Point Group C28



C28 E C28 C14 (C28)3 C7 (C28)5 (C14)3 C4 (C7)2 (C28)9 (C14)5 (C28)11 (C7)3 (C28)13 C2 (C28)15 (C7)4 (C28)17 (C14)9 (C28)19 (C7)5 (C4)3 (C14)11 (C28)23 (C7)6 (C28)25 (C14)13 (C28)27
A 1 1 1 1 1 1 1 1 1 1 1 1 1 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 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1* 2 1.9499 1.8019 1.5637 1.2470 0.8678 0.4450 0 -0.4450 -0.8678 -1.2470 -1.5637 -1.8019 -1.9499 -2 -1.9499 -1.8019 -1.5637 -1.2470 -0.8678 -0.4450 0 0.4450 0.8678 1.2470 1.5637 1.8019 1.9499
E2* 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019
E3* 2 1.5637 0.4450 -0.8678 -1.8019 -1.9499 -1.2470 0 1.2470 1.9499 1.8019 0.8678 -0.4450 -1.5637 -2 -1.5637 -0.4450 0.8678 1.8019 1.9499 1.2470 0 -1.2470 -1.9499 -1.8019 -0.8678 0.4450 1.5637
E4* 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470
E5* 2 0.8678 -1.2470 -1.9499 -0.4450 1.5637 1.8019 0 -1.8019 -1.5637 0.4450 1.9499 1.2470 -0.8678 -2 -0.8678 1.2470 1.9499 0.4450 -1.5637 -1.8019 0 1.8019 1.5637 -0.4450 -1.9499 -1.2470 0.8678
E6* 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450
E7* 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0
E8* 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450
E9* 2 -0.8678 -1.2470 1.9499 -0.4450 -1.5637 1.8019 0 -1.8019 1.5637 0.4450 -1.9499 1.2470 0.8678 -2 0.8678 1.2470 -1.9499 0.4450 1.5637 -1.8019 0 1.8019 -1.5637 -0.4450 1.9499 -1.2470 -0.8678
E10* 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470
E11* 2 -1.5637 0.4450 0.8678 -1.8019 1.9499 -1.2470 0 1.2470 -1.9499 1.8019 -0.8678 -0.4450 1.5637 -2 1.5637 -0.4450 -0.8678 1.8019 -1.9499 1.2470 0 -1.2470 1.9499 -1.8019 0.8678 0.4450 -1.5637
E12* 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019
E13* 2 -1.9499 1.8019 -1.5637 1.2470 -0.8678 0.4450 0 -0.4450 0.8678 -1.2470 1.5637 -1.8019 1.9499 -2 1.9499 -1.8019 1.5637 -1.2470 0.8678 -0.4450 0 0.4450 -0.8678 1.2470 -1.5637 1.8019 -1.9499
Math View
Phys View A
Phys View B
✅
Chem View A
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) yes
Polar yes
Parity no


Reduce representation to irreducible representations


E C28 C14 (C28)3 C7 (C28)5 (C14)3 C4 (C7)2 (C28)9 (C14)5 (C28)11 (C7)3 (C28)13 C2 (C28)15 (C7)4 (C28)17 (C14)9 (C28)19 (C7)5 (C4)3 (C14)11 (C28)23 (C7)6 (C28)25 (C14)13 (C28)27



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 A⊕E1 3 A⊕E1
d (l=2) 5 Quadrupole A⊕E1⊕E2 6 2A⊕E1⊕E2
f (l=3) 7 Octupole A⊕E1⊕E2⊕E3 10 2A⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A⊕E1⊕E2⊕E3⊕E4 15 3A⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6 28 4A⊕3E1⊕3E2⊕2E3⊕2E4⊕E5⊕E6
j (l=7) 15 Octacosahectapole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4A⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 45 5A⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7⊕E8
l (l=9) 19 512-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 55 5A⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕2E7⊕E8⊕E9
m (l=10) 21 1024-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10 66 6A⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕3E6⊕2E7⊕2E8⊕E9⊕E10
n (l=11) 23 2048-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11 78 6A⊕6E1⊕5E2⊕5E3⊕4E4⊕4E5⊕3E6⊕3E7⊕2E8⊕2E9⊕E10⊕E11
o (l=12) 25 4096-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12 91 7A⊕6E1⊕6E2⊕5E3⊕5E4⊕4E5⊕4E6⊕3E7⊕3E8⊕2E9⊕2E10⊕E11⊕E12
More

First nonvanshing multipole: Dipole

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 C28
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E1
D (L=2) 5 A⊕E1⊕E2
F (L=3) 7 A⊕E1⊕E2⊕E3
G (L=4) 9 A⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A⊕E1⊕E2⊕E3⊕E4⊕E5
I (L=6) 13 A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6

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