Character table for point group D2h

D2h E C2 (z) C2 (y) C2 (x) i (xy) (xz) (yz)
linear functions,
rotations
quadratic
functions
cubic
functions
Ag +1 +1 +1 +1 +1 +1 +1 +1 - x2, y2, z2 -
B1g +1 +1 -1 -1 +1 +1 -1 -1 Rz xy -
B2g +1 -1 +1 -1 +1 -1 +1 -1 Ry xz -
B3g +1 -1 -1 +1 +1 -1 -1 +1 Rx yz -
Au +1 +1 +1 +1 -1 -1 -1 -1 - - xyz
B1u +1 +1 -1 -1 -1 -1 +1 +1 z - z3, y2z, x2z
B2u +1 -1 +1 -1 -1 +1 -1 +1 y - yz2, y3, x2y
B3u +1 -1 -1 +1 -1 +1 +1 -1 x - xz2, xy2, x3


Additional information

Number of symmetry elements h = 8
Number of irreducible representations n = 8
Abelian group yes
Number of subgroups14
Number of distinct subgroups6
Subgroups
(Number of different orientations)
Cs (3) , Ci , C2 (3) , D2 , C2v (3) , C2h (3)
Optical Isomerism (Chirality) no


Reduction formula for point group D2h

Type of representation

general 3N vib

E C2 (z) C2 (y) C2 (x) i (xy) (xz) (yz)




Examples

Ethylene Diborane Naphthalene



Multipoles

dipole (p) B1u+B2u+B3u
quadrupole (d) 2Ag+B1g+B2g+B3g
octopole (f) Au+2B1u+2B2u+2B3u
hexadecapole (g) 3Ag+2B1g+2B2g+2B3g
32-pole (h) 2Au+3B1u+3B2u+3B3u
64-pole (i) 4Ag+3B1g+3B2g+3B3g
128-pole (j) 3Au+4B1u+4B2u+4B3u
256-pole(k) 5Ag+4B1g+4B2g+4B3g
512-pole (l) 4Au+5B1u+5B2u+5B3u

First nonvanishing multipole: quadrupole

Literature



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