Point Group I



I E 12C5 12(C5)2 20C3 15C2
A 1 1 1 1 1
T1 3 -2cos(4π/5) -2cos(2π/5) 0 -1
T2 3 -2cos(2π/5) -2cos(4π/5) 0 -1
G 4 -1 -1 1 0
H 5 0 0 -1 1


Additional information

Number of symmetry elements h = 60
Number of classes, irreps n = 5
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 12C5 12(C5)2 20C3 15C2



Genrate representation from irreducible representations


A T1 T2 G H




Direct products of irreducible representations


Binary products
A T1 T2 G H
A A
T1 T1A⊕T1⊕H
T2 T2G⊕HA⊕T2⊕H
G GT2⊕G⊕HT1⊕G⊕HA⊕T1⊕T2⊕G⊕H
H HT1⊕T2⊕G⊕HT1⊕T2⊕G⊕HT1⊕T2⊕G⊕2HA⊕T1⊕T2⊕2G⊕2H

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
T1 A⊕HT1⊕T2⊕GA⊕G⊕2H2T1⊕2T2⊕G⊕H2A⊕T1⊕2G⊕3HMore
T2 A⊕HT1⊕T2⊕GA⊕G⊕2H2T1⊕2T2⊕G⊕H2A⊕T2⊕2G⊕3HMore
G A⊕G⊕HA⊕T1⊕T2⊕2G⊕H2A⊕T1⊕T2⊕3G⊕3H2A⊕3T1⊕3T2⊕4G⊕4H3A⊕3T1⊕3T2⊕7G⊕7HMore
H A⊕G⊕2H2A⊕T1⊕T2⊕3G⊕3H2A⊕2T1⊕2T2⊕4G⊕8H4A⊕5T1⊕5T2⊕8G⊕12H7A⊕8T1⊕8T2⊕15G⊕19HMore



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 A 1 A
p (l=1) 3 Dipole T1 3 T1
d (l=2) 5 Quadrupole H 6 A⊕H
f (l=3) 7 Octupole T2⊕G 10 T1⊕T2⊕G
g (l=4) 9 Hexadecapole G⊕H 15 A⊕G⊕2H
h (l=5) 11 Dotricontapole T1⊕T2⊕H 21 2T1⊕2T2⊕G⊕H
i (l=6) 13 Tetrahexacontapole A⊕T1⊕G⊕H 28 2A⊕T1⊕2G⊕3H
j (l=7) 15 Octacosahectapole T1⊕T2⊕G⊕H 36 3T1⊕3T2⊕2G⊕2H
k (l=8) 17 256-pole T2⊕G⊕2H 45 2A⊕T1⊕T2⊕3G⊕5H
l (l=9) 19 512-pole T1⊕T2⊕2G⊕H 55 4T1⊕4T2⊕4G⊕3H
m (l=10) 21 1024-pole A⊕T1⊕T2⊕G⊕2H 66 3A⊕2T1⊕2T2⊕4G⊕7H
n (l=11) 23 2048-pole 2T1⊕T2⊕G⊕2H 78 6T1⊕5T2⊕5G⊕5H
o (l=12) 25 4096-pole A⊕T1⊕T2⊕2G⊕2H 91 4A⊕3T1⊕3T2⊕6G⊕9H
More

First nonvanshing multipole: Tetrahexacontapole

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 I
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 T1
D (L=2) 5 H
F (L=3) 7 T2⊕G
G (L=4) 9 G⊕H
H (L=5) 11 T1⊕T2⊕H
I (L=6) 13 A⊕T1⊕G⊕H


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