Character table for point group O

O E 8C3 6C'2 6C4 3C2 =(C4)2
linear functions,
rotations
quadratic
functions
cubic
functions
A1 +1 +1 +1 +1 +1 - x2+y2+z2 -
A2 +1 +1 -1 -1 +1 - - xyz
E +2 -1 0 0 +2 - (x2-y2, 2z2-x2-y2) -
T1 +3 0 -1 +1 -1 (x, y, z) (Rx, Ry, Rz) - (x3, y3, z3) [x(z2+y2), y(z2+x2), z(x2+y2)]
T2 +3 0 +1 -1 -1 - (xy, xz, yz) [x(z2-y2), y(z2-x2), z(x2-y2)]


Additional information

Number of symmetry elements h = 24
Number of irreducible representations n = 5
Abelian group no
Number of subgroups9
Number of distinct subgroups7
Subgroups
(Number of different orientations)
C2 (2) , C3 , C4 , D2 (2) , D3 , D4 , T
Optical Isomerism (Chirality) yes


Reduction formula for point group O

Type of representation

general 3N vib

E 8C3 6C'2 6C4 3C2 =(C4)2




Multipoles

dipole (p) T1
quadrupole (d) E+T2
octopole (f) A2+T1+T2
hexadecapole (g) A1+E+T1+T2
32-pole (h) E+2T1+T2
64-pole (i) A1+A2+E+T1+2T2
128-pole (j) A2+E+2T1+2T2
256-pole(k) A1+2E+2T1+2T2
512-pole (l) A1+A2+E+3T1+2T2

First nonvanishing multipole: hexadecapole

Literature



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