Point Group C3h



ε=exp(2πi/3)
C3h E C3 (C3)2 σh S3 (S3)5
A' 1 1 1 1 1 1
E'* 1
1		 ε*
ε*		 ε*
ε*		 1
1		 ε*
ε*		 ε*
ε*
A'' 1 1 1 -1 -1 -1
E''* 1
1		 ε*
ε*		 ε*
ε*		 -1
-1		 -ε*
-ε*		 -ε*
-ε*
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Additional information

Number of symmetry elements h = 6
Number of classes, irreps n = 6
Number of real-valued irreducible representations n = 4
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E C3 (C3)2 σh S3 (S3)5



Genrate representation from irreducible representations


A' E'* A'' E''*



Examples

Boric Acid 1,3,5-Trihydroxybenzene Tricyclo[3.3.3.0(1,5)]undecane
Bicyclo[3.3.3]undecane


Direct products of irreducible representations


Binary products
⊙ A' E'* A'' E''*
A' A'
E'* E'2A'⊕E'
A'' A''E''A'
E''* E''2A''⊕E''E'2A'⊕E'

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E'* A'⊕E'2A'⊕E'A'⊕2E'2A'⊕2E'3A'⊕2E'More
E''* A'⊕E'2A''⊕E''A'⊕2E'2A''⊕2E''3A'⊕2E'More


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 E'⊕A'' 3 E'⊕A''
d (l=2) 5 Quadrupole A'⊕E'⊕E'' 6 2A'⊕E'⊕E''
f (l=3) 7 Octupole 2A'⊕E'⊕A''⊕E'' 10 2A'⊕2E'⊕2A''⊕E''
g (l=4) 9 Hexadecapole A'⊕2E'⊕2A''⊕E'' 15 3A'⊕3E'⊕2A''⊕2E''
h (l=5) 11 Dotricontapole 2A'⊕2E'⊕A''⊕2E'' 21 4A'⊕4E'⊕3A''⊕3E''
i (l=6) 13 Tetrahexacontapole 3A'⊕2E'⊕2A''⊕2E'' 28 6A'⊕5E'⊕4A''⊕4E''
j (l=7) 15 Octacosahectapole 2A'⊕3E'⊕3A''⊕2E'' 36 6A'⊕7E'⊕6A''⊕5E''
k (l=8) 17 256-pole 3A'⊕3E'⊕2A''⊕3E'' 45 9A'⊕8E'⊕6A''⊕7E''
l (l=9) 19 512-pole 4A'⊕3E'⊕3A''⊕3E'' 55 10A'⊕10E'⊕9A''⊕8E''
m (l=10) 21 1024-pole 3A'⊕4E'⊕4A''⊕3E'' 66 12A'⊕12E'⊕10A''⊕10E''
n (l=11) 23 2048-pole 4A'⊕4E'⊕3A''⊕4E'' 78 14A'⊕14E'⊕12A''⊕12E''
o (l=12) 25 4096-pole 5A'⊕4E'⊕4A''⊕4E'' 91 17A'⊕16E'⊕14A''⊕14E''
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 C3h
L 2L+1 Term Splitting
S (L=0) 1 A'
P (L=1) 3 A'⊕E''
D (L=2) 5 A'⊕E'⊕E''
F (L=3) 7 A'⊕E'⊕2A''⊕E''
G (L=4) 9 A'⊕2E'⊕2A''⊕E''
H (L=5) 11 A'⊕2E'⊕2A''⊕2E''
I (L=6) 13 3A'⊕2E'⊕2A''⊕2E''

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