Point Group D18d



D18d E 2S36 2C18 2S12 2C9 2(S36)5 2C6 2(S36)7 2(C9)2 2S4 2(C18)5 2(S36)11 2C3 2(S36)13 2(C18)7 2(S12)5 2(C9)4 2(S36)17 C2 18C'2 18σd
A1 1 1 1 1 1 1 1 1 1 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 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 -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 1 -1 1 -1 1 -1 1 -1 1
E1 2 2cos(π/18) 2cos(π/9) 2cos(π/6) 2cos(2π/9) 2cos(5π/18) 1 2cos(7π/18) 2cos(4π/9) 0 -2cos(4π/9) -2cos(7π/18) -1 -2cos(5π/18) -2cos(2π/9) -2cos(π/6) -2cos(π/9) -2cos(π/18) -2 0 0
E2 2 2cos(π/9) 2cos(2π/9) 1 2cos(4π/9) -2cos(4π/9) -1 -2cos(2π/9) -2cos(π/9) -2 -2cos(π/9) -2cos(2π/9) -1 -2cos(4π/9) 2cos(4π/9) 1 2cos(2π/9) 2cos(π/9) 2 0 0
E3 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 -2cos(π/6) -1 0 1 2cos(π/6) 2 2cos(π/6) 1 0 -1 -2cos(π/6) -2 0 0
E4 2 2cos(2π/9) 2cos(4π/9) -1 -2cos(π/9) -2cos(π/9) -1 2cos(4π/9) 2cos(2π/9) 2 2cos(2π/9) 2cos(4π/9) -1 -2cos(π/9) -2cos(π/9) -1 2cos(4π/9) 2cos(2π/9) 2 0 0
E5 2 2cos(5π/18) -2cos(4π/9) -2cos(π/6) -2cos(π/9) -2cos(7π/18) 1 2cos(π/18) 2cos(2π/9) 0 -2cos(2π/9) -2cos(π/18) -1 2cos(7π/18) 2cos(π/9) 2cos(π/6) 2cos(4π/9) -2cos(5π/18) -2 0 0
E6 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1 2 1 -1 -2 -1 1 2 0 0
E7 2 2cos(7π/18) -2cos(2π/9) -2cos(π/6) 2cos(4π/9) 2cos(π/18) 1 -2cos(5π/18) -2cos(π/9) 0 2cos(π/9) 2cos(5π/18) -1 -2cos(π/18) -2cos(4π/9) 2cos(π/6) 2cos(2π/9) -2cos(7π/18) -2 0 0
E8 2 2cos(4π/9) -2cos(π/9) -1 2cos(2π/9) 2cos(2π/9) -1 -2cos(π/9) 2cos(4π/9) 2 2cos(4π/9) -2cos(π/9) -1 2cos(2π/9) 2cos(2π/9) -1 -2cos(π/9) 2cos(4π/9) 2 0 0
E9 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 0
E10 2 -2cos(4π/9) -2cos(π/9) 1 2cos(2π/9) -2cos(2π/9) -1 2cos(π/9) 2cos(4π/9) -2 2cos(4π/9) 2cos(π/9) -1 -2cos(2π/9) 2cos(2π/9) 1 -2cos(π/9) -2cos(4π/9) 2 0 0
E11 2 -2cos(7π/18) -2cos(2π/9) 2cos(π/6) 2cos(4π/9) -2cos(π/18) 1 2cos(5π/18) -2cos(π/9) 0 2cos(π/9) -2cos(5π/18) -1 2cos(π/18) -2cos(4π/9) -2cos(π/6) 2cos(2π/9) 2cos(7π/18) -2 0 0
E12 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 0 0
E13 2 -2cos(5π/18) -2cos(4π/9) 2cos(π/6) -2cos(π/9) 2cos(7π/18) 1 -2cos(π/18) 2cos(2π/9) 0 -2cos(2π/9) 2cos(π/18) -1 -2cos(7π/18) 2cos(π/9) -2cos(π/6) 2cos(4π/9) 2cos(5π/18) -2 0 0
E14 2 -2cos(2π/9) 2cos(4π/9) 1 -2cos(π/9) 2cos(π/9) -1 -2cos(4π/9) 2cos(2π/9) -2 2cos(2π/9) -2cos(4π/9) -1 2cos(π/9) -2cos(π/9) 1 2cos(4π/9) -2cos(2π/9) 2 0 0
E15 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 2cos(π/6) -1 0 1 -2cos(π/6) 2 -2cos(π/6) 1 0 -1 2cos(π/6) -2 0 0
E16 2 -2cos(π/9) 2cos(2π/9) -1 2cos(4π/9) 2cos(4π/9) -1 2cos(2π/9) -2cos(π/9) 2 -2cos(π/9) 2cos(2π/9) -1 2cos(4π/9) 2cos(4π/9) -1 2cos(2π/9) -2cos(π/9) 2 0 0
E17 2 -2cos(π/18) 2cos(π/9) -2cos(π/6) 2cos(2π/9) -2cos(5π/18) 1 -2cos(7π/18) 2cos(4π/9) 0 -2cos(4π/9) 2cos(7π/18) -1 2cos(5π/18) -2cos(2π/9) 2cos(π/6) -2cos(π/9) 2cos(π/18) -2 0 0


Additional information

Number of symmetry elements h = 72
Number of classes, irreps n = 21
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2S36 2C18 2S12 2C9 2(S36)5 2C6 2(S36)7 2(C9)2 2S4 2(C18)5 2(S36)11 2C3 2(S36)13 2(C18)7 2(S12)5 2(C9)4 2(S36)17 C2 18C'2 18σd



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17




Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E17E17A1⊕A2⊕E2
E2 E2E2E16E16E1⊕E3A1⊕A2⊕E4
E3 E3E3E15E15E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E14E14E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕E8
E5 E5E5E13E13E4⊕E6E3⊕E7E2⊕E8E1⊕E9A1⊕A2⊕E10
E6 E6E6E12E12E5⊕E7E4⊕E8E3⊕E9E2⊕E10E1⊕E11A1⊕A2⊕E12
E7 E7E7E11E11E6⊕E8E5⊕E9E4⊕E10E3⊕E11E2⊕E12E1⊕E13A1⊕A2⊕E14
E8 E8E8E10E10E7⊕E9E6⊕E10E5⊕E11E4⊕E12E3⊕E13E2⊕E14E1⊕E15A1⊕A2⊕E16
E9 E9E9E9E9E8⊕E10E7⊕E11E6⊕E12E5⊕E13E4⊕E14E3⊕E15E2⊕E16E1⊕E17A1⊕A2⊕B1⊕B2
E10 E10E10E8E8E9⊕E11E8⊕E12E7⊕E13E6⊕E14E5⊕E15E4⊕E16E3⊕E17B1⊕B2⊕E2E1⊕E17A1⊕A2⊕E16
E11 E11E11E7E7E10⊕E12E9⊕E13E8⊕E14E7⊕E15E6⊕E16E5⊕E17B1⊕B2⊕E4E3⊕E17E2⊕E16E1⊕E15A1⊕A2⊕E14
E12 E12E12E6E6E11⊕E13E10⊕E14E9⊕E15E8⊕E16E7⊕E17B1⊕B2⊕E6E5⊕E17E4⊕E16E3⊕E15E2⊕E14E1⊕E13A1⊕A2⊕E12
E13 E13E13E5E5E12⊕E14E11⊕E15E10⊕E16E9⊕E17B1⊕B2⊕E8E7⊕E17E6⊕E16E5⊕E15E4⊕E14E3⊕E13E2⊕E12E1⊕E11A1⊕A2⊕E10
E14 E14E14E4E4E13⊕E15E12⊕E16E11⊕E17B1⊕B2⊕E10E9⊕E17E8⊕E16E7⊕E15E6⊕E14E5⊕E13E4⊕E12E3⊕E11E2⊕E10E1⊕E9A1⊕A2⊕E8
E15 E15E15E3E3E14⊕E16E13⊕E17B1⊕B2⊕E12E11⊕E17E10⊕E16E9⊕E15E8⊕E14E7⊕E13E6⊕E12E5⊕E11E4⊕E10E3⊕E9E2⊕E8E1⊕E7A1⊕A2⊕E6
E16 E16E16E2E2E15⊕E17B1⊕B2⊕E14E13⊕E17E12⊕E16E11⊕E15E10⊕E14E9⊕E13E8⊕E12E7⊕E11E6⊕E10E5⊕E9E4⊕E8E3⊕E7E2⊕E6E1⊕E5A1⊕A2⊕E4
E17 E17E17E1E1B1⊕B2⊕E16E15⊕E17E14⊕E16E13⊕E15E12⊕E14E11⊕E13E10⊕E12E9⊕E11E8⊕E10E7⊕E9E6⊕E8E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕E3A1⊕A2⊕E2

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⊕E5A1⊕E2⊕E4⊕E6More
E2 A1⊕E4E2⊕E6A1⊕E4⊕E8E2⊕E6⊕E10A1⊕E4⊕E8⊕E12More
E3 A1⊕E6E3⊕E9A1⊕E6⊕E12E3⊕E9⊕E15A1⊕B1⊕B2⊕E6⊕E12More
E4 A1⊕E8E4⊕E12A1⊕E8⊕E16E4⊕E12⊕E16A1⊕E8⊕E12⊕E16More
E5 A1⊕E10E5⊕E15A1⊕E10⊕E16E5⊕E11⊕E15A1⊕E6⊕E10⊕E16More
E6 A1⊕E12B1⊕B2⊕E6A1⊕2E12B1⊕B2⊕2E62A1⊕A2⊕2E12More
E7 A1⊕E14E7⊕E15A1⊕E8⊕E14E1⊕E7⊕E15A1⊕E6⊕E8⊕E14More
E8 A1⊕E16E8⊕E12A1⊕E4⊕E16E4⊕E8⊕E12A1⊕E4⊕E12⊕E16More
E9 A1⊕B1⊕B22E92A1⊕A2⊕B1⊕B23E92A1⊕A2⊕2B1⊕2B2More
E10 A1⊕E16E6⊕E10A1⊕E4⊕E16E6⊕E10⊕E14A1⊕E4⊕E12⊕E16More
E11 A1⊕E14E3⊕E11A1⊕E8⊕E14E3⊕E11⊕E17A1⊕E6⊕E8⊕E14More
E12 A1⊕E12A1⊕A2⊕E12A1⊕2E12A1⊕A2⊕2E122A1⊕A2⊕2E12More
E13 A1⊕E10E3⊕E13A1⊕E10⊕E16E3⊕E7⊕E13A1⊕E6⊕E10⊕E16More
E14 A1⊕E8E6⊕E14A1⊕E8⊕E16E2⊕E6⊕E14A1⊕E8⊕E12⊕E16More
E15 A1⊕E6E9⊕E15A1⊕E6⊕E12E3⊕E9⊕E15A1⊕B1⊕B2⊕E6⊕E12More
E16 A1⊕E4E12⊕E16A1⊕E4⊕E8E8⊕E12⊕E16A1⊕E4⊕E8⊕E12More
E17 A1⊕E2E15⊕E17A1⊕E2⊕E4E13⊕E15⊕E17A1⊕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 l(xyz)]
s (l=0) 1 Monopole A1 1 A1
p (l=1) 3 Dipole B2⊕E1 3 B2⊕E1
d (l=2) 5 Quadrupole A1⊕E2⊕E17 6 2A1⊕E2⊕E17
f (l=3) 7 Octupole B2⊕E1⊕E3⊕E16 10 2B2⊕2E1⊕E3⊕E16
g (l=4) 9 Hexadecapole A1⊕E2⊕E4⊕E15⊕E17 15 3A1⊕2E2⊕E4⊕E15⊕2E17
h (l=5) 11 Dotricontapole B2⊕E1⊕E3⊕E5⊕E14⊕E16 21 3B2⊕3E1⊕2E3⊕E5⊕E14⊕2E16
i (l=6) 13 Tetrahexacontapole A1⊕E2⊕E4⊕E6⊕E13⊕E15⊕E17 28 4A1⊕3E2⊕2E4⊕E6⊕E13⊕2E15⊕3E17
j (l=7) 15 Octacosahectapole B2⊕E1⊕E3⊕E5⊕E7⊕E12⊕E14⊕E16 36 4B2⊕4E1⊕3E3⊕2E5⊕E7⊕E12⊕2E14⊕3E16
k (l=8) 17 256-pole A1⊕E2⊕E4⊕E6⊕E8⊕E11⊕E13⊕E15⊕E17 45 5A1⊕4E2⊕3E4⊕2E6⊕E8⊕E11⊕2E13⊕3E15⊕4E17
l (l=9) 19 512-pole B2⊕E1⊕E3⊕E5⊕E7⊕E9⊕E10⊕E12⊕E14⊕E16 55 5B2⊕5E1⊕4E3⊕3E5⊕2E7⊕E9⊕E10⊕2E12⊕3E14⊕4E16
m (l=10) 21 1024-pole A1⊕E2⊕E4⊕E6⊕E8⊕E9⊕E10⊕E11⊕E13⊕E15⊕E17 66 6A1⊕5E2⊕4E4⊕3E6⊕2E8⊕E9⊕E10⊕2E11⊕3E13⊕4E15⊕5E17
n (l=11) 23 2048-pole B2⊕E1⊕E3⊕E5⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12⊕E14⊕E16 78 6B2⊕6E1⊕5E3⊕4E5⊕3E7⊕E8⊕2E9⊕2E10⊕E11⊕3E12⊕4E14⊕5E16
o (l=12) 25 4096-pole A1⊕E2⊕E4⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12⊕E13⊕E15⊕E17 91 7A1⊕6E2⊕5E4⊕4E6⊕E7⊕3E8⊕2E9⊕2E10⊕3E11⊕E12⊕4E13⊕5E15⊕6E17
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First nonvanshing multipole: Quadrupole

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 D18d
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E17
D (L=2) 5 A1⊕E2⊕E17
F (L=3) 7 A2⊕E2⊕E15⊕E17
G (L=4) 9 A1⊕E2⊕E4⊕E15⊕E17
H (L=5) 11 A2⊕E2⊕E4⊕E13⊕E15⊕E17
I (L=6) 13 A1⊕E2⊕E4⊕E6⊕E13⊕E15⊕E17


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