Reduction formula for point group D3d
Characters for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3 d |
| Cartesian 3N |
216 |
0 |
0 |
0 |
0 |
8 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
| Vibration |
210 |
0 |
2 |
0 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
20 |
16 |
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 |
20 |
15 |
35 |
16 |
19 |
35 |
140 |
Molecule Parameter
| Number of Atoms (N) |
72 |
| Number of internal coordinates |
210 |
| Number of independant internal coordinates |
20 |
| Number of vibrational modes |
140 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
20 |
15 |
35 |
16 |
19 |
35 |
54 / 86 |
| Quadratic (Raman) |
20 |
15 |
35 |
16 |
19 |
35 |
55 / 85 |
| IR + Raman |
- |
15 |
- |
16 |
- |
- |
0* / 31 |
* 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 |
2 |
0 |
0 |
8 |
| quadratic |
2 |
22.155 |
0 |
107 |
105 |
0 |
137 |
| cubic |
3 |
1.565.620 |
70 |
212 |
0 |
0 |
928 |
| quartic |
4 |
83.369.265 |
0 |
5.777 |
5.565 |
0 |
9.117 |
| quintic |
5 |
3.568.204.542 |
0 |
11.342 |
0 |
0 |
54.120 |
| sextic |
6 |
127.860.662.755 |
2.485 |
209.827 |
198.485 |
35 |
397.333 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
1 |
20 |
15 |
35 |
16 |
19 |
35 |
| quadratic |
2 |
1.916 |
1.794 |
3.710 |
1.830 |
1.845 |
3.675 |
| cubic |
3 |
130.765 |
130.195 |
260.925 |
130.301 |
130.659 |
260.925 |
| quartic |
4 |
6.951.626 |
6.944.179 |
13.895.805 |
6.946.140 |
6.947.810 |
13.893.950 |
| quintic |
5 |
297.366.744 |
297.334.013 |
594.700.757 |
297.339.684 |
297.361.073 |
594.700.757 |
| sextic |
6 |
10.655.223.980 |
10.654.920.400 |
21.310.143.120 |
10.654.992.221 |
10.655.085.974 |
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