Reduction formula for point group C6h
Characters for molecular motions
| Motion |
E |
C6(z) |
C3 |
C2 |
(C3)2 |
(C6)5 |
i |
(S3)5 |
(S6)5 |
 h |
S6 |
S3 |
| Cartesian 3N |
144 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
0 |
2 |
-3 |
-2 |
0 |
1 |
0 |
-2 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
0 |
2 |
3 |
2 |
0 |
-1 |
0 |
2 |
| Vibration |
138 |
-4 |
0 |
2 |
0 |
-4 |
0 |
0 |
0 |
0 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Cartesian 3N |
12 |
12 |
12 |
12 |
12 |
12 |
12 |
12 |
96 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
2 |
| Vibration |
11 |
12 |
11 |
12 |
11 |
12 |
11 |
12 |
92 |
Molecule Parameter
| Number of Atoms (N) |
48 |
| Number of internal coordinates |
138 |
| Number of independant internal coordinates |
11 |
| Number of vibrational modes |
92 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Linear (IR) |
11 |
12 |
11 |
12 |
11 |
12 |
11 |
12 |
22 / 70 |
| Quadratic (Raman) |
11 |
12 |
11 |
12 |
11 |
12 |
11 |
12 |
34 / 58 |
| IR + Raman |
- |
12 |
- |
- |
- |
12 |
- |
12 |
0* / 36 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
C6(z) |
C3 |
C2 |
(C3)2 |
(C6)5 |
i |
(S3)5 |
(S6)5 |
h |
S6 |
S3 |
| linear |
1 |
138 |
-4 |
0 |
2 |
0 |
-4 |
0 |
0 |
0 |
0 |
0 |
0 |
| quadratic |
2 |
9.591 |
8 |
0 |
71 |
0 |
8 |
69 |
0 |
0 |
69 |
0 |
0 |
| cubic |
3 |
447.580 |
-10 |
46 |
140 |
46 |
-10 |
0 |
0 |
0 |
0 |
0 |
0 |
| quartic |
4 |
15.777.195 |
8 |
0 |
2.555 |
0 |
8 |
2.415 |
0 |
0 |
2.415 |
0 |
0 |
| quintic |
5 |
448.072.338 |
-4 |
0 |
4.970 |
0 |
-4 |
0 |
0 |
0 |
0 |
0 |
0 |
| sextic |
6 |
10.679.057.389 |
25 |
1.081 |
62.125 |
1.081 |
25 |
57.155 |
23 |
23 |
57.155 |
23 |
23 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
| linear |
1 |
11 |
12 |
11 |
12 |
11 |
12 |
11 |
12 |
| quadratic |
2 |
818 |
792 |
794 |
816 |
795 |
792 |
794 |
793 |
| cubic |
3 |
37.316 |
37.296 |
37.282 |
37.307 |
37.316 |
37.296 |
37.282 |
37.307 |
| quartic |
4 |
1.315.383 |
1.314.552 |
1.314.554 |
1.315.381 |
1.314.578 |
1.314.552 |
1.314.554 |
1.314.576 |
| quintic |
5 |
37.339.775 |
37.338.948 |
37.338.947 |
37.339.776 |
37.339.775 |
37.338.948 |
37.338.947 |
37.339.776 |
| sextic |
6 |
889.936.344 |
889.916.448 |
889.916.184 |
889.936.056 |
889.917.277 |
889.916.448 |
889.916.184 |
889.917.012 |
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