Point Group D20



D20 E 2C20 2C10 2(C20)3 2C5 2C4 2(C10)3 2(C20)7 2(C5)2 2(C20)9 C2 10C'2 10C''2
A1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1
B2 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 0 0
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 0 0
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 0 0
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 0 0
E5 2 0 -2 0 2 0 -2 0 2 0 -2 0 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 0 0
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 0 0
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 0 0
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 0 0


Additional information

Number of symmetry elements h = 40
Number of classes, irreps n = 13
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 2C20 2C10 2(C20)3 2C5 2C4 2(C10)3 2(C20)7 2(C5)2 2(C20)9 C2 10C'2 10C''2



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9




Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E9E9A1⊕A2⊕E2
E2 E2E2E8E8E1⊕E3A1⊕A2⊕E4
E3 E3E3E7E7E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E6E6E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕E8
E5 E5E5E5E5E4⊕E6E3⊕E7E2⊕E8E1⊕E9A1⊕A2⊕B1⊕B2
E6 E6E6E4E4E5⊕E7E4⊕E8E3⊕E9B1⊕B2⊕E2E1⊕E9A1⊕A2⊕E8
E7 E7E7E3E3E6⊕E8E5⊕E9B1⊕B2⊕E4E3⊕E9E2⊕E8E1⊕E7A1⊕A2⊕E6
E8 E8E8E2E2E7⊕E9B1⊕B2⊕E6E5⊕E9E4⊕E8E3⊕E7E2⊕E6E1⊕E5A1⊕A2⊕E4
E9 E9E9E1E1B1⊕B2⊕E8E7⊕E9E6⊕E8E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕E3A1⊕A2⊕E2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E4E1⊕E3⊕E5A1⊕E2⊕E4⊕E6More
E2 A1⊕E4E2⊕E6A1⊕E4⊕E8B1⊕B2⊕E2⊕E6A1⊕E4⊕2E8More
E3 A1⊕E6E3⊕E9A1⊕E6⊕E8E3⊕E5⊕E9A1⊕E2⊕E6⊕E8More
E4 A1⊕E8E4⊕E8A1⊕E4⊕E8A1⊕A2⊕E4⊕E8A1⊕2E4⊕E8More
E5 A1⊕B1⊕B22E52A1⊕A2⊕B1⊕B23E52A1⊕A2⊕2B1⊕2B2More
E6 A1⊕E8E2⊕E6A1⊕E4⊕E8B1⊕B2⊕E2⊕E6A1⊕2E4⊕E8More
E7 A1⊕E6E1⊕E7A1⊕E6⊕E8E1⊕E5⊕E7A1⊕E2⊕E6⊕E8More
E8 A1⊕E4E4⊕E8A1⊕E4⊕E8A1⊕A2⊕E4⊕E8A1⊕E4⊕2E8More
E9 A1⊕E2E7⊕E9A1⊕E2⊕E4E5⊕E7⊕E9A1⊕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 l(xyz)]
s (l=0) 1 Monopole A1 1 A1
p (l=1) 3 Dipole A2⊕E1 3 A2⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A2⊕E1⊕E2⊕E3 10 2A2⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕E1⊕E2⊕E3⊕E4 15 3A1⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A2⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A2⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6 28 4A1⊕3E1⊕3E2⊕2E3⊕2E4⊕E5⊕E6
j (l=7) 15 Octacosahectapole A2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4A2⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 45 5A1⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7⊕E8
l (l=9) 19 512-pole A2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 55 5A2⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕2E7⊕E8⊕E9
m (l=10) 21 1024-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 66 6A1⊕B1⊕B2⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕3E6⊕2E7⊕2E8⊕E9
n (l=11) 23 2048-pole A2⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕2E9 78 6A2⊕B1⊕B2⊕6E1⊕5E2⊕5E3⊕4E4⊕4E5⊕3E6⊕3E7⊕2E8⊕3E9
o (l=12) 25 4096-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕2E8⊕2E9 91 7A1⊕2B1⊕2B2⊕6E1⊕6E2⊕5E3⊕5E4⊕4E5⊕4E6⊕3E7⊕4E8⊕3E9
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 D20
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E1
D (L=2) 5 A1⊕E1⊕E2
F (L=3) 7 A2⊕E1⊕E2⊕E3
G (L=4) 9 A1⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A2⊕E1⊕E2⊕E3⊕E4⊕E5
I (L=6) 13 A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6


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