Point Group D∞h



D∞h E 2C∞ … ∞σv i 2S∞ … ∞C'2
A1g Σ+g 1 1 1 1 1 1 1 1
A2g Σ-g -1 -1 -1 -1 -1 -1 -1 -1
E1g Πg -1 1 1 1 1 1 1 1
E2g Δg 0 0 2 2 2 2 2 2
E3g Φg 0 0 2 2 2 2 2 2
… … 0 0 2 2 2 2 2 2
Eng 0 0 2 2 2 2 2 2
A1u Σ+u 0 0 2 2 2 2 2 2
A2u Σ-u 0 0 2 2 2 2 2 2
E1u Πu 0 0 2 2 2 2 2 2
E2u Δu 0 0 2 2 2 2 2 2
E3u Φu 0 0 2 2 2 2 2 2
… … 0 0 2 2 2 2 2 2
Enu 0 0 2 2 2 2 2 2
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Additional information

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


Reduce representation to irreducible representations


E 2C∞ … ∞σv i 2S∞ … ∞C'2



Genrate representation from irreducible representations


A1g A2g E1g E2g E3g … Eng A1u A2u E1u E2u E3u … Enu



Examples

Hydrogen Carbondioxide Acetylene
Carbonsuboxide Diacetylene


Direct products of irreducible representations


Binary products
⊙ A1g A2g E1g E2g E3g … Eng A1u A2u E1u E2u E3u … Enu
A1g 8A2g
A2g 8A1g⊕6E1g⊕12E2g⊕12E3g⊕12…⊕12Eng⊕12A1u⊕12A2u⊕12E1u⊕12E2u⊕12E3u⊕12…⊕12Enu8A2g
E1g 6A2g6A1g⊕8E1g⊕12E2g⊕12E3g⊕12…⊕12Eng⊕12A1u⊕12A2u⊕12E1u⊕12E2u⊕12E3u⊕12…⊕12Enu8A2g
E2g 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g
E3g 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g
… 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g
Eng 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g24A2g
A1u 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g24A2g24A2g
A2u 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g24A2g24A2g24A2g
E1u 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g
E2u 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g
E3u 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g
… 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g
Enu 12A2g12A1g⊕12E1g⊕24E2g⊕24E3g⊕24…⊕24Eng⊕24A1u⊕24A2u⊕24E1u⊕24E2u⊕24E3u⊕24…⊕24Enu12A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g24A2g

Ternary Products
Quaternary Products


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 A1g 1 A1g
p (l=1) 3 Dipole E1g 3 E1g
d (l=2) 5 Quadrupole 6
f (l=3) 7 Octupole 10
g (l=4) 9 Hexadecapole 15
h (l=5) 11 Dotricontapole 21
i (l=6) 13 Tetrahexacontapole 28
j (l=7) 15 Octacosahectapole 36
k (l=8) 17 256-pole 45
l (l=9) 19 512-pole 55
m (l=10) 21 1024-pole 66
n (l=11) 23 2048-pole 78
o (l=12) 25 4096-pole 91
More

First nonvanshing multipole: Monopole

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 D∞h
L 2L+1 Term Splitting
S (L=0) 1 A1g
P (L=1) 3 8A2g
D (L=2) 5 8A2g
F (L=3) 7
G (L=4) 9
H (L=5) 11
I (L=6) 13

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