Point Group D9d



D9d E 2C9 2(C9)2 2C3 2(C9)4 9C'2 i 2(S18)7 2(S18)5 2S6 2S18 d
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


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 d



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 2] 3] 4] 5] 6]
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 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⊕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 August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement