Point Group D3d



D3d E 2C3 3C'2 i 2S6 d
A1g 1 1 1 1 1 1
A2g 1 1 -1 1 1 -1
Eg 2 -1 0 2 -1 0
A1u 1 1 1 -1 -1 -1
A2u 1 1 -1 -1 -1 1
Eu 2 -1 0 -2 1 0


Additional information

Number of symmetry elements h = 12
Number of classes, irreps n = 6
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C3 3C'2 i 2S6 d



Genrate representation from irreducible representations


A1g A2g Eg A1u A2u Eu




Examples

Sulfur S6 Ethane (staggered) Cyclohexane (chair)
Hexamethylbenzene



Direct products of irreducible representations


Binary products
A1g A2g Eg A1u A2u Eu
A1g A1g
A2g A2gA1g
Eg EgEgA1g⊕A2g⊕Eg
A1u A1uA2uEuA1g
A2u A2uA1uEuA2gA1g
Eu EuEuA1u⊕A2u⊕EuEgEgA1g⊕A2g⊕Eg

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
Eg A1g⊕EgA1g⊕A2g⊕EgA1g⊕2EgA1g⊕A2g⊕2Eg2A1g⊕A2g⊕2EgMore
Eu A1g⊕EgA1u⊕A2u⊕EuA1g⊕2EgA1u⊕A2u⊕2Eu2A1g⊕A2g⊕2EgMore



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⊕Eu 3 A2u⊕Eu
d (l=2) 5 Quadrupole A1g⊕2Eg 6 2A1g⊕2Eg
f (l=3) 7 Octupole A1u⊕2A2u⊕2Eu 10 A1u⊕3A2u⊕3Eu
g (l=4) 9 Hexadecapole 2A1g⊕A2g⊕3Eg 15 4A1g⊕A2g⊕5Eg
h (l=5) 11 Dotricontapole A1u⊕2A2u⊕4Eu 21 2A1u⊕5A2u⊕7Eu
i (l=6) 13 Tetrahexacontapole 3A1g⊕2A2g⊕4Eg 28 7A1g⊕3A2g⊕9Eg
j (l=7) 15 Octacosahectapole 2A1u⊕3A2u⊕5Eu 36 4A1u⊕8A2u⊕12Eu
k (l=8) 17 256-pole 3A1g⊕2A2g⊕6Eg 45 10A1g⊕5A2g⊕15Eg
l (l=9) 19 512-pole 3A1u⊕4A2u⊕6Eu 55 7A1u⊕12A2u⊕18Eu
m (l=10) 21 1024-pole 4A1g⊕3A2g⊕7Eg 66 14A1g⊕8A2g⊕22Eg
n (l=11) 23 2048-pole 3A1u⊕4A2u⊕8Eu 78 10A1u⊕16A2u⊕26Eu
o (l=12) 25 4096-pole 5A1g⊕4A2g⊕8Eg 91 19A1g⊕12A2g⊕30Eg
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 D3d
L 2L+1 Term Splitting
S (L=0) 1 A1g
P (L=1) 3 A2g⊕Eg
D (L=2) 5 A1g⊕2Eg
F (L=3) 7 A1g⊕2A2g⊕2Eg
G (L=4) 9 2A1g⊕A2g⊕3Eg
H (L=5) 11 A1g⊕2A2g⊕4Eg
I (L=6) 13 3A1g⊕2A2g⊕4Eg


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