Reduction formula for point group C6h
Characters for molecular motions
| Motion |
E |
C6(z) |
C3 |
C2 |
(C3)2 |
(C6)5 |
i |
(S3)5 |
(S6)5 |
 h |
S6 |
S3 |
| Cartesian 3N |
324 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
24 |
0 |
0 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
0 |
2 |
-3 |
-2 |
0 |
1 |
0 |
-2 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
0 |
2 |
3 |
2 |
0 |
-1 |
0 |
2 |
| Vibration |
318 |
-4 |
0 |
2 |
0 |
-4 |
0 |
0 |
0 |
24 |
0 |
0 |
Decomposition into Irreducible representations
| Motion |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Cartesian 3N |
29 |
25 |
25 |
29 |
25 |
29 |
29 |
25 |
216 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
2 |
| Vibration |
28 |
25 |
24 |
29 |
24 |
29 |
28 |
25 |
212 |
Molecule Parameter
| Number of Atoms (N) |
108 |
| Number of internal coordinates |
318 |
| Number of independant internal coordinates |
28 |
| Number of vibrational modes |
212 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
Total |
| Linear (IR) |
28 |
25 |
24 |
29 |
24 |
29 |
28 |
25 |
52 / 160 |
| Quadratic (Raman) |
28 |
25 |
24 |
29 |
24 |
29 |
28 |
25 |
81 / 131 |
| IR + Raman |
- |
25 |
- |
- |
- |
29 |
- |
25 |
0* / 79 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
| Force field |
Tensor
Order |
E |
C6(z) |
C3 |
C2 |
(C3)2 |
(C6)5 |
i |
(S3)5 |
(S6)5 |
h |
S6 |
S3 |
| linear |
1 |
318 |
-4 |
0 |
2 |
0 |
-4 |
0 |
0 |
0 |
24 |
0 |
0 |
| quadratic |
2 |
50.721 |
8 |
0 |
161 |
0 |
8 |
159 |
0 |
0 |
447 |
0 |
0 |
| cubic |
3 |
5.410.240 |
-10 |
106 |
320 |
106 |
-10 |
0 |
8 |
0 |
6.128 |
0 |
8 |
| quartic |
4 |
434.171.760 |
8 |
0 |
13.040 |
0 |
8 |
12.720 |
0 |
0 |
72.528 |
0 |
0 |
| quintic |
5 |
27.960.661.344 |
-4 |
0 |
25.760 |
0 |
-4 |
0 |
0 |
0 |
741.552 |
0 |
0 |
| sextic |
6 |
1.505.215.602.352 |
55 |
5.671 |
708.400 |
5.671 |
55 |
682.640 |
85 |
53 |
6.858.544 |
53 |
85 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
| Force field |
Tensor
Order |
Ag |
Bg |
E1g |
E2g |
Au |
Bu |
E1u |
E2u |
| linear |
1 |
28 |
25 |
24 |
29 |
24 |
29 |
28 |
25 |
| quadratic |
2 |
4.292 |
4.188 |
4.190 |
4.290 |
4.191 |
4.236 |
4.238 |
4.189 |
| cubic |
3 |
451.408 |
450.334 |
450.307 |
451.382 |
450.384 |
451.358 |
451.327 |
450.362 |
| quartic |
4 |
36.189.172 |
36.174.908 |
36.174.910 |
36.189.170 |
36.174.964 |
36.184.876 |
36.184.878 |
36.174.962 |
| quintic |
5 |
2.330.119.054 |
2.329.991.170 |
2.329.991.169 |
2.330.119.055 |
2.329.995.462 |
2.330.114.762 |
2.330.114.761 |
2.329.995.463 |
| sextic |
6 |
125.435.321.972 |
125.434.060.768 |
125.434.059.372 |
125.435.320.506 |
125.434.065.062 |
125.435.090.096 |
125.435.088.684 |
125.434.063.665 |
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