Point Group D10



D10 E 2C10 2C5 2(C10)3 2(C5)2 C2 5C'2 5C''2
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) yes
Polar no
Parity no


Reduce representation to irreducible representations


E 2C10 2C5 2(C10)3 2(C5)2 C2 5C'2 5C''2



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 A2⊕E1 3 A2⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A2⊕E1⊕E2⊕E3 10 2A2⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕E1⊕E2⊕E3⊕E4 15 3A1⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A2⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4 21 3A2⊕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 A2⊕B1⊕B2⊕E1⊕E2⊕2E3⊕2E4 36 4A2⊕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 A2⊕B1⊕B2⊕2E1⊕2E2⊕2E3⊕2E4 55 5A2⊕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 A1⊕2A2⊕B1⊕B2⊕3E1⊕2E2⊕2E3⊕2E4 78 A1⊕7A2⊕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: 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 D10
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