Reduction formula for point group D3d



Characters for molecular motions
Motion E 2C3 3C'2 i 2S6 3d
Cartesian 3N 60 0 0 0 0 4
Translation (x,y,z) 3 0 -1 -3 0 1
Rotation (Rx,Ry,Rz) 3 0 -1 3 0 -1
Vibration 54 0 2 0 0 4



Decomposition into Irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 6 4 10 4 6 10 40
Translation (x,y,z) 0 0 0 0 1 1 2
Rotation (Rx,Ry,Rz) 0 1 1 0 0 0 2
Vibration 6 3 9 4 5 9 36



Molecule Parameter
Number of Atoms (N) 20
Number of internal coordinates 54
Number of independant internal coordinates 6
Number of vibrational modes 36





Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 6 3 9 4 5 9 14 / 22
Quadratic (Raman) 6 3 9 4 5 9 15 / 21
IR + Raman - 3 - 4 - - 0* / 7
* Center of inversion: Mutual Exclusion Principle



Characters of symmetric powers for vibration representation
Force field Tensor
Order
E 2C3 3C'2 i 2S6 3d
linear 1 54 0 2 0 0 4
quadratic 2 1.485 0 29 27 0 35
cubic 3 27.720 18 56 0 0 120
quartic 4 395.010 0 434 378 0 610
quintic 5 4.582.116 0 812 0 0 1.856
sextic 6 45.057.474 171 4.466 3.654 9 7.134


Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field Tensor
Order
A1g A2g Eg A1u A2u Eu
linear 1 6 3 9 4 5 9
quadratic 2 142 110 252 120 123 243
cubic 3 2.357 2.269 4.617 2.297 2.329 4.617
quartic 4 33.210 32.688 65.898 32.842 32.930 65.772
quintic 5 382.510 381.176 763.686 381.582 382.104 763.686
sextic 6 3.758.024 3.752.224 7.510.158 3.753.845 3.755.179 7.508.943


Literature




Character tables for chemically important point groups Character table for point group D3d Constructor University Bremen

Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement