Point Group S6



S6 E C3 (C3)2 i (S6)5 S6
Ag 1 1 1 1 1 1
Eg* 2 -1 -1 2 -1 -1
Au 1 1 1 -1 -1 -1
Eu* 2 -1 -1 -2 1 1


Additional information

Number of symmetry elements h = 6
Number of classes, irreps n = 6
Number of real-valued irreducible representations n = 4
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E C3 (C3)2 i (S6)5 S6



Genrate representation from irreducible representations


Ag Eg* Au Eu*




Direct products of irreducible representations


Binary products
Ag Eg* Au Eu*
Ag Ag
Eg* Eg2Ag⊕Eg
Au AuEuAg
Eu* Eu2Au⊕EuEg2Ag⊕Eg

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
Eu* Ag⊕Eg2Au⊕EuAg⊕2Eg2Au⊕2Eu3Ag⊕2EgMore



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 Au⊕Eu 3 Au⊕Eu
d (l=2) 5 Quadrupole Ag⊕2Eg 6 2Ag⊕2Eg
f (l=3) 7 Octupole 3Au⊕2Eu 10 4Au⊕3Eu
g (l=4) 9 Hexadecapole 3Ag⊕3Eg 15 5Ag⊕5Eg
h (l=5) 11 Dotricontapole 3Au⊕4Eu 21 7Au⊕7Eu
i (l=6) 13 Tetrahexacontapole 5Ag⊕4Eg 28 10Ag⊕9Eg
j (l=7) 15 Octacosahectapole 5Au⊕5Eu 36 12Au⊕12Eu
k (l=8) 17 256-pole 5Ag⊕6Eg 45 15Ag⊕15Eg
l (l=9) 19 512-pole 7Au⊕6Eu 55 19Au⊕18Eu
m (l=10) 21 1024-pole 7Ag⊕7Eg 66 22Ag⊕22Eg
n (l=11) 23 2048-pole 7Au⊕8Eu 78 26Au⊕26Eu
o (l=12) 25 4096-pole 9Ag⊕8Eg 91 31Ag⊕30Eg
More

First nonvanshing multipole: Quadrupole

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 S6
L 2L+1 Term Splitting
S (L=0) 1 Ag
P (L=1) 3 Ag⊕Eg
D (L=2) 5 Ag⊕2Eg
F (L=3) 7 3Ag⊕2Eg
G (L=4) 9 3Ag⊕3Eg
H (L=5) 11 3Ag⊕4Eg
I (L=6) 13 5Ag⊕4Eg


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