Character table for point group D3h

(x axis coincident with C'2 axis)
D3h E 2C3 (z) 3C'2 h (xy) 2S3 3v
linear functions,
rotations
quadratic
functions
cubic
functions
A'1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 x(x2-3y2)
A'2 +1 +1 -1 +1 +1 -1 Rz - y(3x2-y2)
E' +2 -1 0 +2 -1 0 (x, y) (x2-y2, xy) (xz2, yz2) [x(x2+y2), y(x2+y2)]
A''1 +1 +1 +1 -1 -1 -1 - - -
A''2 +1 +1 -1 -1 -1 +1 z - z3, z(x2+y2)
E'' +2 -1 0 -2 +1 0 (Rx, Ry) (xz, yz) [xyz, z(x2-y2)]


Additional information

Number of symmetry elements h = 12
Number of irreducible representations n = 6
Abelian group no
Subgroups Cs , C2 , C3 , D3 , C2v , C3v , C3h
Optical Isomerism (Chirality) no


Reduction formula for point group D3h

Type of representation

general 3N vib

E 2C3 (z) 3C'2 h (xy) 2S3 3v




Multipoles

dipole (p) E'+A''2
quadrupole (d) A'1+E'+E''
octopole (f) A'1+A'2+E'+A''2+E''
hexadecapole (g) A'1+2E'+A''1+A''2+E''
32-pole (h) A'1+A'2+2E'+A''2+2E''
64-pole (i) 2A'1+A'2+2E'+A''1+A''2+2E''
128-pole (j) A'1+A'2+3E'+A''1+2A''2+2E''
256-pole(k) 2A'1+A'2+3E'+A''1+A''2+3E''
512-pole (l) 2A'1+2A'2+3E'+A''1+2A''2+3E''

First nonvanishing multipole: quadrupole

Literature



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