Point Group C8v



C8v E 2C8 2C4 2(C8)3 C2 4σv 4σd
A1 1 1 1 1 1 1 1
A2 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1
E1 2 1.4142 0 -1.4142 -2 0 0
E2 2 0 -2 0 2 0 0
E3 2 -1.4142 0 1.4142 -2 0 0
View A
View B
✅


Additional information

Number of symmetry elements h = 16
Number of classes, irreps n = 7
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C8 2C4 2(C8)3 C2 4σv 4σd



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3



Direct products of irreducible representations


Binary products
⊙ A1 A2 B1 B2 E1 E2 E3
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E3E3A1⊕A2⊕E2
E2 E2E2E2E2E1⊕E3A1⊕A2⊕B1⊕B2
E3 E3E3E1E1B1⊕B2⊕E2E1⊕E3A1⊕A2⊕E2

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1 A1⊕E2E1⊕E3A1⊕B1⊕B2⊕E2E1⊕2E3A1⊕B1⊕B2⊕2E2More
E2 A1⊕B1⊕B22E22A1⊕A2⊕B1⊕B23E22A1⊕A2⊕2B1⊕2B2More
E3 A1⊕E2E1⊕E3A1⊕B1⊕B2⊕E22E1⊕E3A1⊕B1⊕B2⊕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⊕E1⊕E2⊕E3 10 2A1⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕B1⊕B2⊕E1⊕E2⊕E3 15 3A1⊕B1⊕B2⊕2E1⊕2E2⊕E3
h (l=5) 11 Dotricontapole A1⊕B1⊕B2⊕E1⊕E2⊕2E3 21 3A1⊕B1⊕B2⊕3E1⊕2E2⊕3E3
i (l=6) 13 Tetrahexacontapole A1⊕B1⊕B2⊕E1⊕2E2⊕2E3 28 4A1⊕2B1⊕2B2⊕3E1⊕4E2⊕3E3
j (l=7) 15 Octacosahectapole A1⊕B1⊕B2⊕2E1⊕2E2⊕2E3 36 4A1⊕2B1⊕2B2⊕5E1⊕4E2⊕5E3
k (l=8) 17 256-pole 2A1⊕A2⊕B1⊕B2⊕2E1⊕2E2⊕2E3 45 6A1⊕A2⊕3B1⊕3B2⊕5E1⊕6E2⊕5E3
l (l=9) 19 512-pole 2A1⊕A2⊕B1⊕B2⊕3E1⊕2E2⊕2E3 55 6A1⊕A2⊕3B1⊕3B2⊕8E1⊕6E2⊕7E3
m (l=10) 21 1024-pole 2A1⊕A2⊕B1⊕B2⊕3E1⊕3E2⊕2E3 66 8A1⊕2A2⊕4B1⊕4B2⊕8E1⊕9E2⊕7E3
n (l=11) 23 2048-pole 2A1⊕A2⊕B1⊕B2⊕3E1⊕3E2⊕3E3 78 8A1⊕2A2⊕4B1⊕4B2⊕11E1⊕9E2⊕10E3
o (l=12) 25 4096-pole 2A1⊕A2⊕2B1⊕2B2⊕3E1⊕3E2⊕3E3 91 10A1⊕3A2⊕6B1⊕6B2⊕11E1⊕12E2⊕10E3
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 C8v
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⊕E1⊕E2⊕E3
G (L=4) 9 A1⊕B1⊕B2⊕E1⊕E2⊕E3
H (L=5) 11 A2⊕B1⊕B2⊕E1⊕E2⊕2E3
I (L=6) 13 A1⊕B1⊕B2⊕E1⊕2E2⊕2E3

Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement