Reduction formula for point group D3h
Characters for molecular motions
| Motion |
E |
2C3 (z) |
3C'2 |
 h (xy) |
2S3 |
3v |
| Cartesian 3N |
27 |
0 |
-1 |
3 |
0 |
3 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-2 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
-1 |
2 |
-1 |
| Vibration |
21 |
0 |
1 |
3 |
0 |
3 |
Decomposition into Irreducible representations
| Motion |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
| Cartesian 3N |
3 |
2 |
5 |
1 |
3 |
4 |
18 |
| Translation (x,y,z) |
0 |
0 |
1 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
0 |
1 |
2 |
| Vibration |
3 |
1 |
4 |
1 |
2 |
3 |
14 |
Molecule Parameter
| Number of Atoms (N) |
9 |
| Number of internal coordinates |
21 |
| Number of independant internal coordinates |
3 |
| Number of vibrational modes |
14 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
| Linear (IR) |
3 |
1 |
4 |
1 |
2 |
3 |
6 / 8 |
| Quadratic (Raman) |
3 |
1 |
4 |
1 |
2 |
3 |
10 / 4 |
| IR + Raman |
- |
1 |
4 |
1 |
- |
- |
4 / 2 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C3 (z) |
3C'2 |
h (xy) |
2S3 |
3v |
| linear |
1 |
21 |
0 |
1 |
3 |
0 |
3 |
| quadratic |
2 |
231 |
0 |
11 |
15 |
0 |
15 |
| cubic |
3 |
1.771 |
7 |
11 |
37 |
1 |
37 |
| quartic |
4 |
10.626 |
0 |
66 |
114 |
0 |
114 |
| quintic |
5 |
53.130 |
0 |
66 |
246 |
0 |
246 |
| sextic |
6 |
230.230 |
28 |
286 |
598 |
4 |
598 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
| linear |
1 |
3 |
1 |
4 |
1 |
2 |
3 |
| quadratic |
2 |
27 |
14 |
41 |
17 |
19 |
36 |
| cubic |
3 |
164 |
140 |
300 |
139 |
152 |
288 |
| quartic |
4 |
940 |
850 |
1.790 |
864 |
888 |
1.752 |
| quintic |
5 |
4.526 |
4.370 |
8.896 |
4.362 |
4.452 |
8.814 |
| sextic |
6 |
19.462 |
19.020 |
38.466 |
19.062 |
19.218 |
38.268 |
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