Point Group D4d



D4d E 2S8 2C4 2(S8)3 C2 4C'2 d
A1 1 1 1 1 1 1 1
A2 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1
E1 2 2cos(π/4) 0 -2cos(π/4) -2 0 0
E2 2 0 -2 0 2 0 0
E3 2 -2cos(π/4) 0 2cos(π/4) -2 0 0


Additional information

Number of symmetry elements h = 16
Number of classes, irreps n = 7
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2S8 2C4 2(S8)3 C2 4C'2 d



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3




Examples

Sulfur S8



Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2 E3
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E3E3A1⊕A2⊕E2
E2 E2E2E2E2E1⊕E3A1⊕A2⊕B1⊕B2
E3 E3E3E1E1B1⊕B2⊕E2E1⊕E3A1⊕A2⊕E2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1 A1⊕E2E1⊕E3A1⊕B1⊕B2⊕E2E1⊕2E3A1⊕B1⊕B2⊕2E2More
E2 A1⊕B1⊕B22E22A1⊕A2⊕B1⊕B23E22A1⊕A2⊕2B1⊕2B2More
E3 A1⊕E2E1⊕E3A1⊕B1⊕B2⊕E22E1⊕E3A1⊕B1⊕B2⊕2E2More



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 A1 1 A1
p (l=1) 3 Dipole B2⊕E1 3 B2⊕E1
d (l=2) 5 Quadrupole A1⊕E2⊕E3 6 2A1⊕E2⊕E3
f (l=3) 7 Octupole B2⊕E1⊕E2⊕E3 10 2B2⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕B1⊕B2⊕E1⊕E2⊕E3 15 3A1⊕B1⊕B2⊕E1⊕2E2⊕2E3
h (l=5) 11 Dotricontapole A1⊕A2⊕B2⊕E1⊕E2⊕2E3 21 A1⊕A2⊕3B2⊕3E1⊕2E2⊕3E3
i (l=6) 13 Tetrahexacontapole A1⊕B1⊕B2⊕2E1⊕2E2⊕E3 28 4A1⊕2B1⊕2B2⊕3E1⊕4E2⊕3E3
j (l=7) 15 Octacosahectapole A1⊕A2⊕B2⊕2E1⊕2E2⊕2E3 36 2A1⊕2A2⊕4B2⊕5E1⊕4E2⊕5E3
k (l=8) 17 256-pole 2A1⊕A2⊕B1⊕B2⊕2E1⊕2E2⊕2E3 45 6A1⊕A2⊕3B1⊕3B2⊕5E1⊕6E2⊕5E3
l (l=9) 19 512-pole A1⊕A2⊕B1⊕2B2⊕3E1⊕2E2⊕2E3 55 3A1⊕3A2⊕B1⊕6B2⊕8E1⊕6E2⊕7E3
m (l=10) 21 1024-pole 2A1⊕A2⊕B1⊕B2⊕2E1⊕3E2⊕3E3 66 8A1⊕2A2⊕4B1⊕4B2⊕7E1⊕9E2⊕8E3
n (l=11) 23 2048-pole A1⊕A2⊕B1⊕2B2⊕3E1⊕3E2⊕3E3 78 4A1⊕4A2⊕2B1⊕8B2⊕11E1⊕9E2⊕10E3
o (l=12) 25 4096-pole 2A1⊕A2⊕2B1⊕2B2⊕3E1⊕3E2⊕3E3 91 10A1⊕3A2⊕6B1⊕6B2⊕10E1⊕12E2⊕11E3
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 D4d
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E3
D (L=2) 5 A1⊕E2⊕E3
F (L=3) 7 A2⊕E1⊕E2⊕E3
G (L=4) 9 A1⊕B1⊕B2⊕E1⊕E2⊕E3
H (L=5) 11 A2⊕B1⊕B2⊕2E1⊕E2⊕E3
I (L=6) 13 A1⊕B1⊕B2⊕2E1⊕2E2⊕E3


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