Reduction formula for point group Td
Characters for molecular motions
| Motion |
E |
8C3 |
3C2 |
6S4 |
6 d |
| Cartesian 3N |
24 |
0 |
0 |
0 |
4 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
18 |
0 |
2 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
2 |
0 |
2 |
2 |
4 |
10 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
2 |
0 |
2 |
1 |
3 |
8 |
Molecule Parameter
| Number of Atoms (N) |
8 |
| Number of internal coordinates |
18 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
8 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
2 |
0 |
2 |
1 |
3 |
3 / 5 |
| Quadratic (Raman) |
2 |
0 |
2 |
1 |
3 |
7 / 1 |
| IR + Raman |
- |
0 |
- |
1 |
3 |
3 / 1 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
8C3 |
3C2 |
6S4 |
6d |
| linear |
1 |
18 |
0 |
2 |
0 |
4 |
| quadratic |
2 |
171 |
0 |
11 |
1 |
17 |
| cubic |
3 |
1.140 |
6 |
20 |
0 |
48 |
| quartic |
4 |
5.985 |
0 |
65 |
5 |
133 |
| quintic |
5 |
26.334 |
0 |
110 |
0 |
308 |
| sextic |
6 |
100.947 |
21 |
275 |
5 |
693 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
1 |
2 |
0 |
2 |
1 |
3 |
| quadratic |
2 |
13 |
4 |
17 |
16 |
24 |
| cubic |
3 |
64 |
40 |
98 |
128 |
152 |
| quartic |
4 |
292 |
223 |
515 |
708 |
772 |
| quintic |
5 |
1.188 |
1.034 |
2.222 |
3.201 |
3.355 |
| sextic |
6 |
4.422 |
4.073 |
8.474 |
12.412 |
12.756 |
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