Point Group D28



D28 E 2C28 2C14 2(C28)3 2C7 2(C28)5 2(C14)3 2C4 2(C7)2 2(C28)9 2(C14)5 2(C28)11 2(C7)3 2(C28)13 C2 14C'2 14C''2
A1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1 2 0 -2 0 0
E2 2 -2 2 0 0
E3 2 0 -2 0 0
E4 2 2 2 0 0
E5 2 0 -2 0 0
E6 2 -2 2 0 0
E7 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 0
E8 2 2 2 0 0
E9 2 0 -2 0 0
E10 2 -2 2 0 0
E11 2 0 -2 0 0
E12 2 2 2 0 0
E13 2 0 -2 0 0
View A
View B


Additional information

Number of symmetry elements h = 56
Number of classes, irreps n = 17
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 2C28 2C14 2(C28)3 2C7 2(C28)5 2(C14)3 2C4 2(C7)2 2(C28)9 2(C14)5 2(C28)11 2(C7)3 2(C28)13 C2 14C'2 14C''2



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13



Direct products of irreducible representations


Binary products
⊙ A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E13E13A1⊕A2⊕E2
E2 E2E2E12E12E1⊕E3A1⊕A2⊕E4
E3 E3E3E11E11E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E10E10E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕E8
E5 E5E5E9E9E4⊕E6E3⊕E7E2⊕E8E1⊕E9A1⊕A2⊕E10
E6 E6E6E8E8E5⊕E7E4⊕E8E3⊕E9E2⊕E10E1⊕E11A1⊕A2⊕E12
E7 E7E7E7E7E6⊕E8E5⊕E9E4⊕E10E3⊕E11E2⊕E12E1⊕E13A1⊕A2⊕B1⊕B2
E8 E8E8E6E6E7⊕E9E6⊕E10E5⊕E11E4⊕E12E3⊕E13B1⊕B2⊕E2E1⊕E13A1⊕A2⊕E12
E9 E9E9E5E5E8⊕E10E7⊕E11E6⊕E12E5⊕E13B1⊕B2⊕E4E3⊕E13E2⊕E12E1⊕E11A1⊕A2⊕E10
E10 E10E10E4E4E9⊕E11E8⊕E12E7⊕E13B1⊕B2⊕E6E5⊕E13E4⊕E12E3⊕E11E2⊕E10E1⊕E9A1⊕A2⊕E8
E11 E11E11E3E3E10⊕E12E9⊕E13B1⊕B2⊕E8E7⊕E13E6⊕E12E5⊕E11E4⊕E10E3⊕E9E2⊕E8E1⊕E7A1⊕A2⊕E6
E12 E12E12E2E2E11⊕E13B1⊕B2⊕E10E9⊕E13E8⊕E12E7⊕E11E6⊕E10E5⊕E9E4⊕E8E3⊕E7E2⊕E6E1⊕E5A1⊕A2⊕E4
E13 E13E13E1E1B1⊕B2⊕E12E11⊕E13E10⊕E12E9⊕E11E8⊕E10E7⊕E9E6⊕E8E5⊕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⊕E4⊕E8E2⊕E6⊕E10A1⊕E4⊕E8⊕E12More
E3 A1⊕E6E3⊕E9A1⊕E6⊕E12E3⊕E9⊕E13A1⊕E6⊕E10⊕E12More
E4 A1⊕E8E4⊕E12A1⊕E8⊕E12E4⊕E8⊕E12A1⊕E4⊕E8⊕E12More
E5 A1⊕E10E5⊕E13A1⊕E8⊕E10E3⊕E5⊕E13A1⊕E2⊕E8⊕E10More
E6 A1⊕E12E6⊕E10A1⊕E4⊕E12E2⊕E6⊕E10A1⊕E4⊕E8⊕E12More
E7 A1⊕B1⊕B22E72A1⊕A2⊕B1⊕B23E72A1⊕A2⊕2B1⊕2B2More
E8 A1⊕E12E4⊕E8A1⊕E4⊕E12E4⊕E8⊕E12A1⊕E4⊕E8⊕E12More
E9 A1⊕E10E1⊕E9A1⊕E8⊕E10E1⊕E9⊕E11A1⊕E2⊕E8⊕E10More
E10 A1⊕E8E2⊕E10A1⊕E8⊕E12E2⊕E6⊕E10A1⊕E4⊕E8⊕E12More
E11 A1⊕E6E5⊕E11A1⊕E6⊕E12E1⊕E5⊕E11A1⊕E6⊕E10⊕E12More
E12 A1⊕E4E8⊕E12A1⊕E4⊕E8E4⊕E8⊕E12A1⊕E4⊕E8⊕E12More
E13 A1⊕E2E11⊕E13A1⊕E2⊕E4E9⊕E11⊕E13A1⊕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 A2⊕E1 3 A2⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A2⊕E1⊕E2⊕E3 10 2A2⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕E1⊕E2⊕E3⊕E4 15 3A1⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A2⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A2⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6 28 4A1⊕3E1⊕3E2⊕2E3⊕2E4⊕E5⊕E6
j (l=7) 15 Octacosahectapole A2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4A2⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 45 5A1⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7⊕E8
l (l=9) 19 512-pole A2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 55 5A2⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕2E7⊕E8⊕E9
m (l=10) 21 1024-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10 66 6A1⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕3E6⊕2E7⊕2E8⊕E9⊕E10
n (l=11) 23 2048-pole A2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11 78 6A2⊕6E1⊕5E2⊕5E3⊕4E4⊕4E5⊕3E6⊕3E7⊕2E8⊕2E9⊕E10⊕E11
o (l=12) 25 4096-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12 91 7A1⊕6E1⊕6E2⊕5E3⊕5E4⊕4E5⊕4E6⊕3E7⊕3E8⊕2E9⊕2E10⊕E11⊕E12
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 D28
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E1
D (L=2) 5 A1⊕E1⊕E2
F (L=3) 7 A2⊕E1⊕E2⊕E3
G (L=4) 9 A1⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A2⊕E1⊕E2⊕E3⊕E4⊕E5
I (L=6) 13 A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6

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