Point Group Cs



Cs E σh
A' 1 1
A'' 1 -1


Additional information

Number of symmetry elements h = 2
Number of classes, irreps n = 2
Abelian group yes
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E σh



Genrate representation from irreducible representations


A' A''



Examples

Hypochloric Acid Formylfluoride Sulfur S7
Acrolein Cyclopentane (envelope) Quinoline


Direct products of irreducible representations


Binary products
⊙ A' A''
A' A'
A'' A''A'

Ternary Products
Quaternary Products


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 2A'⊕A'' 3 2A'⊕A''
d (l=2) 5 Quadrupole 3A'⊕2A'' 6 4A'⊕2A''
f (l=3) 7 Octupole 4A'⊕3A'' 10 6A'⊕4A''
g (l=4) 9 Hexadecapole 5A'⊕4A'' 15 9A'⊕6A''
h (l=5) 11 Dotricontapole 6A'⊕5A'' 21 12A'⊕9A''
i (l=6) 13 Tetrahexacontapole 7A'⊕6A'' 28 16A'⊕12A''
j (l=7) 15 Octacosahectapole 8A'⊕7A'' 36 20A'⊕16A''
k (l=8) 17 256-pole 9A'⊕8A'' 45 25A'⊕20A''
l (l=9) 19 512-pole 10A'⊕9A'' 55 30A'⊕25A''
m (l=10) 21 1024-pole 11A'⊕10A'' 66 36A'⊕30A''
n (l=11) 23 2048-pole 12A'⊕11A'' 78 42A'⊕36A''
o (l=12) 25 4096-pole 13A'⊕12A'' 91 49A'⊕42A''
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 Cs
L 2L+1 Term Splitting
S (L=0) 1 A'
P (L=1) 3 A'⊕2A''
D (L=2) 5 3A'⊕2A''
F (L=3) 7 3A'⊕4A''
G (L=4) 9 5A'⊕4A''
H (L=5) 11 5A'⊕6A''
I (L=6) 13 7A'⊕6A''

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