Point Group D8d



D8d E 2S16 2C8 2(S16)3 2C4 2(S16)5 2(C8)3 2(S16)7 C2 8C'2 8σd
A1 1 1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1 2 2cos(π/8) 2cos(π/4) 2cos(3π/8) 0 -2cos(3π/8) -2cos(π/4) -2cos(π/8) -2 0 0
E2 2 2cos(π/4) 0 -2cos(π/4) -2 -2cos(π/4) 0 2cos(π/4) 2 0 0
E3 2 2cos(3π/8) -2cos(π/4) -2cos(π/8) 0 2cos(π/8) 2cos(π/4) -2cos(3π/8) -2 0 0
E4 2 0 -2 0 2 0 -2 0 2 0 0
E5 2 -2cos(3π/8) -2cos(π/4) 2cos(π/8) 0 -2cos(π/8) 2cos(π/4) 2cos(3π/8) -2 0 0
E6 2 -2cos(π/4) 0 2cos(π/4) -2 2cos(π/4) 0 -2cos(π/4) 2 0 0
E7 2 -2cos(π/8) 2cos(π/4) -2cos(3π/8) 0 2cos(3π/8) -2cos(π/4) 2cos(π/8) -2 0 0
View A
✅
View B


Additional information

Number of symmetry elements h = 32
Number of classes, irreps n = 11
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2S16 2C8 2(S16)3 2C4 2(S16)5 2(C8)3 2(S16)7 C2 8C'2 8σd



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7



Examples

Uranocene (staggered)


Direct products of irreducible representations


Binary products
⊙ A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E7E7A1⊕A2⊕E2
E2 E2E2E6E6E1⊕E3A1⊕A2⊕E4
E3 E3E3E5E5E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E4E4E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕B1⊕B2
E5 E5E5E3E3E4⊕E6E3⊕E7B1⊕B2⊕E2E1⊕E7A1⊕A2⊕E6
E6 E6E6E2E2E5⊕E7B1⊕B2⊕E4E3⊕E7E2⊕E6E1⊕E5A1⊕A2⊕E4
E7 E7E7E1E1B1⊕B2⊕E6E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕E3A1⊕A2⊕E2

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E4E1⊕E3⊕E5A1⊕E2⊕E4⊕E6More
E2 A1⊕E4E2⊕E6A1⊕B1⊕B2⊕E4E2⊕2E6A1⊕B1⊕B2⊕2E4More
E3 A1⊕E6E3⊕E7A1⊕E4⊕E6E1⊕E3⊕E7A1⊕E2⊕E4⊕E6More
E4 A1⊕B1⊕B22E42A1⊕A2⊕B1⊕B23E42A1⊕A2⊕2B1⊕2B2More
E5 A1⊕E6E1⊕E5A1⊕E4⊕E6E1⊕E5⊕E7A1⊕E2⊕E4⊕E6More
E6 A1⊕E4E2⊕E6A1⊕B1⊕B2⊕E42E2⊕E6A1⊕B1⊕B2⊕2E4More
E7 A1⊕E2E5⊕E7A1⊕E2⊕E4E3⊕E5⊕E7A1⊕E2⊕E4⊕E6More


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 A1 1 A1
p (l=1) 3 Dipole B2⊕E1 3 B2⊕E1
d (l=2) 5 Quadrupole A1⊕E2⊕E7 6 2A1⊕E2⊕E7
f (l=3) 7 Octupole B2⊕E1⊕E3⊕E6 10 2B2⊕2E1⊕E3⊕E6
g (l=4) 9 Hexadecapole A1⊕E2⊕E4⊕E5⊕E7 15 3A1⊕2E2⊕E4⊕E5⊕2E7
h (l=5) 11 Dotricontapole B2⊕E1⊕E3⊕E4⊕E5⊕E6 21 3B2⊕3E1⊕2E3⊕E4⊕E5⊕2E6
i (l=6) 13 Tetrahexacontapole A1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 28 4A1⊕3E2⊕E3⊕2E4⊕2E5⊕E6⊕3E7
j (l=7) 15 Octacosahectapole B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4B2⊕4E1⊕E2⊕3E3⊕2E4⊕2E5⊕3E6⊕E7
k (l=8) 17 256-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 45 5A1⊕B1⊕B2⊕E1⊕4E2⊕2E3⊕3E4⊕3E5⊕2E6⊕4E7
l (l=9) 19 512-pole A1⊕A2⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕2E7 55 A1⊕A2⊕5B2⊕5E1⊕2E2⊕4E3⊕3E4⊕3E5⊕4E6⊕3E7
m (l=10) 21 1024-pole A1⊕B1⊕B2⊕2E1⊕E2⊕E3⊕E4⊕E5⊕2E6⊕E7 66 6A1⊕2B1⊕2B2⊕3E1⊕5E2⊕3E3⊕4E4⊕4E5⊕4E6⊕5E7
n (l=11) 23 2048-pole A1⊕A2⊕B2⊕E1⊕2E2⊕E3⊕E4⊕2E5⊕E6⊕2E7 78 2A1⊕2A2⊕6B2⊕6E1⊕4E2⊕5E3⊕4E4⊕5E5⊕5E6⊕5E7
o (l=12) 25 4096-pole A1⊕B1⊕B2⊕2E1⊕E2⊕2E3⊕2E4⊕E5⊕2E6⊕E7 91 7A1⊕3B1⊕3B2⊕5E1⊕6E2⊕5E3⊕6E4⊕5E5⊕6E6⊕6E7
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 D8d
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E7
D (L=2) 5 A1⊕E2⊕E7
F (L=3) 7 A2⊕E2⊕E5⊕E7
G (L=4) 9 A1⊕E2⊕E4⊕E5⊕E7
H (L=5) 11 A2⊕E2⊕E3⊕E4⊕E5⊕E7
I (L=6) 13 A1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7

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