Reduction formula for point group D3h
Characters for molecular motions
Motion |
E |
2C3 (z) |
3C'2 |
h (xy) |
2S3 |
3v |
Cartesian 3N |
18 |
0 |
-2 |
4 |
-2 |
4 |
Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-2 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
-1 |
2 |
-1 |
Vibration |
12 |
0 |
0 |
4 |
-2 |
4 |
Decomposition into Irreducible representations
Motion |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
Cartesian 3N |
2 |
1 |
4 |
0 |
3 |
2 |
12 |
Translation (x,y,z) |
0 |
0 |
1 |
0 |
1 |
0 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
0 |
1 |
2 |
Vibration |
2 |
0 |
3 |
0 |
2 |
1 |
8 |
Molecule Parameter
Number of Atoms (N) |
6 |
Number of internal coordinates |
12 |
Number of independant internal coordinates |
2 |
Number of vibrational modes |
8 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
Total |
Linear (IR) |
2 |
0 |
3 |
0 |
2 |
1 |
5 / 3 |
Quadratic (Raman) |
2 |
0 |
3 |
0 |
2 |
1 |
6 / 2 |
IR + Raman |
- |
0 |
3 |
0 |
- |
- |
3 / 0 |
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
2C3 (z) |
3C'2 |
h (xy) |
2S3 |
3v |
linear |
1 |
12 |
0 |
0 |
4 |
-2 |
4 |
quadratic |
2 |
78 |
0 |
6 |
14 |
2 |
14 |
cubic |
3 |
364 |
4 |
0 |
36 |
0 |
36 |
quartic |
4 |
1.365 |
0 |
21 |
85 |
-2 |
85 |
quintic |
5 |
4.368 |
0 |
0 |
176 |
2 |
176 |
sextic |
6 |
12.376 |
10 |
56 |
344 |
2 |
344 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A'1 |
A'2 |
E' |
A''1 |
A''2 |
E'' |
linear |
1 |
2 |
0 |
3 |
0 |
2 |
1 |
quadratic |
2 |
13 |
3 |
15 |
3 |
7 |
11 |
cubic |
3 |
43 |
25 |
66 |
19 |
37 |
54 |
quartic |
4 |
147 |
94 |
242 |
91 |
123 |
213 |
quintic |
5 |
423 |
335 |
757 |
305 |
393 |
699 |
sextic |
6 |
1.162 |
962 |
2.118 |
932 |
1.076 |
2.004 |
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