Point Group S20



S20 E S20 C10 (S20)3 C5 S4 (C10)3 (S20)7 (C5)2 (S20)9 C2 (S20)11 (C5)3 (S20)13 (C10)7 (S4)3 (C5)4 (S20)17 (C10)9 (S20)19
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1* 2 2cos(π/10) 2cos(π/5) 2cos(3π/10) 2cos(2π/5) 0 -2cos(2π/5) -2cos(3π/10) -2cos(π/5) -2cos(π/10) -2 -2cos(π/10) -2cos(π/5) -2cos(3π/10) -2cos(2π/5) 0 2cos(2π/5) 2cos(3π/10) 2cos(π/5) 2cos(π/10)
E2* 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 -2cos(π/5) -2cos(2π/5) 2cos(2π/5) 2cos(π/5) 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 -2cos(π/5) -2cos(2π/5) 2cos(2π/5) 2cos(π/5)
E3* 2 2cos(3π/10) -2cos(2π/5) -2cos(π/10) -2cos(π/5) 0 2cos(π/5) 2cos(π/10) 2cos(2π/5) -2cos(3π/10) -2 -2cos(3π/10) 2cos(2π/5) 2cos(π/10) 2cos(π/5) 0 -2cos(π/5) -2cos(π/10) -2cos(2π/5) 2cos(3π/10)
E4* 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5)
E5* 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0
E6* 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 2cos(2π/5) 2cos(π/5) -2cos(π/5) -2cos(2π/5) 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 2cos(2π/5) 2cos(π/5) -2cos(π/5) -2cos(2π/5)
E7* 2 -2cos(3π/10) -2cos(2π/5) 2cos(π/10) -2cos(π/5) 0 2cos(π/5) -2cos(π/10) 2cos(2π/5) 2cos(3π/10) -2 2cos(3π/10) 2cos(2π/5) -2cos(π/10) 2cos(π/5) 0 -2cos(π/5) 2cos(π/10) -2cos(2π/5) -2cos(3π/10)
E8* 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5)
E9* 2 -2cos(π/10) 2cos(π/5) -2cos(3π/10) 2cos(2π/5) 0 -2cos(2π/5) 2cos(3π/10) -2cos(π/5) 2cos(π/10) -2 2cos(π/10) -2cos(π/5) 2cos(3π/10) -2cos(2π/5) 0 2cos(2π/5) -2cos(3π/10) 2cos(π/5) -2cos(π/10)
Math View
Phys View A
✅
Phys View B
Chem View A
Chem View B


Additional information

Number of symmetry elements h = 20
Number of classes, irreps n = 20
Number of real-valued irreducible representations n = 11
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E S20 C10 (S20)3 C5 S4 (C10)3 (S20)7 (C5)2 (S20)9 C2 (S20)11 (C5)3 (S20)13 (C10)7 (S4)3 (C5)4 (S20)17 (C10)9 (S20)19



Genrate representation from irreducible representations


A B E1* E2* E3* E4* E5* E6* E7* E8* E9*



Direct products of irreducible representations


Binary products
⊙ A B E1* E2* E3* E4* E5* E6* E7* E8* E9*
A A
B BA
E1* E1E92A⊕E2
E2* E2E8E1⊕E32A⊕E4
E3* E3E7E2⊕E4E1⊕E52A⊕E6
E4* E4E6E3⊕E5E2⊕E6E1⊕E72A⊕E8
E5* E5E5E4⊕E6E3⊕E7E2⊕E8E1⊕E92A⊕2B
E6* E6E4E5⊕E7E4⊕E8E3⊕E92B⊕E2E1⊕E92A⊕E8
E7* E7E3E6⊕E8E5⊕E92B⊕E4E3⊕E9E2⊕E8E1⊕E72A⊕E6
E8* E8E2E7⊕E92B⊕E6E5⊕E9E4⊕E8E3⊕E7E2⊕E6E1⊕E52A⊕E4
E9* E9E12B⊕E8E7⊕E9E6⊕E8E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕E32A⊕E2

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1* A⊕E2E1⊕E3A⊕E2⊕E4E1⊕E3⊕E5A⊕E2⊕E4⊕E6More
E2* A⊕E4E2⊕E6A⊕E4⊕E82B⊕E2⊕E6A⊕E4⊕2E8More
E3* A⊕E6E3⊕E9A⊕E6⊕E8E3⊕E5⊕E9A⊕E2⊕E6⊕E8More
E4* A⊕E8E4⊕E8A⊕E4⊕E82A⊕E4⊕E8A⊕2E4⊕E8More
E5* A⊕2B2E53A⊕2B3E53A⊕4BMore
E6* A⊕E8E2⊕E6A⊕E4⊕E82B⊕E2⊕E6A⊕2E4⊕E8More
E7* A⊕E6E1⊕E7A⊕E6⊕E8E1⊕E5⊕E7A⊕E2⊕E6⊕E8More
E8* A⊕E4E4⊕E8A⊕E4⊕E82A⊕E4⊕E8A⊕E4⊕2E8More
E9* A⊕E2E7⊕E9A⊕E2⊕E4E5⊕E7⊕E9A⊕E2⊕E4⊕E6More


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 A 1 A
p (l=1) 3 Dipole B⊕E1 3 B⊕E1
d (l=2) 5 Quadrupole A⊕E2⊕E9 6 2A⊕E2⊕E9
f (l=3) 7 Octupole B⊕E1⊕E3⊕E8 10 2B⊕2E1⊕E3⊕E8
g (l=4) 9 Hexadecapole A⊕E2⊕E4⊕E7⊕E9 15 3A⊕2E2⊕E4⊕E7⊕2E9
h (l=5) 11 Dotricontapole B⊕E1⊕E3⊕E5⊕E6⊕E8 21 3B⊕3E1⊕2E3⊕E5⊕E6⊕2E8
i (l=6) 13 Tetrahexacontapole A⊕E2⊕E4⊕E5⊕E6⊕E7⊕E9 28 4A⊕3E2⊕2E4⊕E5⊕E6⊕2E7⊕3E9
j (l=7) 15 Octacosahectapole B⊕E1⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 36 4B⊕4E1⊕3E3⊕E4⊕2E5⊕2E6⊕E7⊕3E8
k (l=8) 17 256-pole A⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 45 5A⊕4E2⊕E3⊕3E4⊕2E5⊕2E6⊕3E7⊕E8⊕4E9
l (l=9) 19 512-pole B⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 55 5B⊕5E1⊕E2⊕4E3⊕2E4⊕3E5⊕3E6⊕2E7⊕4E8⊕E9
m (l=10) 21 1024-pole A⊕2B⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 66 6A⊕2B⊕E1⊕5E2⊕2E3⊕4E4⊕3E5⊕3E6⊕4E7⊕2E8⊕5E9
n (l=11) 23 2048-pole 2A⊕B⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕2E9 78 2A⊕6B⊕6E1⊕2E2⊕5E3⊕3E4⊕4E5⊕4E6⊕3E7⊕5E8⊕3E9
o (l=12) 25 4096-pole A⊕2B⊕2E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕2E8⊕E9 91 7A⊕4B⊕3E1⊕6E2⊕3E3⊕5E4⊕4E5⊕4E6⊕5E7⊕4E8⊕6E9
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 S20
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E9
D (L=2) 5 A⊕E2⊕E9
F (L=3) 7 A⊕E2⊕E7⊕E9
G (L=4) 9 A⊕E2⊕E4⊕E7⊕E9
H (L=5) 11 A⊕E2⊕E4⊕E5⊕E7⊕E9
I (L=6) 13 A⊕E2⊕E4⊕E5⊕E6⊕E7⊕E9

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