Point Group C6v



C6v E 2C6 2C3 C2 v d
A1 1 1 1 1 1 1
A2 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1
B2 1 -1 1 -1 -1 1
E1 2 1 -1 -2 0 0
E2 2 -1 -1 2 0 0


Additional information

Number of symmetry elements h = 12
Number of classes, irreps n = 6
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C6 2C3 C2 v d



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2




Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E2E2A1⊕A2⊕E2
E2 E2E2E1E1B1⊕B2⊕E1A1⊕A2⊕E2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1 A1⊕E2B1⊕B2⊕E1A1⊕2E2B1⊕B2⊕2E12A1⊕A2⊕2E2More
E2 A1⊕E2A1⊕A2⊕E2A1⊕2E2A1⊕A2⊕2E22A1⊕A2⊕2E2More



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⊕E1 3 A1⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A1⊕B1⊕B2⊕E1⊕E2 10 2A1⊕B1⊕B2⊕2E1⊕E2
g (l=4) 9 Hexadecapole A1⊕B1⊕B2⊕E1⊕2E2 15 3A1⊕B1⊕B2⊕2E1⊕3E2
h (l=5) 11 Dotricontapole A1⊕B1⊕B2⊕2E1⊕2E2 21 3A1⊕2B1⊕2B2⊕4E1⊕3E2
i (l=6) 13 Tetrahexacontapole 2A1⊕A2⊕B1⊕B2⊕2E1⊕2E2 28 5A1⊕A2⊕2B1⊕2B2⊕4E1⊕5E2
j (l=7) 15 Octacosahectapole 2A1⊕A2⊕B1⊕B2⊕3E1⊕2E2 36 5A1⊕A2⊕3B1⊕3B2⊕7E1⊕5E2
k (l=8) 17 256-pole 2A1⊕A2⊕B1⊕B2⊕3E1⊕3E2 45 7A1⊕2A2⊕3B1⊕3B2⊕7E1⊕8E2
l (l=9) 19 512-pole 2A1⊕A2⊕2B1⊕2B2⊕3E1⊕3E2 55 7A1⊕2A2⊕5B1⊕5B2⊕10E1⊕8E2
m (l=10) 21 1024-pole 2A1⊕A2⊕2B1⊕2B2⊕3E1⊕4E2 66 9A1⊕3A2⊕5B1⊕5B2⊕10E1⊕12E2
n (l=11) 23 2048-pole 2A1⊕A2⊕2B1⊕2B2⊕4E1⊕4E2 78 9A1⊕3A2⊕7B1⊕7B2⊕14E1⊕12E2
o (l=12) 25 4096-pole 3A1⊕2A2⊕2B1⊕2B2⊕4E1⊕4E2 91 12A1⊕5A2⊕7B1⊕7B2⊕14E1⊕16E2
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 C6v
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E1
D (L=2) 5 A1⊕E1⊕E2
F (L=3) 7 A2⊕B1⊕B2⊕E1⊕E2
G (L=4) 9 A1⊕B1⊕B2⊕E1⊕2E2
H (L=5) 11 A2⊕B1⊕B2⊕2E1⊕2E2
I (L=6) 13 2A1⊕A2⊕B1⊕B2⊕2E1⊕2E2


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