Point Group D12



D12 E 2C12 2C6 2C4 2C3 2(C12)5 C2 6C'2 6C''2
A1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1 -1 1
E1 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 0 0
E2 2 1 -1 -2 -1 1 2 0 0
E3 2 0 -2 0 2 0 -2 0 0
E4 2 -1 -1 2 -1 -1 2 0 0
E5 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 0 0


Additional information

Number of symmetry elements h = 24
Number of classes, irreps n = 9
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 2C12 2C6 2C4 2C3 2(C12)5 C2 6C'2 6C''2



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5




Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2 E3 E4 E5
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E5E5A1⊕A2⊕E2
E2 E2E2E4E4E1⊕E3A1⊕A2⊕E4
E3 E3E3E3E3E2⊕E4E1⊕E5A1⊕A2⊕B1⊕B2
E4 E4E4E2E2E3⊕E5B1⊕B2⊕E2E1⊕E5A1⊕A2⊕E4
E5 E5E5E1E1B1⊕B2⊕E4E3⊕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⊕B1⊕B2⊕E2⊕E4More
E2 A1⊕E4B1⊕B2⊕E2A1⊕2E4B1⊕B2⊕2E22A1⊕A2⊕2E4More
E3 A1⊕B1⊕B22E32A1⊕A2⊕B1⊕B23E32A1⊕A2⊕2B1⊕2B2More
E4 A1⊕E4A1⊕A2⊕E4A1⊕2E4A1⊕A2⊕2E42A1⊕A2⊕2E4More
E5 A1⊕E2E3⊕E5A1⊕E2⊕E4E1⊕E3⊕E5A1⊕B1⊕B2⊕E2⊕E4More



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⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5 28 4A1⊕B1⊕B2⊕3E1⊕3E2⊕2E3⊕2E4⊕E5
j (l=7) 15 Octacosahectapole A2⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕2E5 36 4A2⊕B1⊕B2⊕4E1⊕3E2⊕3E3⊕2E4⊕3E5
k (l=8) 17 256-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕2E4⊕2E5 45 5A1⊕2B1⊕2B2⊕4E1⊕4E2⊕3E3⊕4E4⊕3E5
l (l=9) 19 512-pole A2⊕B1⊕B2⊕E1⊕E2⊕2E3⊕2E4⊕2E5 55 5A2⊕2B1⊕2B2⊕5E1⊕4E2⊕5E3⊕4E4⊕5E5
m (l=10) 21 1024-pole A1⊕B1⊕B2⊕E1⊕2E2⊕2E3⊕2E4⊕2E5 66 6A1⊕3B1⊕3B2⊕5E1⊕6E2⊕5E3⊕6E4⊕5E5
n (l=11) 23 2048-pole A2⊕B1⊕B2⊕2E1⊕2E2⊕2E3⊕2E4⊕2E5 78 6A2⊕3B1⊕3B2⊕7E1⊕6E2⊕7E3⊕6E4⊕7E5
o (l=12) 25 4096-pole 2A1⊕A2⊕B1⊕B2⊕2E1⊕2E2⊕2E3⊕2E4⊕2E5 91 8A1⊕A2⊕4B1⊕4B2⊕7E1⊕8E2⊕7E3⊕8E4⊕7E5
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 D12
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⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5


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