Reduction formula for point group D6h
Characters for molecular motions
| Motion |
E |
2C6 (z) |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
 h (xy) |
3d |
3v |
| Cartesian 3N |
36 |
0 |
0 |
0 |
-4 |
0 |
0 |
0 |
0 |
12 |
0 |
4 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
-3 |
-2 |
0 |
1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
3 |
2 |
0 |
-1 |
-1 |
-1 |
| Vibration |
30 |
-4 |
0 |
2 |
-2 |
2 |
0 |
0 |
0 |
12 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Cartesian 3N |
2 |
2 |
0 |
2 |
2 |
4 |
0 |
2 |
2 |
2 |
4 |
2 |
24 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| Vibration |
2 |
1 |
0 |
2 |
1 |
4 |
0 |
1 |
2 |
2 |
3 |
2 |
20 |
Molecule Parameter
| Number of Atoms (N) |
12 |
| Number of internal coordinates |
30 |
| Number of independant internal coordinates |
2 |
| Number of vibrational modes |
20 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Linear (IR) |
2 |
1 |
0 |
2 |
1 |
4 |
0 |
1 |
2 |
2 |
3 |
2 |
4 / 16 |
| Quadratic (Raman) |
2 |
1 |
0 |
2 |
1 |
4 |
0 |
1 |
2 |
2 |
3 |
2 |
7 / 13 |
| IR + Raman |
- |
1 |
0 |
2 |
- |
- |
0 |
- |
2 |
2 |
- |
2 |
0* / 9 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C6 (z) |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
h (xy) |
3d |
3v |
| linear |
1 |
30 |
-4 |
0 |
2 |
-2 |
2 |
0 |
0 |
0 |
12 |
0 |
4 |
| quadratic |
2 |
465 |
8 |
0 |
17 |
17 |
17 |
15 |
0 |
0 |
87 |
15 |
23 |
| cubic |
3 |
4.960 |
-10 |
10 |
32 |
-32 |
32 |
0 |
4 |
0 |
472 |
0 |
72 |
| quartic |
4 |
40.920 |
8 |
0 |
152 |
152 |
152 |
120 |
0 |
0 |
2.112 |
120 |
256 |
| quintic |
5 |
278.256 |
-4 |
0 |
272 |
-272 |
272 |
0 |
0 |
0 |
8.184 |
0 |
680 |
| sextic |
6 |
1.623.160 |
7 |
55 |
952 |
952 |
952 |
680 |
13 |
5 |
28.336 |
680 |
1.904 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
| linear |
1 |
2 |
1 |
0 |
2 |
1 |
4 |
0 |
1 |
2 |
2 |
3 |
2 |
| quadratic |
2 |
34 |
16 |
14 |
16 |
32 |
48 |
16 |
17 |
22 |
20 |
44 |
31 |
| cubic |
3 |
237 |
219 |
170 |
204 |
370 |
455 |
179 |
197 |
228 |
226 |
448 |
377 |
| quartic |
4 |
1.890 |
1.720 |
1.598 |
1.632 |
3.232 |
3.608 |
1.610 |
1.628 |
1.798 |
1.764 |
3.564 |
3.236 |
| quintic |
5 |
12.031 |
11.861 |
11.089 |
11.395 |
22.483 |
23.893 |
11.179 |
11.349 |
11.941 |
11.907 |
23.847 |
22.529 |
| sextic |
6 |
69.448 |
68.326 |
66.290 |
66.596 |
132.876 |
137.754 |
66.381 |
66.551 |
68.902 |
68.596 |
137.484 |
132.921 |
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