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''


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 l(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''

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