Point Group C9v



C9v E 2C9 2(C9)2 2C3 2(C9)4 v
A1 1 1 1 1 1 1
A2 1 1 1 1 1 -1
E1 2 2cos(2π/9) 2cos(4π/9) -1 2cos(8π/9) 0
E2 2 2cos(4π/9) 2cos(8π/9) -1 2cos(2π/9) 0
E3 2 -1 -1 2 -1 0
E4 2 2cos(8π/9) 2cos(2π/9) -1 2cos(4π/9) 0


Additional information

Number of symmetry elements h = 18
Number of classes, irreps n = 6
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C9 2(C9)2 2C3 2(C9)4 v



Genrate representation from irreducible representations


A1 A2 E1 E2 E3 E4




Direct products of irreducible representations


Binary products
A1 A2 E1 E2 E3 E4
A1 A1
A2 A2A1
E1 E1E1A1⊕A2⊕E2
E2 E2E2E1⊕E3A1⊕A2⊕E4
E3 E3E3E2⊕E4E1⊕E4A1⊕A2⊕E3
E4 E4E4E3⊕E4E2⊕E3E1⊕E2A1⊕A2⊕E1

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E4E1⊕E3⊕E4A1⊕E2⊕E3⊕E4More
E2 A1⊕E4E2⊕E3A1⊕E1⊕E4E1⊕E2⊕E3A1⊕E1⊕E3⊕E4More
E3 A1⊕E3A1⊕A2⊕E3A1⊕2E3A1⊕A2⊕2E32A1⊕A2⊕2E3More
E4 A1⊕E1E3⊕E4A1⊕E1⊕E2E2⊕E3⊕E4A1⊕E1⊕E2⊕E3More



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⊕E1⊕E2⊕E3 10 2A1⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕E1⊕E2⊕E3⊕E4 15 3A1⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A1⊕E1⊕E2⊕E3⊕2E4 21 3A1⊕3E1⊕2E2⊕2E3⊕2E4
i (l=6) 13 Tetrahexacontapole A1⊕E1⊕E2⊕2E3⊕2E4 28 4A1⊕3E1⊕3E2⊕3E3⊕3E4
j (l=7) 15 Octacosahectapole A1⊕E1⊕2E2⊕2E3⊕2E4 36 4A1⊕4E1⊕4E2⊕4E3⊕4E4
k (l=8) 17 256-pole A1⊕2E1⊕2E2⊕2E3⊕2E4 45 5A1⊕5E1⊕5E2⊕5E3⊕5E4
l (l=9) 19 512-pole 2A1⊕A2⊕2E1⊕2E2⊕2E3⊕2E4 55 6A1⊕A2⊕6E1⊕6E2⊕6E3⊕6E4
m (l=10) 21 1024-pole 2A1⊕A2⊕3E1⊕2E2⊕2E3⊕2E4 66 7A1⊕A2⊕8E1⊕7E2⊕7E3⊕7E4
n (l=11) 23 2048-pole 2A1⊕A2⊕3E1⊕3E2⊕2E3⊕2E4 78 8A1⊕2A2⊕9E1⊕9E2⊕8E3⊕8E4
o (l=12) 25 4096-pole 2A1⊕A2⊕3E1⊕3E2⊕3E3⊕2E4 91 9A1⊕2A2⊕11E1⊕10E2⊕10E3⊕9E4
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 C9v
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⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A2⊕E1⊕E2⊕E3⊕2E4
I (L=6) 13 A1⊕E1⊕E2⊕2E3⊕2E4


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