Character table for point group Oh

Oh E 8C3 6C2 6C4 3C2 =(C4)2 i 6S4 8S6 3h 6d
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 - - -
Eg +2 -1 0 0 +2 +2 0 -1 +2 0 - (2z2-x2-y2, x2-y2) -
T1g +3 0 -1 +1 -1 +3 +1 0 -1 -1 (Rx, Ry, Rz) - -
T2g +3 0 +1 -1 -1 +3 -1 0 -1 +1 - (xz, yz, xy) -
A1u +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 - - -
A2u +1 +1 -1 -1 +1 -1 +1 -1 -1 +1 - - xyz
Eu +2 -1 0 0 +2 -2 0 +1 -2 0 - - -
T1u +3 0 -1 +1 -1 -3 -1 0 +1 +1 (x, y, z) - (x3, y3, z3) [x(z2+y2), y(z2+x2), z(x2+y2)]
T2u +3 0 +1 -1 -1 -3 +1 0 +1 -1 - - [x(z2-y2), y(z2-x2), z(x2-y2)]


Additional information

Number of symmetry elements h = 48
Number of irreducible representations n = 10
Abelian group no
Subgroups Cs , Ci , C2 , C3 , C4 , D2 , D3 , D4 , C2v , C3v , C4v , C2h , C4h , D2h , D4h , D2d , D3d , S4 , S6 , T , Th , Td , O
Optical Isomerism (Chirality) no


Reduction formula for point group Oh

Type of representation

general 3N vib

E 8C3 6C2 6C4 3C2 =(C4)2 i 6S4 8S6 3h 6d




Multipoles

dipole (p) T1u
quadrupole (d) Eg+T2g
octopole (f) A2u+T1u+T2u
hexadecapole (g) A1g+Eg+T1g+T2g
32-pole (h) Eu+2T1u+T2u
64-pole (i) A1g+A2g+Eg+T1g+2T2g
128-pole (j) A2u+Eu+2T1u+2T2u
256-pole(k) A1g+2Eg+2T1g+2T2g
512-pole (l) A1u+A2u+Eu+3T1u+2T2u

First nonvanishing multipole: hexadecapole

Literature



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