Point Group D3h



D3h E 2C3 3C'2 σh 2S3 3σv
A'1 1 1 1 1 1 1
A'2 1 1 -1 1 1 -1
E' 2 -1 0 2 -1 0
A''1 1 1 1 -1 -1 -1
A''2 1 1 -1 -1 -1 1
E'' 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 no


Reduce representation to irreducible representations


E 2C3 3C'2 σh 2S3 3σv



Genrate representation from irreducible representations


A'1 A'2 E' A''1 A''2 E''



Examples

Borontrifluoride Phosphoruspentafluoride Ethane (eclipsed)
Cyclopropane Ironpentacarbonyl Borazine
Prismane Diironnonacarbonyl Triphenylene
Triptycene Fullerene C74


Direct products of irreducible representations


Binary products
⊙ A'1 A'2 E' A''1 A''2 E''
A'1 A'1
A'2 A'2A'1
E' E'E'A'1⊕A'2⊕E'
A''1 A''1A''2E''A'1
A''2 A''2A''1E''A'2A'1
E'' E''E''A''1⊕A''2⊕E''E'E'A'1⊕A'2⊕E'

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E' A'1⊕E'A'1⊕A'2⊕E'A'1⊕2E'A'1⊕A'2⊕2E'2A'1⊕A'2⊕2E'More
E'' A'1⊕E'A''1⊕A''2⊕E''A'1⊕2E'A''1⊕A''2⊕2E''2A'1⊕A'2⊕2E'More


Spherical harmonics and Multipoles
Symmetric Powers of Γxyz


Spherical Harmonics Yl / Multipole Symmetric Power [Γl(xyz)]
l 2l+1 Multipole Symmetry Rank [Γl(xyz)]
s (l=0) 1 Monopole A'1 1 A'1
p (l=1) 3 Dipole E'⊕A''2 3 E'⊕A''2
d (l=2) 5 Quadrupole A'1⊕E'⊕E'' 6 2A'1⊕E'⊕E''
f (l=3) 7 Octupole A'1⊕A'2⊕E'⊕A''2⊕E'' 10 A'1⊕A'2⊕2E'⊕2A''2⊕E''
g (l=4) 9 Hexadecapole A'1⊕2E'⊕A''1⊕A''2⊕E'' 15 3A'1⊕3E'⊕A''1⊕A''2⊕2E''
h (l=5) 11 Dotricontapole A'1⊕A'2⊕2E'⊕A''2⊕2E'' 21 2A'1⊕2A'2⊕4E'⊕3A''2⊕3E''
i (l=6) 13 Tetrahexacontapole 2A'1⊕A'2⊕2E'⊕A''1⊕A''2⊕2E'' 28 5A'1⊕A'2⊕5E'⊕2A''1⊕2A''2⊕4E''
j (l=7) 15 Octacosahectapole A'1⊕A'2⊕3E'⊕A''1⊕2A''2⊕2E'' 36 3A'1⊕3A'2⊕7E'⊕A''1⊕5A''2⊕5E''
k (l=8) 17 256-pole 2A'1⊕A'2⊕3E'⊕A''1⊕A''2⊕3E'' 45 7A'1⊕2A'2⊕8E'⊕3A''1⊕3A''2⊕7E''
l (l=9) 19 512-pole 2A'1⊕2A'2⊕3E'⊕A''1⊕2A''2⊕3E'' 55 5A'1⊕5A'2⊕10E'⊕2A''1⊕7A''2⊕8E''
m (l=10) 21 1024-pole 2A'1⊕A'2⊕4E'⊕2A''1⊕2A''2⊕3E'' 66 9A'1⊕3A'2⊕12E'⊕5A''1⊕5A''2⊕10E''
n (l=11) 23 2048-pole 2A'1⊕2A'2⊕4E'⊕A''1⊕2A''2⊕4E'' 78 7A'1⊕7A'2⊕14E'⊕3A''1⊕9A''2⊕12E''
o (l=12) 25 4096-pole 3A'1⊕2A'2⊕4E'⊕2A''1⊕2A''2⊕4E'' 91 12A'1⊕5A'2⊕16E'⊕7A''1⊕7A''2⊕14E''
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 D3h
L 2L+1 Term Splitting
S (L=0) 1 A'1
P (L=1) 3 A'2⊕E''
D (L=2) 5 A'1⊕E'⊕E''
F (L=3) 7 A'2⊕E'⊕A''1⊕A''2⊕E''
G (L=4) 9 A'1⊕2E'⊕A''1⊕A''2⊕E''
H (L=5) 11 A'2⊕2E'⊕A''1⊕A''2⊕2E''
I (L=6) 13 2A'1⊕A'2⊕2E'⊕A''1⊕A''2⊕2E''

Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement