Reduction formula for point group D3d
Characters for molecular motions
Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3d |
Cartesian 3N |
54 |
0 |
0 |
0 |
0 |
6 |
Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
Vibration |
48 |
0 |
2 |
0 |
0 |
6 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Cartesian 3N |
6 |
3 |
9 |
3 |
6 |
9 |
36 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
Vibration |
6 |
2 |
8 |
3 |
5 |
8 |
32 |
Molecule Parameter
Number of Atoms (N) |
18 |
Number of internal coordinates |
48 |
Number of independant internal coordinates |
6 |
Number of vibrational modes |
32 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Linear (IR) |
6 |
2 |
8 |
3 |
5 |
8 |
13 / 19 |
Quadratic (Raman) |
6 |
2 |
8 |
3 |
5 |
8 |
14 / 18 |
IR + Raman |
- |
2 |
- |
3 |
- |
- |
0* / 5 |
* 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 |
48 |
0 |
2 |
0 |
0 |
6 |
quadratic |
2 |
1.176 |
0 |
26 |
24 |
0 |
42 |
cubic |
3 |
19.600 |
16 |
50 |
0 |
0 |
182 |
quartic |
4 |
249.900 |
0 |
350 |
300 |
0 |
798 |
quintic |
5 |
2.598.960 |
0 |
650 |
0 |
0 |
2.814 |
sextic |
6 |
22.957.480 |
136 |
3.250 |
2.600 |
8 |
9.730 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
linear |
1 |
6 |
2 |
8 |
3 |
5 |
8 |
quadratic |
2 |
117 |
83 |
200 |
92 |
100 |
192 |
cubic |
3 |
1.694 |
1.578 |
3.264 |
1.603 |
1.669 |
3.264 |
quartic |
4 |
21.137 |
20.563 |
41.700 |
20.688 |
20.912 |
41.600 |
quintic |
5 |
217.446 |
215.714 |
433.160 |
216.039 |
217.121 |
433.160 |
sextic |
6 |
1.916.609 |
1.910.119 |
3.826.656 |
1.911.308 |
1.914.548 |
3.825.792 |
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