Character table for point group Ih

Ih E 12C5 12(C5)2 20C3 15C2 i 12S10 12(S10)3 20S6 15
linear functions,
rotations
quadratic
functions
cubic
functions
Ag +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2+z2 -
T1g +3 -2cos(4/5) -2cos(2/5) 0 -1 +3 -2cos(2/5) -2cos(4/5) 0 -1 (Rx, Ry, Rz) - -
T2g +3 -2cos(2/5) -2cos(4/5) 0 -1 +3 -2cos(4/5) -2cos(2/5) 0 -1 - - -
Gg +4 -1 -1 +1 0 +4 -1 -1 +1 0 - - -
Hg +5 0 0 -1 +1 +5 0 0 -1 +1 - [2z2-x2-y2, x2-y2, xy, xz, yz] -
Au +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 - - -
T1u +3 -2cos(4/5) -2cos(2/5) 0 -1 -3 +2cos(2/5) +2cos(4/5) 0 +1 (x, y, z) - [x(z2+y2), y(z2+x2), z(x2+y2)]
T2u +3 -2cos(2/5) -2cos(4/5) 0 -1 -3 +2cos(4/5) +2cos(2/5) 0 +1 - - [x3, y3, z3]
Gu +4 -1 -1 +1 0 -4 +1 +1 -1 0 - - [x(z2-y2), y(z2-x2), z(x2-y2), xyz]
Hu +5 0 0 -1 +1 -5 0 0 +1 -1 - - -

Information for point groups with fivefold rotational axis


Additional information

Number of symmetry elements h = 120
Number of irreducible representations n = 10
Abelian group no
Number of subgroups20
Subgroups Cs , Ci , C2 , C3 , C5 , D2 , D3 , D5 , C2v , C3v , C5v , C2h , D2h , D3d , D5d , S6 , S10 , T , Th , I
Optical Isomerism (Chirality) no


Reduction formula for point group Ih

Type of representation

general 3N vib

E 12C5 12(C5)2 20C3 15C2 i 12S10 12(S10)3 20S6 15




Examples

[Pb12]2--Ion Fullerene C20 [B12H12]2--Ion
Dodecahedrane Fullerene C60



Multipoles

dipole (p) T1u
quadrupole (d) Hg
octopole (f) T2u+Gu
hexadecapole (g) Gg+Hg
32-pole (h) T1u+T2u+Hu
64-pole (i) Ag+T1g+Gg+Hg
128-pole (j) T1u+T2u+Gu+Hu
256-pole(k) T2g+Gg+2Hg
512-pole (l) T1u+T2u+2Gu+Hu

First nonvanishing multipole: 64-pole

Literature



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