Reduction formula for point group C6
Characters for molecular motions
Motion |
E |
C6 |
C3 |
C2 |
(C3)2 |
(C6)5 |
Cartesian 3N |
18 |
12 |
0 |
-6 |
0 |
12 |
Translation (x,y,z) |
3 |
2 |
0 |
-1 |
0 |
2 |
Rotation (Rx,Ry)* |
2 |
1 |
-1 |
-2 |
-1 |
1 |
Vibration |
13 |
9 |
1 |
-3 |
1 |
9 |
* Linear molecule
Decomposition into Irreducible representations
Motion |
A |
B |
E1 |
E2 |
Total |
Cartesian 3N |
6 |
0 |
6 |
0 |
12 |
Translation (x,y,z) |
1 |
0 |
1 |
0 |
2 |
Rotation (Rx,Ry)* |
0 |
0 |
1 |
0 |
1 |
Vibration |
5 |
0 |
4 |
0 |
9 |
* Linear molecule
Molecule Parameter
Number of Atoms (N) |
6 |
Number of internal coordinates |
13 |
Number of independant internal coordinates |
5 |
Number of vibrational modes |
9 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A |
B |
E1 |
E2 |
Total |
Linear (IR) |
5 |
0 |
4 |
0 |
9 / 0 |
Quadratic (Raman) |
5 |
0 |
4 |
0 |
9 / 0 |
IR + Raman |
5 |
0 |
4 |
- |
9 / 0 |
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
C6 |
C3 |
C2 |
(C3)2 |
(C6)5 |
linear |
1 |
13 |
9 |
1 |
-3 |
1 |
9 |
quadratic |
2 |
91 |
41 |
1 |
11 |
1 |
41 |
cubic |
3 |
455 |
125 |
5 |
-25 |
5 |
125 |
quartic |
4 |
1.820 |
285 |
5 |
60 |
5 |
285 |
quintic |
5 |
6.188 |
517 |
5 |
-116 |
5 |
517 |
sextic |
6 |
18.564 |
783 |
15 |
228 |
15 |
783 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A |
B |
E1 |
E2 |
linear |
1 |
5 |
0 |
4 |
0 |
quadratic |
2 |
31 |
0 |
20 |
10 |
cubic |
3 |
115 |
40 |
100 |
50 |
quartic |
4 |
410 |
200 |
340 |
265 |
quintic |
5 |
1.186 |
880 |
1.136 |
925 |
sextic |
6 |
3.398 |
2.800 |
3.184 |
2.999 |
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