Character table for point group D3d

(x axis coincident with C'2 axis)
D3d E 2C3 3C'2 i 2S6 3d
linear functions,
rotations
quadratic
functions
cubic
functions
A1g +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A2g +1 +1 -1 +1 +1 -1 Rz - -
Eg +2 -1 0 +2 -1 0 (Rx, Ry) (x2-y2, xy) (xz, yz) -
A1u +1 +1 +1 -1 -1 -1 - - x(x2-3y2)
A2u +1 +1 -1 -1 -1 +1 z - y(3x2-y2), z3, z(x2+y2)
Eu +2 -1 0 -2 +1 0 (x, y) - (xz2, yz2) [xyz, z(x2-y2)] [x(x2+y2), y(x2+y2)]


Additional information

Number of symmetry elements h = 12
Number of irreducible representations n = 6
Abelian group no
Subgroups Cs , Ci , C2 , C3 , D3 , C3v , S6
Optical Isomerism (Chirality) no


Reduction formula for point group D3d

Type of representation

general 3N vib

E 2C3 3C'2 i 2S6 3d




Multipoles

dipole (p) A2u+Eu
quadrupole (d) A1g+2Eg
octopole (f) A1u+2A2u+2Eu
hexadecapole (g) 2A1g+A2g+3Eg
32-pole (h) A1u+2A2u+4Eu
64-pole (i) 3A1g+2A2g+4Eg
128-pole (j) 2A1u+3A2u+5Eu
256-pole(k) 3A1g+2A2g+6Eg
512-pole (l) 3A1u+4A2u+6Eu

First nonvanishing multipole: quadrupole

Literature



Character tables for chemically important point groups Computational Laboratory for Analysis, Modeling and Visualization Jacobs University Bremen