Reduction formula for point group Td
Characters for molecular motions
Motion |
E |
8C3 |
3C2 |
6S4 |
6d |
Cartesian 3N |
15 |
0 |
-1 |
-1 |
3 |
Translation (x,y,z) |
3 |
0 |
-1 |
-1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
Vibration |
9 |
0 |
1 |
-1 |
3 |
Decomposition into Irreducible representations
Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
Cartesian 3N |
1 |
0 |
1 |
1 |
3 |
6 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
Vibration |
1 |
0 |
1 |
0 |
2 |
4 |
Molecule Parameter
Number of Atoms (N) |
5 |
Number of internal coordinates |
9 |
Number of independant internal coordinates |
1 |
Number of vibrational modes |
4 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
Linear (IR) |
1 |
0 |
1 |
0 |
2 |
2 / 2 |
Quadratic (Raman) |
1 |
0 |
1 |
0 |
2 |
4 / 0 |
IR + Raman |
- |
0 |
- |
0 |
2 |
2 / 0 |
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
8C3 |
3C2 |
6S4 |
6d |
linear |
1 |
9 |
0 |
1 |
-1 |
3 |
quadratic |
2 |
45 |
0 |
5 |
1 |
9 |
cubic |
3 |
165 |
3 |
5 |
-1 |
19 |
quartic |
4 |
495 |
0 |
15 |
3 |
39 |
quintic |
5 |
1.287 |
0 |
15 |
-3 |
69 |
sextic |
6 |
3.003 |
6 |
35 |
3 |
119 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1 |
A2 |
E |
T1 |
T2 |
linear |
1 |
1 |
0 |
1 |
0 |
2 |
quadratic |
2 |
5 |
0 |
5 |
3 |
7 |
cubic |
3 |
13 |
4 |
14 |
15 |
25 |
quartic |
4 |
33 |
12 |
45 |
51 |
69 |
quintic |
5 |
72 |
39 |
111 |
141 |
177 |
sextic |
6 |
162 |
101 |
257 |
342 |
400 |
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