Point Group C13v



C13v E 2C13 2(C13)2 2(C13)3 2(C13)4 2(C13)5 2(C13)6 13σv
A1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 -1
E1 2 1.7709 1.1361 0.2411 -0.7092 -1.4970 -1.9419 0
E2 2 1.1361 -0.7092 -1.9419 -1.4970 0.2411 1.7709 0
E3 2 0.2411 -1.9419 -0.7092 1.7709 1.1361 -1.4970 0
E4 2 -0.7092 -1.4970 1.7709 0.2411 -1.9419 1.1361 0
E5 2 -1.4970 0.2411 1.1361 -1.9419 1.7709 -0.7092 0
E6 2 -1.9419 1.7709 -1.4970 1.1361 -0.7092 0.2411 0
View A
View B
✅


Additional information

Number of symmetry elements h = 26
Number of classes, irreps n = 8
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C13 2(C13)2 2(C13)3 2(C13)4 2(C13)5 2(C13)6 13σv



Genrate representation from irreducible representations


A1 A2 E1 E2 E3 E4 E5 E6



Direct products of irreducible representations


Binary products
⊙ A1 A2 E1 E2 E3 E4 E5 E6
A1 A1
A2 A2A1
E1 E1E1A1⊕A2⊕E2
E2 E2E2E1⊕E3A1⊕A2⊕E4
E3 E3E3E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E3⊕E5E2⊕E6E1⊕E6A1⊕A2⊕E5
E5 E5E5E4⊕E6E3⊕E6E2⊕E5E1⊕E4A1⊕A2⊕E3
E6 E6E6E5⊕E6E4⊕E5E3⊕E4E2⊕E3E1⊕E2A1⊕A2⊕E1

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E4E1⊕E3⊕E5A1⊕E2⊕E4⊕E6More
E2 A1⊕E4E2⊕E6A1⊕E4⊕E5E2⊕E3⊕E6A1⊕E1⊕E4⊕E5More
E3 A1⊕E6E3⊕E4A1⊕E1⊕E6E2⊕E3⊕E4A1⊕E1⊕E5⊕E6More
E4 A1⊕E5E1⊕E4A1⊕E3⊕E5E1⊕E4⊕E6A1⊕E2⊕E3⊕E5More
E5 A1⊕E3E2⊕E5A1⊕E3⊕E6E1⊕E2⊕E5A1⊕E3⊕E4⊕E6More
E6 A1⊕E1E5⊕E6A1⊕E1⊕E2E4⊕E5⊕E6A1⊕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 [Γl(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⊕2E6 36 4A1⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕2E6
k (l=8) 17 256-pole A1⊕E1⊕E2⊕E3⊕E4⊕2E5⊕2E6 45 5A1⊕4E1⊕4E2⊕3E3⊕3E4⊕3E5⊕3E6
l (l=9) 19 512-pole A1⊕E1⊕E2⊕E3⊕2E4⊕2E5⊕2E6 55 5A1⊕5E1⊕4E2⊕4E3⊕4E4⊕4E5⊕4E6
m (l=10) 21 1024-pole A1⊕E1⊕E2⊕2E3⊕2E4⊕2E5⊕2E6 66 6A1⊕5E1⊕5E2⊕5E3⊕5E4⊕5E5⊕5E6
n (l=11) 23 2048-pole A1⊕E1⊕2E2⊕2E3⊕2E4⊕2E5⊕2E6 78 6A1⊕6E1⊕6E2⊕6E3⊕6E4⊕6E5⊕6E6
o (l=12) 25 4096-pole A1⊕2E1⊕2E2⊕2E3⊕2E4⊕2E5⊕2E6 91 7A1⊕7E1⊕7E2⊕7E3⊕7E4⊕7E5⊕7E6
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 C13v
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 November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement