Point Group D18h



D18h E 2C18 2C9 2C6 2(C9)2 2(C18)5 2C3 2(C18)7 2(C9)4 C2 9C'2 9C''2 i 2(S9)5 2(S18)7 2S3 2(S18)5 2(S9)7 2S6 2S9 2S18 σh d v
A1g 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2g 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1
B1g 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
B2g 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1
E1g 2 2cos(π/9) 2cos(2π/9) 1 2cos(4π/9) -2cos(4π/9) -1 -2cos(2π/9) -2cos(π/9) -2 0 0 2 2cos(π/9) 2cos(2π/9) 1 2cos(4π/9) -2cos(4π/9) -1 -2cos(2π/9) -2cos(π/9) -2 0 0
E2g 2 2cos(2π/9) 2cos(4π/9) -1 -2cos(π/9) -2cos(π/9) -1 2cos(4π/9) 2cos(2π/9) 2 0 0 2 2cos(2π/9) 2cos(4π/9) -1 -2cos(π/9) -2cos(π/9) -1 2cos(4π/9) 2cos(2π/9) 2 0 0
E3g 2 1 -1 -2 -1 1 2 1 -1 -2 0 0 2 1 -1 -2 -1 1 2 1 -1 -2 0 0
E4g 2 2cos(4π/9) -2cos(π/9) -1 2cos(2π/9) 2cos(2π/9) -1 -2cos(π/9) 2cos(4π/9) 2 0 0 2 2cos(4π/9) -2cos(π/9) -1 2cos(2π/9) 2cos(2π/9) -1 -2cos(π/9) 2cos(4π/9) 2 0 0
E5g 2 -2cos(4π/9) -2cos(π/9) 1 2cos(2π/9) -2cos(2π/9) -1 2cos(π/9) 2cos(4π/9) -2 0 0 2 -2cos(4π/9) -2cos(π/9) 1 2cos(2π/9) -2cos(2π/9) -1 2cos(π/9) 2cos(4π/9) -2 0 0
E6g 2 -1 -1 2 -1 -1 2 -1 -1 2 0 0 2 -1 -1 2 -1 -1 2 -1 -1 2 0 0
E7g 2 -2cos(2π/9) 2cos(4π/9) 1 -2cos(π/9) 2cos(π/9) -1 -2cos(4π/9) 2cos(2π/9) -2 0 0 2 -2cos(2π/9) 2cos(4π/9) 1 -2cos(π/9) 2cos(π/9) -1 -2cos(4π/9) 2cos(2π/9) -2 0 0
E8g 2 -2cos(π/9) 2cos(2π/9) -1 2cos(4π/9) 2cos(4π/9) -1 2cos(2π/9) -2cos(π/9) 2 0 0 2 -2cos(π/9) 2cos(2π/9) -1 2cos(4π/9) 2cos(4π/9) -1 2cos(2π/9) -2cos(π/9) 2 0 0
A1u 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
A2u 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1
B1u 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
B2u 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1
E1u 2 2cos(π/9) 2cos(2π/9) 1 2cos(4π/9) -2cos(4π/9) -1 -2cos(2π/9) -2cos(π/9) -2 0 0 -2 -2cos(π/9) -2cos(2π/9) -1 -2cos(4π/9) 2cos(4π/9) 1 2cos(2π/9) 2cos(π/9) 2 0 0
E2u 2 2cos(2π/9) 2cos(4π/9) -1 -2cos(π/9) -2cos(π/9) -1 2cos(4π/9) 2cos(2π/9) 2 0 0 -2 -2cos(2π/9) -2cos(4π/9) 1 2cos(π/9) 2cos(π/9) 1 -2cos(4π/9) -2cos(2π/9) -2 0 0
E3u 2 1 -1 -2 -1 1 2 1 -1 -2 0 0 -2 -1 1 2 1 -1 -2 -1 1 2 0 0
E4u 2 2cos(4π/9) -2cos(π/9) -1 2cos(2π/9) 2cos(2π/9) -1 -2cos(π/9) 2cos(4π/9) 2 0 0 -2 -2cos(4π/9) 2cos(π/9) 1 -2cos(2π/9) -2cos(2π/9) 1 2cos(π/9) -2cos(4π/9) -2 0 0
E5u 2 -2cos(4π/9) -2cos(π/9) 1 2cos(2π/9) -2cos(2π/9) -1 2cos(π/9) 2cos(4π/9) -2 0 0 -2 2cos(4π/9) 2cos(π/9) -1 -2cos(2π/9) 2cos(2π/9) 1 -2cos(π/9) -2cos(4π/9) 2 0 0
E6u 2 -1 -1 2 -1 -1 2 -1 -1 2 0 0 -2 1 1 -2 1 1 -2 1 1 -2 0 0
E7u 2 -2cos(2π/9) 2cos(4π/9) 1 -2cos(π/9) 2cos(π/9) -1 -2cos(4π/9) 2cos(2π/9) -2 0 0 -2 2cos(2π/9) -2cos(4π/9) -1 2cos(π/9) -2cos(π/9) 1 2cos(4π/9) -2cos(2π/9) 2 0 0
E8u 2 -2cos(π/9) 2cos(2π/9) -1 2cos(4π/9) 2cos(4π/9) -1 2cos(2π/9) -2cos(π/9) 2 0 0 -2 2cos(π/9) -2cos(2π/9) 1 -2cos(4π/9) -2cos(4π/9) 1 -2cos(2π/9) 2cos(π/9) -2 0 0


Additional information

Number of symmetry elements h = 72
Number of classes, irreps n = 24
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C18 2C9 2C6 2(C9)2 2(C18)5 2C3 2(C18)7 2(C9)4 C2 9C'2 9C''2 i 2(S9)5 2(S18)7 2S3 2(S18)5 2(S9)7 2S6 2S9 2S18 σh d v



Genrate representation from irreducible representations


A1g A2g B1g B2g E1g E2g E3g E4g E5g E6g E7g E8g A1u A2u B1u B2u E1u E2u E3u E4u E5u E6u E7u E8u




Direct products of irreducible representations


Binary products
A1g A2g B1g B2g E1g E2g E3g E4g E5g E6g E7g E8g A1u A2u B1u B2u E1u E2u E3u E4u E5u E6u E7u E8u
A1g A1g
A2g A2gA1g
B1g B1gB2gA1g
B2g B2gB1gA2gA1g
E1g E1gE1gE8gE8gA1g⊕A2g⊕E2g
E2g E2gE2gE7gE7gE1g⊕E3gA1g⊕A2g⊕E4g
E3g E3gE3gE6gE6gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕E6g
E4g E4gE4gE5gE5gE3g⊕E5gE2g⊕E6gE1g⊕E7gA1g⊕A2g⊕E8g
E5g E5gE5gE4gE4gE4g⊕E6gE3g⊕E7gE2g⊕E8gB1g⊕B2g⊕E1gA1g⊕A2g⊕E8g
E6g E6gE6gE3gE3gE5g⊕E7gE4g⊕E8gB1g⊕B2g⊕E3gE2g⊕E8gE1g⊕E7gA1g⊕A2g⊕E6g
E7g E7gE7gE2gE2gE6g⊕E8gB1g⊕B2g⊕E5gE4g⊕E8gE3g⊕E7gE2g⊕E6gE1g⊕E5gA1g⊕A2g⊕E4g
E8g E8gE8gE1gE1gB1g⊕B2g⊕E7gE6g⊕E8gE5g⊕E7gE4g⊕E6gE3g⊕E5gE2g⊕E4gE1g⊕E3gA1g⊕A2g⊕E2g
A1u A1uA2uB1uB2uE1uE2uE3uE4uE5uE6uE7uE8uA1g
A2u A2uA1uB2uB1uE1uE2uE3uE4uE5uE6uE7uE8uA2gA1g
B1u B1uB2uA1uA2uE8uE7uE6uE5uE4uE3uE2uE1uB1gB2gA1g
B2u B2uB1uA2uA1uE8uE7uE6uE5uE4uE3uE2uE1uB2gB1gA2gA1g
E1u E1uE1uE8uE8uA1u⊕A2u⊕E2uE1u⊕E3uE2u⊕E4uE3u⊕E5uE4u⊕E6uE5u⊕E7uE6u⊕E8uB1u⊕B2u⊕E7uE1gE1gE8gE8gA1g⊕A2g⊕E2g
E2u E2uE2uE7uE7uE1u⊕E3uA1u⊕A2u⊕E4uE1u⊕E5uE2u⊕E6uE3u⊕E7uE4u⊕E8uB1u⊕B2u⊕E5uE6u⊕E8uE2gE2gE7gE7gE1g⊕E3gA1g⊕A2g⊕E4g
E3u E3uE3uE6uE6uE2u⊕E4uE1u⊕E5uA1u⊕A2u⊕E6uE1u⊕E7uE2u⊕E8uB1u⊕B2u⊕E3uE4u⊕E8uE5u⊕E7uE3gE3gE6gE6gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕E6g
E4u E4uE4uE5uE5uE3u⊕E5uE2u⊕E6uE1u⊕E7uA1u⊕A2u⊕E8uB1u⊕B2u⊕E1uE2u⊕E8uE3u⊕E7uE4u⊕E6uE4gE4gE5gE5gE3g⊕E5gE2g⊕E6gE1g⊕E7gA1g⊕A2g⊕E8g
E5u E5uE5uE4uE4uE4u⊕E6uE3u⊕E7uE2u⊕E8uB1u⊕B2u⊕E1uA1u⊕A2u⊕E8uE1u⊕E7uE2u⊕E6uE3u⊕E5uE5gE5gE4gE4gE4g⊕E6gE3g⊕E7gE2g⊕E8gB1g⊕B2g⊕E1gA1g⊕A2g⊕E8g
E6u E6uE6uE3uE3uE5u⊕E7uE4u⊕E8uB1u⊕B2u⊕E3uE2u⊕E8uE1u⊕E7uA1u⊕A2u⊕E6uE1u⊕E5uE2u⊕E4uE6gE6gE3gE3gE5g⊕E7gE4g⊕E8gB1g⊕B2g⊕E3gE2g⊕E8gE1g⊕E7gA1g⊕A2g⊕E6g
E7u E7uE7uE2uE2uE6u⊕E8uB1u⊕B2u⊕E5uE4u⊕E8uE3u⊕E7uE2u⊕E6uE1u⊕E5uA1u⊕A2u⊕E4uE1u⊕E3uE7gE7gE2gE2gE6g⊕E8gB1g⊕B2g⊕E5gE4g⊕E8gE3g⊕E7gE2g⊕E6gE1g⊕E5gA1g⊕A2g⊕E4g
E8u E8uE8uE1uE1uB1u⊕B2u⊕E7uE6u⊕E8uE5u⊕E7uE4u⊕E6uE3u⊕E5uE2u⊕E4uE1u⊕E3uA1u⊕A2u⊕E2uE8gE8gE1gE1gB1g⊕B2g⊕E7gE6g⊕E8gE5g⊕E7gE4g⊕E6gE3g⊕E5gE2g⊕E4gE1g⊕E3gA1g⊕A2g⊕E2g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g A1g⊕E2gE1g⊕E3gA1g⊕E2g⊕E4gE1g⊕E3g⊕E5gA1g⊕E2g⊕E4g⊕E6gMore
E2g A1g⊕E4gE2g⊕E6gA1g⊕E4g⊕E8gE2g⊕E6g⊕E8gA1g⊕E4g⊕E6g⊕E8gMore
E3g A1g⊕E6gB1g⊕B2g⊕E3gA1g⊕2E6gB1g⊕B2g⊕2E3g2A1g⊕A2g⊕2E6gMore
E4g A1g⊕E8gE4g⊕E6gA1g⊕E2g⊕E8gE2g⊕E4g⊕E6gA1g⊕E2g⊕E6g⊕E8gMore
E5g A1g⊕E8gE3g⊕E5gA1g⊕E2g⊕E8gE3g⊕E5g⊕E7gA1g⊕E2g⊕E6g⊕E8gMore
E6g A1g⊕E6gA1g⊕A2g⊕E6gA1g⊕2E6gA1g⊕A2g⊕2E6g2A1g⊕A2g⊕2E6gMore
E7g A1g⊕E4gE3g⊕E7gA1g⊕E4g⊕E8gE1g⊕E3g⊕E7gA1g⊕E4g⊕E6g⊕E8gMore
E8g A1g⊕E2gE6g⊕E8gA1g⊕E2g⊕E4gE4g⊕E6g⊕E8gA1g⊕E2g⊕E4g⊕E6gMore
E1u A1g⊕E2gE1u⊕E3uA1g⊕E2g⊕E4gE1u⊕E3u⊕E5uA1g⊕E2g⊕E4g⊕E6gMore
E2u A1g⊕E4gE2u⊕E6uA1g⊕E4g⊕E8gE2u⊕E6u⊕E8uA1g⊕E4g⊕E6g⊕E8gMore
E3u A1g⊕E6gB1u⊕B2u⊕E3uA1g⊕2E6gB1u⊕B2u⊕2E3u2A1g⊕A2g⊕2E6gMore
E4u A1g⊕E8gE4u⊕E6uA1g⊕E2g⊕E8gE2u⊕E4u⊕E6uA1g⊕E2g⊕E6g⊕E8gMore
E5u A1g⊕E8gE3u⊕E5uA1g⊕E2g⊕E8gE3u⊕E5u⊕E7uA1g⊕E2g⊕E6g⊕E8gMore
E6u A1g⊕E6gA1u⊕A2u⊕E6uA1g⊕2E6gA1u⊕A2u⊕2E6u2A1g⊕A2g⊕2E6gMore
E7u A1g⊕E4gE3u⊕E7uA1g⊕E4g⊕E8gE1u⊕E3u⊕E7uA1g⊕E4g⊕E6g⊕E8gMore
E8u A1g⊕E2gE6u⊕E8uA1g⊕E2g⊕E4gE4u⊕E6u⊕E8uA1g⊕E2g⊕E4g⊕E6gMore



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 A1g 1 A1g
p (l=1) 3 Dipole A2u⊕E1u 3 A2u⊕E1u
d (l=2) 5 Quadrupole A1g⊕E1g⊕E2g 6 2A1g⊕E1g⊕E2g
f (l=3) 7 Octupole A2u⊕E1u⊕E2u⊕E3u 10 2A2u⊕2E1u⊕E2u⊕E3u
g (l=4) 9 Hexadecapole A1g⊕E1g⊕E2g⊕E3g⊕E4g 15 3A1g⊕2E1g⊕2E2g⊕E3g⊕E4g
h (l=5) 11 Dotricontapole A2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u 21 3A2u⊕3E1u⊕2E2u⊕2E3u⊕E4u⊕E5u
i (l=6) 13 Tetrahexacontapole A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g 28 4A1g⊕3E1g⊕3E2g⊕2E3g⊕2E4g⊕E5g⊕E6g
j (l=7) 15 Octacosahectapole A2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕E6u⊕E7u 36 4A2u⊕4E1u⊕3E2u⊕3E3u⊕2E4u⊕2E5u⊕E6u⊕E7u
k (l=8) 17 256-pole A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g⊕E7g⊕E8g 45 5A1g⊕4E1g⊕4E2g⊕3E3g⊕3E4g⊕2E5g⊕2E6g⊕E7g⊕E8g
l (l=9) 19 512-pole A2u⊕B1u⊕B2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕E6u⊕E7u⊕E8u 55 5A2u⊕B1u⊕B2u⊕5E1u⊕4E2u⊕4E3u⊕3E4u⊕3E5u⊕2E6u⊕2E7u⊕E8u
m (l=10) 21 1024-pole A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g⊕E7g⊕2E8g 66 6A1g⊕B1g⊕B2g⊕5E1g⊕5E2g⊕4E3g⊕4E4g⊕3E5g⊕3E6g⊕2E7g⊕3E8g
n (l=11) 23 2048-pole A2u⊕B1u⊕B2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕E6u⊕2E7u⊕2E8u 78 6A2u⊕2B1u⊕2B2u⊕6E1u⊕5E2u⊕5E3u⊕4E4u⊕4E5u⊕3E6u⊕4E7u⊕3E8u
o (l=12) 25 4096-pole A1g⊕B1g⊕B2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕2E6g⊕2E7g⊕2E8g 91 7A1g⊕2B1g⊕2B2g⊕6E1g⊕6E2g⊕5E3g⊕5E4g⊕4E5g⊕5E6g⊕4E7g⊕5E8g
More

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 D18h
L 2L+1 Term Splitting
S (L=0) 1 A1g
P (L=1) 3 A2g⊕E1g
D (L=2) 5 A1g⊕E1g⊕E2g
F (L=3) 7 A2g⊕E1g⊕E2g⊕E3g
G (L=4) 9 A1g⊕E1g⊕E2g⊕E3g⊕E4g
H (L=5) 11 A2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g
I (L=6) 13 A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g


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