Reduction formula for point group D3h
Characters for molecular motions
| Motion |
E |
2C3 (z) |
3C'2 |
 h (xy) |
2S3 |
3v |
| Cartesian 3N |
90 |
0 |
0 |
30 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-2 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
-1 |
2 |
-1 |
| Vibration |
84 |
0 |
2 |
30 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
| Cartesian 3N |
10 |
10 |
20 |
5 |
5 |
10 |
60 |
| Translation (x,y,z) |
0 |
0 |
1 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
0 |
1 |
2 |
| Vibration |
10 |
9 |
19 |
5 |
4 |
9 |
56 |
Molecule Parameter
| Number of Atoms (N) |
30 |
| Number of internal coordinates |
84 |
| Number of independant internal coordinates |
10 |
| Number of vibrational modes |
56 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
| Linear (IR) |
10 |
9 |
19 |
5 |
4 |
9 |
23 / 33 |
| Quadratic (Raman) |
10 |
9 |
19 |
5 |
4 |
9 |
38 / 18 |
| IR + Raman |
- |
9 |
19 |
5 |
- |
- |
19 / 14 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C3 (z) |
3C'2 |
h (xy) |
2S3 |
3v |
| linear |
1 |
84 |
0 |
2 |
30 |
0 |
0 |
| quadratic |
2 |
3.570 |
0 |
44 |
492 |
0 |
42 |
| cubic |
3 |
102.340 |
28 |
86 |
5.770 |
10 |
0 |
| quartic |
4 |
2.225.895 |
0 |
989 |
53.853 |
0 |
903 |
| quintic |
5 |
39.175.752 |
0 |
1.892 |
423.516 |
0 |
0 |
| sextic |
6 |
581.106.988 |
406 |
15.136 |
2.907.424 |
64 |
13.244 |
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 |
10 |
9 |
19 |
5 |
4 |
9 |
| quadratic |
2 |
360 |
317 |
677 |
257 |
256 |
513 |
| cubic |
3 |
9.037 |
8.994 |
18.012 |
8.072 |
8.029 |
16.092 |
| quartic |
4 |
190.452 |
189.506 |
379.958 |
181.025 |
180.982 |
362.007 |
| quintic |
5 |
3.300.412 |
3.299.466 |
6.599.878 |
3.229.826 |
3.228.880 |
6.458.706 |
| sextic |
6 |
48.675.041 |
48.660.851 |
97.335.657 |
48.183.827 |
48.182.881 |
96.366.537 |
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