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 |
144 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
8 |
| 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 |
138 |
-4 |
0 |
2 |
2 |
2 |
0 |
0 |
0 |
0 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Cartesian 3N |
7 |
5 |
5 |
7 |
12 |
12 |
5 |
7 |
7 |
5 |
12 |
12 |
96 |
| 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 |
7 |
4 |
5 |
7 |
11 |
12 |
5 |
6 |
7 |
5 |
11 |
12 |
92 |
Molecule Parameter
| Number of Atoms (N) |
48 |
| Number of internal coordinates |
138 |
| Number of independant internal coordinates |
7 |
| Number of vibrational modes |
92 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Linear (IR) |
7 |
4 |
5 |
7 |
11 |
12 |
5 |
6 |
7 |
5 |
11 |
12 |
17 / 75 |
| Quadratic (Raman) |
7 |
4 |
5 |
7 |
11 |
12 |
5 |
6 |
7 |
5 |
11 |
12 |
30 / 62 |
| IR + Raman |
- |
4 |
5 |
7 |
- |
- |
5 |
- |
7 |
5 |
- |
12 |
0* / 45 |
* 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 |
138 |
-4 |
0 |
2 |
2 |
2 |
0 |
0 |
0 |
0 |
0 |
8 |
| quadratic |
2 |
9.591 |
8 |
0 |
71 |
71 |
71 |
69 |
0 |
0 |
69 |
69 |
101 |
| cubic |
3 |
447.580 |
-10 |
46 |
140 |
140 |
140 |
0 |
0 |
0 |
0 |
0 |
640 |
| quartic |
4 |
15.777.195 |
8 |
0 |
2.555 |
2.555 |
2.555 |
2.415 |
0 |
0 |
2.415 |
2.415 |
4.815 |
| quintic |
5 |
448.072.338 |
-4 |
0 |
4.970 |
4.970 |
4.970 |
0 |
0 |
0 |
0 |
0 |
25.752 |
| sextic |
6 |
10.679.057.389 |
25 |
1.081 |
62.125 |
62.125 |
62.125 |
57.155 |
23 |
23 |
57.155 |
57.155 |
148.291 |
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 |
7 |
4 |
5 |
7 |
11 |
12 |
5 |
6 |
7 |
5 |
11 |
12 |
| quadratic |
2 |
448 |
370 |
392 |
400 |
794 |
816 |
394 |
401 |
400 |
392 |
794 |
793 |
| cubic |
3 |
18.773 |
18.543 |
18.568 |
18.728 |
37.282 |
37.307 |
18.613 |
18.703 |
18.728 |
18.568 |
37.282 |
37.307 |
| quartic |
4 |
659.234 |
656.149 |
656.976 |
657.576 |
1.314.554 |
1.315.381 |
657.024 |
657.554 |
657.576 |
656.976 |
1.314.554 |
1.314.576 |
| quintic |
5 |
18.674.349 |
18.665.426 |
18.666.255 |
18.672.693 |
37.338.947 |
37.339.776 |
18.667.911 |
18.671.864 |
18.672.693 |
18.666.255 |
37.338.947 |
37.339.776 |
| sextic |
6 |
445.009.384 |
444.926.960 |
444.946.832 |
444.969.616 |
889.916.184 |
889.936.056 |
444.948.489 |
444.968.788 |
444.969.616 |
444.946.832 |
889.916.184 |
889.917.012 |
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