Reduction formula for point group D3h
Characters for molecular motions
| Motion |
E |
2C3 (z) |
3C'2 |
 h (xy) |
2S3 |
3v |
| Cartesian 3N |
36 |
0 |
0 |
12 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-2 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
-1 |
2 |
-1 |
| Vibration |
30 |
0 |
2 |
12 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
| Cartesian 3N |
4 |
4 |
8 |
2 |
2 |
4 |
24 |
| Translation (x,y,z) |
0 |
0 |
1 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
0 |
1 |
2 |
| Vibration |
4 |
3 |
7 |
2 |
1 |
3 |
20 |
Molecule Parameter
| Number of Atoms (N) |
12 |
| Number of internal coordinates |
30 |
| Number of independant internal coordinates |
4 |
| Number of vibrational modes |
20 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
| Linear (IR) |
4 |
3 |
7 |
2 |
1 |
3 |
8 / 12 |
| Quadratic (Raman) |
4 |
3 |
7 |
2 |
1 |
3 |
14 / 6 |
| IR + Raman |
- |
3 |
7 |
2 |
- |
- |
7 / 5 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C3 (z) |
3C'2 |
h (xy) |
2S3 |
3v |
| linear |
1 |
30 |
0 |
2 |
12 |
0 |
0 |
| quadratic |
2 |
465 |
0 |
17 |
87 |
0 |
15 |
| cubic |
3 |
4.960 |
10 |
32 |
472 |
4 |
0 |
| quartic |
4 |
40.920 |
0 |
152 |
2.112 |
0 |
120 |
| quintic |
5 |
278.256 |
0 |
272 |
8.184 |
0 |
0 |
| sextic |
6 |
1.623.160 |
55 |
952 |
28.336 |
13 |
680 |
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 |
4 |
3 |
7 |
2 |
1 |
3 |
| quadratic |
2 |
54 |
38 |
92 |
32 |
31 |
63 |
| cubic |
3 |
463 |
447 |
903 |
383 |
367 |
747 |
| quartic |
4 |
3.654 |
3.518 |
7.172 |
3.242 |
3.226 |
6.468 |
| quintic |
5 |
23.938 |
23.802 |
47.740 |
22.574 |
22.438 |
45.012 |
| sextic |
6 |
138.044 |
137.228 |
275.238 |
132.977 |
132.841 |
265.797 |
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