Reduction formula for point group C6v
Characters for molecular motions
| Motion |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3 v |
3d |
| Cartesian 3N |
144 |
0 |
0 |
0 |
0 |
8 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
| Vibration |
138 |
-4 |
0 |
2 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Cartesian 3N |
14 |
10 |
10 |
14 |
24 |
24 |
96 |
| Translation (x,y,z) |
1 |
0 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Vibration |
13 |
9 |
10 |
14 |
22 |
24 |
92 |
Molecule Parameter
| Number of Atoms (N) |
48 |
| Number of internal coordinates |
138 |
| Number of independant internal coordinates |
13 |
| Number of vibrational modes |
92 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Linear (IR) |
13 |
9 |
10 |
14 |
22 |
24 |
35 / 57 |
| Quadratic (Raman) |
13 |
9 |
10 |
14 |
22 |
24 |
59 / 33 |
| IR + Raman |
13 |
9 |
10 |
14 |
22 |
- |
35 / 33 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3v |
3d |
| linear |
1 |
138 |
-4 |
0 |
2 |
0 |
8 |
| quadratic |
2 |
9.591 |
8 |
0 |
71 |
69 |
101 |
| cubic |
3 |
447.580 |
-10 |
46 |
140 |
0 |
640 |
| quartic |
4 |
15.777.195 |
8 |
0 |
2.555 |
2.415 |
4.815 |
| quintic |
5 |
448.072.338 |
-4 |
0 |
4.970 |
0 |
25.752 |
| sextic |
6 |
10.679.057.389 |
25 |
1.081 |
62.125 |
57.155 |
148.291 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
| linear |
1 |
13 |
9 |
10 |
14 |
22 |
24 |
| quadratic |
2 |
849 |
764 |
784 |
800 |
1.588 |
1.609 |
| cubic |
3 |
37.476 |
37.156 |
37.136 |
37.456 |
74.564 |
74.614 |
| quartic |
4 |
1.316.788 |
1.313.173 |
1.313.952 |
1.315.152 |
2.629.108 |
2.629.957 |
| quintic |
5 |
37.346.213 |
37.333.337 |
37.332.510 |
37.345.386 |
74.677.894 |
74.679.552 |
| sextic |
6 |
889.978.172 |
889.875.449 |
889.893.664 |
889.939.232 |
1.779.832.368 |
1.779.853.068 |
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