Point Group C18v



C18v E 2C18 2C9 2C6 2(C9)2 2(C18)5 2C3 2(C18)7 2(C9)4 C2 9σv 9σd
A1 1 1 1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
B2 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1
E1 2 1.8794 1.5321 1 0.3473 -0.3473 -1 -1.5321 -1.8794 -2 0 0
E2 2 1.5321 0.3473 -1 -1.8794 -1.8794 -1 0.3473 1.5321 2 0 0
E3 2 1 -1 -2 -1 1 2 1 -1 -2 0 0
E4 2 0.3473 -1.8794 -1 1.5321 1.5321 -1 -1.8794 0.3473 2 0 0
E5 2 -0.3473 -1.8794 1 1.5321 -1.5321 -1 1.8794 0.3473 -2 0 0
E6 2 -1 -1 2 -1 -1 2 -1 -1 2 0 0
E7 2 -1.5321 0.3473 1 -1.8794 1.8794 -1 -0.3473 1.5321 -2 0 0
E8 2 -1.8794 1.5321 -1 0.3473 0.3473 -1 1.5321 -1.8794 2 0 0
View A
View B
✅


Additional information

Number of symmetry elements h = 36
Number of classes, irreps n = 12
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C18 2C9 2C6 2(C9)2 2(C18)5 2C3 2(C18)7 2(C9)4 C2 9σv 9σd



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8



Direct products of irreducible representations


Binary products
⊙ A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E8E8A1⊕A2⊕E2
E2 E2E2E7E7E1⊕E3A1⊕A2⊕E4
E3 E3E3E6E6E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E5E5E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕E8
E5 E5E5E4E4E4⊕E6E3⊕E7E2⊕E8B1⊕B2⊕E1A1⊕A2⊕E8
E6 E6E6E3E3E5⊕E7E4⊕E8B1⊕B2⊕E3E2⊕E8E1⊕E7A1⊕A2⊕E6
E7 E7E7E2E2E6⊕E8B1⊕B2⊕E5E4⊕E8E3⊕E7E2⊕E6E1⊕E5A1⊕A2⊕E4
E8 E8E8E1E1B1⊕B2⊕E7E6⊕E8E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕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⊕E2⊕E4E1⊕E3⊕E5A1⊕E2⊕E4⊕E6More
E2 A1⊕E4E2⊕E6A1⊕E4⊕E8E2⊕E6⊕E8A1⊕E4⊕E6⊕E8More
E3 A1⊕E6B1⊕B2⊕E3A1⊕2E6B1⊕B2⊕2E32A1⊕A2⊕2E6More
E4 A1⊕E8E4⊕E6A1⊕E2⊕E8E2⊕E4⊕E6A1⊕E2⊕E6⊕E8More
E5 A1⊕E8E3⊕E5A1⊕E2⊕E8E3⊕E5⊕E7A1⊕E2⊕E6⊕E8More
E6 A1⊕E6A1⊕A2⊕E6A1⊕2E6A1⊕A2⊕2E62A1⊕A2⊕2E6More
E7 A1⊕E4E3⊕E7A1⊕E4⊕E8E1⊕E3⊕E7A1⊕E4⊕E6⊕E8More
E8 A1⊕E2E6⊕E8A1⊕E2⊕E4E4⊕E6⊕E8A1⊕E2⊕E4⊕E6More


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⊕E1⊕E2⊕E3⊕E4 15 3A1⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A1⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A1⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6 28 4A1⊕3E1⊕3E2⊕2E3⊕2E4⊕E5⊕E6
j (l=7) 15 Octacosahectapole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4A1⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 45 5A1⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7⊕E8
l (l=9) 19 512-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 55 5A1⊕B1⊕B2⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕2E7⊕E8
m (l=10) 21 1024-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕2E8 66 6A1⊕B1⊕B2⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕3E6⊕2E7⊕3E8
n (l=11) 23 2048-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕2E7⊕2E8 78 6A1⊕2B1⊕2B2⊕6E1⊕5E2⊕5E3⊕4E4⊕4E5⊕3E6⊕4E7⊕3E8
o (l=12) 25 4096-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕2E6⊕2E7⊕2E8 91 7A1⊕2B1⊕2B2⊕6E1⊕6E2⊕5E3⊕5E4⊕4E5⊕5E6⊕4E7⊕5E8
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 C18v
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⊕E4⊕E5
I (L=6) 13 A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6

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