Reduction formula for point group C6v
Characters for molecular motions
| Motion |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3 v |
3d |
| Cartesian 3N |
216 |
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 |
210 |
-4 |
0 |
2 |
0 |
8 |
Decomposition into Irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Cartesian 3N |
20 |
16 |
16 |
20 |
36 |
36 |
144 |
| Translation (x,y,z) |
1 |
0 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Vibration |
19 |
15 |
16 |
20 |
34 |
36 |
140 |
Molecule Parameter
| Number of Atoms (N) |
72 |
| Number of internal coordinates |
210 |
| Number of independant internal coordinates |
19 |
| Number of vibrational modes |
140 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Linear (IR) |
19 |
15 |
16 |
20 |
34 |
36 |
53 / 87 |
| Quadratic (Raman) |
19 |
15 |
16 |
20 |
34 |
36 |
89 / 51 |
| IR + Raman |
19 |
15 |
16 |
20 |
34 |
- |
53 / 51 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
2C6 (z) |
2C3 (z) |
C2 (z) |
3v |
3d |
| linear |
1 |
210 |
-4 |
0 |
2 |
0 |
8 |
| quadratic |
2 |
22.155 |
8 |
0 |
107 |
105 |
137 |
| cubic |
3 |
1.565.620 |
-10 |
70 |
212 |
0 |
928 |
| quartic |
4 |
83.369.265 |
8 |
0 |
5.777 |
5.565 |
9.117 |
| quintic |
5 |
3.568.204.542 |
-4 |
0 |
11.342 |
0 |
54.120 |
| sextic |
6 |
127.860.662.755 |
37 |
2.485 |
209.827 |
198.485 |
397.333 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
| linear |
1 |
19 |
15 |
16 |
20 |
34 |
36 |
| quadratic |
2 |
1.917 |
1.796 |
1.828 |
1.844 |
3.676 |
3.709 |
| cubic |
3 |
130.728 |
130.264 |
130.232 |
130.696 |
260.888 |
260.962 |
| quartic |
4 |
6.951.592 |
6.944.251 |
6.946.068 |
6.947.844 |
13.893.916 |
13.895.839 |
| quintic |
5 |
297.364.853 |
297.337.793 |
297.335.904 |
297.362.964 |
594.698.866 |
594.702.648 |
| sextic |
6 |
10.655.222.090 |
10.654.924.181 |
10.654.988.440 |
10.655.087.864 |
21.310.075.080 |
21.310.145.010 |
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