Point Group S28



S28 E 2S28 2C14 2(S28)3 2C7 2(S28)5 2(C14)3 2S4 2(C7)2 2(S28)9 2(C14)5 2(S28)11 2(C7)3 2(S28)13 C2
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1* 2 2cos(π/14) 2cos(π/7) 2cos(3π/14) 2cos(2π/7) 2cos(5π/14) 2cos(3π/7) 0 -2cos(3π/7) -2cos(5π/14) -2cos(2π/7) -2cos(3π/14) -2cos(π/7) -2cos(π/14) -2
E2* 2 2cos(π/7) 2cos(2π/7) 2cos(3π/7) -2cos(3π/7) -2cos(2π/7) -2cos(π/7) -2 -2cos(π/7) -2cos(2π/7) -2cos(3π/7) 2cos(3π/7) 2cos(2π/7) 2cos(π/7) 2
E3* 2 2cos(3π/14) 2cos(3π/7) -2cos(5π/14) -2cos(π/7) -2cos(π/14) -2cos(2π/7) 0 2cos(2π/7) 2cos(π/14) 2cos(π/7) 2cos(5π/14) -2cos(3π/7) -2cos(3π/14) -2
E4* 2 2cos(2π/7) -2cos(3π/7) -2cos(π/7) -2cos(π/7) -2cos(3π/7) 2cos(2π/7) 2 2cos(2π/7) -2cos(3π/7) -2cos(π/7) -2cos(π/7) -2cos(3π/7) 2cos(2π/7) 2
E5* 2 2cos(5π/14) -2cos(2π/7) -2cos(π/14) -2cos(3π/7) 2cos(3π/14) 2cos(π/7) 0 -2cos(π/7) -2cos(3π/14) 2cos(3π/7) 2cos(π/14) 2cos(2π/7) -2cos(5π/14) -2
E6* 2 2cos(3π/7) -2cos(π/7) -2cos(2π/7) 2cos(2π/7) 2cos(π/7) -2cos(3π/7) -2 -2cos(3π/7) 2cos(π/7) 2cos(2π/7) -2cos(2π/7) -2cos(π/7) 2cos(3π/7) 2
E7* 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2
E8* 2 -2cos(3π/7) -2cos(π/7) 2cos(2π/7) 2cos(2π/7) -2cos(π/7) -2cos(3π/7) 2 -2cos(3π/7) -2cos(π/7) 2cos(2π/7) 2cos(2π/7) -2cos(π/7) -2cos(3π/7) 2
E9* 2 -2cos(5π/14) -2cos(2π/7) 2cos(π/14) -2cos(3π/7) -2cos(3π/14) 2cos(π/7) 0 -2cos(π/7) 2cos(3π/14) 2cos(3π/7) -2cos(π/14) 2cos(2π/7) 2cos(5π/14) -2
E10* 2 -2cos(2π/7) -2cos(3π/7) 2cos(π/7) -2cos(π/7) 2cos(3π/7) 2cos(2π/7) -2 2cos(2π/7) 2cos(3π/7) -2cos(π/7) 2cos(π/7) -2cos(3π/7) -2cos(2π/7) 2
E11* 2 -2cos(3π/14) 2cos(3π/7) 2cos(5π/14) -2cos(π/7) 2cos(π/14) -2cos(2π/7) 0 2cos(2π/7) -2cos(π/14) 2cos(π/7) -2cos(5π/14) -2cos(3π/7) 2cos(3π/14) -2
E12* 2 -2cos(π/7) 2cos(2π/7) -2cos(3π/7) -2cos(3π/7) 2cos(2π/7) -2cos(π/7) 2 -2cos(π/7) 2cos(2π/7) -2cos(3π/7) -2cos(3π/7) 2cos(2π/7) -2cos(π/7) 2
E13* 2 -2cos(π/14) 2cos(π/7) -2cos(3π/14) 2cos(2π/7) -2cos(5π/14) 2cos(3π/7) 0 -2cos(3π/7) 2cos(5π/14) -2cos(2π/7) 2cos(3π/14) -2cos(π/7) 2cos(π/14) -2
Math View
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Chem View A
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Chem View B


Additional information

Number of symmetry elements h = 28
Number of classes, irreps n = 28
Number of real-valued irreducible representations n = 15
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2S28 2C14 2(S28)3 2C7 2(S28)5 2(C14)3 2S4 2(C7)2 2(S28)9 2(C14)5 2(S28)11 2(C7)3 2(S28)13 C2



Genrate representation from irreducible representations


A B E1* E2* E3* E4* E5* E6* E7* E8* E9* E10* E11* E12* E13*



Direct products of irreducible representations


Binary products
⊙ A B E1* E2* E3* E4* E5* E6* E7* E8* E9* E10* E11* E12* E13*
A A
B BA
E1* E1E132A⊕E2
E2* E2E12E1⊕E32A⊕E4
E3* E3E11E2⊕E4E1⊕E52A⊕E6
E4* E4E10E3⊕E5E2⊕E6E1⊕E72A⊕E8
E5* E5E9E4⊕E6E3⊕E7E2⊕E8E1⊕E92A⊕E10
E6* E6E8E5⊕E7E4⊕E8E3⊕E9E2⊕E10E1⊕E112A⊕E12
E7* E7E7E6⊕E8E5⊕E9E4⊕E10E3⊕E11E2⊕E12E1⊕E132A⊕2B
E8* E8E6E7⊕E9E6⊕E10E5⊕E11E4⊕E12E3⊕E132B⊕E2E1⊕E132A⊕E12
E9* E9E5E8⊕E10E7⊕E11E6⊕E12E5⊕E132B⊕E4E3⊕E13E2⊕E12E1⊕E112A⊕E10
E10* E10E4E9⊕E11E8⊕E12E7⊕E132B⊕E6E5⊕E13E4⊕E12E3⊕E11E2⊕E10E1⊕E92A⊕E8
E11* E11E3E10⊕E12E9⊕E132B⊕E8E7⊕E13E6⊕E12E5⊕E11E4⊕E10E3⊕E9E2⊕E8E1⊕E72A⊕E6
E12* E12E2E11⊕E132B⊕E10E9⊕E13E8⊕E12E7⊕E11E6⊕E10E5⊕E9E4⊕E8E3⊕E7E2⊕E6E1⊕E52A⊕E4
E13* E13E12B⊕E12E11⊕E13E10⊕E12E9⊕E11E8⊕E10E7⊕E9E6⊕E8E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕E32A⊕E2

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1* A⊕E2E1⊕E3A⊕E2⊕E4E1⊕E3⊕E5A⊕E2⊕E4⊕E6More
E2* A⊕E4E2⊕E6A⊕E4⊕E8E2⊕E6⊕E10A⊕E4⊕E8⊕E12More
E3* A⊕E6E3⊕E9A⊕E6⊕E12E3⊕E9⊕E13A⊕E6⊕E10⊕E12More
E4* A⊕E8E4⊕E12A⊕E8⊕E12E4⊕E8⊕E12A⊕E4⊕E8⊕E12More
E5* A⊕E10E5⊕E13A⊕E8⊕E10E3⊕E5⊕E13A⊕E2⊕E8⊕E10More
E6* A⊕E12E6⊕E10A⊕E4⊕E12E2⊕E6⊕E10A⊕E4⊕E8⊕E12More
E7* A⊕2B2E73A⊕2B3E73A⊕4BMore
E8* A⊕E12E4⊕E8A⊕E4⊕E12E4⊕E8⊕E12A⊕E4⊕E8⊕E12More
E9* A⊕E10E1⊕E9A⊕E8⊕E10E1⊕E9⊕E11A⊕E2⊕E8⊕E10More
E10* A⊕E8E2⊕E10A⊕E8⊕E12E2⊕E6⊕E10A⊕E4⊕E8⊕E12More
E11* A⊕E6E5⊕E11A⊕E6⊕E12E1⊕E5⊕E11A⊕E6⊕E10⊕E12More
E12* A⊕E4E8⊕E12A⊕E4⊕E8E4⊕E8⊕E12A⊕E4⊕E8⊕E12More
E13* A⊕E2E11⊕E13A⊕E2⊕E4E9⊕E11⊕E13A⊕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 A 1 A
p (l=1) 3 Dipole B⊕E1 3 B⊕E1
d (l=2) 5 Quadrupole A⊕E2⊕E13 6 2A⊕E2⊕E13
f (l=3) 7 Octupole B⊕E1⊕E3⊕E12 10 2B⊕2E1⊕E3⊕E12
g (l=4) 9 Hexadecapole A⊕E2⊕E4⊕E11⊕E13 15 3A⊕2E2⊕E4⊕E11⊕2E13
h (l=5) 11 Dotricontapole B⊕E1⊕E3⊕E5⊕E10⊕E12 21 3B⊕3E1⊕2E3⊕E5⊕E10⊕2E12
i (l=6) 13 Tetrahexacontapole A⊕E2⊕E4⊕E6⊕E9⊕E11⊕E13 28 4A⊕3E2⊕2E4⊕E6⊕E9⊕2E11⊕3E13
j (l=7) 15 Octacosahectapole B⊕E1⊕E3⊕E5⊕E7⊕E8⊕E10⊕E12 36 4B⊕4E1⊕3E3⊕2E5⊕E7⊕E8⊕2E10⊕3E12
k (l=8) 17 256-pole A⊕E2⊕E4⊕E6⊕E7⊕E8⊕E9⊕E11⊕E13 45 5A⊕4E2⊕3E4⊕2E6⊕E7⊕E8⊕2E9⊕3E11⊕4E13
l (l=9) 19 512-pole B⊕E1⊕E3⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E12 55 5B⊕5E1⊕4E3⊕3E5⊕E6⊕2E7⊕2E8⊕E9⊕3E10⊕4E12
m (l=10) 21 1024-pole A⊕E2⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E13 66 6A⊕5E2⊕4E4⊕E5⊕3E6⊕2E7⊕2E8⊕3E9⊕E10⊕4E11⊕5E13
n (l=11) 23 2048-pole B⊕E1⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12 78 6B⊕6E1⊕5E3⊕E4⊕4E5⊕2E6⊕3E7⊕3E8⊕2E9⊕4E10⊕E11⊕5E12
o (l=12) 25 4096-pole A⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12⊕E13 91 7A⊕6E2⊕E3⊕5E4⊕2E5⊕4E6⊕3E7⊕3E8⊕4E9⊕2E10⊕5E11⊕E12⊕6E13
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 S28
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E13
D (L=2) 5 A⊕E2⊕E13
F (L=3) 7 A⊕E2⊕E11⊕E13
G (L=4) 9 A⊕E2⊕E4⊕E11⊕E13
H (L=5) 11 A⊕E2⊕E4⊕E9⊕E11⊕E13
I (L=6) 13 A⊕E2⊕E4⊕E6⊕E9⊕E11⊕E13

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