Point Group Ih



Ih E 12C5 12(C5)2 20C3 15C2 i 12S10 12(S10)3 20S6 15σd
Ag 1 1 1 1 1 1 1 1 1 1
T1g 3 -2cos(4π/5) -2cos(2π/5) 0 -1 3 -2cos(2π/5) -2cos(4π/5) 0 -1
T2g 3 -2cos(2π/5) -2cos(4π/5) 0 -1 3 -2cos(4π/5) -2cos(2π/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
Au 1 1 1 1 1 -1 -1 -1 -1 -1
T1u 3 -2cos(4π/5) -2cos(2π/5) 0 -1 -3 2cos(2π/5) 2cos(4π/5) 0 1
T2u 3 -2cos(2π/5) -2cos(4π/5) 0 -1 -3 2cos(4π/5) 2cos(2π/5) 0 1
Gu 4 -1 -1 1 0 -4 1 1 -1 0
Hu 5 0 0 -1 1 -5 0 0 1 -1
View A
✅
View B


Additional information

Number of symmetry elements h = 120
Number of classes, irreps n = 10
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


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



Genrate representation from irreducible representations


Ag T1g T2g Gg Hg Au T1u T2u Gu Hu



Examples

[Pb12]2--Ion Fullerene C20 [B12H12]2--Ion
Dodecahedrane Fullerene C60 Fullerene C80 : 7


Direct products of irreducible representations


Binary products
⊙ Ag T1g T2g Gg Hg Au T1u T2u Gu Hu
Ag Ag
T1g T1gAg⊕T1g⊕Hg
T2g T2gGg⊕HgAg⊕T2g⊕Hg
Gg GgT2g⊕Gg⊕HgT1g⊕Gg⊕HgAg⊕T1g⊕T2g⊕Gg⊕Hg
Hg HgT1g⊕T2g⊕Gg⊕HgT1g⊕T2g⊕Gg⊕HgT1g⊕T2g⊕Gg⊕2HgAg⊕T1g⊕T2g⊕2Gg⊕2Hg
Au AuT1uT2uGuHuAg
T1u T1uAu⊕T1u⊕HuGu⊕HuT2u⊕Gu⊕HuT1u⊕T2u⊕Gu⊕HuT1gAg⊕T1g⊕Hg
T2u T2uGu⊕HuAu⊕T2u⊕HuT1u⊕Gu⊕HuT1u⊕T2u⊕Gu⊕HuT2gGg⊕HgAg⊕T2g⊕Hg
Gu GuT2u⊕Gu⊕HuT1u⊕Gu⊕HuAu⊕T1u⊕T2u⊕Gu⊕HuT1u⊕T2u⊕Gu⊕2HuGgT2g⊕Gg⊕HgT1g⊕Gg⊕HgAg⊕T1g⊕T2g⊕Gg⊕Hg
Hu HuT1u⊕T2u⊕Gu⊕HuT1u⊕T2u⊕Gu⊕HuT1u⊕T2u⊕Gu⊕2HuAu⊕T1u⊕T2u⊕2Gu⊕2HuHgT1g⊕T2g⊕Gg⊕HgT1g⊕T2g⊕Gg⊕HgT1g⊕T2g⊕Gg⊕2HgAg⊕T1g⊕T2g⊕2Gg⊕2Hg

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
T1g Ag⊕HgT1g⊕T2g⊕GgAg⊕Gg⊕2Hg2T1g⊕2T2g⊕Gg⊕Hg2Ag⊕T1g⊕2Gg⊕3HgMore
T2g Ag⊕HgT1g⊕T2g⊕GgAg⊕Gg⊕2Hg2T1g⊕2T2g⊕Gg⊕Hg2Ag⊕T2g⊕2Gg⊕3HgMore
Gg Ag⊕Gg⊕HgAg⊕T1g⊕T2g⊕2Gg⊕Hg2Ag⊕T1g⊕T2g⊕3Gg⊕3Hg2Ag⊕3T1g⊕3T2g⊕4Gg⊕4Hg3Ag⊕3T1g⊕3T2g⊕7Gg⊕7HgMore
Hg Ag⊕Gg⊕2Hg2Ag⊕T1g⊕T2g⊕3Gg⊕3Hg2Ag⊕2T1g⊕2T2g⊕4Gg⊕8Hg4Ag⊕5T1g⊕5T2g⊕8Gg⊕12Hg7Ag⊕8T1g⊕8T2g⊕15Gg⊕19HgMore
T1u Ag⊕HgT1u⊕T2u⊕GuAg⊕Gg⊕2Hg2T1u⊕2T2u⊕Gu⊕Hu2Ag⊕T1g⊕2Gg⊕3HgMore
T2u Ag⊕HgT1u⊕T2u⊕GuAg⊕Gg⊕2Hg2T1u⊕2T2u⊕Gu⊕Hu2Ag⊕T2g⊕2Gg⊕3HgMore
Gu Ag⊕Gg⊕HgAu⊕T1u⊕T2u⊕2Gu⊕Hu2Ag⊕T1g⊕T2g⊕3Gg⊕3Hg2Au⊕3T1u⊕3T2u⊕4Gu⊕4Hu3Ag⊕3T1g⊕3T2g⊕7Gg⊕7HgMore
Hu Ag⊕Gg⊕2Hg2Au⊕T1u⊕T2u⊕3Gu⊕3Hu2Ag⊕2T1g⊕2T2g⊕4Gg⊕8Hg4Au⊕5T1u⊕5T2u⊕8Gu⊕12Hu7Ag⊕8T1g⊕8T2g⊕15Gg⊕19HgMore


Spherical harmonics and Multipoles
Symmetric Powers of Γxyz


Spherical Harmonics Yl / Multipole Symmetric Power [Γl(xyz)]
l 2l+1 Multipole Symmetry Rank [Γl(xyz)]
s (l=0) 1 Monopole Ag 1 Ag
p (l=1) 3 Dipole T1u 3 T1u
d (l=2) 5 Quadrupole Hg 6 Ag⊕Hg
f (l=3) 7 Octupole T2u⊕Gu 10 T1u⊕T2u⊕Gu
g (l=4) 9 Hexadecapole Gg⊕Hg 15 Ag⊕Gg⊕2Hg
h (l=5) 11 Dotricontapole T1u⊕T2u⊕Hu 21 2T1u⊕2T2u⊕Gu⊕Hu
i (l=6) 13 Tetrahexacontapole Ag⊕T1g⊕Gg⊕Hg 28 2Ag⊕T1g⊕2Gg⊕3Hg
j (l=7) 15 Octacosahectapole T1u⊕T2u⊕Gu⊕Hu 36 3T1u⊕3T2u⊕2Gu⊕2Hu
k (l=8) 17 256-pole T2g⊕Gg⊕2Hg 45 2Ag⊕T1g⊕T2g⊕3Gg⊕5Hg
l (l=9) 19 512-pole T1u⊕T2u⊕2Gu⊕Hu 55 4T1u⊕4T2u⊕4Gu⊕3Hu
m (l=10) 21 1024-pole Ag⊕T1g⊕T2g⊕Gg⊕2Hg 66 3Ag⊕2T1g⊕2T2g⊕4Gg⊕7Hg
n (l=11) 23 2048-pole 2T1u⊕T2u⊕Gu⊕2Hu 78 6T1u⊕5T2u⊕5Gu⊕5Hu
o (l=12) 25 4096-pole Ag⊕T1g⊕T2g⊕2Gg⊕2Hg 91 4Ag⊕3T1g⊕3T2g⊕6Gg⊕9Hg
More

First nonvanshing multipole: Tetrahexacontapole

Further Reading

  • A. Gelessus, W. Thiel, W. Weber. J. Chem. Educ. 72 505 (1995)
     Multipoles and symmetry



Ligand Field, dn term splitting


Term symbols for electronic configurations dn
dn Term Symbols
d1 = d9 2D
d2 = d8 1S, 1D, 1G, 3P, 3F
d3 = d7 2P, 2D (2), 2F, 2G, 2H, 4P, 4F
d4 = d6 1S (2), 1D (2), 1F, 1G (2), 1I, 3P (2), 3D, 3F (2), 3G, 3H, 5D
d5 2S, 2P, 2D (3), 2F (2), 2G (2), 2H, 2I, 4P, 4D, 4F, 4G, 6S


Term splitting in point group Ih
L 2L+1 Term Splitting
S (L=0) 1 Ag
P (L=1) 3 T1g
D (L=2) 5 Hg
F (L=3) 7 T2g⊕Gg
G (L=4) 9 Gg⊕Hg
H (L=5) 11 T1g⊕T2g⊕Hg
I (L=6) 13 Ag⊕T1g⊕Gg⊕Hg

Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement