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 |
324 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
24 |
12 |
8 |
| 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 |
318 |
-4 |
0 |
2 |
2 |
2 |
0 |
0 |
0 |
24 |
12 |
8 |
Decomposition into Irreducible representations
| Motion |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Cartesian 3N |
17 |
12 |
13 |
12 |
25 |
29 |
10 |
15 |
14 |
15 |
29 |
25 |
216 |
| 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 |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |
212 |
Molecule Parameter
| Number of Atoms (N) |
108 |
| Number of internal coordinates |
318 |
| Number of independant internal coordinates |
17 |
| Number of vibrational modes |
212 |
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Linear (IR) |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |
42 / 170 |
| Quadratic (Raman) |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |
70 / 142 |
| IR + Raman |
- |
11 |
13 |
12 |
- |
- |
10 |
- |
14 |
15 |
- |
25 |
0* / 100 |
* 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 |
318 |
-4 |
0 |
2 |
2 |
2 |
0 |
0 |
0 |
24 |
12 |
8 |
| quadratic |
2 |
50.721 |
8 |
0 |
161 |
161 |
161 |
159 |
0 |
0 |
447 |
231 |
191 |
| cubic |
3 |
5.410.240 |
-10 |
106 |
320 |
320 |
320 |
0 |
8 |
0 |
6.128 |
2.200 |
1.360 |
| quartic |
4 |
434.171.760 |
8 |
0 |
13.040 |
13.040 |
13.040 |
12.720 |
0 |
0 |
72.528 |
25.080 |
18.000 |
| quintic |
5 |
27.960.661.344 |
-4 |
0 |
25.760 |
25.760 |
25.760 |
0 |
0 |
0 |
741.552 |
201.432 |
116.112 |
| sextic |
6 |
1.505.215.602.352 |
55 |
5.671 |
708.400 |
708.400 |
708.400 |
682.640 |
85 |
53 |
6.858.544 |
1.748.824 |
1.120.816 |
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 |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |
| quadratic |
2 |
2.239 |
2.053 |
2.099 |
2.089 |
4.190 |
4.290 |
2.083 |
2.108 |
2.113 |
2.123 |
4.238 |
4.189 |
| cubic |
3 |
226.229 |
225.179 |
225.272 |
225.062 |
450.307 |
451.382 |
224.827 |
225.557 |
225.574 |
225.784 |
451.327 |
450.362 |
| quartic |
4 |
18.103.231 |
18.085.941 |
18.088.339 |
18.086.569 |
36.174.910 |
36.189.170 |
18.085.357 |
18.089.607 |
18.091.553 |
18.093.323 |
36.184.878 |
36.174.962 |
| quintic |
5 |
1.165.105.660 |
1.165.013.394 |
1.165.006.250 |
1.164.984.920 |
2.329.991.169 |
2.330.119.055 |
1.164.964.478 |
1.165.030.984 |
1.165.046.716 |
1.165.068.046 |
2.330.114.761 |
2.329.995.463 |
| sextic |
6 |
62.718.196.791 |
62.717.125.181 |
62.717.108.885 |
62.716.951.883 |
125.434.059.372 |
125.435.320.506 |
62.716.850.926 |
62.717.214.136 |
62.717.466.547 |
62.717.623.549 |
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