Point Group D29



D29 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 29C'2
A1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2 1 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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
View A
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View B


Additional information

Number of symmetry elements h = 58
Number of classes, irreps n = 16
Abelian group no
Optical Isomerism (Chirality) yes
Polar no
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 29C'2



Genrate representation from irreducible representations


A1 A2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14



Direct products of irreducible representations


Binary products
⊙ A1 A2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14
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⊕E8
E5 E5E5E4⊕E6E3⊕E7E2⊕E8E1⊕E9A1⊕A2⊕E10
E6 E6E6E5⊕E7E4⊕E8E3⊕E9E2⊕E10E1⊕E11A1⊕A2⊕E12
E7 E7E7E6⊕E8E5⊕E9E4⊕E10E3⊕E11E2⊕E12E1⊕E13A1⊕A2⊕E14
E8 E8E8E7⊕E9E6⊕E10E5⊕E11E4⊕E12E3⊕E13E2⊕E14E1⊕E14A1⊕A2⊕E13
E9 E9E9E8⊕E10E7⊕E11E6⊕E12E5⊕E13E4⊕E14E3⊕E14E2⊕E13E1⊕E12A1⊕A2⊕E11
E10 E10E10E9⊕E11E8⊕E12E7⊕E13E6⊕E14E5⊕E14E4⊕E13E3⊕E12E2⊕E11E1⊕E10A1⊕A2⊕E9
E11 E11E11E10⊕E12E9⊕E13E8⊕E14E7⊕E14E6⊕E13E5⊕E12E4⊕E11E3⊕E10E2⊕E9E1⊕E8A1⊕A2⊕E7
E12 E12E12E11⊕E13E10⊕E14E9⊕E14E8⊕E13E7⊕E12E6⊕E11E5⊕E10E4⊕E9E3⊕E8E2⊕E7E1⊕E6A1⊕A2⊕E5
E13 E13E13E12⊕E14E11⊕E14E10⊕E13E9⊕E12E8⊕E11E7⊕E10E6⊕E9E5⊕E8E4⊕E7E3⊕E6E2⊕E5E1⊕E4A1⊕A2⊕E3
E14 E14E14E13⊕E14E12⊕E13E11⊕E12E10⊕E11E9⊕E10E8⊕E9E7⊕E8E6⊕E7E5⊕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⊕E8E2⊕E6⊕E10A1⊕E4⊕E8⊕E12More
E3 A1⊕E6E3⊕E9A1⊕E6⊕E12E3⊕E9⊕E14A1⊕E6⊕E11⊕E12More
E4 A1⊕E8E4⊕E12A1⊕E8⊕E13E4⊕E9⊕E12A1⊕E5⊕E8⊕E13More
E5 A1⊕E10E5⊕E14A1⊕E9⊕E10E4⊕E5⊕E14A1⊕E1⊕E9⊕E10More
E6 A1⊕E12E6⊕E11A1⊕E5⊕E12E1⊕E6⊕E11A1⊕E5⊕E7⊕E12More
E7 A1⊕E14E7⊕E8A1⊕E1⊕E14E6⊕E7⊕E8A1⊕E1⊕E13⊕E14More
E8 A1⊕E13E5⊕E8A1⊕E3⊕E13E5⊕E8⊕E11A1⊕E3⊕E10⊕E13More
E9 A1⊕E11E2⊕E9A1⊕E7⊕E11E2⊕E9⊕E13A1⊕E4⊕E7⊕E11More
E10 A1⊕E9E1⊕E10A1⊕E9⊕E11E1⊕E8⊕E10A1⊕E2⊕E9⊕E11More
E11 A1⊕E7E4⊕E11A1⊕E7⊕E14E3⊕E4⊕E11A1⊕E7⊕E8⊕E14More
E12 A1⊕E5E7⊕E12A1⊕E5⊕E10E2⊕E7⊕E12A1⊕E5⊕E10⊕E14More
E13 A1⊕E3E10⊕E13A1⊕E3⊕E6E7⊕E10⊕E13A1⊕E3⊕E6⊕E9More
E14 A1⊕E1E13⊕E14A1⊕E1⊕E2E12⊕E13⊕E14A1⊕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 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⊕E6⊕E7 36 4A2⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 45 5A1⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7⊕E8
l (l=9) 19 512-pole A2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 55 5A2⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕2E7⊕E8⊕E9
m (l=10) 21 1024-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10 66 6A1⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕3E6⊕2E7⊕2E8⊕E9⊕E10
n (l=11) 23 2048-pole A2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11 78 6A2⊕6E1⊕5E2⊕5E3⊕4E4⊕4E5⊕3E6⊕3E7⊕2E8⊕2E9⊕E10⊕E11
o (l=12) 25 4096-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12 91 7A1⊕6E1⊕6E2⊕5E3⊕5E4⊕4E5⊕4E6⊕3E7⊕3E8⊕2E9⊕2E10⊕E11⊕E12
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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 D29
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