Point Group C5



C5 E 2C5 2(C5)2
A 1 1 1
E1* 2 2cos(2π/5) 2cos(4π/5)
E2* 2 2cos(4π/5) 2cos(2π/5)


Additional information

Number of symmetry elements h = 5
Number of classes, irreps n = 5
Number of real-valued irreducible representations n = 3
Abelian group yes
Optical Isomerism (Chirality) yes
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C5 2(C5)2



Genrate representation from irreducible representations


A E1* E2*




Direct products of irreducible representations


Binary products
A E1* E2*
A A
E1* E12A⊕E2
E2* E2E1⊕E22A⊕E1

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1* A⊕E2E1⊕E2A⊕E1⊕E22A⊕E1⊕E2A⊕2E1⊕E2More
E2* A⊕E1E1⊕E2A⊕E1⊕E22A⊕E1⊕E2A⊕E1⊕2E2More



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 A⊕E1 3 A⊕E1
d (l=2) 5 Quadrupole A⊕E1⊕E2 6 2A⊕E1⊕E2
f (l=3) 7 Octupole A⊕E1⊕2E2 10 2A⊕2E1⊕2E2
g (l=4) 9 Hexadecapole A⊕2E1⊕2E2 15 3A⊕3E1⊕3E2
h (l=5) 11 Dotricontapole 3A⊕2E1⊕2E2 21 5A⊕4E1⊕4E2
i (l=6) 13 Tetrahexacontapole 3A⊕3E1⊕2E2 28 6A⊕6E1⊕5E2
j (l=7) 15 Octacosahectapole 3A⊕3E1⊕3E2 36 8A⊕7E1⊕7E2
k (l=8) 17 256-pole 3A⊕3E1⊕4E2 45 9A⊕9E1⊕9E2
l (l=9) 19 512-pole 3A⊕4E1⊕4E2 55 11A⊕11E1⊕11E2
m (l=10) 21 1024-pole 5A⊕4E1⊕4E2 66 14A⊕13E1⊕13E2
n (l=11) 23 2048-pole 5A⊕5E1⊕4E2 78 16A⊕16E1⊕15E2
o (l=12) 25 4096-pole 5A⊕5E1⊕5E2 91 19A⊕18E1⊕18E2
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 C5
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⊕2E2
G (L=4) 9 A⊕2E1⊕2E2
H (L=5) 11 3A⊕2E1⊕2E2
I (L=6) 13 3A⊕3E1⊕2E2


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