Reduction formula for point group C6v
Characters for molecular motions
| Motion |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3 v |
3d |
| Cartesian 3N |
75 |
2 |
0 |
-1 |
1 |
9 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
| Vibration |
69 |
-2 |
0 |
1 |
1 |
9 |
Decomposition into Irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Cartesian 3N |
9 |
4 |
4 |
8 |
13 |
12 |
50 |
| Translation (x,y,z) |
1 |
0 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Vibration |
8 |
3 |
4 |
8 |
11 |
12 |
46 |
Molecule Parameter
| Number of Atoms (N) |
25 |
| Number of internal coordinates |
69 |
| Number of independant internal coordinates |
8 |
| Number of vibrational modes |
46 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Linear (IR) |
8 |
3 |
4 |
8 |
11 |
12 |
19 / 27 |
| Quadratic (Raman) |
8 |
3 |
4 |
8 |
11 |
12 |
31 / 15 |
| IR + Raman |
8 |
3 |
4 |
8 |
11 |
- |
19 / 15 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3v |
3d |
| linear |
1 |
69 |
-2 |
0 |
1 |
1 |
9 |
| quadratic |
2 |
2.415 |
2 |
0 |
35 |
35 |
75 |
| cubic |
3 |
57.155 |
-1 |
23 |
35 |
35 |
435 |
| quartic |
4 |
1.028.790 |
0 |
0 |
630 |
630 |
2.310 |
| quintic |
5 |
15.020.334 |
0 |
0 |
630 |
630 |
10.422 |
| sextic |
6 |
185.250.786 |
12 |
276 |
7.770 |
7.770 |
43.738 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
| linear |
1 |
8 |
3 |
4 |
8 |
11 |
12 |
| quadratic |
2 |
232 |
177 |
188 |
208 |
397 |
408 |
| cubic |
3 |
4.887 |
4.652 |
4.664 |
4.864 |
9.516 |
9.528 |
| quartic |
4 |
86.520 |
85.050 |
85.260 |
86.100 |
171.360 |
171.570 |
| quintic |
5 |
1.254.510 |
1.248.984 |
1.249.194 |
1.254.090 |
2.503.284 |
2.503.494 |
| sextic |
6 |
15.451.138 |
15.425.384 |
15.427.970 |
15.445.954 |
30.873.792 |
30.876.378 |
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