Reduction formula for point group C6v
Characters for molecular motions
| Motion |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3 v |
3d |
| Cartesian 3N |
108 |
0 |
0 |
0 |
0 |
4 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
| Vibration |
102 |
-4 |
0 |
2 |
0 |
4 |
Decomposition into Irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Cartesian 3N |
10 |
8 |
8 |
10 |
18 |
18 |
72 |
| Translation (x,y,z) |
1 |
0 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Vibration |
9 |
7 |
8 |
10 |
16 |
18 |
68 |
Molecule Parameter
| Number of Atoms (N) |
36 |
| Number of internal coordinates |
102 |
| Number of independant internal coordinates |
9 |
| Number of vibrational modes |
68 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Linear (IR) |
9 |
7 |
8 |
10 |
16 |
18 |
25 / 43 |
| Quadratic (Raman) |
9 |
7 |
8 |
10 |
16 |
18 |
43 / 25 |
| IR + Raman |
9 |
7 |
8 |
10 |
16 |
- |
25 / 25 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3v |
3d |
| linear |
1 |
102 |
-4 |
0 |
2 |
0 |
4 |
| quadratic |
2 |
5.253 |
8 |
0 |
53 |
51 |
59 |
| cubic |
3 |
182.104 |
-10 |
34 |
104 |
0 |
216 |
| quartic |
4 |
4.780.230 |
8 |
0 |
1.430 |
1.326 |
1.750 |
| quintic |
5 |
101.340.876 |
-4 |
0 |
2.756 |
0 |
5.936 |
| sextic |
6 |
1.807.245.622 |
19 |
595 |
26.182 |
23.426 |
34.874 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
| linear |
1 |
9 |
7 |
8 |
10 |
16 |
18 |
| quadratic |
2 |
471 |
416 |
430 |
434 |
868 |
883 |
| cubic |
3 |
15.242 |
15.134 |
15.120 |
15.228 |
30.326 |
30.364 |
| quartic |
4 |
399.242 |
397.704 |
398.126 |
398.338 |
796.468 |
796.942 |
| quintic |
5 |
8.446.786 |
8.443.818 |
8.443.360 |
8.446.328 |
16.889.686 |
16.890.606 |
| sextic |
6 |
150.620.661 |
150.591.511 |
150.598.854 |
150.604.578 |
301.203.144 |
301.211.865 |
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