Point Group C15v



C15v E 2C15 2(C15)2 2C5 2(C15)4 2C3 2(C5)2 2(C15)7 15σv
A1 1 1 1 1 1 1 1 1 1
A2 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) 0
E2 2 2cos(4π/15) 2cos(8π/15) 2cos(4π/5) 2cos(14π/15) -1 2cos(2π/5) 2cos(2π/15) 0
E3 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) 2 2cos(2π/5) 2cos(4π/5) 0
E4 2 2cos(8π/15) 2cos(14π/15) 2cos(2π/5) 2cos(2π/15) -1 2cos(4π/5) 2cos(4π/15) 0
E5 2 -1 -1 2 -1 -1 2 -1 0
E6 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) 2 2cos(4π/5) 2cos(2π/5) 0
E7 2 2cos(14π/15) 2cos(2π/15) 2cos(4π/5) 2cos(4π/15) -1 2cos(2π/5) 2cos(8π/15) 0


Additional information

Number of symmetry elements h = 30
Number of classes, irreps n = 9
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C15 2(C15)2 2C5 2(C15)4 2C3 2(C5)2 2(C15)7 15σv



Genrate representation from irreducible representations


A1 A2 E1 E2 E3 E4 E5 E6 E7




Direct products of irreducible representations


Binary products
A1 A2 E1 E2 E3 E4 E5 E6 E7
A1 A1
A2 A2A1
E1 E1E1A1⊕A2⊕E2
E2 E2E2E1⊕E3A1⊕A2⊕E4
E3 E3E3E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕E7
E5 E5E5E4⊕E6E3⊕E7E2⊕E7E1⊕E6A1⊕A2⊕E5
E6 E6E6E5⊕E7E4⊕E7E3⊕E6E2⊕E5E1⊕E4A1⊕A2⊕E3
E7 E7E7E6⊕E7E5⊕E6E4⊕E5E3⊕E4E2⊕E3E1⊕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⊕E3A1⊕E2⊕E4E1⊕E3⊕E5A1⊕E2⊕E4⊕E6More
E2 A1⊕E4E2⊕E6A1⊕E4⊕E7E2⊕E5⊕E6A1⊕E3⊕E4⊕E7More
E3 A1⊕E6E3⊕E6A1⊕E3⊕E6A1⊕A2⊕E3⊕E6A1⊕2E3⊕E6More
E4 A1⊕E7E3⊕E4A1⊕E1⊕E7E3⊕E4⊕E5A1⊕E1⊕E6⊕E7More
E5 A1⊕E5A1⊕A2⊕E5A1⊕2E5A1⊕A2⊕2E52A1⊕A2⊕2E5More
E6 A1⊕E3E3⊕E6A1⊕E3⊕E6A1⊕A2⊕E3⊕E6A1⊕E3⊕2E6More
E7 A1⊕E1E6⊕E7A1⊕E1⊕E2E5⊕E6⊕E7A1⊕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 A1 1 A1
p (l=1) 3 Dipole A1⊕E1 3 A1⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A1⊕E1⊕E2⊕E3 10 2A1⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕E1⊕E2⊕E3⊕E4 15 3A1⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A1⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A1⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6 28 4A1⊕3E1⊕3E2⊕2E3⊕2E4⊕E5⊕E6
j (l=7) 15 Octacosahectapole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4A1⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕2E7 45 5A1⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕2E7
l (l=9) 19 512-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕2E6⊕2E7 55 5A1⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕3E6⊕3E7
m (l=10) 21 1024-pole A1⊕E1⊕E2⊕E3⊕E4⊕2E5⊕2E6⊕2E7 66 6A1⊕5E1⊕5E2⊕4E3⊕4E4⊕4E5⊕4E6⊕4E7
n (l=11) 23 2048-pole A1⊕E1⊕E2⊕E3⊕2E4⊕2E5⊕2E6⊕2E7 78 6A1⊕6E1⊕5E2⊕5E3⊕5E4⊕5E5⊕5E6⊕5E7
o (l=12) 25 4096-pole A1⊕E1⊕E2⊕2E3⊕2E4⊕2E5⊕2E6⊕2E7 91 7A1⊕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 C15v
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⊕E1⊕E2⊕E3⊕E4⊕E5
I (L=6) 13 A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6


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