Reduction formula for point group D3d



Characters for molecular motions
Motion E 2C3 3C'2 i 2S6 3d
Cartesian 3N 54 0 0 0 0 6
Translation (x,y,z) 3 0 -1 -3 0 1
Rotation (Rx,Ry,Rz) 3 0 -1 3 0 -1
Vibration 48 0 2 0 0 6



Decomposition into Irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 6 3 9 3 6 9 36
Translation (x,y,z) 0 0 0 0 1 1 2
Rotation (Rx,Ry,Rz) 0 1 1 0 0 0 2
Vibration 6 2 8 3 5 8 32



Molecule Parameter
Number of Atoms (N) 18
Number of internal coordinates 48
Number of independant internal coordinates 6
Number of vibrational modes 32





Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 6 2 8 3 5 8 13 / 19
Quadratic (Raman) 6 2 8 3 5 8 14 / 18
IR + Raman - 2 - 3 - - 0* / 5
* 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 48 0 2 0 0 6
quadratic 2 1.176 0 26 24 0 42
cubic 3 19.600 16 50 0 0 182
quartic 4 249.900 0 350 300 0 798
quintic 5 2.598.960 0 650 0 0 2.814
sextic 6 22.957.480 136 3.250 2.600 8 9.730


Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field Tensor
Order
A1g A2g Eg A1u A2u Eu
linear 1 6 2 8 3 5 8
quadratic 2 117 83 200 92 100 192
cubic 3 1.694 1.578 3.264 1.603 1.669 3.264
quartic 4 21.137 20.563 41.700 20.688 20.912 41.600
quintic 5 217.446 215.714 433.160 216.039 217.121 433.160
sextic 6 1.916.609 1.910.119 3.826.656 1.911.308 1.914.548 3.825.792


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