Character table for point group D4h

(x axis coincident with C'2 axis)
D4h E 2C4 (z) C2 2C'2 2C''2 i 2S4 h 2v 2d
linear functions,
rotations
quadratic
functions
cubic
functions
A1g +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A2g +1 +1 +1 -1 -1 +1 +1 +1 -1 -1 Rz - -
B1g +1 -1 +1 +1 -1 +1 -1 +1 +1 -1 - x2-y2 -
B2g +1 -1 +1 -1 +1 +1 -1 +1 -1 +1 - xy -
Eg +2 0 -2 0 0 +2 0 -2 0 0 (Rx, Ry) (xz, yz) -
A1u +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 - - -
A2u +1 +1 +1 -1 -1 -1 -1 -1 +1 +1 z - z3, z(x2+y2)
B1u +1 -1 +1 +1 -1 -1 +1 -1 -1 +1 - - xyz
B2u +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 - - z(x2-y2)
Eu +2 0 -2 0 0 -2 0 +2 0 0 (x, y) - (xz2, yz2) (xy2, x2y), (x3, y3)


Additional information

Number of symmetry elements h = 16
Number of irreducible representations n = 10
Abelian group no
Number of subgroups25
Number of distinct subgroups13
Subgroups
(Number of different orientations)
Cs (3) , Ci , C2 (3) , C4 , D2 (2) , D4 , C2v (4) , C4v , C2h (3) , C4h , D2h (2) , D2d (2) , S4
Optical Isomerism (Chirality) no


Reduction formula for point group D4h

Type of representation

general 3N vib

E 2C4 (z) C2 2C'2 2C''2 i 2S4 h 2v 2d




Multipoles

dipole (p) A2u+Eu
quadrupole (d) A1g+B1g+B2g+Eg
octopole (f) A2u+B1u+B2u+2Eu
hexadecapole (g) 2A1g+A2g+B1g+B2g+2Eg
32-pole (h) A1u+2A2u+B1u+B2u+3Eu
64-pole (i) 2A1g+A2g+2B1g+2B2g+3Eg
128-pole (j) A1u+2A2u+2B1u+2B2u+4Eu
256-pole(k) 3A1g+2A2g+2B1g+2B2g+4Eg
512-pole (l) 2A1u+3A2u+2B1u+2B2u+5Eu

First nonvanishing multipole: quadrupole

Literature



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