Reduction formula for point group D3d
Characters for molecular motions
Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3d |
Cartesian 3N |
108 |
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 |
102 |
0 |
2 |
0 |
0 |
4 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Cartesian 3N |
10 |
8 |
18 |
8 |
10 |
18 |
72 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
Vibration |
10 |
7 |
17 |
8 |
9 |
17 |
68 |
Molecule Parameter
Number of Atoms (N) |
36 |
Number of internal coordinates |
102 |
Number of independant internal coordinates |
10 |
Number of vibrational modes |
68 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Linear (IR) |
10 |
7 |
17 |
8 |
9 |
17 |
26 / 42 |
Quadratic (Raman) |
10 |
7 |
17 |
8 |
9 |
17 |
27 / 41 |
IR + Raman |
- |
7 |
- |
8 |
- |
- |
0* / 15 |
* 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 |
102 |
0 |
2 |
0 |
0 |
4 |
quadratic |
2 |
5.253 |
0 |
53 |
51 |
0 |
59 |
cubic |
3 |
182.104 |
34 |
104 |
0 |
0 |
216 |
quartic |
4 |
4.780.230 |
0 |
1.430 |
1.326 |
0 |
1.750 |
quintic |
5 |
101.340.876 |
0 |
2.756 |
0 |
0 |
5.936 |
sextic |
6 |
1.807.245.622 |
595 |
26.182 |
23.426 |
17 |
34.874 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
linear |
1 |
10 |
7 |
17 |
8 |
9 |
17 |
quadratic |
2 |
470 |
414 |
884 |
432 |
435 |
867 |
cubic |
3 |
15.261 |
15.101 |
30.345 |
15.153 |
15.209 |
30.345 |
quartic |
4 |
399.258 |
397.668 |
796.926 |
398.162 |
398.322 |
796.484 |
quintic |
5 |
8.447.246 |
8.442.900 |
16.890.146 |
8.444.278 |
8.445.868 |
16.890.146 |
sextic |
6 |
150.621.120 |
150.590.592 |
301.211.406 |
150.599.773 |
150.604.119 |
301.203.603 |
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