Point Group D2



D2 E C2.(z) C2.(y) C2.(x)
A 1 1 1 1
B1 1 1 -1 -1
B2 1 -1 1 -1
B3 1 -1 -1 1


Additional information

Number of symmetry elements h = 4
Number of classes, irreps n = 4
Abelian group yes
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E C2.(z) C2.(y) C2.(x)



Genrate representation from irreducible representations


A B1 B2 B3




Examples

Biphenyl Twistane



Direct products of irreducible representations


Binary products
A B1 B2 B3
A A
B1 B1A
B2 B2B3A
B3 B3B2B1A

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 A 1 A
p (l=1) 3 Dipole B1⊕B2⊕B3 3 B1⊕B2⊕B3
d (l=2) 5 Quadrupole 2A⊕B1⊕B2⊕B3 6 3A⊕B1⊕B2⊕B3
f (l=3) 7 Octupole A⊕2B1⊕2B2⊕2B3 10 A⊕3B1⊕3B2⊕3B3
g (l=4) 9 Hexadecapole 3A⊕2B1⊕2B2⊕2B3 15 6A⊕3B1⊕3B2⊕3B3
h (l=5) 11 Dotricontapole 2A⊕3B1⊕3B2⊕3B3 21 3A⊕6B1⊕6B2⊕6B3
i (l=6) 13 Tetrahexacontapole 4A⊕3B1⊕3B2⊕3B3 28 10A⊕6B1⊕6B2⊕6B3
j (l=7) 15 Octacosahectapole 3A⊕4B1⊕4B2⊕4B3 36 6A⊕10B1⊕10B2⊕10B3
k (l=8) 17 256-pole 5A⊕4B1⊕4B2⊕4B3 45 15A⊕10B1⊕10B2⊕10B3
l (l=9) 19 512-pole 4A⊕5B1⊕5B2⊕5B3 55 10A⊕15B1⊕15B2⊕15B3
m (l=10) 21 1024-pole 6A⊕5B1⊕5B2⊕5B3 66 21A⊕15B1⊕15B2⊕15B3
n (l=11) 23 2048-pole 5A⊕6B1⊕6B2⊕6B3 78 15A⊕21B1⊕21B2⊕21B3
o (l=12) 25 4096-pole 7A⊕6B1⊕6B2⊕6B3 91 28A⊕21B1⊕21B2⊕21B3
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 D2
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 B1⊕B2⊕B3
D (L=2) 5 2A⊕B1⊕B2⊕B3
F (L=3) 7 A⊕2B1⊕2B2⊕2B3
G (L=4) 9 3A⊕2B1⊕2B2⊕2B3
H (L=5) 11 2A⊕3B1⊕3B2⊕3B3
I (L=6) 13 4A⊕3B1⊕3B2⊕3B3


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