Reduction formula for point group D3d



Characters for molecular motions
Motion E 2C3 3C'2 i 2S6 3d
Cartesian 3N 36 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 30 0 2 0 0 4



Decomposition into Irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 4 2 6 2 4 6 24
Translation (x,y,z) 0 0 0 0 1 1 2
Rotation (Rx,Ry,Rz) 0 1 1 0 0 0 2
Vibration 4 1 5 2 3 5 20



Molecule Parameter
Number of Atoms (N) 12
Number of internal coordinates 30
Number of independant internal coordinates 4
Number of vibrational modes 20





Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 4 1 5 2 3 5 8 / 12
Quadratic (Raman) 4 1 5 2 3 5 9 / 11
IR + Raman - 1 - 2 - - 0* / 3
* 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 30 0 2 0 0 4
quadratic 2 465 0 17 15 0 23
cubic 3 4.960 10 32 0 0 72
quartic 4 40.920 0 152 120 0 256
quintic 5 278.256 0 272 0 0 680
sextic 6 1.623.160 55 952 680 5 1.904


Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field Tensor
Order
A1g A2g Eg A1u A2u Eu
linear 1 4 1 5 2 3 5
quadratic 2 50 30 80 36 39 75
cubic 3 441 389 825 405 425 825
quartic 4 3.522 3.318 6.840 3.374 3.426 6.800
quintic 5 23.426 22.950 46.376 23.086 23.290 46.376
sextic 6 136.044 134.616 270.630 134.977 135.453 270.405


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