Point Group D13

D13 E 2C13 2(C13)2 2(C13)3 2(C13)4 2(C13)5 2(C13)6 13C'2
A1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 -1
E1 2 2cos(2π/13) 2cos(4π/13) 2cos(6π/13) 2cos(8π/13) 2cos(10π/13) 2cos(12π/13) 0
E2 2 2cos(4π/13) 2cos(8π/13) 2cos(12π/13) 2cos(10π/13) 2cos(6π/13) 2cos(2π/13) 0
E3 2 2cos(6π/13) 2cos(12π/13) 2cos(8π/13) 2cos(2π/13) 2cos(4π/13) 2cos(10π/13) 0
E4 2 2cos(8π/13) 2cos(10π/13) 2cos(2π/13) 2cos(6π/13) 2cos(12π/13) 2cos(4π/13) 0
E5 2 2cos(10π/13) 2cos(6π/13) 2cos(4π/13) 2cos(12π/13) 2cos(2π/13) 2cos(8π/13) 0
E6 2 2cos(12π/13) 2cos(2π/13) 2cos(10π/13) 2cos(4π/13) 2cos(8π/13) 2cos(6π/13) 0

Additional information

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

Reduce representation to irreducible representations

E 2C13 2(C13)2 2(C13)3 2(C13)4 2(C13)5 2(C13)6 13C'2

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 2] 3] 4] 5] 6]
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 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⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A2⊕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 A2⊕E1⊕E2⊕E3⊕E4⊕E5⊕2E6 36 4A2⊕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 A2⊕E1⊕E2⊕E3⊕2E4⊕2E5⊕2E6 55 5A2⊕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 A2⊕E1⊕2E2⊕2E3⊕2E4⊕2E5⊕2E6 78 6A2⊕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

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 D13
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