Point Group O



O E 8C3 6C2 6C4 3C2
A1 1 1 1 1 1
A2 1 1 -1 -1 1
E 2 -1 0 0 2
T1 3 0 -1 1 -1
T2 3 0 1 -1 -1


Additional information

Number of symmetry elements h = 24
Number of classes, irreps n = 5
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 8C3 6C2 6C4 3C2



Genrate representation from irreducible representations


A1 A2 E T1 T2




Examples

Octamethylsilsesquioxane (rotated) Dodecaethyleneoctamine



Direct products of irreducible representations


Binary products
A1 A2 E T1 T2
A1 A1
A2 A2A1
E EEA1⊕A2⊕E
T1 T1T2T1⊕T2A1⊕E⊕T1⊕T2
T2 T2T1T1⊕T2A2⊕E⊕T1⊕T2A1⊕E⊕T1⊕T2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E A1⊕EA1⊕A2⊕EA1⊕2EA1⊕A2⊕2E2A1⊕A2⊕2EMore
T1 A1⊕E⊕T2A2⊕2T1⊕T22A1⊕2E⊕T1⊕2T2A2⊕E⊕4T1⊕2T23A1⊕A2⊕3E⊕2T1⊕4T2More
T2 A1⊕E⊕T2A1⊕T1⊕2T22A1⊕2E⊕T1⊕2T2A1⊕E⊕2T1⊕4T23A1⊕A2⊕3E⊕2T1⊕4T2More



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 A1 1 A1
p (l=1) 3 Dipole T1 3 T1
d (l=2) 5 Quadrupole E⊕T2 6 A1⊕E⊕T2
f (l=3) 7 Octupole A2⊕T1⊕T2 10 A2⊕2T1⊕T2
g (l=4) 9 Hexadecapole A1⊕E⊕T1⊕T2 15 2A1⊕2E⊕T1⊕2T2
h (l=5) 11 Dotricontapole E⊕2T1⊕T2 21 A2⊕E⊕4T1⊕2T2
i (l=6) 13 Tetrahexacontapole A1⊕A2⊕E⊕T1⊕2T2 28 3A1⊕A2⊕3E⊕2T1⊕4T2
j (l=7) 15 Octacosahectapole A2⊕E⊕2T1⊕2T2 36 2A2⊕2E⊕6T1⊕4T2
k (l=8) 17 256-pole A1⊕2E⊕2T1⊕2T2 45 4A1⊕A2⊕5E⊕4T1⊕6T2
l (l=9) 19 512-pole A1⊕A2⊕E⊕3T1⊕2T2 55 A1⊕3A2⊕3E⊕9T1⊕6T2
m (l=10) 21 1024-pole A1⊕A2⊕2E⊕2T1⊕3T2 66 5A1⊕2A2⊕7E⊕6T1⊕9T2
n (l=11) 23 2048-pole A2⊕2E⊕3T1⊕3T2 78 A1⊕4A2⊕5E⊕12T1⊕9T2
o (l=12) 25 4096-pole 2A1⊕A2⊕2E⊕3T1⊕3T2 91 7A1⊕3A2⊕9E⊕9T1⊕12T2
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 O
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 T1
D (L=2) 5 E⊕T2
F (L=3) 7 A2⊕T1⊕T2
G (L=4) 9 A1⊕E⊕T1⊕T2
H (L=5) 11 E⊕2T1⊕T2
I (L=6) 13 A1⊕A2⊕E⊕T1⊕2T2


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