Point Group D5

D5 E 2C5 2(C5)2 5C'2
A1 1 1 1 1
A2 1 1 1 -1
E1 2 2cos(2π/5) 2cos(4π/5) 0
E2 2 2cos(4π/5) 2cos(2π/5) 0

Additional information

Number of symmetry elements h = 10
Number of classes, irreps n = 4
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
Parity no

Reduce representation to irreducible representations

E 2C5 2(C5)2 5C'2

Genrate representation from irreducible representations

A1 A2 E1 E2


Ferrocene (rotated)

Direct products of irreducible representations

Binary products
A1 A2 E1 E2
A1 A1
A2 A2A1
E1 E1E1A1⊕A2⊕E2
E2 E2E2E1⊕E2A1⊕A2⊕E1

Ternary Products
Quaternary Products

Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones

irrep 2] 3] 4] 5] 6]
E1 A1⊕E2E1⊕E2A1⊕E1⊕E2A1⊕A2⊕E1⊕E2A1⊕2E1⊕E2More
E2 A1⊕E1E1⊕E2A1⊕E1⊕E2A1⊕A2⊕E1⊕E2A1⊕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 A1 1 A1
p (l=1) 3 Dipole A2⊕E1 3 A2⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A2⊕E1⊕2E2 10 2A2⊕2E1⊕2E2
g (l=4) 9 Hexadecapole A1⊕2E1⊕2E2 15 3A1⊕3E1⊕3E2
h (l=5) 11 Dotricontapole A1⊕2A2⊕2E1⊕2E2 21 A1⊕4A2⊕4E1⊕4E2
i (l=6) 13 Tetrahexacontapole 2A1⊕A2⊕3E1⊕2E2 28 5A1⊕A2⊕6E1⊕5E2
j (l=7) 15 Octacosahectapole A1⊕2A2⊕3E1⊕3E2 36 2A1⊕6A2⊕7E1⊕7E2
k (l=8) 17 256-pole 2A1⊕A2⊕3E1⊕4E2 45 7A1⊕2A2⊕9E1⊕9E2
l (l=9) 19 512-pole A1⊕2A2⊕4E1⊕4E2 55 3A1⊕8A2⊕11E1⊕11E2
m (l=10) 21 1024-pole 3A1⊕2A2⊕4E1⊕4E2 66 10A1⊕4A2⊕13E1⊕13E2
n (l=11) 23 2048-pole 2A1⊕3A2⊕5E1⊕4E2 78 5A1⊕11A2⊕16E1⊕15E2
o (l=12) 25 4096-pole 3A1⊕2A2⊕5E1⊕5E2 91 13A1⊕6A2⊕18E1⊕18E2

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 D5
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⊕2E2
G (L=4) 9 A1⊕2E1⊕2E2
H (L=5) 11 A1⊕2A2⊕2E1⊕2E2
I (L=6) 13 2A1⊕A2⊕3E1⊕2E2

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