Point Group C10h



C10h E C10 C5 (C10)3 (C5)2 C2 (C5)3 (C10)7 (C5)4 (C10)9 i (S5)3 (S10)7 (S5)9 (S10)9 σh S10 S5 (S10)3 (S5)7
Ag 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Bg 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1g* 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 -2cos(π/5) -2cos(2π/5) 2cos(2π/5) 2cos(π/5) 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 -2cos(π/5) -2cos(2π/5) 2cos(2π/5) 2cos(π/5)
E2g* 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5)
E3g* 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 2cos(2π/5) 2cos(π/5) -2cos(π/5) -2cos(2π/5) 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 2cos(2π/5) 2cos(π/5) -2cos(π/5) -2cos(2π/5)
E4g* 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5)
Au 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
Bu 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1
E1u* 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 -2cos(π/5) -2cos(2π/5) 2cos(2π/5) 2cos(π/5) -2 -2cos(π/5) -2cos(2π/5) 2cos(2π/5) 2cos(π/5) 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5)
E2u* 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) -2 -2cos(2π/5) 2cos(π/5) 2cos(π/5) -2cos(2π/5) -2 -2cos(2π/5) 2cos(π/5) 2cos(π/5) -2cos(2π/5)
E3u* 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 2cos(2π/5) 2cos(π/5) -2cos(π/5) -2cos(2π/5) -2 2cos(2π/5) 2cos(π/5) -2cos(π/5) -2cos(2π/5) 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5)
E4u* 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) -2 2cos(π/5) -2cos(2π/5) -2cos(2π/5) 2cos(π/5) -2 2cos(π/5) -2cos(2π/5) -2cos(2π/5) 2cos(π/5)


Additional information

Number of symmetry elements h = 20
Number of classes, irreps n = 20
Number of real-valued irreducible representations n = 12
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E C10 C5 (C10)3 (C5)2 C2 (C5)3 (C10)7 (C5)4 (C10)9 i (S5)3 (S10)7 (S5)9 (S10)9 σh S10 S5 (S10)3 (S5)7



Genrate representation from irreducible representations


Ag Bg E1g* E2g* E3g* E4g* Au Bu E1u* E2u* E3u* E4u*




Direct products of irreducible representations


Binary products
Ag Bg E1g* E2g* E3g* E4g* Au Bu E1u* E2u* E3u* E4u*
Ag Ag
Bg BgAg
E1g* E1gE4g2Ag⊕E2g
E2g* E2gE3gE1g⊕E3g2Ag⊕E4g
E3g* E3gE2gE2g⊕E4g2Bg⊕E1g2Ag⊕E4g
E4g* E4gE1g2Bg⊕E3gE2g⊕E4gE1g⊕E3g2Ag⊕E2g
Au AuBuE1uE2uE3uE4uAg
Bu BuAuE4uE3uE2uE1uBgAg
E1u* E1uE4u2Au⊕E2uE1u⊕E3uE2u⊕E4u2Bu⊕E3uE1gE4g2Ag⊕E2g
E2u* E2uE3uE1u⊕E3u2Au⊕E4u2Bu⊕E1uE2u⊕E4uE2gE3gE1g⊕E3g2Ag⊕E4g
E3u* E3uE2uE2u⊕E4u2Bu⊕E1u2Au⊕E4uE1u⊕E3uE3gE2gE2g⊕E4g2Bg⊕E1g2Ag⊕E4g
E4u* E4uE1u2Bu⊕E3uE2u⊕E4uE1u⊕E3u2Au⊕E2uE4gE1g2Bg⊕E3gE2g⊕E4gE1g⊕E3g2Ag⊕E2g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g* Ag⊕E2gE1g⊕E3gAg⊕E2g⊕E4g2Bg⊕E1g⊕E3gAg⊕E2g⊕2E4gMore
E2g* Ag⊕E4gE2g⊕E4gAg⊕E2g⊕E4g2Ag⊕E2g⊕E4gAg⊕2E2g⊕E4gMore
E3g* Ag⊕E4gE1g⊕E3gAg⊕E2g⊕E4g2Bg⊕E1g⊕E3gAg⊕2E2g⊕E4gMore
E4g* Ag⊕E2gE2g⊕E4gAg⊕E2g⊕E4g2Ag⊕E2g⊕E4gAg⊕E2g⊕2E4gMore
E1u* Ag⊕E2gE1u⊕E3uAg⊕E2g⊕E4g2Bu⊕E1u⊕E3uAg⊕E2g⊕2E4gMore
E2u* Ag⊕E4gE2u⊕E4uAg⊕E2g⊕E4g2Au⊕E2u⊕E4uAg⊕2E2g⊕E4gMore
E3u* Ag⊕E4gE1u⊕E3uAg⊕E2g⊕E4g2Bu⊕E1u⊕E3uAg⊕2E2g⊕E4gMore
E4u* Ag⊕E2gE2u⊕E4uAg⊕E2g⊕E4g2Au⊕E2u⊕E4uAg⊕E2g⊕2E4gMore



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 Ag 1 Ag
p (l=1) 3 Dipole Au⊕E1u 3 Au⊕E1u
d (l=2) 5 Quadrupole Ag⊕E1g⊕E2g 6 2Ag⊕E1g⊕E2g
f (l=3) 7 Octupole Au⊕E1u⊕E2u⊕E3u 10 2Au⊕2E1u⊕E2u⊕E3u
g (l=4) 9 Hexadecapole Ag⊕E1g⊕E2g⊕E3g⊕E4g 15 3Ag⊕2E1g⊕2E2g⊕E3g⊕E4g
h (l=5) 11 Dotricontapole Au⊕2Bu⊕E1u⊕E2u⊕E3u⊕E4u 21 3Au⊕2Bu⊕3E1u⊕2E2u⊕2E3u⊕E4u
i (l=6) 13 Tetrahexacontapole Ag⊕2Bg⊕E1g⊕E2g⊕E3g⊕2E4g 28 4Ag⊕2Bg⊕3E1g⊕3E2g⊕2E3g⊕3E4g
j (l=7) 15 Octacosahectapole Au⊕2Bu⊕E1u⊕E2u⊕2E3u⊕2E4u 36 4Au⊕4Bu⊕4E1u⊕3E2u⊕4E3u⊕3E4u
k (l=8) 17 256-pole Ag⊕2Bg⊕E1g⊕2E2g⊕2E3g⊕2E4g 45 5Ag⊕4Bg⊕4E1g⊕5E2g⊕4E3g⊕5E4g
l (l=9) 19 512-pole Au⊕2Bu⊕2E1u⊕2E2u⊕2E3u⊕2E4u 55 5Au⊕6Bu⊕6E1u⊕5E2u⊕6E3u⊕5E4u
m (l=10) 21 1024-pole 3Ag⊕2Bg⊕2E1g⊕2E2g⊕2E3g⊕2E4g 66 8Ag⊕6Bg⊕6E1g⊕7E2g⊕6E3g⊕7E4g
n (l=11) 23 2048-pole 3Au⊕2Bu⊕3E1u⊕2E2u⊕2E3u⊕2E4u 78 8Au⊕8Bu⊕9E1u⊕7E2u⊕8E3u⊕7E4u
o (l=12) 25 4096-pole 3Ag⊕2Bg⊕3E1g⊕3E2g⊕2E3g⊕2E4g 91 11Ag⊕8Bg⊕9E1g⊕10E2g⊕8E3g⊕9E4g
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 C10h
L 2L+1 Term Splitting
S (L=0) 1 Ag
P (L=1) 3 Ag⊕E1g
D (L=2) 5 Ag⊕E1g⊕E2g
F (L=3) 7 Ag⊕E1g⊕E2g⊕E3g
G (L=4) 9 Ag⊕E1g⊕E2g⊕E3g⊕E4g
H (L=5) 11 Ag⊕2Bg⊕E1g⊕E2g⊕E3g⊕E4g
I (L=6) 13 Ag⊕2Bg⊕E1g⊕E2g⊕E3g⊕2E4g


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