Reduction formula for point group S6
Characters for molecular motions
| Motion |
E |
C3(z) |
(C3)2 |
i |
(S6)5 |
S6 |
| Cartesian 3N |
120 |
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 |
114 |
0 |
0 |
0 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Eg |
Au |
Eu |
Total |
| Cartesian 3N |
20 |
20 |
20 |
20 |
80 |
| Translation (x,y,z) |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
1 |
0 |
0 |
2 |
| Vibration |
19 |
19 |
19 |
19 |
76 |
Molecule Parameter
| Number of Atoms (N) |
40 |
| Number of internal coordinates |
114 |
| Number of independant internal coordinates |
19 |
| Number of vibrational modes |
76 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg |
Au |
Eu |
Total |
| Linear (IR) |
19 |
19 |
19 |
19 |
38 / 38 |
| Quadratic (Raman) |
19 |
19 |
19 |
19 |
38 / 38 |
| 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 |
114 |
0 |
0 |
0 |
0 |
0 |
| quadratic |
2 |
6.555 |
0 |
0 |
57 |
0 |
0 |
| cubic |
3 |
253.460 |
38 |
38 |
0 |
0 |
0 |
| quartic |
4 |
7.413.705 |
0 |
0 |
1.653 |
0 |
0 |
| quintic |
5 |
174.963.438 |
0 |
0 |
0 |
0 |
0 |
| sextic |
6 |
3.470.108.187 |
741 |
741 |
32.509 |
19 |
19 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Eg |
Au |
Eu |
| linear |
1 |
19 |
19 |
19 |
19 |
| quadratic |
2 |
1.102 |
1.102 |
1.083 |
1.083 |
| cubic |
3 |
42.256 |
42.237 |
42.256 |
42.237 |
| quartic |
4 |
1.235.893 |
1.235.893 |
1.235.342 |
1.235.342 |
| quintic |
5 |
29.160.573 |
29.160.573 |
29.160.573 |
29.160.573 |
| sextic |
6 |
578.357.036 |
578.356.656 |
578.346.187 |
578.345.826 |
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