Point Group Th



Th E 4C3 4(C3)2 3C2 i 4(S6)5 4S6 h
Ag 1 1 1 1 1 1 1 1
Eg* 2 -1 -1 2 2 -1 -1 2
Tg 3 0 0 -1 3 0 0 -1
Au 1 1 1 1 -1 -1 -1 -1
Eu* 2 -1 -1 2 -2 1 1 -2
Tu 3 0 0 -1 -3 0 0 1


Additional information

Number of symmetry elements h = 24
Number of classes, irreps n = 8
Number of real-valued irreducible representations n = 6
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 4C3 4(C3)2 3C2 i 4(S6)5 4S6 h



Genrate representation from irreducible representations


Ag Eg* Tg Au Eu* Tu




Examples

[Mg(H2O)6]2+-Ion Bromofullerene C60Br24



Direct products of irreducible representations


Binary products
Ag Eg* Tg Au Eu* Tu
Ag Ag
Eg* Eg2Ag⊕Eg
Tg Tg2TgAg⊕Eg⊕2Tg
Au AuEuTuAg
Eu* Eu2Au⊕Eu2TuEg2Ag⊕Eg
Tu Tu2TuAu⊕Eu⊕2TuTg2TgAg⊕Eg⊕2Tg

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
Eg* Ag⊕Eg2Ag⊕EgAg⊕2Eg2Ag⊕2Eg3Ag⊕2EgMore
Tg Ag⊕Eg⊕TgAg⊕3Tg2Ag⊕2Eg⊕3TgAg⊕Eg⊕6Tg4Ag⊕3Eg⊕6TgMore
Eu* Ag⊕Eg2Au⊕EuAg⊕2Eg2Au⊕2Eu3Ag⊕2EgMore
Tu Ag⊕Eg⊕TgAu⊕3Tu2Ag⊕2Eg⊕3TgAu⊕Eu⊕6Tu4Ag⊕3Eg⊕6TgMore



Spherical harmonics and Multipoles
Symmetric Powers of Γxyz


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

First nonvanshing multipole: Hexadecapole

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 Th
L 2L+1 Term Splitting
S (L=0) 1 Ag
P (L=1) 3 Tg
D (L=2) 5 Eg⊕Tg
F (L=3) 7 Ag⊕2Tg
G (L=4) 9 Ag⊕Eg⊕2Tg
H (L=5) 11 Eg⊕3Tg
I (L=6) 13 2Ag⊕Eg⊕3Tg


Last update August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement