Reduction formula for point group D6h
Characters for molecular motions
Motion |
E |
2C6 (z) |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
h (xy) |
3d |
3v |
Cartesian 3N |
108 |
0 |
0 |
0 |
-4 |
0 |
0 |
0 |
0 |
36 |
0 |
4 |
Translation (x,y,z) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
-3 |
-2 |
0 |
1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
3 |
2 |
0 |
-1 |
-1 |
-1 |
Vibration |
102 |
-4 |
0 |
2 |
-2 |
2 |
0 |
0 |
0 |
36 |
0 |
4 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
Cartesian 3N |
6 |
6 |
2 |
4 |
6 |
12 |
2 |
4 |
6 |
6 |
12 |
6 |
72 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
Vibration |
6 |
5 |
2 |
4 |
5 |
12 |
2 |
3 |
6 |
6 |
11 |
6 |
68 |
Molecule Parameter
Number of Atoms (N) |
36 |
Number of internal coordinates |
102 |
Number of independant internal coordinates |
6 |
Number of vibrational modes |
68 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
Linear (IR) |
6 |
5 |
2 |
4 |
5 |
12 |
2 |
3 |
6 |
6 |
11 |
6 |
14 / 54 |
Quadratic (Raman) |
6 |
5 |
2 |
4 |
5 |
12 |
2 |
3 |
6 |
6 |
11 |
6 |
23 / 45 |
IR + Raman |
- |
5 |
2 |
4 |
- |
- |
2 |
- |
6 |
6 |
- |
6 |
0* / 31 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
2C6 (z) |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
h (xy) |
3d |
3v |
linear |
1 |
102 |
-4 |
0 |
2 |
-2 |
2 |
0 |
0 |
0 |
36 |
0 |
4 |
quadratic |
2 |
5.253 |
8 |
0 |
53 |
53 |
53 |
51 |
0 |
0 |
699 |
51 |
59 |
cubic |
3 |
182.104 |
-10 |
34 |
104 |
-104 |
104 |
0 |
12 |
0 |
9.624 |
0 |
216 |
quartic |
4 |
4.780.230 |
8 |
0 |
1.430 |
1.430 |
1.430 |
1.326 |
0 |
0 |
104.790 |
1.326 |
1.750 |
quintic |
5 |
101.340.876 |
-4 |
0 |
2.756 |
-2.756 |
2.756 |
0 |
0 |
0 |
956.592 |
0 |
5.936 |
sextic |
6 |
1.807.245.622 |
19 |
595 |
26.182 |
26.182 |
26.182 |
23.426 |
89 |
17 |
7.590.842 |
23.426 |
34.874 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
linear |
1 |
6 |
5 |
2 |
4 |
5 |
12 |
2 |
3 |
6 |
6 |
11 |
6 |
quadratic |
2 |
280 |
226 |
188 |
190 |
380 |
504 |
190 |
191 |
244 |
242 |
488 |
379 |
cubic |
3 |
8.023 |
7.969 |
7.132 |
7.238 |
14.362 |
15.983 |
7.165 |
7.219 |
7.990 |
7.988 |
15.964 |
14.381 |
quartic |
4 |
204.400 |
202.916 |
194.752 |
194.858 |
389.612 |
407.314 |
194.788 |
194.842 |
203.480 |
203.374 |
406.856 |
389.628 |
quintic |
5 |
4.263.251 |
4.261.767 |
4.181.133 |
4.183.995 |
8.365.127 |
8.525.019 |
4.182.051 |
4.183.535 |
4.262.333 |
4.262.227 |
8.524.559 |
8.365.587 |
sextic |
6 |
75.634.146 |
75.606.480 |
74.984.112 |
74.986.974 |
149.970.960 |
151.240.446 |
74.985.031 |
74.986.515 |
75.617.604 |
75.614.742 |
151.232.184 |
149.971.419 |
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