Reduction formula for point group D3d
Characters for molecular motions
Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3d |
Cartesian 3N |
60 |
0 |
0 |
0 |
0 |
4 |
Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
Vibration |
54 |
0 |
2 |
0 |
0 |
4 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Cartesian 3N |
6 |
4 |
10 |
4 |
6 |
10 |
40 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
Vibration |
6 |
3 |
9 |
4 |
5 |
9 |
36 |
Molecule Parameter
Number of Atoms (N) |
20 |
Number of internal coordinates |
54 |
Number of independant internal coordinates |
6 |
Number of vibrational modes |
36 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Linear (IR) |
6 |
3 |
9 |
4 |
5 |
9 |
14 / 22 |
Quadratic (Raman) |
6 |
3 |
9 |
4 |
5 |
9 |
15 / 21 |
IR + Raman |
- |
3 |
- |
4 |
- |
- |
0* / 7 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
2C3 |
3C'2 |
i |
2S6 |
3d |
linear |
1 |
54 |
0 |
2 |
0 |
0 |
4 |
quadratic |
2 |
1.485 |
0 |
29 |
27 |
0 |
35 |
cubic |
3 |
27.720 |
18 |
56 |
0 |
0 |
120 |
quartic |
4 |
395.010 |
0 |
434 |
378 |
0 |
610 |
quintic |
5 |
4.582.116 |
0 |
812 |
0 |
0 |
1.856 |
sextic |
6 |
45.057.474 |
171 |
4.466 |
3.654 |
9 |
7.134 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
linear |
1 |
6 |
3 |
9 |
4 |
5 |
9 |
quadratic |
2 |
142 |
110 |
252 |
120 |
123 |
243 |
cubic |
3 |
2.357 |
2.269 |
4.617 |
2.297 |
2.329 |
4.617 |
quartic |
4 |
33.210 |
32.688 |
65.898 |
32.842 |
32.930 |
65.772 |
quintic |
5 |
382.510 |
381.176 |
763.686 |
381.582 |
382.104 |
763.686 |
sextic |
6 |
3.758.024 |
3.752.224 |
7.510.158 |
3.753.845 |
3.755.179 |
7.508.943 |
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