Reduction formula for point group D3d
Characters for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3 d |
| Cartesian 3N |
144 |
0 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
| Vibration |
138 |
0 |
2 |
0 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
12 |
12 |
24 |
12 |
12 |
24 |
96 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
12 |
11 |
23 |
12 |
11 |
23 |
92 |
Molecule Parameter
| Number of Atoms (N) |
48 |
| Number of internal coordinates |
138 |
| Number of independant internal coordinates |
12 |
| Number of vibrational modes |
92 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
12 |
11 |
23 |
12 |
11 |
23 |
34 / 58 |
| Quadratic (Raman) |
12 |
11 |
23 |
12 |
11 |
23 |
35 / 57 |
| IR + Raman |
- |
11 |
- |
12 |
- |
- |
0* / 23 |
* 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 |
138 |
0 |
2 |
0 |
0 |
0 |
| quadratic |
2 |
9.591 |
0 |
71 |
69 |
0 |
69 |
| cubic |
3 |
447.580 |
46 |
140 |
0 |
0 |
0 |
| quartic |
4 |
15.777.195 |
0 |
2.555 |
2.415 |
0 |
2.415 |
| quintic |
5 |
448.072.338 |
0 |
4.970 |
0 |
0 |
0 |
| sextic |
6 |
10.679.057.389 |
1.081 |
62.125 |
57.155 |
23 |
57.155 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
1 |
12 |
11 |
23 |
12 |
11 |
23 |
| quadratic |
2 |
840 |
770 |
1.610 |
794 |
793 |
1.587 |
| cubic |
3 |
37.341 |
37.271 |
74.589 |
37.341 |
37.271 |
74.589 |
| quartic |
4 |
1.316.210 |
1.313.725 |
2.629.935 |
1.314.600 |
1.314.530 |
2.629.130 |
| quintic |
5 |
37.340.604 |
37.338.119 |
74.678.723 |
37.340.604 |
37.338.119 |
74.678.723 |
| sextic |
6 |
889.956.216 |
889.896.576 |
1.779.852.240 |
889.918.105 |
889.915.620 |
1.779.833.196 |
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