Reduction formula for point group S6
Characters for molecular motions
| Motion |
E |
C3(z) |
(C3)2 |
i |
(S6)5 |
S6 |
| Cartesian 3N |
60 |
0 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
0 |
-3 |
0 |
0 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
3 |
0 |
0 |
| Vibration |
54 |
0 |
0 |
0 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Au |
Eu |
Total |
| Cartesian 3N |
10 |
10 |
10 |
10 |
40 |
| Translation (x,y,z) |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
1 |
0 |
0 |
2 |
| Vibration |
9 |
9 |
9 |
9 |
36 |
Molecule Parameter
| Number of Atoms (N) |
20 |
| Number of internal coordinates |
54 |
| Number of independant internal coordinates |
9 |
| Number of vibrational modes |
36 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Au |
Eu |
Total |
| Linear (IR) |
9 |
9 |
9 |
9 |
18 / 18 |
| Quadratic (Raman) |
9 |
9 |
9 |
9 |
18 / 18 |
| IR + Raman |
- |
- |
- |
- |
0* / 0 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
C3(z) |
(C3)2 |
i |
(S6)5 |
S6 |
| linear |
1 |
54 |
0 |
0 |
0 |
0 |
0 |
| quadratic |
2 |
1.485 |
0 |
0 |
27 |
0 |
0 |
| cubic |
3 |
27.720 |
18 |
18 |
0 |
0 |
0 |
| quartic |
4 |
395.010 |
0 |
0 |
378 |
0 |
0 |
| quintic |
5 |
4.582.116 |
0 |
0 |
0 |
0 |
0 |
| sextic |
6 |
45.057.474 |
171 |
171 |
3.654 |
9 |
9 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Au |
Eu |
| linear |
1 |
9 |
9 |
9 |
9 |
| quadratic |
2 |
252 |
252 |
243 |
243 |
| cubic |
3 |
4.626 |
4.617 |
4.626 |
4.617 |
| quartic |
4 |
65.898 |
65.898 |
65.772 |
65.772 |
| quintic |
5 |
763.686 |
763.686 |
763.686 |
763.686 |
| sextic |
6 |
7.510.248 |
7.510.158 |
7.509.024 |
7.508.943 |
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