Point Group Oh



Oh E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
A1g 1 1 1 1 1 1 1 1 1 1
A2g 1 1 -1 -1 1 1 -1 1 1 -1
Eg 2 -1 0 0 2 2 0 -1 2 0
T1g 3 0 -1 1 -1 3 1 0 -1 -1
T2g 3 0 1 -1 -1 3 -1 0 -1 1
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 -1 -1 1 -1 1 -1 -1 1
Eu 2 -1 0 0 2 -2 0 1 -2 0
T1u 3 0 -1 1 -1 -3 -1 0 1 1
T2u 3 0 1 -1 -1 -3 1 0 1 -1


Additional information

Number of symmetry elements h = 48
Number of classes, irreps n = 10
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d



Genrate representation from irreducible representations


A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u




Examples

Sulfurhexafluoride [B6H6]2--Ion Tungstenhexacarbonyl
Cubane Octamethylsilsesquioxane



Direct products of irreducible representations


Binary products
A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u
A1g A1g
A2g A2gA1g
Eg EgEgA1g⊕A2g⊕Eg
T1g T1gT2gT1g⊕T2gA1g⊕Eg⊕T1g⊕T2g
T2g T2gT1gT1g⊕T2gA2g⊕Eg⊕T1g⊕T2gA1g⊕Eg⊕T1g⊕T2g
A1u A1uA2uEuT1uT2uA1g
A2u A2uA1uEuT2uT1uA2gA1g
Eu EuEuA1u⊕A2u⊕EuT1u⊕T2uT1u⊕T2uEgEgA1g⊕A2g⊕Eg
T1u T1uT2uT1u⊕T2uA1u⊕Eu⊕T1u⊕T2uA2u⊕Eu⊕T1u⊕T2uT1gT2gT1g⊕T2gA1g⊕Eg⊕T1g⊕T2g
T2u T2uT1uT1u⊕T2uA2u⊕Eu⊕T1u⊕T2uA1u⊕Eu⊕T1u⊕T2uT2gT1gT1g⊕T2gA2g⊕Eg⊕T1g⊕T2gA1g⊕Eg⊕T1g⊕T2g

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
T1g A1g⊕Eg⊕T2gA2g⊕2T1g⊕T2g2A1g⊕2Eg⊕T1g⊕2T2gA2g⊕Eg⊕4T1g⊕2T2g3A1g⊕A2g⊕3Eg⊕2T1g⊕4T2gMore
T2g A1g⊕Eg⊕T2gA1g⊕T1g⊕2T2g2A1g⊕2Eg⊕T1g⊕2T2gA1g⊕Eg⊕2T1g⊕4T2g3A1g⊕A2g⊕3Eg⊕2T1g⊕4T2gMore
Eu A1g⊕EgA1u⊕A2u⊕EuA1g⊕2EgA1u⊕A2u⊕2Eu2A1g⊕A2g⊕2EgMore
T1u A1g⊕Eg⊕T2gA2u⊕2T1u⊕T2u2A1g⊕2Eg⊕T1g⊕2T2gA2u⊕Eu⊕4T1u⊕2T2u3A1g⊕A2g⊕3Eg⊕2T1g⊕4T2gMore
T2u A1g⊕Eg⊕T2gA1u⊕T1u⊕2T2u2A1g⊕2Eg⊕T1g⊕2T2gA1u⊕Eu⊕2T1u⊕4T2u3A1g⊕A2g⊕3Eg⊕2T1g⊕4T2gMore



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 T1u 3 T1u
d (l=2) 5 Quadrupole Eg⊕T2g 6 A1g⊕Eg⊕T2g
f (l=3) 7 Octupole A2u⊕T1u⊕T2u 10 A2u⊕2T1u⊕T2u
g (l=4) 9 Hexadecapole A1g⊕Eg⊕T1g⊕T2g 15 2A1g⊕2Eg⊕T1g⊕2T2g
h (l=5) 11 Dotricontapole Eu⊕2T1u⊕T2u 21 A2u⊕Eu⊕4T1u⊕2T2u
i (l=6) 13 Tetrahexacontapole A1g⊕A2g⊕Eg⊕T1g⊕2T2g 28 3A1g⊕A2g⊕3Eg⊕2T1g⊕4T2g
j (l=7) 15 Octacosahectapole A2u⊕Eu⊕2T1u⊕2T2u 36 2A2u⊕2Eu⊕6T1u⊕4T2u
k (l=8) 17 256-pole A1g⊕2Eg⊕2T1g⊕2T2g 45 4A1g⊕A2g⊕5Eg⊕4T1g⊕6T2g
l (l=9) 19 512-pole A1u⊕A2u⊕Eu⊕3T1u⊕2T2u 55 A1u⊕3A2u⊕3Eu⊕9T1u⊕6T2u
m (l=10) 21 1024-pole A1g⊕A2g⊕2Eg⊕2T1g⊕3T2g 66 5A1g⊕2A2g⊕7Eg⊕6T1g⊕9T2g
n (l=11) 23 2048-pole A2u⊕2Eu⊕3T1u⊕3T2u 78 A1u⊕4A2u⊕5Eu⊕12T1u⊕9T2u
o (l=12) 25 4096-pole 2A1g⊕A2g⊕2Eg⊕3T1g⊕3T2g 91 7A1g⊕3A2g⊕9Eg⊕9T1g⊕12T2g
More

First nonvanshing multipole: Hexadecapole

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 Oh
L 2L+1 Term Splitting
S (L=0) 1 A1g
P (L=1) 3 T1g
D (L=2) 5 Eg⊕T2g
F (L=3) 7 A2g⊕T1g⊕T2g
G (L=4) 9 A1g⊕Eg⊕T1g⊕T2g
H (L=5) 11 Eg⊕2T1g⊕T2g
I (L=6) 13 A1g⊕A2g⊕Eg⊕T1g⊕2T2g


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