Point Group C2v



C2v E C2.(z) σv.(xz) σd.(yz)
A1 1 1 1 1
A2 1 1 -1 -1
B1 1 -1 1 -1
B2 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 yes
Parity no


Reduce representation to irreducible representations


E C2.(z) σv.(xz) σd.(yz)



Genrate representation from irreducible representations


A1 A2 B1 B2




Examples

Water Formaldehyde Diazomethane
1,2-Dichloroethylene (cis) Trisulfurdinitrogendioxide Squaric Acid Difluoride
Butadiene (s-cis) Propane Cyclohexane (boat)
Azulene Phenanthrene



Direct products of irreducible representations


Binary products
A1 A2 B1 B2
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1

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 A1 1 A1
p (l=1) 3 Dipole A1⊕B1⊕B2 3 A1⊕B1⊕B2
d (l=2) 5 Quadrupole 2A1⊕A2⊕B1⊕B2 6 3A1⊕A2⊕B1⊕B2
f (l=3) 7 Octupole 2A1⊕A2⊕2B1⊕2B2 10 3A1⊕A2⊕3B1⊕3B2
g (l=4) 9 Hexadecapole 3A1⊕2A2⊕2B1⊕2B2 15 6A1⊕3A2⊕3B1⊕3B2
h (l=5) 11 Dotricontapole 3A1⊕2A2⊕3B1⊕3B2 21 6A1⊕3A2⊕6B1⊕6B2
i (l=6) 13 Tetrahexacontapole 4A1⊕3A2⊕3B1⊕3B2 28 10A1⊕6A2⊕6B1⊕6B2
j (l=7) 15 Octacosahectapole 4A1⊕3A2⊕4B1⊕4B2 36 10A1⊕6A2⊕10B1⊕10B2
k (l=8) 17 256-pole 5A1⊕4A2⊕4B1⊕4B2 45 15A1⊕10A2⊕10B1⊕10B2
l (l=9) 19 512-pole 5A1⊕4A2⊕5B1⊕5B2 55 15A1⊕10A2⊕15B1⊕15B2
m (l=10) 21 1024-pole 6A1⊕5A2⊕5B1⊕5B2 66 21A1⊕15A2⊕15B1⊕15B2
n (l=11) 23 2048-pole 6A1⊕5A2⊕6B1⊕6B2 78 21A1⊕15A2⊕21B1⊕21B2
o (l=12) 25 4096-pole 7A1⊕6A2⊕6B1⊕6B2 91 28A1⊕21A2⊕21B1⊕21B2
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 C2v
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕B1⊕B2
D (L=2) 5 2A1⊕A2⊕B1⊕B2
F (L=3) 7 A1⊕2A2⊕2B1⊕2B2
G (L=4) 9 3A1⊕2A2⊕2B1⊕2B2
H (L=5) 11 2A1⊕3A2⊕3B1⊕3B2
I (L=6) 13 4A1⊕3A2⊕3B1⊕3B2


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