Reduction formula for point group C3h
Characters for molecular motions
| Motion |
E |
C3(z) |
(C3)2 |
 h |
S3 |
(S3)5 |
| Cartesian 3N |
21 |
0 |
0 |
7 |
-2 |
-2 |
| Translation (x,y,z) |
3 |
0 |
0 |
1 |
-2 |
-2 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
-1 |
2 |
2 |
| Vibration |
15 |
0 |
0 |
7 |
-2 |
-2 |
Decomposition into Irreducible representations
| Motion |
A' |
E' |
A'' |
E'' |
Total |
| Cartesian 3N |
4 |
5 |
3 |
2 |
14 |
| Translation (x,y,z) |
0 |
1 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
0 |
1 |
2 |
| Vibration |
3 |
4 |
2 |
1 |
10 |
Molecule Parameter
| Number of Atoms (N) |
7 |
| Number of internal coordinates |
15 |
| Number of independant internal coordinates |
3 |
| Number of vibrational modes |
10 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A' |
E' |
A'' |
E'' |
Total |
| Linear (IR) |
3 |
4 |
2 |
1 |
6 / 4 |
| Quadratic (Raman) |
3 |
4 |
2 |
1 |
8 / 2 |
| IR + Raman |
- |
4 |
- |
- |
4 / 0 |
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
C3(z) |
(C3)2 |
h |
S3 |
(S3)5 |
| linear |
1 |
15 |
0 |
0 |
7 |
-2 |
-2 |
| quadratic |
2 |
120 |
0 |
0 |
32 |
2 |
2 |
| cubic |
3 |
680 |
5 |
5 |
112 |
1 |
1 |
| quartic |
4 |
3.060 |
0 |
0 |
332 |
-4 |
-4 |
| quintic |
5 |
11.628 |
0 |
0 |
868 |
4 |
4 |
| sextic |
6 |
38.760 |
15 |
15 |
2.064 |
3 |
3 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
A' |
E' |
A'' |
E'' |
| linear |
1 |
3 |
4 |
2 |
1 |
| quadratic |
2 |
26 |
25 |
14 |
15 |
| cubic |
3 |
134 |
131 |
96 |
94 |
| quartic |
4 |
564 |
566 |
456 |
454 |
| quintic |
5 |
2.084 |
2.082 |
1.792 |
1.794 |
| sextic |
6 |
6.810 |
6.801 |
6.120 |
6.114 |
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