Point Group C3



C3 E C3 (C3)2
A 1 1 1
E* 2 -1 -1
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Additional information

Number of symmetry elements h = 3
Number of classes, irreps n = 3
Number of real-valued irreducible representations n = 2
Abelian group yes
Optical Isomerism (Chirality) yes
Polar yes
Parity no


Reduce representation to irreducible representations


E C3 (C3)2



Genrate representation from irreducible representations


A E*



Examples

1,1,1-Trichloroethane (rotated) Triphenylmethane


Direct products of irreducible representations


Binary products
⊙ A E*
A A
E* E2A⊕E

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E* A⊕E2A⊕EA⊕2E2A⊕2E3A⊕2EMore


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 A⊕E 3 A⊕E
d (l=2) 5 Quadrupole A⊕2E 6 2A⊕2E
f (l=3) 7 Octupole 3A⊕2E 10 4A⊕3E
g (l=4) 9 Hexadecapole 3A⊕3E 15 5A⊕5E
h (l=5) 11 Dotricontapole 3A⊕4E 21 7A⊕7E
i (l=6) 13 Tetrahexacontapole 5A⊕4E 28 10A⊕9E
j (l=7) 15 Octacosahectapole 5A⊕5E 36 12A⊕12E
k (l=8) 17 256-pole 5A⊕6E 45 15A⊕15E
l (l=9) 19 512-pole 7A⊕6E 55 19A⊕18E
m (l=10) 21 1024-pole 7A⊕7E 66 22A⊕22E
n (l=11) 23 2048-pole 7A⊕8E 78 26A⊕26E
o (l=12) 25 4096-pole 9A⊕8E 91 31A⊕30E
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 C3
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E
D (L=2) 5 A⊕2E
F (L=3) 7 3A⊕2E
G (L=4) 9 3A⊕3E
H (L=5) 11 3A⊕4E
I (L=6) 13 5A⊕4E

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