Point Group C29



C29 E 2C29 2(C29)2 2(C29)3 2(C29)4 2(C29)5 2(C29)6 2(C29)7 2(C29)8 2(C29)9 2(C29)10 2(C29)11 2(C29)12 2(C29)13 2(C29)14
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E1* 2 2cos(2π/29) 2cos(4π/29) 2cos(6π/29) 2cos(8π/29) 2cos(10π/29) 2cos(12π/29) 2cos(14π/29) 2cos(16π/29) 2cos(18π/29) 2cos(20π/29) 2cos(22π/29) 2cos(24π/29) 2cos(26π/29) 2cos(28π/29)
E2* 2 2cos(4π/29) 2cos(8π/29) 2cos(12π/29) 2cos(16π/29) 2cos(20π/29) 2cos(24π/29) 2cos(28π/29) 2cos(26π/29) 2cos(22π/29) 2cos(18π/29) 2cos(14π/29) 2cos(10π/29) 2cos(6π/29) 2cos(2π/29)
E3* 2 2cos(6π/29) 2cos(12π/29) 2cos(18π/29) 2cos(24π/29) 2cos(28π/29) 2cos(22π/29) 2cos(16π/29) 2cos(10π/29) 2cos(4π/29) 2cos(2π/29) 2cos(8π/29) 2cos(14π/29) 2cos(20π/29) 2cos(26π/29)
E4* 2 2cos(8π/29) 2cos(16π/29) 2cos(24π/29) 2cos(26π/29) 2cos(18π/29) 2cos(10π/29) 2cos(2π/29) 2cos(6π/29) 2cos(14π/29) 2cos(22π/29) 2cos(28π/29) 2cos(20π/29) 2cos(12π/29) 2cos(4π/29)
E5* 2 2cos(10π/29) 2cos(20π/29) 2cos(28π/29) 2cos(18π/29) 2cos(8π/29) 2cos(2π/29) 2cos(12π/29) 2cos(22π/29) 2cos(26π/29) 2cos(16π/29) 2cos(6π/29) 2cos(4π/29) 2cos(14π/29) 2cos(24π/29)
E6* 2 2cos(12π/29) 2cos(24π/29) 2cos(22π/29) 2cos(10π/29) 2cos(2π/29) 2cos(14π/29) 2cos(26π/29) 2cos(20π/29) 2cos(8π/29) 2cos(4π/29) 2cos(16π/29) 2cos(28π/29) 2cos(18π/29) 2cos(6π/29)
E7* 2 2cos(14π/29) 2cos(28π/29) 2cos(16π/29) 2cos(2π/29) 2cos(12π/29) 2cos(26π/29) 2cos(18π/29) 2cos(4π/29) 2cos(10π/29) 2cos(24π/29) 2cos(20π/29) 2cos(6π/29) 2cos(8π/29) 2cos(22π/29)
E8* 2 2cos(16π/29) 2cos(26π/29) 2cos(10π/29) 2cos(6π/29) 2cos(22π/29) 2cos(20π/29) 2cos(4π/29) 2cos(12π/29) 2cos(28π/29) 2cos(14π/29) 2cos(2π/29) 2cos(18π/29) 2cos(24π/29) 2cos(8π/29)
E9* 2 2cos(18π/29) 2cos(22π/29) 2cos(4π/29) 2cos(14π/29) 2cos(26π/29) 2cos(8π/29) 2cos(10π/29) 2cos(28π/29) 2cos(12π/29) 2cos(6π/29) 2cos(24π/29) 2cos(16π/29) 2cos(2π/29) 2cos(20π/29)
E10* 2 2cos(20π/29) 2cos(18π/29) 2cos(2π/29) 2cos(22π/29) 2cos(16π/29) 2cos(4π/29) 2cos(24π/29) 2cos(14π/29) 2cos(6π/29) 2cos(26π/29) 2cos(12π/29) 2cos(8π/29) 2cos(28π/29) 2cos(10π/29)
E11* 2 2cos(22π/29) 2cos(14π/29) 2cos(8π/29) 2cos(28π/29) 2cos(6π/29) 2cos(16π/29) 2cos(20π/29) 2cos(2π/29) 2cos(24π/29) 2cos(12π/29) 2cos(10π/29) 2cos(26π/29) 2cos(4π/29) 2cos(18π/29)
E12* 2 2cos(24π/29) 2cos(10π/29) 2cos(14π/29) 2cos(20π/29) 2cos(4π/29) 2cos(28π/29) 2cos(6π/29) 2cos(18π/29) 2cos(16π/29) 2cos(8π/29) 2cos(26π/29) 2cos(2π/29) 2cos(22π/29) 2cos(12π/29)
E13* 2 2cos(26π/29) 2cos(6π/29) 2cos(20π/29) 2cos(12π/29) 2cos(14π/29) 2cos(18π/29) 2cos(8π/29) 2cos(24π/29) 2cos(2π/29) 2cos(28π/29) 2cos(4π/29) 2cos(22π/29) 2cos(10π/29) 2cos(16π/29)
E14* 2 2cos(28π/29) 2cos(2π/29) 2cos(26π/29) 2cos(4π/29) 2cos(24π/29) 2cos(6π/29) 2cos(22π/29) 2cos(8π/29) 2cos(20π/29) 2cos(10π/29) 2cos(18π/29) 2cos(12π/29) 2cos(16π/29) 2cos(14π/29)
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Additional information

Number of symmetry elements h = 29
Number of classes, irreps n = 29
Number of real-valued irreducible representations n = 15
Abelian group yes
Optical Isomerism (Chirality) yes
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C29 2(C29)2 2(C29)3 2(C29)4 2(C29)5 2(C29)6 2(C29)7 2(C29)8 2(C29)9 2(C29)10 2(C29)11 2(C29)12 2(C29)13 2(C29)14



Genrate representation from irreducible representations


A E1* E2* E3* E4* E5* E6* E7* E8* E9* E10* E11* E12* E13* E14*



Direct products of irreducible representations


Binary products
⊙ A E1* E2* E3* E4* E5* E6* E7* E8* E9* E10* E11* E12* E13* E14*
A A
E1* E12A⊕E2
E2* E2E1⊕E32A⊕E4
E3* E3E2⊕E4E1⊕E52A⊕E6
E4* E4E3⊕E5E2⊕E6E1⊕E72A⊕E8
E5* E5E4⊕E6E3⊕E7E2⊕E8E1⊕E92A⊕E10
E6* E6E5⊕E7E4⊕E8E3⊕E9E2⊕E10E1⊕E112A⊕E12
E7* E7E6⊕E8E5⊕E9E4⊕E10E3⊕E11E2⊕E12E1⊕E132A⊕E14
E8* E8E7⊕E9E6⊕E10E5⊕E11E4⊕E12E3⊕E13E2⊕E14E1⊕E142A⊕E13
E9* E9E8⊕E10E7⊕E11E6⊕E12E5⊕E13E4⊕E14E3⊕E14E2⊕E13E1⊕E122A⊕E11
E10* E10E9⊕E11E8⊕E12E7⊕E13E6⊕E14E5⊕E14E4⊕E13E3⊕E12E2⊕E11E1⊕E102A⊕E9
E11* E11E10⊕E12E9⊕E13E8⊕E14E7⊕E14E6⊕E13E5⊕E12E4⊕E11E3⊕E10E2⊕E9E1⊕E82A⊕E7
E12* E12E11⊕E13E10⊕E14E9⊕E14E8⊕E13E7⊕E12E6⊕E11E5⊕E10E4⊕E9E3⊕E8E2⊕E7E1⊕E62A⊕E5
E13* E13E12⊕E14E11⊕E14E10⊕E13E9⊕E12E8⊕E11E7⊕E10E6⊕E9E5⊕E8E4⊕E7E3⊕E6E2⊕E5E1⊕E42A⊕E3
E14* E14E13⊕E14E12⊕E13E11⊕E12E10⊕E11E9⊕E10E8⊕E9E7⊕E8E6⊕E7E5⊕E6E4⊕E5E3⊕E4E2⊕E3E1⊕E22A⊕E1

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E1* A⊕E2E1⊕E3A⊕E2⊕E4E1⊕E3⊕E5A⊕E2⊕E4⊕E6More
E2* A⊕E4E2⊕E6A⊕E4⊕E8E2⊕E6⊕E10A⊕E4⊕E8⊕E12More
E3* A⊕E6E3⊕E9A⊕E6⊕E12E3⊕E9⊕E14A⊕E6⊕E11⊕E12More
E4* A⊕E8E4⊕E12A⊕E8⊕E13E4⊕E9⊕E12A⊕E5⊕E8⊕E13More
E5* A⊕E10E5⊕E14A⊕E9⊕E10E4⊕E5⊕E14A⊕E1⊕E9⊕E10More
E6* A⊕E12E6⊕E11A⊕E5⊕E12E1⊕E6⊕E11A⊕E5⊕E7⊕E12More
E7* A⊕E14E7⊕E8A⊕E1⊕E14E6⊕E7⊕E8A⊕E1⊕E13⊕E14More
E8* A⊕E13E5⊕E8A⊕E3⊕E13E5⊕E8⊕E11A⊕E3⊕E10⊕E13More
E9* A⊕E11E2⊕E9A⊕E7⊕E11E2⊕E9⊕E13A⊕E4⊕E7⊕E11More
E10* A⊕E9E1⊕E10A⊕E9⊕E11E1⊕E8⊕E10A⊕E2⊕E9⊕E11More
E11* A⊕E7E4⊕E11A⊕E7⊕E14E3⊕E4⊕E11A⊕E7⊕E8⊕E14More
E12* A⊕E5E7⊕E12A⊕E5⊕E10E2⊕E7⊕E12A⊕E5⊕E10⊕E14More
E13* A⊕E3E10⊕E13A⊕E3⊕E6E7⊕E10⊕E13A⊕E3⊕E6⊕E9More
E14* A⊕E1E13⊕E14A⊕E1⊕E2E12⊕E13⊕E14A⊕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 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⊕E7⊕E8 45 5A⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7⊕E8
l (l=9) 19 512-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 55 5A⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕2E7⊕E8⊕E9
m (l=10) 21 1024-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10 66 6A⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕3E6⊕2E7⊕2E8⊕E9⊕E10
n (l=11) 23 2048-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11 78 6A⊕6E1⊕5E2⊕5E3⊕4E4⊕4E5⊕3E6⊕3E7⊕2E8⊕2E9⊕E10⊕E11
o (l=12) 25 4096-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12 91 7A⊕6E1⊕6E2⊕5E3⊕5E4⊕4E5⊕4E6⊕3E7⊕3E8⊕2E9⊕2E10⊕E11⊕E12
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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 C29
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 November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement