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 |
216 |
0 |
0 |
0 |
-8 |
0 |
0 |
0 |
0 |
72 |
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 |
210 |
-4 |
0 |
2 |
-6 |
2 |
0 |
0 |
0 |
72 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Cartesian 3N |
12 |
12 |
4 |
8 |
12 |
24 |
4 |
8 |
12 |
12 |
24 |
12 |
144 |
| 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 |
12 |
11 |
4 |
8 |
11 |
24 |
4 |
7 |
12 |
12 |
23 |
12 |
140 |
Molecule Parameter
| Number of Atoms (N) |
72 |
| Number of internal coordinates |
210 |
| Number of independant internal coordinates |
12 |
| Number of vibrational modes |
140 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Linear (IR) |
12 |
11 |
4 |
8 |
11 |
24 |
4 |
7 |
12 |
12 |
23 |
12 |
30 / 110 |
| Quadratic (Raman) |
12 |
11 |
4 |
8 |
11 |
24 |
4 |
7 |
12 |
12 |
23 |
12 |
47 / 93 |
| IR + Raman |
- |
11 |
4 |
8 |
- |
- |
4 |
- |
12 |
12 |
- |
12 |
0* / 63 |
* 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 |
210 |
-4 |
0 |
2 |
-6 |
2 |
0 |
0 |
0 |
72 |
0 |
8 |
| quadratic |
2 |
22.155 |
8 |
0 |
107 |
123 |
107 |
105 |
0 |
0 |
2.697 |
105 |
137 |
| cubic |
3 |
1.565.620 |
-10 |
70 |
212 |
-668 |
212 |
0 |
24 |
0 |
69.792 |
0 |
928 |
| quartic |
4 |
83.369.265 |
8 |
0 |
5.777 |
7.521 |
5.777 |
5.565 |
0 |
0 |
1.399.197 |
5.565 |
9.117 |
| quintic |
5 |
3.568.204.542 |
-4 |
0 |
11.342 |
-37.482 |
11.342 |
0 |
0 |
0 |
23.121.576 |
0 |
54.120 |
| sextic |
6 |
127.860.662.755 |
37 |
2.485 |
209.827 |
305.731 |
209.827 |
198.485 |
323 |
35 |
327.363.605 |
198.485 |
397.333 |
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 |
12 |
11 |
4 |
8 |
11 |
24 |
4 |
7 |
12 |
12 |
23 |
12 |
| quadratic |
2 |
1.104 |
986 |
808 |
812 |
1.622 |
2.088 |
810 |
813 |
1.032 |
1.020 |
2.054 |
1.621 |
| cubic |
3 |
68.217 |
68.099 |
62.096 |
62.548 |
124.630 |
136.295 |
62.165 |
62.511 |
68.148 |
68.136 |
136.258 |
124.667 |
| quartic |
4 |
3.535.990 |
3.528.995 |
3.415.184 |
3.415.636 |
6.830.822 |
7.064.983 |
3.415.256 |
3.415.602 |
3.532.208 |
3.530.884 |
7.063.094 |
6.830.856 |
| quintic |
5 |
149.642.558 |
149.635.563 |
147.698.450 |
147.724.186 |
295.422.635 |
299.278.122 |
147.702.230 |
147.722.295 |
149.638.778 |
149.637.454 |
299.276.231 |
295.424.526 |
| sextic |
6 |
5.341.323.940 |
5.341.046.096 |
5.313.874.304 |
5.313.900.040 |
10.627.773.804 |
10.682.369.316 |
5.313.878.085 |
5.313.898.150 |
5.341.187.824 |
5.341.114.136 |
10.682.301.276 |
10.627.775.694 |
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