Point Group C6v



C6v E 2C6 2C3 C2 3σv 3σd
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 3σv 3σd



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 [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
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 [Γl(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 November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement