Point Group C2h

C2h E C2 i σh
Ag 1 1 1 1
Bg 1 -1 1 -1
Au 1 1 -1 -1
Bu 1 -1 -1 1

Additional information

Number of symmetry elements h = 4
Number of classes, irreps n = 4
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes

Reduce representation to irreducible representations

E C2 i σh

Genrate representation from irreducible representations

Ag Bg Au Bu


1,2-Dichloroethylene (trans) Oxalic Acid Butadiene (s-trans)
Butane 1,5-Diazanaphthalene Chrysene

Direct products of irreducible representations

Binary products
Ag Bg Au Bu
Ag Ag
Bg BgAg
Au AuBuAg
Bu BuAuBgAg

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 Ag 1 Ag
p (l=1) 3 Dipole Au⊕2Bu 3 Au⊕2Bu
d (l=2) 5 Quadrupole 3Ag⊕2Bg 6 4Ag⊕2Bg
f (l=3) 7 Octupole 3Au⊕4Bu 10 4Au⊕6Bu
g (l=4) 9 Hexadecapole 5Ag⊕4Bg 15 9Ag⊕6Bg
h (l=5) 11 Dotricontapole 5Au⊕6Bu 21 9Au⊕12Bu
i (l=6) 13 Tetrahexacontapole 7Ag⊕6Bg 28 16Ag⊕12Bg
j (l=7) 15 Octacosahectapole 7Au⊕8Bu 36 16Au⊕20Bu
k (l=8) 17 256-pole 9Ag⊕8Bg 45 25Ag⊕20Bg
l (l=9) 19 512-pole 9Au⊕10Bu 55 25Au⊕30Bu
m (l=10) 21 1024-pole 11Ag⊕10Bg 66 36Ag⊕30Bg
n (l=11) 23 2048-pole 11Au⊕12Bu 78 36Au⊕42Bu
o (l=12) 25 4096-pole 13Ag⊕12Bg 91 49Ag⊕42Bg

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 C2h
L 2L+1 Term Splitting
S (L=0) 1 Ag
P (L=1) 3 Ag⊕2Bg
D (L=2) 5 3Ag⊕2Bg
F (L=3) 7 3Ag⊕4Bg
G (L=4) 9 5Ag⊕4Bg
H (L=5) 11 5Ag⊕6Bg
I (L=6) 13 7Ag⊕6Bg

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