Point Group C15



C15 E C15 (C15)2 C5 (C15)4 C3 (C5)2 (C15)7 (C15)8 (C5)3 (C3)2 (C15)11 (C5)4 (C15)13 (C15)14
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E1* 2 2cos(2π/15) 2cos(4π/15) 2cos(2π/5) 2cos(8π/15) -1 2cos(4π/5) 2cos(14π/15) 2cos(14π/15) 2cos(4π/5) -1 2cos(8π/15) 2cos(2π/5) 2cos(4π/15) 2cos(2π/15)
E2* 2 2cos(4π/15) 2cos(8π/15) 2cos(4π/5) 2cos(14π/15) -1 2cos(2π/5) 2cos(2π/15) 2cos(2π/15) 2cos(2π/5) -1 2cos(14π/15) 2cos(4π/5) 2cos(8π/15) 2cos(4π/15)
E3* 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) 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5)
E4* 2 2cos(8π/15) 2cos(14π/15) 2cos(2π/5) 2cos(2π/15) -1 2cos(4π/5) 2cos(4π/15) 2cos(4π/15) 2cos(4π/5) -1 2cos(2π/15) 2cos(2π/5) 2cos(14π/15) 2cos(8π/15)
E5* 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1
E6* 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) 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5)
E7* 2 2cos(14π/15) 2cos(2π/15) 2cos(4π/5) 2cos(4π/15) -1 2cos(2π/5) 2cos(8π/15) 2cos(8π/15) 2cos(2π/5) -1 2cos(4π/15) 2cos(4π/5) 2cos(2π/15) 2cos(14π/15)


Additional information

Number of symmetry elements h = 15
Number of classes, irreps n = 15
Number of real-valued irreducible representations n = 8
Abelian group yes
Optical Isomerism (Chirality) yes
Polar yes
Parity no


Reduce representation to irreducible representations


E C15 (C15)2 C5 (C15)4 C3 (C5)2 (C15)7 (C15)8 (C5)3 (C3)2 (C15)11 (C5)4 (C15)13 (C15)14



Genrate representation from irreducible representations


A E1* E2* E3* E4* E5* E6* E7*




Direct products of irreducible representations


Binary products
A E1* E2* E3* E4* E5* E6* E7*
A A
E1* E12A⊕E2
E2* E2E1⊕E32A⊕E4
E3* E3E2⊕E4E1⊕E52A⊕E6
E4* E4E3⊕E5E2⊕E6E1⊕E72A⊕E7
E5* E5E4⊕E6E3⊕E7E2⊕E7E1⊕E62A⊕E5
E6* E6E5⊕E7E4⊕E7E3⊕E6E2⊕E5E1⊕E42A⊕E3
E7* E7E6⊕E7E5⊕E6E4⊕E5E3⊕E4E2⊕E3E1⊕E22A⊕E1

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1* A⊕E2E1⊕E3A⊕E2⊕E4E1⊕E3⊕E5A⊕E2⊕E4⊕E6More
E2* A⊕E4E2⊕E6A⊕E4⊕E7E2⊕E5⊕E6A⊕E3⊕E4⊕E7More
E3* A⊕E6E3⊕E6A⊕E3⊕E62A⊕E3⊕E6A⊕2E3⊕E6More
E4* A⊕E7E3⊕E4A⊕E1⊕E7E3⊕E4⊕E5A⊕E1⊕E6⊕E7More
E5* A⊕E52A⊕E5A⊕2E52A⊕2E53A⊕2E5More
E6* A⊕E3E3⊕E6A⊕E3⊕E62A⊕E3⊕E6A⊕E3⊕2E6More
E7* A⊕E1E6⊕E7A⊕E1⊕E2E5⊕E6⊕E7A⊕E1⊕E2⊕E3More



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⊕E2⊕E3 10 2A⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A⊕E1⊕E2⊕E3⊕E4 15 3A⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6 28 4A⊕3E1⊕3E2⊕2E3⊕2E4⊕E5⊕E6
j (l=7) 15 Octacosahectapole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4A⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕2E7 45 5A⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕2E7
l (l=9) 19 512-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕2E6⊕2E7 55 5A⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕3E6⊕3E7
m (l=10) 21 1024-pole A⊕E1⊕E2⊕E3⊕E4⊕2E5⊕2E6⊕2E7 66 6A⊕5E1⊕5E2⊕4E3⊕4E4⊕4E5⊕4E6⊕4E7
n (l=11) 23 2048-pole A⊕E1⊕E2⊕E3⊕2E4⊕2E5⊕2E6⊕2E7 78 6A⊕6E1⊕5E2⊕5E3⊕5E4⊕5E5⊕5E6⊕5E7
o (l=12) 25 4096-pole A⊕E1⊕E2⊕2E3⊕2E4⊕2E5⊕2E6⊕2E7 91 7A⊕6E1⊕6E2⊕6E3⊕6E4⊕6E5⊕6E6⊕6E7
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 C15
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⊕E2⊕E3
G (L=4) 9 A⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A⊕E1⊕E2⊕E3⊕E4⊕E5
I (L=6) 13 A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6


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