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 |
75 |
2 |
0 |
-1 |
0 |
2 |
-3 |
-2 |
0 |
1 |
0 |
-2 |
| 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 |
69 |
-2 |
0 |
1 |
0 |
-2 |
-3 |
-2 |
0 |
1 |
0 |
-2 |
Decomposition into Irreducible representations
| Motion |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Cartesian 3N |
6 |
6 |
6 |
6 |
7 |
6 |
7 |
6 |
50 |
| 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 |
5 |
6 |
5 |
6 |
6 |
6 |
6 |
6 |
46 |
Molecule Parameter
| Number of Atoms (N) |
25 |
| Number of internal coordinates |
69 |
| Number of independant internal coordinates |
5 |
| Number of vibrational modes |
46 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Linear (IR) |
5 |
6 |
5 |
6 |
6 |
6 |
6 |
6 |
12 / 34 |
| Quadratic (Raman) |
5 |
6 |
5 |
6 |
6 |
6 |
6 |
6 |
16 / 30 |
| IR + Raman |
- |
6 |
- |
- |
- |
6 |
- |
6 |
0* / 18 |
* 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 |
69 |
-2 |
0 |
1 |
0 |
-2 |
-3 |
-2 |
0 |
1 |
0 |
-2 |
| quadratic |
2 |
2.415 |
2 |
0 |
35 |
0 |
2 |
39 |
2 |
0 |
35 |
0 |
2 |
| cubic |
3 |
57.155 |
-1 |
23 |
35 |
23 |
-1 |
-109 |
-1 |
-1 |
35 |
-1 |
-1 |
| quartic |
4 |
1.028.790 |
0 |
0 |
630 |
0 |
0 |
774 |
0 |
0 |
630 |
0 |
0 |
| quintic |
5 |
15.020.334 |
0 |
0 |
630 |
0 |
0 |
-2.034 |
0 |
0 |
630 |
0 |
0 |
| sextic |
6 |
185.250.786 |
12 |
276 |
7.770 |
276 |
12 |
10.434 |
12 |
12 |
7.770 |
12 |
12 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
| linear |
1 |
5 |
6 |
5 |
6 |
6 |
6 |
6 |
6 |
| quadratic |
2 |
211 |
198 |
199 |
210 |
198 |
198 |
198 |
198 |
| cubic |
3 |
4.763 |
4.752 |
4.746 |
4.758 |
4.776 |
4.776 |
4.770 |
4.770 |
| quartic |
4 |
85.902 |
85.692 |
85.692 |
85.902 |
85.668 |
85.668 |
85.668 |
85.668 |
| quintic |
5 |
1.251.630 |
1.251.420 |
1.251.420 |
1.251.630 |
1.251.864 |
1.251.864 |
1.251.864 |
1.251.864 |
| sextic |
6 |
15.439.782 |
15.437.184 |
15.437.118 |
15.439.704 |
15.436.740 |
15.436.740 |
15.436.674 |
15.436.674 |
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