Point Group C9



ε=exp(2πi/9)
C9 E C9 (C9)2 C3 (C9)4 (C9)5 (C3)2 (C9)7 (C9)8
A 1 1 1 1 1 1 1 1 1
E1* 1
1		 ε*
ε*		 ε2*
ε2*		 ε3*
ε3*		 ε4*
ε4*		 ε4*
ε4*		 ε3*
ε3*		 ε2*
ε2*		 ε*
ε*
E2* 1
1		 ε2*
ε2*		 ε4*
ε4*		 ε3*
ε3*		 ε*
ε*		 ε*
ε*		 ε3*
ε3*		 ε4*
ε4*		 ε2*
ε2*
E3* 1
1		 ε3*
ε3*		 ε3*
ε3*		 1
1		 ε3*
ε3*		 ε3*
ε3*		 1
1		 ε3*
ε3*		 ε3*
ε3*
E4* 1
1		 ε4*
ε4*		 ε*
ε*		 ε3*
ε3*		 ε2*
ε2*		 ε2*
ε2*		 ε3*
ε3*		 ε*
ε*		 ε4*
ε4*
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Additional information

Number of symmetry elements h = 9
Number of classes, irreps n = 9
Number of real-valued irreducible representations n = 5
Abelian group yes
Optical Isomerism (Chirality) yes
Polar yes
Parity no


Reduce representation to irreducible representations


E C9 (C9)2 C3 (C9)4 (C9)5 (C3)2 (C9)7 (C9)8



Genrate representation from irreducible representations


A E1* E2* E3* E4*



Direct products of irreducible representations


Binary products
⊙ A E1* E2* E3* E4*
A A
E1* E12A⊕E2
E2* E2E1⊕E32A⊕E4
E3* E3E2⊕E4E1⊕E42A⊕E3
E4* E4E3⊕E4E2⊕E3E1⊕E22A⊕E1

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1* A⊕E2E1⊕E3A⊕E2⊕E4E1⊕E3⊕E4A⊕E2⊕E3⊕E4More
E2* A⊕E4E2⊕E3A⊕E1⊕E4E1⊕E2⊕E3A⊕E1⊕E3⊕E4More
E3* A⊕E32A⊕E3A⊕2E32A⊕2E33A⊕2E3More
E4* A⊕E1E3⊕E4A⊕E1⊕E2E2⊕E3⊕E4A⊕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 [Γl(xyz)]
s (l=0) 1 Monopole A 1 A
p (l=1) 3 Dipole A⊕E1 3 A⊕E1
d (l=2) 5 Quadrupole A⊕E1⊕E2 6 2A⊕E1⊕E2
f (l=3) 7 Octupole A⊕E1⊕E2⊕E3 10 2A⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A⊕E1⊕E2⊕E3⊕E4 15 3A⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A⊕E1⊕E2⊕E3⊕2E4 21 3A⊕3E1⊕2E2⊕2E3⊕2E4
i (l=6) 13 Tetrahexacontapole A⊕E1⊕E2⊕2E3⊕2E4 28 4A⊕3E1⊕3E2⊕3E3⊕3E4
j (l=7) 15 Octacosahectapole A⊕E1⊕2E2⊕2E3⊕2E4 36 4A⊕4E1⊕4E2⊕4E3⊕4E4
k (l=8) 17 256-pole A⊕2E1⊕2E2⊕2E3⊕2E4 45 5A⊕5E1⊕5E2⊕5E3⊕5E4
l (l=9) 19 512-pole 3A⊕2E1⊕2E2⊕2E3⊕2E4 55 7A⊕6E1⊕6E2⊕6E3⊕6E4
m (l=10) 21 1024-pole 3A⊕3E1⊕2E2⊕2E3⊕2E4 66 8A⊕8E1⊕7E2⊕7E3⊕7E4
n (l=11) 23 2048-pole 3A⊕3E1⊕3E2⊕2E3⊕2E4 78 10A⊕9E1⊕9E2⊕8E3⊕8E4
o (l=12) 25 4096-pole 3A⊕3E1⊕3E2⊕3E3⊕2E4 91 11A⊕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 C9
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E1
D (L=2) 5 A⊕E1⊕E2
F (L=3) 7 A⊕E1⊕E2⊕E3
G (L=4) 9 A⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A⊕E1⊕E2⊕E3⊕2E4
I (L=6) 13 A⊕E1⊕E2⊕2E3⊕2E4

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