Point Group D11



D11 E 2C11 2(C11)2 2(C11)3 2(C11)4 2(C11)5 11C'2
A1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 -1
E1 2 2cos(2π/11) 2cos(4π/11) 2cos(6π/11) 2cos(8π/11) 2cos(10π/11) 0
E2 2 2cos(4π/11) 2cos(8π/11) 2cos(10π/11) 2cos(6π/11) 2cos(2π/11) 0
E3 2 2cos(6π/11) 2cos(10π/11) 2cos(4π/11) 2cos(2π/11) 2cos(8π/11) 0
E4 2 2cos(8π/11) 2cos(6π/11) 2cos(2π/11) 2cos(10π/11) 2cos(4π/11) 0
E5 2 2cos(10π/11) 2cos(2π/11) 2cos(8π/11) 2cos(4π/11) 2cos(6π/11) 0
View A
✅
View B


Additional information

Number of symmetry elements h = 22
Number of classes, irreps n = 7
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 2C11 2(C11)2 2(C11)3 2(C11)4 2(C11)5 11C'2



Genrate representation from irreducible representations


A1 A2 E1 E2 E3 E4 E5



Direct products of irreducible representations


Binary products
⊙ A1 A2 E1 E2 E3 E4 E5
A1 A1
A2 A2A1
E1 E1E1A1⊕A2⊕E2
E2 E2E2E1⊕E3A1⊕A2⊕E4
E3 E3E3E2⊕E4E1⊕E5A1⊕A2⊕E5
E4 E4E4E3⊕E5E2⊕E5E1⊕E4A1⊕A2⊕E3
E5 E5E5E4⊕E5E3⊕E4E2⊕E3E1⊕E2A1⊕A2⊕E1

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E4E1⊕E3⊕E5A1⊕E2⊕E4⊕E5More
E2 A1⊕E4E2⊕E5A1⊕E3⊕E4E1⊕E2⊕E5A1⊕E1⊕E3⊕E4More
E3 A1⊕E5E2⊕E3A1⊕E1⊕E5E2⊕E3⊕E4A1⊕E1⊕E4⊕E5More
E4 A1⊕E3E1⊕E4A1⊕E3⊕E5E1⊕E2⊕E4A1⊕E2⊕E3⊕E5More
E5 A1⊕E1E4⊕E5A1⊕E1⊕E2E3⊕E4⊕E5A1⊕E1⊕E2⊕E3More


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 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⊕2E5 28 4A1⊕3E1⊕3E2⊕2E3⊕2E4⊕2E5
j (l=7) 15 Octacosahectapole A2⊕E1⊕E2⊕E3⊕2E4⊕2E5 36 4A2⊕4E1⊕3E2⊕3E3⊕3E4⊕3E5
k (l=8) 17 256-pole A1⊕E1⊕E2⊕2E3⊕2E4⊕2E5 45 5A1⊕4E1⊕4E2⊕4E3⊕4E4⊕4E5
l (l=9) 19 512-pole A2⊕E1⊕2E2⊕2E3⊕2E4⊕2E5 55 5A2⊕5E1⊕5E2⊕5E3⊕5E4⊕5E5
m (l=10) 21 1024-pole A1⊕2E1⊕2E2⊕2E3⊕2E4⊕2E5 66 6A1⊕6E1⊕6E2⊕6E3⊕6E4⊕6E5
n (l=11) 23 2048-pole A1⊕2A2⊕2E1⊕2E2⊕2E3⊕2E4⊕2E5 78 A1⊕7A2⊕7E1⊕7E2⊕7E3⊕7E4⊕7E5
o (l=12) 25 4096-pole 2A1⊕A2⊕3E1⊕2E2⊕2E3⊕2E4⊕2E5 91 8A1⊕A2⊕9E1⊕8E2⊕8E3⊕8E4⊕8E5
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 D11
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⊕2E5

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