Point Group C5h



C5h E C5 (C5)2 (C5)3 (C5)4 σh S5 (S5)7 (S5)3 (S5)9
A' 1 1 1 1 1 1 1 1 1 1
E'1* 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5)
E'2* 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5)
A'' 1 1 1 1 1 -1 -1 -1 -1 -1
E''1* 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) -2 -2cos(2π/5) -2cos(4π/5) -2cos(4π/5) -2cos(2π/5)
E''2* 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) -2 -2cos(4π/5) -2cos(2π/5) -2cos(2π/5) -2cos(4π/5)


Additional information

Number of symmetry elements h = 10
Number of classes, irreps n = 10
Number of real-valued irreducible representations n = 6
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E C5 (C5)2 (C5)3 (C5)4 σh S5 (S5)7 (S5)3 (S5)9



Genrate representation from irreducible representations


A' E'1* E'2* A'' E''1* E''2*




Direct products of irreducible representations


Binary products
A' E'1* E'2* A'' E''1* E''2*
A' A'
E'1* E'12A'⊕E'2
E'2* E'2E'1⊕E'22A'⊕E'1
A'' A''E''1E''2A'
E''1* E''12A''⊕E''2E''1⊕E''2E'12A'⊕E'2
E''2* E''2E''1⊕E''22A''⊕E''1E'2E'1⊕E'22A'⊕E'1

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E'1* A'⊕E'2E'1⊕E'2A'⊕E'1⊕E'22A'⊕E'1⊕E'2A'⊕2E'1⊕E'2More
E'2* A'⊕E'1E'1⊕E'2A'⊕E'1⊕E'22A'⊕E'1⊕E'2A'⊕E'1⊕2E'2More
E''1* A'⊕E'2E''1⊕E''2A'⊕E'1⊕E'22A''⊕E''1⊕E''2A'⊕2E'1⊕E'2More
E''2* A'⊕E'1E''1⊕E''2A'⊕E'1⊕E'22A''⊕E''1⊕E''2A'⊕E'1⊕2E'2More



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 E'1⊕A'' 3 E'1⊕A''
d (l=2) 5 Quadrupole A'⊕E'2⊕E''1 6 2A'⊕E'2⊕E''1
f (l=3) 7 Octupole E'1⊕E'2⊕A''⊕E''2 10 2E'1⊕E'2⊕2A''⊕E''2
g (l=4) 9 Hexadecapole A'⊕E'1⊕E'2⊕E''1⊕E''2 15 3A'⊕E'1⊕2E'2⊕2E''1⊕E''2
h (l=5) 11 Dotricontapole 2A'⊕E'1⊕E'2⊕A''⊕E''1⊕E''2 21 2A'⊕3E'1⊕2E'2⊕3A''⊕E''1⊕2E''2
i (l=6) 13 Tetrahexacontapole A'⊕2E'1⊕E'2⊕2A''⊕E''1⊕E''2 28 4A'⊕3E'1⊕3E'2⊕2A''⊕3E''1⊕2E''2
j (l=7) 15 Octacosahectapole 2A'⊕E'1⊕2E'2⊕A''⊕2E''1⊕E''2 36 4A'⊕4E'1⊕4E'2⊕4A''⊕3E''1⊕3E''2
k (l=8) 17 256-pole A'⊕2E'1⊕2E'2⊕2A''⊕E''1⊕2E''2 45 5A'⊕5E'1⊕5E'2⊕4A''⊕4E''1⊕4E''2
l (l=9) 19 512-pole 2A'⊕2E'1⊕2E'2⊕A''⊕2E''1⊕2E''2 55 6A'⊕6E'1⊕6E'2⊕5A''⊕5E''1⊕5E''2
m (l=10) 21 1024-pole 3A'⊕2E'1⊕2E'2⊕2A''⊕2E''1⊕2E''2 66 8A'⊕7E'1⊕7E'2⊕6A''⊕6E''1⊕6E''2
n (l=11) 23 2048-pole 2A'⊕3E'1⊕2E'2⊕3A''⊕2E''1⊕2E''2 78 8A'⊕9E'1⊕8E'2⊕8A''⊕7E''1⊕7E''2
o (l=12) 25 4096-pole 3A'⊕2E'1⊕3E'2⊕2A''⊕3E''1⊕2E''2 91 11A'⊕9E'1⊕10E'2⊕8A''⊕9E''1⊕8E''2
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 C5h
L 2L+1 Term Splitting
S (L=0) 1 A'
P (L=1) 3 A'⊕E''1
D (L=2) 5 A'⊕E'2⊕E''1
F (L=3) 7 A'⊕E'2⊕E''1⊕E''2
G (L=4) 9 A'⊕E'1⊕E'2⊕E''1⊕E''2
H (L=5) 11 A'⊕E'1⊕E'2⊕2A''⊕E''1⊕E''2
I (L=6) 13 A'⊕2E'1⊕E'2⊕2A''⊕E''1⊕E''2


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