Point Group D3



D3 E 2C3 3C'2
A1 1 1 1
A2 1 1 -1
E 2 -1 0


Additional information

Number of symmetry elements h = 6
Number of classes, irreps n = 3
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 2C3 3C'2



Genrate representation from irreducible representations


A1 A2 E




Examples

Ethane (rotated)



Direct products of irreducible representations


Binary products
A1 A2 E
A1 A1
A2 A2A1
E EEA1⊕A2⊕E

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



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 A2⊕E 3 A2⊕E
d (l=2) 5 Quadrupole A1⊕2E 6 2A1⊕2E
f (l=3) 7 Octupole A1⊕2A2⊕2E 10 A1⊕3A2⊕3E
g (l=4) 9 Hexadecapole 2A1⊕A2⊕3E 15 4A1⊕A2⊕5E
h (l=5) 11 Dotricontapole A1⊕2A2⊕4E 21 2A1⊕5A2⊕7E
i (l=6) 13 Tetrahexacontapole 3A1⊕2A2⊕4E 28 7A1⊕3A2⊕9E
j (l=7) 15 Octacosahectapole 2A1⊕3A2⊕5E 36 4A1⊕8A2⊕12E
k (l=8) 17 256-pole 3A1⊕2A2⊕6E 45 10A1⊕5A2⊕15E
l (l=9) 19 512-pole 3A1⊕4A2⊕6E 55 7A1⊕12A2⊕18E
m (l=10) 21 1024-pole 4A1⊕3A2⊕7E 66 14A1⊕8A2⊕22E
n (l=11) 23 2048-pole 3A1⊕4A2⊕8E 78 10A1⊕16A2⊕26E
o (l=12) 25 4096-pole 5A1⊕4A2⊕8E 91 19A1⊕12A2⊕30E
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 D3
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E
D (L=2) 5 A1⊕2E
F (L=3) 7 A1⊕2A2⊕2E
G (L=4) 9 2A1⊕A2⊕3E
H (L=5) 11 A1⊕2A2⊕4E
I (L=6) 13 3A1⊕2A2⊕4E


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