Reduction formula for point group D3h
Characters for molecular motions
| Motion |
E |
2C3 (z) |
3C'2 |
 h (xy) |
2S3 |
3v |
| Cartesian 3N |
60 |
0 |
-2 |
6 |
0 |
8 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-2 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
-1 |
2 |
-1 |
| Vibration |
54 |
0 |
0 |
6 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
| Cartesian 3N |
7 |
4 |
11 |
2 |
7 |
9 |
40 |
| Translation (x,y,z) |
0 |
0 |
1 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
0 |
1 |
2 |
| Vibration |
7 |
3 |
10 |
2 |
6 |
8 |
36 |
Molecule Parameter
| Number of Atoms (N) |
20 |
| Number of internal coordinates |
54 |
| Number of independant internal coordinates |
7 |
| Number of vibrational modes |
36 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
| Linear (IR) |
7 |
3 |
10 |
2 |
6 |
8 |
16 / 20 |
| Quadratic (Raman) |
7 |
3 |
10 |
2 |
6 |
8 |
25 / 11 |
| IR + Raman |
- |
3 |
10 |
2 |
- |
- |
10 / 5 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C3 (z) |
3C'2 |
h (xy) |
2S3 |
3v |
| linear |
1 |
54 |
0 |
0 |
6 |
0 |
8 |
| quadratic |
2 |
1.485 |
0 |
27 |
45 |
0 |
59 |
| cubic |
3 |
27.720 |
18 |
0 |
200 |
2 |
304 |
| quartic |
4 |
395.010 |
0 |
378 |
930 |
0 |
1.434 |
| quintic |
5 |
4.582.116 |
0 |
0 |
3.396 |
0 |
5.760 |
| sextic |
6 |
45.057.474 |
171 |
3.654 |
12.386 |
11 |
21.542 |
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 |
7 |
3 |
10 |
2 |
6 |
8 |
| quadratic |
2 |
149 |
106 |
255 |
112 |
128 |
240 |
| cubic |
3 |
2.406 |
2.254 |
4.650 |
2.220 |
2.372 |
4.584 |
| quartic |
4 |
33.448 |
32.542 |
65.990 |
32.576 |
33.104 |
65.680 |
| quintic |
5 |
383.566 |
380.686 |
764.252 |
380.120 |
383.000 |
763.120 |
| sextic |
6 |
3.762.151 |
3.749.553 |
7.511.613 |
3.749.312 |
3.758.256 |
7.507.488 |
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