Point Group S8



S8 E S8 C4 (S8)3 C2 (S8)5 (C4)3 (S8)7
A 1 1 1 1 1 1 1 1
B 1 -1 1 -1 1 -1 1 -1
E1* 2 2cos(π/4) 0 -2cos(π/4) -2 -2cos(π/4) 0 2cos(π/4)
E2* 2 0 -2 0 2 0 -2 0
E3* 2 -2cos(π/4) 0 2cos(π/4) -2 2cos(π/4) 0 -2cos(π/4)


Additional information

Number of symmetry elements h = 8
Number of classes, irreps n = 8
Number of real-valued irreducible representations n = 5
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E S8 C4 (S8)3 C2 (S8)5 (C4)3 (S8)7



Genrate representation from irreducible representations


A B E1* E2* E3*




Direct products of irreducible representations


Binary products
A B E1* E2* E3*
A A
B BA
E1* E1E32A⊕E2
E2* E2E2E1⊕E32A⊕2B
E3* E3E12B⊕E2E1⊕E32A⊕E2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1* A⊕E2E1⊕E3A⊕2B⊕E2E1⊕2E3A⊕2B⊕2E2More
E2* A⊕2B2E23A⊕2B3E23A⊕4BMore
E3* A⊕E2E1⊕E3A⊕2B⊕E22E1⊕E3A⊕2B⊕2E2More



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 A 1 A
p (l=1) 3 Dipole B⊕E1 3 B⊕E1
d (l=2) 5 Quadrupole A⊕E2⊕E3 6 2A⊕E2⊕E3
f (l=3) 7 Octupole B⊕E1⊕E2⊕E3 10 2B⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A⊕2B⊕E1⊕E2⊕E3 15 3A⊕2B⊕E1⊕2E2⊕2E3
h (l=5) 11 Dotricontapole 2A⊕B⊕E1⊕E2⊕2E3 21 2A⊕3B⊕3E1⊕2E2⊕3E3
i (l=6) 13 Tetrahexacontapole A⊕2B⊕2E1⊕2E2⊕E3 28 4A⊕4B⊕3E1⊕4E2⊕3E3
j (l=7) 15 Octacosahectapole 2A⊕B⊕2E1⊕2E2⊕2E3 36 4A⊕4B⊕5E1⊕4E2⊕5E3
k (l=8) 17 256-pole 3A⊕2B⊕2E1⊕2E2⊕2E3 45 7A⊕6B⊕5E1⊕6E2⊕5E3
l (l=9) 19 512-pole 2A⊕3B⊕3E1⊕2E2⊕2E3 55 6A⊕7B⊕8E1⊕6E2⊕7E3
m (l=10) 21 1024-pole 3A⊕2B⊕2E1⊕3E2⊕3E3 66 10A⊕8B⊕7E1⊕9E2⊕8E3
n (l=11) 23 2048-pole 2A⊕3B⊕3E1⊕3E2⊕3E3 78 8A⊕10B⊕11E1⊕9E2⊕10E3
o (l=12) 25 4096-pole 3A⊕4B⊕3E1⊕3E2⊕3E3 91 13A⊕12B⊕10E1⊕12E2⊕11E3
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 S8
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E3
D (L=2) 5 A⊕E2⊕E3
F (L=3) 7 A⊕E1⊕E2⊕E3
G (L=4) 9 A⊕2B⊕E1⊕E2⊕E3
H (L=5) 11 A⊕2B⊕2E1⊕E2⊕E3
I (L=6) 13 A⊕2B⊕2E1⊕2E2⊕E3


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