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 |
216 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
72 |
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 |
210 |
-4 |
0 |
2 |
0 |
-4 |
0 |
0 |
0 |
72 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Cartesian 3N |
24 |
12 |
12 |
24 |
12 |
24 |
24 |
12 |
144 |
| 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 |
23 |
12 |
11 |
24 |
11 |
24 |
23 |
12 |
140 |
Molecule Parameter
| Number of Atoms (N) |
72 |
| Number of internal coordinates |
210 |
| Number of independant internal coordinates |
23 |
| Number of vibrational modes |
140 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Linear (IR) |
23 |
12 |
11 |
24 |
11 |
24 |
23 |
12 |
34 / 106 |
| Quadratic (Raman) |
23 |
12 |
11 |
24 |
11 |
24 |
23 |
12 |
58 / 82 |
| IR + Raman |
- |
12 |
- |
- |
- |
24 |
- |
12 |
0* / 48 |
* 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 |
210 |
-4 |
0 |
2 |
0 |
-4 |
0 |
0 |
0 |
72 |
0 |
0 |
| quadratic |
2 |
22.155 |
8 |
0 |
107 |
0 |
8 |
105 |
0 |
0 |
2.697 |
0 |
0 |
| cubic |
3 |
1.565.620 |
-10 |
70 |
212 |
70 |
-10 |
0 |
24 |
0 |
69.792 |
0 |
24 |
| quartic |
4 |
83.369.265 |
8 |
0 |
5.777 |
0 |
8 |
5.565 |
0 |
0 |
1.399.197 |
0 |
0 |
| quintic |
5 |
3.568.204.542 |
-4 |
0 |
11.342 |
0 |
-4 |
0 |
0 |
0 |
23.121.576 |
0 |
0 |
| sextic |
6 |
127.860.662.755 |
37 |
2.485 |
209.827 |
2.485 |
37 |
198.485 |
323 |
35 |
327.363.605 |
35 |
323 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
| linear |
1 |
23 |
12 |
11 |
24 |
11 |
24 |
23 |
12 |
| quadratic |
2 |
2.090 |
1.620 |
1.622 |
2.088 |
1.623 |
2.052 |
2.054 |
1.621 |
| cubic |
3 |
136.316 |
124.644 |
124.630 |
136.295 |
124.676 |
136.284 |
136.258 |
124.667 |
| quartic |
4 |
7.064.985 |
6.830.820 |
6.830.822 |
7.064.983 |
6.830.858 |
7.063.092 |
7.063.094 |
6.830.856 |
| quintic |
5 |
299.278.121 |
295.422.636 |
295.422.635 |
299.278.122 |
295.424.525 |
299.276.232 |
299.276.231 |
295.424.526 |
| sextic |
6 |
10.682.370.036 |
10.627.774.344 |
10.627.773.804 |
10.682.369.316 |
10.627.776.235 |
10.682.301.960 |
10.682.301.276 |
10.627.775.694 |
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