## 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 quadraticfunctions cubicfunctions 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)

 Cs (3) , Ci , C2 (3) , C4 , D2 (2) , D4 , C2v (4) , C4v , C2h (3) , C4h , D2h (2) , D2d (2) , S4 Number of symmetry elements h = 16 Number of irreducible representations n = 10 Abelian group no Number of subgroups 25 Number of distinct subgroups 13 Subgroups (Number of different orientations) Optical Isomerism (Chirality) no Polar 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 A1g+B1g+B2g+Eg A2u+B1u+B2u+2Eu 2A1g+A2g+B1g+B2g+2Eg A1u+2A2u+B1u+B2u+3Eu 2A1g+A2g+2B1g+2B2g+3Eg A1u+2A2u+2B1u+2B2u+4Eu 3A1g+2A2g+2B1g+2B2g+4Eg 2A1u+3A2u+2B1u+2B2u+5Eu