Reduction formula for point group D6h



Characters for molecular motions
Motion E 2C6 (z) 2C3 C2 3C'2 3C''2 i 2S3 2S6 h (xy) 3d 3v
Cartesian 3N 108 0 0 0 -4 0 0 0 0 36 0 4
Translation (x,y,z) 3 2 0 -1 -1 -1 -3 -2 0 1 1 1
Rotation (Rx,Ry,Rz) 3 2 0 -1 -1 -1 3 2 0 -1 -1 -1
Vibration 102 -4 0 2 -2 2 0 0 0 36 0 4



Decomposition into Irreducible representations
Motion A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u Total
Cartesian 3N 6 6 2 4 6 12 2 4 6 6 12 6 72
Translation (x,y,z) 0 0 0 0 0 0 0 1 0 0 1 0 2
Rotation (Rx,Ry,Rz) 0 1 0 0 1 0 0 0 0 0 0 0 2
Vibration 6 5 2 4 5 12 2 3 6 6 11 6 68



Molecule Parameter
Number of Atoms (N) 36
Number of internal coordinates 102
Number of independant internal coordinates 6
Number of vibrational modes 68





Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u Total
Linear (IR) 6 5 2 4 5 12 2 3 6 6 11 6 14 / 54
Quadratic (Raman) 6 5 2 4 5 12 2 3 6 6 11 6 23 / 45
IR + Raman - 5 2 4 - - 2 - 6 6 - 6 0* / 31
* Center of inversion: Mutual Exclusion Principle



Characters of symmetric powers for vibration representation
Force field Tensor
Order
E 2C6 (z) 2C3 C2 3C'2 3C''2 i 2S3 2S6 h (xy) 3d 3v
linear 1 102 -4 0 2 -2 2 0 0 0 36 0 4
quadratic 2 5.253 8 0 53 53 53 51 0 0 699 51 59
cubic 3 182.104 -10 34 104 -104 104 0 12 0 9.624 0 216
quartic 4 4.780.230 8 0 1.430 1.430 1.430 1.326 0 0 104.790 1.326 1.750
quintic 5 101.340.876 -4 0 2.756 -2.756 2.756 0 0 0 956.592 0 5.936
sextic 6 1.807.245.622 19 595 26.182 26.182 26.182 23.426 89 17 7.590.842 23.426 34.874


Decomposition into Irreducible representations
Number of nonvanshing force constants
Force field Tensor
Order
A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u
linear 1 6 5 2 4 5 12 2 3 6 6 11 6
quadratic 2 280 226 188 190 380 504 190 191 244 242 488 379
cubic 3 8.023 7.969 7.132 7.238 14.362 15.983 7.165 7.219 7.990 7.988 15.964 14.381
quartic 4 204.400 202.916 194.752 194.858 389.612 407.314 194.788 194.842 203.480 203.374 406.856 389.628
quintic 5 4.263.251 4.261.767 4.181.133 4.183.995 8.365.127 8.525.019 4.182.051 4.183.535 4.262.333 4.262.227 8.524.559 8.365.587
sextic 6 75.634.146 75.606.480 74.984.112 74.986.974 149.970.960 151.240.446 74.985.031 74.986.515 75.617.604 75.614.742 151.232.184 149.971.419


Literature




Character tables for chemically important point groups Character table for point group D6h Constructor University Bremen

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