Reduction formula for point group C4h
Characters for molecular motions
Motion |
E |
C4(z) |
C2 |
(C4)3 |
i |
(S4)3 |
h |
S4 |
Cartesian 3N |
15 |
1 |
-1 |
1 |
-3 |
-1 |
5 |
-1 |
Translation (x,y,z) |
3 |
1 |
-1 |
1 |
-3 |
-1 |
1 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
1 |
3 |
1 |
-1 |
1 |
Vibration |
9 |
-1 |
1 |
-1 |
-3 |
-1 |
5 |
-1 |
Decomposition into Irreducible representations
Motion |
Ag |
Bg |
Eg |
Au |
Bu |
Eu |
Total |
Cartesian 3N |
2 |
2 |
1 |
2 |
1 |
3 |
11 |
Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
Vibration |
1 |
2 |
0 |
1 |
1 |
2 |
7 |
Molecule Parameter
Number of Atoms (N) |
5 |
Number of internal coordinates |
9 |
Number of independant internal coordinates |
1 |
Number of vibrational modes |
7 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
Ag |
Bg |
Eg |
Au |
Bu |
Eu |
Total |
Linear (IR) |
1 |
2 |
0 |
1 |
1 |
2 |
3 / 4 |
Quadratic (Raman) |
1 |
2 |
0 |
1 |
1 |
2 |
3 / 4 |
IR + Raman |
- |
- |
- |
- |
1 |
- |
0* / 1 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
C4(z) |
C2 |
(C4)3 |
i |
(S4)3 |
h |
S4 |
linear |
1 |
9 |
-1 |
1 |
-1 |
-3 |
-1 |
5 |
-1 |
quadratic |
2 |
45 |
1 |
5 |
1 |
9 |
1 |
17 |
1 |
cubic |
3 |
165 |
-1 |
5 |
-1 |
-19 |
-1 |
45 |
-1 |
quartic |
4 |
495 |
3 |
15 |
3 |
39 |
3 |
103 |
3 |
quintic |
5 |
1.287 |
-3 |
15 |
-3 |
-69 |
-3 |
211 |
-3 |
sextic |
6 |
3.003 |
3 |
35 |
3 |
119 |
3 |
399 |
3 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
Ag |
Bg |
Eg |
Au |
Bu |
Eu |
linear |
1 |
1 |
2 |
0 |
1 |
1 |
2 |
quadratic |
2 |
10 |
9 |
4 |
3 |
3 |
6 |
cubic |
3 |
24 |
25 |
12 |
18 |
18 |
28 |
quartic |
4 |
83 |
80 |
52 |
46 |
46 |
68 |
quintic |
5 |
179 |
182 |
124 |
145 |
145 |
194 |
sextic |
6 |
446 |
443 |
336 |
315 |
315 |
406 |
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