Point Group C12



C12 E C12 C6 C4 C3 (C12)5 C2 (C12)7 (C3)2 (C4)3 (C6)5 (C12)11
A 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
E1* 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 -2cos(π/6) -1 0 1 2cos(π/6)
E2* 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1
E3* 2 0 -2 0 2 0 -2 0 2 0 -2 0
E4* 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1
E5* 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 2cos(π/6) -1 0 1 -2cos(π/6)


Additional information

Number of symmetry elements h = 12
Number of classes, irreps n = 12
Number of real-valued irreducible representations n = 7
Abelian group yes
Optical Isomerism (Chirality) yes
Polar yes
Parity no


Reduce representation to irreducible representations


E C12 C6 C4 C3 (C12)5 C2 (C12)7 (C3)2 (C4)3 (C6)5 (C12)11



Genrate representation from irreducible representations


A B E1* E2* E3* E4* E5*




Direct products of irreducible representations


Binary products
A B E1* E2* E3* E4* E5*
A A
B BA
E1* E1E52A⊕E2
E2* E2E4E1⊕E32A⊕E4
E3* E3E3E2⊕E4E1⊕E52A⊕2B
E4* E4E2E3⊕E52B⊕E2E1⊕E52A⊕E4
E5* E5E12B⊕E4E3⊕E5E2⊕E4E1⊕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⊕E2⊕E4E1⊕E3⊕E5A⊕2B⊕E2⊕E4More
E2* A⊕E42B⊕E2A⊕2E42B⊕2E23A⊕2E4More
E3* A⊕2B2E33A⊕2B3E33A⊕4BMore
E4* A⊕E42A⊕E4A⊕2E42A⊕2E43A⊕2E4More
E5* A⊕E2E3⊕E5A⊕E2⊕E4E1⊕E3⊕E5A⊕2B⊕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 A 1 A
p (l=1) 3 Dipole A⊕E1 3 A⊕E1
d (l=2) 5 Quadrupole A⊕E1⊕E2 6 2A⊕E1⊕E2
f (l=3) 7 Octupole A⊕E1⊕E2⊕E3 10 2A⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A⊕E1⊕E2⊕E3⊕E4 15 3A⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A⊕2B⊕E1⊕E2⊕E3⊕E4⊕E5 28 4A⊕2B⊕3E1⊕3E2⊕2E3⊕2E4⊕E5
j (l=7) 15 Octacosahectapole A⊕2B⊕E1⊕E2⊕E3⊕E4⊕2E5 36 4A⊕2B⊕4E1⊕3E2⊕3E3⊕2E4⊕3E5
k (l=8) 17 256-pole A⊕2B⊕E1⊕E2⊕E3⊕2E4⊕2E5 45 5A⊕4B⊕4E1⊕4E2⊕3E3⊕4E4⊕3E5
l (l=9) 19 512-pole A⊕2B⊕E1⊕E2⊕2E3⊕2E4⊕2E5 55 5A⊕4B⊕5E1⊕4E2⊕5E3⊕4E4⊕5E5
m (l=10) 21 1024-pole A⊕2B⊕E1⊕2E2⊕2E3⊕2E4⊕2E5 66 6A⊕6B⊕5E1⊕6E2⊕5E3⊕6E4⊕5E5
n (l=11) 23 2048-pole A⊕2B⊕2E1⊕2E2⊕2E3⊕2E4⊕2E5 78 6A⊕6B⊕7E1⊕6E2⊕7E3⊕6E4⊕7E5
o (l=12) 25 4096-pole 3A⊕2B⊕2E1⊕2E2⊕2E3⊕2E4⊕2E5 91 9A⊕8B⊕7E1⊕8E2⊕7E3⊕8E4⊕7E5
More

First nonvanshing multipole: Dipole

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 C12
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E1
D (L=2) 5 A⊕E1⊕E2
F (L=3) 7 A⊕E1⊕E2⊕E3
G (L=4) 9 A⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A⊕E1⊕E2⊕E3⊕E4⊕E5
I (L=6) 13 A⊕2B⊕E1⊕E2⊕E3⊕E4⊕E5


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