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 |
108 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
36 |
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 |
102 |
-4 |
0 |
2 |
0 |
-4 |
0 |
0 |
0 |
36 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Cartesian 3N |
12 |
6 |
6 |
12 |
6 |
12 |
12 |
6 |
72 |
| 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 |
6 |
5 |
12 |
5 |
12 |
11 |
6 |
68 |
Molecule Parameter
| Number of Atoms (N) |
36 |
| Number of internal coordinates |
102 |
| Number of independant internal coordinates |
11 |
| Number of vibrational modes |
68 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Linear (IR) |
11 |
6 |
5 |
12 |
5 |
12 |
11 |
6 |
16 / 52 |
| Quadratic (Raman) |
11 |
6 |
5 |
12 |
5 |
12 |
11 |
6 |
28 / 40 |
| IR + Raman |
- |
6 |
- |
- |
- |
12 |
- |
6 |
0* / 24 |
* 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 |
102 |
-4 |
0 |
2 |
0 |
-4 |
0 |
0 |
0 |
36 |
0 |
0 |
| quadratic |
2 |
5.253 |
8 |
0 |
53 |
0 |
8 |
51 |
0 |
0 |
699 |
0 |
0 |
| cubic |
3 |
182.104 |
-10 |
34 |
104 |
34 |
-10 |
0 |
12 |
0 |
9.624 |
0 |
12 |
| quartic |
4 |
4.780.230 |
8 |
0 |
1.430 |
0 |
8 |
1.326 |
0 |
0 |
104.790 |
0 |
0 |
| quintic |
5 |
101.340.876 |
-4 |
0 |
2.756 |
0 |
-4 |
0 |
0 |
0 |
956.592 |
0 |
0 |
| sextic |
6 |
1.807.245.622 |
19 |
595 |
26.182 |
595 |
19 |
23.426 |
89 |
17 |
7.590.842 |
17 |
89 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
| linear |
1 |
11 |
6 |
5 |
12 |
5 |
12 |
11 |
6 |
| quadratic |
2 |
506 |
378 |
380 |
504 |
381 |
486 |
488 |
379 |
| cubic |
3 |
15.992 |
14.370 |
14.362 |
15.983 |
14.384 |
15.978 |
15.964 |
14.381 |
| quartic |
4 |
407.316 |
389.610 |
389.612 |
407.314 |
389.630 |
406.854 |
406.856 |
389.628 |
| quintic |
5 |
8.525.018 |
8.365.128 |
8.365.127 |
8.525.019 |
8.365.586 |
8.524.560 |
8.524.559 |
8.365.587 |
| sextic |
6 |
151.240.626 |
149.971.086 |
149.970.960 |
151.240.446 |
149.971.546 |
151.232.346 |
151.232.184 |
149.971.419 |
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