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 72 0 0 -4 -4 0 0 24 4 4
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 66 -2 2 -2 -2 0 0 24 4 4



Decomposition into Irreducible representations
Motion A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Cartesian 3N 6 6 6 6 6 1 5 3 3 12 54
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 6 5 6 6 5 1 4 3 3 11 50



Molecule Parameter
Number of Atoms (N) 24
Number of internal coordinates 66
Number of independant internal coordinates 6
Number of vibrational modes 50





Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Linear (IR) 6 5 6 6 5 1 4 3 3 11 15 / 35
Quadratic (Raman) 6 5 6 6 5 1 4 3 3 11 23 / 27
IR + Raman - 5 - - - 1 - 3 3 - 0* / 12
* 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 66 -2 2 -2 -2 0 0 24 4 4
quadratic 2 2.211 3 35 35 35 33 1 321 41 41
cubic 3 50.116 -4 68 -68 -68 0 0 3.104 144 144
quartic 4 864.501 21 629 629 629 561 17 24.081 841 841
quintic 5 12.103.014 -38 1.190 -1.190 -1.190 0 0 158.424 2.660 2.660
sextic 6 143.218.999 55 7.735 7.735 7.735 6.545 17 914.641 11.585 11.585


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 6 5 6 6 5 1 4 3 3 11
quadratic 2 182 144 162 162 236 117 120 118 118 308
cubic 3 3.349 3.311 3.331 3.331 5.868 2.889 2.995 2.943 2.943 6.644
quartic 4 55.983 55.248 55.606 55.606 105.044 52.478 52.584 52.530 52.530 110.924
quintic 5 766.777 766.042 766.419 766.419 1.492.925 745.644 747.569 746.616 746.616 1.532.531
sextic 6 9.014.084 9.004.424 9.009.236 9.009.236 17.787.896 8.893.139 8.895.064 8.894.092 8.894.092 18.014.920


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