Point Group D9d



D9d E 2C9 2(C9)2 2C3 2(C9)4 9C'2 i 2(S18)7 2(S18)5 2S6 2S18 9σd
A1g 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
E1g 2 2cos(2π/9) 2cos(4π/9) -1 2cos(8π/9) 0 2 2cos(2π/9) 2cos(4π/9) -1 2cos(8π/9) 0
E2g 2 2cos(4π/9) 2cos(8π/9) -1 2cos(2π/9) 0 2 2cos(4π/9) 2cos(8π/9) -1 2cos(2π/9) 0
E3g 2 -1 -1 2 -1 0 2 -1 -1 2 -1 0
E4g 2 2cos(8π/9) 2cos(2π/9) -1 2cos(4π/9) 0 2 2cos(8π/9) 2cos(2π/9) -1 2cos(4π/9) 0
A1u 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
E1u 2 2cos(2π/9) 2cos(4π/9) -1 2cos(8π/9) 0 -2 -2cos(2π/9) -2cos(4π/9) 1 -2cos(8π/9) 0
E2u 2 2cos(4π/9) 2cos(8π/9) -1 2cos(2π/9) 0 -2 -2cos(4π/9) -2cos(8π/9) 1 -2cos(2π/9) 0
E3u 2 -1 -1 2 -1 0 -2 1 1 -2 1 0
E4u 2 2cos(8π/9) 2cos(2π/9) -1 2cos(4π/9) 0 -2 -2cos(8π/9) -2cos(2π/9) 1 -2cos(4π/9) 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 no
Parity yes


Reduce representation to irreducible representations


E 2C9 2(C9)2 2C3 2(C9)4 9C'2 i 2(S18)7 2(S18)5 2S6 2S18 9σd



Genrate representation from irreducible representations


A1g A2g E1g E2g E3g E4g A1u A2u E1u E2u E3u E4u



Direct products of irreducible representations


Binary products
⊙ A1g A2g E1g E2g E3g E4g A1u A2u E1u E2u E3u E4u
A1g A1g
A2g A2gA1g
E1g E1gE1gA1g⊕A2g⊕E2g
E2g E2gE2gE1g⊕E3gA1g⊕A2g⊕E4g
E3g E3gE3gE2g⊕E4gE1g⊕E4gA1g⊕A2g⊕E3g
E4g E4gE4gE3g⊕E4gE2g⊕E3gE1g⊕E2gA1g⊕A2g⊕E1g
A1u A1uA2uE1uE2uE3uE4uA1g
A2u A2uA1uE1uE2uE3uE4uA2gA1g
E1u E1uE1uA1u⊕A2u⊕E2uE1u⊕E3uE2u⊕E4uE3u⊕E4uE1gE1gA1g⊕A2g⊕E2g
E2u E2uE2uE1u⊕E3uA1u⊕A2u⊕E4uE1u⊕E4uE2u⊕E3uE2gE2gE1g⊕E3gA1g⊕A2g⊕E4g
E3u E3uE3uE2u⊕E4uE1u⊕E4uA1u⊕A2u⊕E3uE1u⊕E2uE3gE3gE2g⊕E4gE1g⊕E4gA1g⊕A2g⊕E3g
E4u E4uE4uE3u⊕E4uE2u⊕E3uE1u⊕E2uA1u⊕A2u⊕E1uE4gE4gE3g⊕E4gE2g⊕E3gE1g⊕E2gA1g⊕A2g⊕E1g

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1g A1g⊕E2gE1g⊕E3gA1g⊕E2g⊕E4gE1g⊕E3g⊕E4gA1g⊕E2g⊕E3g⊕E4gMore
E2g A1g⊕E4gE2g⊕E3gA1g⊕E1g⊕E4gE1g⊕E2g⊕E3gA1g⊕E1g⊕E3g⊕E4gMore
E3g A1g⊕E3gA1g⊕A2g⊕E3gA1g⊕2E3gA1g⊕A2g⊕2E3g2A1g⊕A2g⊕2E3gMore
E4g A1g⊕E1gE3g⊕E4gA1g⊕E1g⊕E2gE2g⊕E3g⊕E4gA1g⊕E1g⊕E2g⊕E3gMore
E1u A1g⊕E2gE1u⊕E3uA1g⊕E2g⊕E4gE1u⊕E3u⊕E4uA1g⊕E2g⊕E3g⊕E4gMore
E2u A1g⊕E4gE2u⊕E3uA1g⊕E1g⊕E4gE1u⊕E2u⊕E3uA1g⊕E1g⊕E3g⊕E4gMore
E3u A1g⊕E3gA1u⊕A2u⊕E3uA1g⊕2E3gA1u⊕A2u⊕2E3u2A1g⊕A2g⊕2E3gMore
E4u A1g⊕E1gE3u⊕E4uA1g⊕E1g⊕E2gE2u⊕E3u⊕E4uA1g⊕E1g⊕E2g⊕E3gMore


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 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⊕2E4u 21 3A2u⊕3E1u⊕2E2u⊕2E3u⊕2E4u
i (l=6) 13 Tetrahexacontapole A1g⊕E1g⊕E2g⊕2E3g⊕2E4g 28 4A1g⊕3E1g⊕3E2g⊕3E3g⊕3E4g
j (l=7) 15 Octacosahectapole A2u⊕E1u⊕2E2u⊕2E3u⊕2E4u 36 4A2u⊕4E1u⊕4E2u⊕4E3u⊕4E4u
k (l=8) 17 256-pole A1g⊕2E1g⊕2E2g⊕2E3g⊕2E4g 45 5A1g⊕5E1g⊕5E2g⊕5E3g⊕5E4g
l (l=9) 19 512-pole A1u⊕2A2u⊕2E1u⊕2E2u⊕2E3u⊕2E4u 55 A1u⊕6A2u⊕6E1u⊕6E2u⊕6E3u⊕6E4u
m (l=10) 21 1024-pole 2A1g⊕A2g⊕3E1g⊕2E2g⊕2E3g⊕2E4g 66 7A1g⊕A2g⊕8E1g⊕7E2g⊕7E3g⊕7E4g
n (l=11) 23 2048-pole A1u⊕2A2u⊕3E1u⊕3E2u⊕2E3u⊕2E4u 78 2A1u⊕8A2u⊕9E1u⊕9E2u⊕8E3u⊕8E4u
o (l=12) 25 4096-pole 2A1g⊕A2g⊕3E1g⊕3E2g⊕3E3g⊕2E4g 91 9A1g⊕2A2g⊕11E1g⊕10E2g⊕10E3g⊕9E4g
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 D9d
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⊕2E4g
I (L=6) 13 A1g⊕E1g⊕E2g⊕2E3g⊕2E4g

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