Point Group C10v



C10v E 2C10 2C5 2(C10)3 2(C5)2 C2 v d
A1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1
B2 1 -1 1 -1 1 -1 -1 1
E1 2 2cos(π/5) 2cos(2π/5) -2cos(2π/5) -2cos(π/5) -2 0 0
E2 2 2cos(2π/5) -2cos(π/5) -2cos(π/5) 2cos(2π/5) 2 0 0
E3 2 -2cos(2π/5) -2cos(π/5) 2cos(π/5) 2cos(2π/5) -2 0 0
E4 2 -2cos(π/5) 2cos(2π/5) 2cos(2π/5) -2cos(π/5) 2 0 0


Additional information

Number of symmetry elements h = 20
Number of classes, irreps n = 8
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C10 2C5 2(C10)3 2(C5)2 C2 v d



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4




Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2 E3 E4
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E4E4A1⊕A2⊕E2
E2 E2E2E3E3E1⊕E3A1⊕A2⊕E4
E3 E3E3E2E2E2⊕E4B1⊕B2⊕E1A1⊕A2⊕E4
E4 E4E4E1E1B1⊕B2⊕E3E2⊕E4E1⊕E3A1⊕A2⊕E2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E4B1⊕B2⊕E1⊕E3A1⊕E2⊕2E4More
E2 A1⊕E4E2⊕E4A1⊕E2⊕E4A1⊕A2⊕E2⊕E4A1⊕2E2⊕E4More
E3 A1⊕E4E1⊕E3A1⊕E2⊕E4B1⊕B2⊕E1⊕E3A1⊕2E2⊕E4More
E4 A1⊕E2E2⊕E4A1⊕E2⊕E4A1⊕A2⊕E2⊕E4A1⊕E2⊕2E4More



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 A1 1 A1
p (l=1) 3 Dipole A1⊕E1 3 A1⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A1⊕E1⊕E2⊕E3 10 2A1⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕E1⊕E2⊕E3⊕E4 15 3A1⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4 21 3A1⊕B1⊕B2⊕3E1⊕2E2⊕2E3⊕E4
i (l=6) 13 Tetrahexacontapole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕2E4 28 4A1⊕B1⊕B2⊕3E1⊕3E2⊕2E3⊕3E4
j (l=7) 15 Octacosahectapole A1⊕B1⊕B2⊕E1⊕E2⊕2E3⊕2E4 36 4A1⊕2B1⊕2B2⊕4E1⊕3E2⊕4E3⊕3E4
k (l=8) 17 256-pole A1⊕B1⊕B2⊕E1⊕2E2⊕2E3⊕2E4 45 5A1⊕2B1⊕2B2⊕4E1⊕5E2⊕4E3⊕5E4
l (l=9) 19 512-pole A1⊕B1⊕B2⊕2E1⊕2E2⊕2E3⊕2E4 55 5A1⊕3B1⊕3B2⊕6E1⊕5E2⊕6E3⊕5E4
m (l=10) 21 1024-pole 2A1⊕A2⊕B1⊕B2⊕2E1⊕2E2⊕2E3⊕2E4 66 7A1⊕A2⊕3B1⊕3B2⊕6E1⊕7E2⊕6E3⊕7E4
n (l=11) 23 2048-pole 2A1⊕A2⊕B1⊕B2⊕3E1⊕2E2⊕2E3⊕2E4 78 7A1⊕A2⊕4B1⊕4B2⊕9E1⊕7E2⊕8E3⊕7E4
o (l=12) 25 4096-pole 2A1⊕A2⊕B1⊕B2⊕3E1⊕3E2⊕2E3⊕2E4 91 9A1⊕2A2⊕4B1⊕4B2⊕9E1⊕10E2⊕8E3⊕9E4
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 C10v
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⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4
I (L=6) 13 A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕2E4


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