
3 |
| Number of internal coordinates |
4 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
3 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Linear (IR) |
2 |
0 |
0 |
0 |
1 |
0 |
3 / 0 |
| Quadratic (Raman) |
2 |
0 |
0 |
0 |
1 |
0 |
3 / 0 |
| IR + Raman |
2 |
0 |
0 |
0 |
1 |
- |
3 / 0 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3v |
3d |
| linear |
1 |
4 |
3 |
1 |
0 |
2 |
2 |
| quadratic |
2 |
10 |
5 |
1 |
2 |
4 |
4 |
| cubic |
3 |
20 |
6 |
2 |
0 |
6 |
6 |
| quartic |
4 |
35 |
6 |
2 |
3 |
9 |
9 |
| quintic |
5 |
56 |
6 |
2 |
0 |
12 |
12 |
| sextic |
6 |
84 |
7 |
3 |
4 |
16 |
16 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
| linear |
1 |
2 |
0 |
0 |
0 |
1 |
0 |
| quadratic |
2 |
4 |
0 |
0 |
0 |
2 |
1 |
| cubic |
3 |
6 |
0 |
1 |
1 |
4 |
2 |
| quartic |
4 |
9 |
0 |
2 |
2 |
6 |
5 |
| quintic |
5 |
12 |
0 |
4 |
4 |
10 |
8 |
| sextic |
6 |
17 |
1 |
6 |
6 |
14 |
13 |
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