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 15 1 -1 -3 -1 -3 -1 5 3 1
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 9 -1 1 -1 1 -3 -1 5 3 1



Decomposition into Irreducible representations
Motion A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Cartesian 3N 1 1 1 1 1 0 2 0 1 3 11
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 1 0 1 1 0 0 1 0 1 2 7



Molecule Parameter
Number of Atoms (N) 5
Number of internal coordinates 9
Number of independant internal coordinates 1
Number of vibrational modes 7





Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Linear (IR) 1 0 1 1 0 0 1 0 1 2 3 / 4
Quadratic (Raman) 1 0 1 1 0 0 1 0 1 2 3 / 4
IR + Raman - 0 - - - 0 - 0 1 - 0* / 1
* 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 9 -1 1 -1 1 -3 -1 5 3 1
quadratic 2 45 1 5 5 5 9 1 17 9 5
cubic 3 165 -1 5 -5 5 -19 -1 45 19 5
quartic 4 495 3 15 15 15 39 3 103 39 15
quintic 5 1.287 -3 15 -15 15 -69 -3 211 69 15
sextic 6 3.003 3 35 35 35 119 3 399 119 35


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 1 0 1 1 0 0 1 0 1 2
quadratic 2 8 2 5 4 4 1 2 1 2 6
cubic 3 15 9 13 12 12 6 12 6 12 28
quartic 4 52 31 43 37 52 20 26 20 26 68
quintic 5 100 79 94 88 124 62 83 62 83 194
sextic 6 251 195 232 211 336 147 168 147 168 406


Literature




Character tables for chemically important point groups Character table for point group D4h Constructor University Bremen

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