Reduction formula for point group D3d
Characters for molecular motions
Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3d |
Cartesian 3N |
216 |
0 |
-8 |
0 |
0 |
0 |
Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
Vibration |
210 |
0 |
-6 |
0 |
0 |
0 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Cartesian 3N |
16 |
20 |
36 |
16 |
20 |
36 |
144 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
Vibration |
16 |
19 |
35 |
16 |
19 |
35 |
140 |
Molecule Parameter
Number of Atoms (N) |
72 |
Number of internal coordinates |
210 |
Number of independant internal coordinates |
16 |
Number of vibrational modes |
140 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Linear (IR) |
16 |
19 |
35 |
16 |
19 |
35 |
54 / 86 |
Quadratic (Raman) |
16 |
19 |
35 |
16 |
19 |
35 |
51 / 89 |
IR + Raman |
- |
19 |
- |
16 |
- |
- |
0* / 35 |
* 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 |
210 |
0 |
-6 |
0 |
0 |
0 |
quadratic |
2 |
22.155 |
0 |
123 |
105 |
0 |
105 |
cubic |
3 |
1.565.620 |
70 |
-668 |
0 |
0 |
0 |
quartic |
4 |
83.369.265 |
0 |
7.521 |
5.565 |
0 |
5.565 |
quintic |
5 |
3.568.204.542 |
0 |
-37.482 |
0 |
0 |
0 |
sextic |
6 |
127.860.662.755 |
2.485 |
305.731 |
198.485 |
35 |
198.485 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
linear |
1 |
16 |
19 |
35 |
16 |
19 |
35 |
quadratic |
2 |
1.912 |
1.798 |
3.710 |
1.842 |
1.833 |
3.675 |
cubic |
3 |
130.313 |
130.647 |
260.925 |
130.313 |
130.647 |
260.925 |
quartic |
4 |
6.951.174 |
6.944.631 |
13.895.805 |
6.947.464 |
6.946.486 |
13.893.950 |
quintic |
5 |
297.341.008 |
297.359.749 |
594.700.757 |
297.341.008 |
297.359.749 |
594.700.757 |
sextic |
6 |
10.655.198.244 |
10.654.946.136 |
21.310.143.120 |
10.655.065.909 |
10.655.012.286 |
21.310.076.970 |
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