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




Examples

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
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 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