Reduction formula for point group C4h
Characters for molecular motions
Motion |
E |
C4(z) |
C2 |
(C4)3 |
i |
(S4)3 |
h |
S4 |
Cartesian 3N |
24 |
0 |
0 |
0 |
0 |
0 |
8 |
0 |
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 |
18 |
-2 |
2 |
-2 |
0 |
0 |
8 |
0 |
Decomposition into Irreducible representations
Motion |
Ag |
Bg |
Eg |
Au |
Bu |
Eu |
Total |
Cartesian 3N |
4 |
4 |
2 |
2 |
2 |
4 |
18 |
Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
Vibration |
3 |
4 |
1 |
1 |
2 |
3 |
14 |
Molecule Parameter
Number of Atoms (N) |
8 |
Number of internal coordinates |
18 |
Number of independant internal coordinates |
3 |
Number of vibrational modes |
14 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
Ag |
Bg |
Eg |
Au |
Bu |
Eu |
Total |
Linear (IR) |
3 |
4 |
1 |
1 |
2 |
3 |
4 / 10 |
Quadratic (Raman) |
3 |
4 |
1 |
1 |
2 |
3 |
8 / 6 |
IR + Raman |
- |
- |
- |
- |
2 |
- |
0* / 2 |
* 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 |
18 |
-2 |
2 |
-2 |
0 |
0 |
8 |
0 |
quadratic |
2 |
171 |
3 |
11 |
3 |
9 |
1 |
41 |
1 |
cubic |
3 |
1.140 |
-4 |
20 |
-4 |
0 |
0 |
160 |
0 |
quartic |
4 |
5.985 |
9 |
65 |
9 |
45 |
5 |
525 |
5 |
quintic |
5 |
26.334 |
-14 |
110 |
-14 |
0 |
0 |
1.512 |
0 |
sextic |
6 |
100.947 |
19 |
275 |
19 |
165 |
5 |
3.941 |
5 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
Ag |
Bg |
Eg |
Au |
Bu |
Eu |
linear |
1 |
3 |
4 |
1 |
1 |
2 |
3 |
quadratic |
2 |
30 |
28 |
16 |
17 |
16 |
24 |
cubic |
3 |
164 |
166 |
120 |
124 |
126 |
160 |
quartic |
4 |
831 |
824 |
680 |
686 |
684 |
800 |
quintic |
5 |
3.491 |
3.498 |
3.089 |
3.113 |
3.120 |
3.467 |
sextic |
6 |
13.172 |
13.160 |
12.112 |
12.143 |
12.136 |
13.056 |
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