Reduction formula for point group D4h
Characters for molecular motions
Motion |
E |
2C4 (z) |
C2 |
2C'2 |
2C''2 |
i |
2S4 |
h |
2v |
2d |
Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
0 |
0 |
16 |
0 |
0 |
Translation (x,y,z) |
3 |
1 |
-1 |
-1 |
-1 |
-3 |
-1 |
1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
3 |
1 |
-1 |
-1 |
-1 |
Vibration |
42 |
-2 |
2 |
2 |
2 |
0 |
0 |
16 |
0 |
0 |
Decomposition into Irreducible representations
Motion |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Cartesian 3N |
4 |
4 |
4 |
4 |
4 |
2 |
2 |
2 |
2 |
8 |
36 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
2 |
Vibration |
4 |
3 |
4 |
4 |
3 |
2 |
1 |
2 |
2 |
7 |
32 |
Molecule Parameter
Number of Atoms (N) |
16 |
Number of internal coordinates |
42 |
Number of independant internal coordinates |
4 |
Number of vibrational modes |
32 |
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Linear (IR) |
4 |
3 |
4 |
4 |
3 |
2 |
1 |
2 |
2 |
7 |
8 / 24 |
Quadratic (Raman) |
4 |
3 |
4 |
4 |
3 |
2 |
1 |
2 |
2 |
7 |
15 / 17 |
IR + Raman |
- |
3 |
- |
- |
- |
2 |
- |
2 |
2 |
- |
0* / 9 |
* Center of inversion: Mutual Exclusion Principle
Characters of symmetric powers for vibration representation
Force field |
Tensor Order |
E |
2C4 (z) |
C2 |
2C'2 |
2C''2 |
i |
2S4 |
h |
2v |
2d |
linear |
1 |
42 |
-2 |
2 |
2 |
2 |
0 |
0 |
16 |
0 |
0 |
quadratic |
2 |
903 |
3 |
23 |
23 |
23 |
21 |
1 |
149 |
21 |
21 |
cubic |
3 |
13.244 |
-4 |
44 |
44 |
44 |
0 |
0 |
1.024 |
0 |
0 |
quartic |
4 |
148.995 |
15 |
275 |
275 |
275 |
231 |
11 |
5.735 |
231 |
231 |
quintic |
5 |
1.370.754 |
-26 |
506 |
506 |
506 |
0 |
0 |
27.568 |
0 |
0 |
sextic |
6 |
10.737.573 |
37 |
2.277 |
2.277 |
2.277 |
1.771 |
11 |
117.483 |
1.771 |
1.771 |
Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field |
Tensor Order |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
linear |
1 |
4 |
3 |
4 |
4 |
3 |
2 |
1 |
2 |
2 |
7 |
quadratic |
2 |
80 |
58 |
68 |
68 |
94 |
48 |
47 |
47 |
47 |
126 |
cubic |
3 |
905 |
883 |
895 |
895 |
1.522 |
777 |
755 |
767 |
767 |
1.778 |
quartic |
4 |
9.832 |
9.579 |
9.699 |
9.699 |
17.902 |
8.968 |
8.946 |
8.956 |
8.956 |
19.278 |
quintic |
5 |
87.550 |
87.297 |
87.430 |
87.430 |
167.835 |
84.104 |
83.851 |
83.984 |
83.984 |
174.727 |
sextic |
6 |
679.712 |
677.688 |
678.688 |
678.688 |
1.327.448 |
663.917 |
663.664 |
663.784 |
663.784 |
1.356.376 |
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