Reduction formula for point group D3d



Characters for molecular motions
Motion E 2C3 3C'2 i 2S6 3d
Cartesian 3N 18 0 0 0 0 2
Translation (x,y,z) 3 0 -1 -3 0 1
Rotation (Rx,Ry,Rz) 3 0 -1 3 0 -1
Vibration 12 0 2 0 0 2



Decomposition into Irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 2 1 3 1 2 3 12
Translation (x,y,z) 0 0 0 0 1 1 2
Rotation (Rx,Ry,Rz) 0 1 1 0 0 0 2
Vibration 2 0 2 1 1 2 8



Molecule Parameter
Number of Atoms (N) 6
Number of internal coordinates 12
Number of independant internal coordinates 2
Number of vibrational modes 8




Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 2 0 2 1 1 2 3 / 5
Quadratic (Raman) 2 0 2 1 1 2 4 / 4
IR + Raman - 0 - 1 - - 0* / 1
* 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 12 0 2 0 0 2
quadratic 2 78 0 8 6 0 8
cubic 3 364 4 14 0 0 14
quartic 4 1.365 0 35 21 0 35
quintic 5 4.368 0 56 0 0 56
sextic 6 12.376 10 112 56 2 112


Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field Tensor
Order		 A1g A2g Eg A1u A2u Eu
linear 1 2 0 2 1 1 2
quadratic 2 11 3 14 6 6 12
cubic 3 38 24 60 31 31 60
quartic 4 133 98 231 112 112 224
quintic 5 392 336 728 364 364 728
sextic 6 1.094 982 2.070 1.028 1.028 2.052


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