Point Group D2h



D2h E C2.(z) C2.(y) C2.(x) i σ.(xy) σ.(xy) σ.(xy)
A1g 1 1 1 1 1 1 1 1
B1g 1 1 -1 -1 1 1 -1 -1
B2g 1 -1 1 -1 1 -1 1 -1
B3g 1 -1 -1 1 1 -1 -1 1
A1u 1 1 1 1 -1 -1 -1 -1
B1u 1 1 -1 -1 -1 -1 1 1
B2u 1 -1 1 -1 -1 1 -1 1
B3u 1 -1 -1 1 -1 1 1 -1


Additional information

Number of symmetry elements h = 8
Number of classes, irreps n = 8
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E C2.(z) C2.(y) C2.(x) i σ.(xy) σ.(xy) σ.(xy)



Genrate representation from irreducible representations


A1g B1g B2g B3g A1u B1u B2u B3u




Examples

Ethylene Diborane Ethylenetetracarboxylic Acid Dianhydride
Tetrahydroxy-1,4-benzoquinone Biscarbonate Naphthalene Fulvalene
Tricyclo[3.1.1.1(2,4)]octane Anthracene Pyrene
Tetracene Pagodane Dianthracene



Direct products of irreducible representations


Binary products
A1g B1g B2g B3g A1u B1u B2u B3u
A1g A1g
B1g B1gA1g
B2g B2gB3gA1g
B3g B3gB2gB1gA1g
A1u A1uB1uB2uB3uA1g
B1u B1uA1uB3uB2uB1gA1g
B2u B2uB3uA1uB1uB2gB3gA1g
B3u B3uB2uB1uA1uB3gB2gB1gA1g

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 l(xyz)]
s (l=0) 1 Monopole A1g 1 A1g
p (l=1) 3 Dipole B1u⊕B2u⊕B3u 3 B1u⊕B2u⊕B3u
d (l=2) 5 Quadrupole 2A1g⊕B1g⊕B2g⊕B3g 6 3A1g⊕B1g⊕B2g⊕B3g
f (l=3) 7 Octupole A1u⊕2B1u⊕2B2u⊕2B3u 10 A1u⊕3B1u⊕3B2u⊕3B3u
g (l=4) 9 Hexadecapole 3A1g⊕2B1g⊕2B2g⊕2B3g 15 6A1g⊕3B1g⊕3B2g⊕3B3g
h (l=5) 11 Dotricontapole 2A1u⊕3B1u⊕3B2u⊕3B3u 21 3A1u⊕6B1u⊕6B2u⊕6B3u
i (l=6) 13 Tetrahexacontapole 4A1g⊕3B1g⊕3B2g⊕3B3g 28 10A1g⊕6B1g⊕6B2g⊕6B3g
j (l=7) 15 Octacosahectapole 3A1u⊕4B1u⊕4B2u⊕4B3u 36 6A1u⊕10B1u⊕10B2u⊕10B3u
k (l=8) 17 256-pole 5A1g⊕4B1g⊕4B2g⊕4B3g 45 15A1g⊕10B1g⊕10B2g⊕10B3g
l (l=9) 19 512-pole 4A1u⊕5B1u⊕5B2u⊕5B3u 55 10A1u⊕15B1u⊕15B2u⊕15B3u
m (l=10) 21 1024-pole 6A1g⊕5B1g⊕5B2g⊕5B3g 66 21A1g⊕15B1g⊕15B2g⊕15B3g
n (l=11) 23 2048-pole 5A1u⊕6B1u⊕6B2u⊕6B3u 78 15A1u⊕21B1u⊕21B2u⊕21B3u
o (l=12) 25 4096-pole 7A1g⊕6B1g⊕6B2g⊕6B3g 91 28A1g⊕21B1g⊕21B2g⊕21B3g
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 D2h
L 2L+1 Term Splitting
S (L=0) 1 A1g
P (L=1) 3 B1g⊕B2g⊕B3g
D (L=2) 5 2A1g⊕B1g⊕B2g⊕B3g
F (L=3) 7 A1g⊕2B1g⊕2B2g⊕2B3g
G (L=4) 9 3A1g⊕2B1g⊕2B2g⊕2B3g
H (L=5) 11 2A1g⊕3B1g⊕3B2g⊕3B3g
I (L=6) 13 4A1g⊕3B1g⊕3B2g⊕3B3g


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