Point Group C13h



C13h E C13 (C13)2 (C13)3 (C13)4 (C13)5 (C13)6 (C13)7 (C13)8 (C13)9 (C13)10 (C13)11 (C13)12 σh S13 (S13)15 (S13)3 (S13)17 (S13)5 (S13)19 (S13)7 (S13)21 (S13)9 (S13)23 (S13)11 (S13)25
A' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E'1* 2 1.7709 1.1361 0.2411 -0.7092 -1.4970 -1.9419 -1.9419 -1.4970 -0.7092 0.2411 1.1361 1.7709 2 1.7709 1.1361 0.2411 -0.7092 -1.4970 -1.9419 -1.9419 -1.4970 -0.7092 0.2411 1.1361 1.7709
E'2* 2 1.1361 -0.7092 -1.9419 -1.4970 0.2411 1.7709 1.7709 0.2411 -1.4970 -1.9419 -0.7092 1.1361 2 1.1361 -0.7092 -1.9419 -1.4970 0.2411 1.7709 1.7709 0.2411 -1.4970 -1.9419 -0.7092 1.1361
E'3* 2 0.2411 -1.9419 -0.7092 1.7709 1.1361 -1.4970 -1.4970 1.1361 1.7709 -0.7092 -1.9419 0.2411 2 0.2411 -1.9419 -0.7092 1.7709 1.1361 -1.4970 -1.4970 1.1361 1.7709 -0.7092 -1.9419 0.2411
E'4* 2 -0.7092 -1.4970 1.7709 0.2411 -1.9419 1.1361 1.1361 -1.9419 0.2411 1.7709 -1.4970 -0.7092 2 -0.7092 -1.4970 1.7709 0.2411 -1.9419 1.1361 1.1361 -1.9419 0.2411 1.7709 -1.4970 -0.7092
E'5* 2 -1.4970 0.2411 1.1361 -1.9419 1.7709 -0.7092 -0.7092 1.7709 -1.9419 1.1361 0.2411 -1.4970 2 -1.4970 0.2411 1.1361 -1.9419 1.7709 -0.7092 -0.7092 1.7709 -1.9419 1.1361 0.2411 -1.4970
E'6* 2 -1.9419 1.7709 -1.4970 1.1361 -0.7092 0.2411 0.2411 -0.7092 1.1361 -1.4970 1.7709 -1.9419 2 -1.9419 1.7709 -1.4970 1.1361 -0.7092 0.2411 0.2411 -0.7092 1.1361 -1.4970 1.7709 -1.9419
A'' 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
E''1* 2 1.7709 1.1361 0.2411 -0.7092 -1.4970 -1.9419 -1.9419 -1.4970 -0.7092 0.2411 1.1361 1.7709 -2 -1.7709 -1.1361 -0.2411 0.7092 1.4970 1.9419 1.9419 1.4970 0.7092 -0.2411 -1.1361 -1.7709
E''2* 2 1.1361 -0.7092 -1.9419 -1.4970 0.2411 1.7709 1.7709 0.2411 -1.4970 -1.9419 -0.7092 1.1361 -2 -1.1361 0.7092 1.9419 1.4970 -0.2411 -1.7709 -1.7709 -0.2411 1.4970 1.9419 0.7092 -1.1361
E''3* 2 0.2411 -1.9419 -0.7092 1.7709 1.1361 -1.4970 -1.4970 1.1361 1.7709 -0.7092 -1.9419 0.2411 -2 -0.2411 1.9419 0.7092 -1.7709 -1.1361 1.4970 1.4970 -1.1361 -1.7709 0.7092 1.9419 -0.2411
E''4* 2 -0.7092 -1.4970 1.7709 0.2411 -1.9419 1.1361 1.1361 -1.9419 0.2411 1.7709 -1.4970 -0.7092 -2 0.7092 1.4970 -1.7709 -0.2411 1.9419 -1.1361 -1.1361 1.9419 -0.2411 -1.7709 1.4970 0.7092
E''5* 2 -1.4970 0.2411 1.1361 -1.9419 1.7709 -0.7092 -0.7092 1.7709 -1.9419 1.1361 0.2411 -1.4970 -2 1.4970 -0.2411 -1.1361 1.9419 -1.7709 0.7092 0.7092 -1.7709 1.9419 -1.1361 -0.2411 1.4970
E''6* 2 -1.9419 1.7709 -1.4970 1.1361 -0.7092 0.2411 0.2411 -0.7092 1.1361 -1.4970 1.7709 -1.9419 -2 1.9419 -1.7709 1.4970 -1.1361 0.7092 -0.2411 -0.2411 0.7092 -1.1361 1.4970 -1.7709 1.9419
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Additional information

Number of symmetry elements h = 26
Number of classes, irreps n = 26
Number of real-valued irreducible representations n = 14
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E C13 (C13)2 (C13)3 (C13)4 (C13)5 (C13)6 (C13)7 (C13)8 (C13)9 (C13)10 (C13)11 (C13)12 σh S13 (S13)15 (S13)3 (S13)17 (S13)5 (S13)19 (S13)7 (S13)21 (S13)9 (S13)23 (S13)11 (S13)25



Genrate representation from irreducible representations


A' E'1* E'2* E'3* E'4* E'5* E'6* A'' E''1* E''2* E''3* E''4* E''5* E''6*



Direct products of irreducible representations


Binary products
⊙ A' E'1* E'2* E'3* E'4* E'5* E'6* A'' E''1* E''2* E''3* E''4* E''5* E''6*
A' A'
E'1* E'12A'⊕E'2
E'2* E'2E'1⊕E'32A'⊕E'4
E'3* E'3E'2⊕E'4E'1⊕E'52A'⊕E'6
E'4* E'4E'3⊕E'5E'2⊕E'6E'1⊕E'62A'⊕E'5
E'5* E'5E'4⊕E'6E'3⊕E'6E'2⊕E'5E'1⊕E'42A'⊕E'3
E'6* E'6E'5⊕E'6E'4⊕E'5E'3⊕E'4E'2⊕E'3E'1⊕E'22A'⊕E'1
A'' A''E''1E''2E''3E''4E''5E''6A'
E''1* E''12A''⊕E''2E''1⊕E''3E''2⊕E''4E''3⊕E''5E''4⊕E''6E''5⊕E''6E'12A'⊕E'2
E''2* E''2E''1⊕E''32A''⊕E''4E''1⊕E''5E''2⊕E''6E''3⊕E''6E''4⊕E''5E'2E'1⊕E'32A'⊕E'4
E''3* E''3E''2⊕E''4E''1⊕E''52A''⊕E''6E''1⊕E''6E''2⊕E''5E''3⊕E''4E'3E'2⊕E'4E'1⊕E'52A'⊕E'6
E''4* E''4E''3⊕E''5E''2⊕E''6E''1⊕E''62A''⊕E''5E''1⊕E''4E''2⊕E''3E'4E'3⊕E'5E'2⊕E'6E'1⊕E'62A'⊕E'5
E''5* E''5E''4⊕E''6E''3⊕E''6E''2⊕E''5E''1⊕E''42A''⊕E''3E''1⊕E''2E'5E'4⊕E'6E'3⊕E'6E'2⊕E'5E'1⊕E'42A'⊕E'3
E''6* E''6E''5⊕E''6E''4⊕E''5E''3⊕E''4E''2⊕E''3E''1⊕E''22A''⊕E''1E'6E'5⊕E'6E'4⊕E'5E'3⊕E'4E'2⊕E'3E'1⊕E'22A'⊕E'1

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E'1* A'⊕E'2E'1⊕E'3A'⊕E'2⊕E'4E'1⊕E'3⊕E'5A'⊕E'2⊕E'4⊕E'6More
E'2* A'⊕E'4E'2⊕E'6A'⊕E'4⊕E'5E'2⊕E'3⊕E'6A'⊕E'1⊕E'4⊕E'5More
E'3* A'⊕E'6E'3⊕E'4A'⊕E'1⊕E'6E'2⊕E'3⊕E'4A'⊕E'1⊕E'5⊕E'6More
E'4* A'⊕E'5E'1⊕E'4A'⊕E'3⊕E'5E'1⊕E'4⊕E'6A'⊕E'2⊕E'3⊕E'5More
E'5* A'⊕E'3E'2⊕E'5A'⊕E'3⊕E'6E'1⊕E'2⊕E'5A'⊕E'3⊕E'4⊕E'6More
E'6* A'⊕E'1E'5⊕E'6A'⊕E'1⊕E'2E'4⊕E'5⊕E'6A'⊕E'1⊕E'2⊕E'3More
E''1* A'⊕E'2E''1⊕E''3A'⊕E'2⊕E'4E''1⊕E''3⊕E''5A'⊕E'2⊕E'4⊕E'6More
E''2* A'⊕E'4E''2⊕E''6A'⊕E'4⊕E'5E''2⊕E''3⊕E''6A'⊕E'1⊕E'4⊕E'5More
E''3* A'⊕E'6E''3⊕E''4A'⊕E'1⊕E'6E''2⊕E''3⊕E''4A'⊕E'1⊕E'5⊕E'6More
E''4* A'⊕E'5E''1⊕E''4A'⊕E'3⊕E'5E''1⊕E''4⊕E''6A'⊕E'2⊕E'3⊕E'5More
E''5* A'⊕E'3E''2⊕E''5A'⊕E'3⊕E'6E''1⊕E''2⊕E''5A'⊕E'3⊕E'4⊕E'6More
E''6* A'⊕E'1E''5⊕E''6A'⊕E'1⊕E'2E''4⊕E''5⊕E''6A'⊕E'1⊕E'2⊕E'3More


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 A' 1 A'
p (l=1) 3 Dipole E'1⊕A'' 3 E'1⊕A''
d (l=2) 5 Quadrupole A'⊕E'2⊕E''1 6 2A'⊕E'2⊕E''1
f (l=3) 7 Octupole E'1⊕E'3⊕A''⊕E''2 10 2E'1⊕E'3⊕2A''⊕E''2
g (l=4) 9 Hexadecapole A'⊕E'2⊕E'4⊕E''1⊕E''3 15 3A'⊕2E'2⊕E'4⊕2E''1⊕E''3
h (l=5) 11 Dotricontapole E'1⊕E'3⊕E'5⊕A''⊕E''2⊕E''4 21 3E'1⊕2E'3⊕E'5⊕3A''⊕2E''2⊕E''4
i (l=6) 13 Tetrahexacontapole A'⊕E'2⊕E'4⊕E'6⊕E''1⊕E''3⊕E''5 28 4A'⊕3E'2⊕2E'4⊕E'6⊕3E''1⊕2E''3⊕E''5
j (l=7) 15 Octacosahectapole E'1⊕E'3⊕E'5⊕E'6⊕A''⊕E''2⊕E''4⊕E''6 36 4E'1⊕3E'3⊕2E'5⊕E'6⊕4A''⊕3E''2⊕2E''4⊕E''6
k (l=8) 17 256-pole A'⊕E'2⊕E'4⊕E'5⊕E'6⊕E''1⊕E''3⊕E''5⊕E''6 45 5A'⊕4E'2⊕3E'4⊕E'5⊕2E'6⊕4E''1⊕3E''3⊕2E''5⊕E''6
l (l=9) 19 512-pole E'1⊕E'3⊕E'4⊕E'5⊕E'6⊕A''⊕E''2⊕E''4⊕E''5⊕E''6 55 5E'1⊕4E'3⊕E'4⊕3E'5⊕2E'6⊕5A''⊕4E''2⊕3E''4⊕E''5⊕2E''6
m (l=10) 21 1024-pole A'⊕E'2⊕E'3⊕E'4⊕E'5⊕E'6⊕E''1⊕E''3⊕E''4⊕E''5⊕E''6 66 6A'⊕5E'2⊕E'3⊕4E'4⊕2E'5⊕3E'6⊕5E''1⊕4E''3⊕E''4⊕3E''5⊕2E''6
n (l=11) 23 2048-pole E'1⊕E'2⊕E'3⊕E'4⊕E'5⊕E'6⊕A''⊕E''2⊕E''3⊕E''4⊕E''5⊕E''6 78 6E'1⊕E'2⊕5E'3⊕2E'4⊕4E'5⊕3E'6⊕6A''⊕5E''2⊕E''3⊕4E''4⊕2E''5⊕3E''6
o (l=12) 25 4096-pole A'⊕E'1⊕E'2⊕E'3⊕E'4⊕E'5⊕E'6⊕E''1⊕E''2⊕E''3⊕E''4⊕E''5⊕E''6 91 7A'⊕E'1⊕6E'2⊕2E'3⊕5E'4⊕3E'5⊕4E'6⊕6E''1⊕E''2⊕5E''3⊕2E''4⊕4E''5⊕3E''6
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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 C13h
L 2L+1 Term Splitting
S (L=0) 1 A'
P (L=1) 3 A'⊕E''1
D (L=2) 5 A'⊕E'2⊕E''1
F (L=3) 7 A'⊕E'2⊕E''1⊕E''3
G (L=4) 9 A'⊕E'2⊕E'4⊕E''1⊕E''3
H (L=5) 11 A'⊕E'2⊕E'4⊕E''1⊕E''3⊕E''5
I (L=6) 13 A'⊕E'2⊕E'4⊕E'6⊕E''1⊕E''3⊕E''5

Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement