Point Group C29



C29 E C29 (C29)2 (C29)3 (C29)4 (C29)5 (C29)6 (C29)7 (C29)8 (C29)9 (C29)10 (C29)11 (C29)12 (C29)13 (C29)14 (C29)15 (C29)16 (C29)17 (C29)18 (C29)19 (C29)20 (C29)21 (C29)22 (C29)23 (C29)24 (C29)25 (C29)26 (C29)27 (C29)28
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 1 1 1
E1* 2 1.9532 1.8152 1.5922 1.2948 0.9368 0.5351 0.1083 -0.3236 -0.7403 -1.1224 -1.4520 -1.7137 -1.8953 -1.9883 -1.9883 -1.8953 -1.7137 -1.4520 -1.1224 -0.7403 -0.3236 0.1083 0.5351 0.9368 1.2948 1.5922 1.8152 1.9532
E2* 2 1.8152 1.2948 0.5351 -0.3236 -1.1224 -1.7137 -1.9883 -1.8953 -1.4520 -0.7403 0.1083 0.9368 1.5922 1.9532 1.9532 1.5922 0.9368 0.1083 -0.7403 -1.4520 -1.8953 -1.9883 -1.7137 -1.1224 -0.3236 0.5351 1.2948 1.8152
E3* 2 1.5922 0.5351 -0.7403 -1.7137 -1.9883 -1.4520 -0.3236 0.9368 1.8152 1.9532 1.2948 0.1083 -1.1224 -1.8953 -1.8953 -1.1224 0.1083 1.2948 1.9532 1.8152 0.9368 -0.3236 -1.4520 -1.9883 -1.7137 -0.7403 0.5351 1.5922
E4* 2 1.2948 -0.3236 -1.7137 -1.8953 -0.7403 0.9368 1.9532 1.5922 0.1083 -1.4520 -1.9883 -1.1224 0.5351 1.8152 1.8152 0.5351 -1.1224 -1.9883 -1.4520 0.1083 1.5922 1.9532 0.9368 -0.7403 -1.8953 -1.7137 -0.3236 1.2948
E5* 2 0.9368 -1.1224 -1.9883 -0.7403 1.2948 1.9532 0.5351 -1.4520 -1.8953 -0.3236 1.5922 1.8152 0.1083 -1.7137 -1.7137 0.1083 1.8152 1.5922 -0.3236 -1.8953 -1.4520 0.5351 1.9532 1.2948 -0.7403 -1.9883 -1.1224 0.9368
E6* 2 0.5351 -1.7137 -1.4520 0.9368 1.9532 0.1083 -1.8953 -1.1224 1.2948 1.8152 -0.3236 -1.9883 -0.7403 1.5922 1.5922 -0.7403 -1.9883 -0.3236 1.8152 1.2948 -1.1224 -1.8953 0.1083 1.9532 0.9368 -1.4520 -1.7137 0.5351
E7* 2 0.1083 -1.9883 -0.3236 1.9532 0.5351 -1.8953 -0.7403 1.8152 0.9368 -1.7137 -1.1224 1.5922 1.2948 -1.4520 -1.4520 1.2948 1.5922 -1.1224 -1.7137 0.9368 1.8152 -0.7403 -1.8953 0.5351 1.9532 -0.3236 -1.9883 0.1083
E8* 2 -0.3236 -1.8953 0.9368 1.5922 -1.4520 -1.1224 1.8152 0.5351 -1.9883 0.1083 1.9532 -0.7403 -1.7137 1.2948 1.2948 -1.7137 -0.7403 1.9532 0.1083 -1.9883 0.5351 1.8152 -1.1224 -1.4520 1.5922 0.9368 -1.8953 -0.3236
E9* 2 -0.7403 -1.4520 1.8152 0.1083 -1.8953 1.2948 0.9368 -1.9883 0.5351 1.5922 -1.7137 -0.3236 1.9532 -1.1224 -1.1224 1.9532 -0.3236 -1.7137 1.5922 0.5351 -1.9883 0.9368 1.2948 -1.8953 0.1083 1.8152 -1.4520 -0.7403
E10* 2 -1.1224 -0.7403 1.9532 -1.4520 -0.3236 1.8152 -1.7137 0.1083 1.5922 -1.8953 0.5351 1.2948 -1.9883 0.9368 0.9368 -1.9883 1.2948 0.5351 -1.8953 1.5922 0.1083 -1.7137 1.8152 -0.3236 -1.4520 1.9532 -0.7403 -1.1224
E11* 2 -1.4520 0.1083 1.2948 -1.9883 1.5922 -0.3236 -1.1224 1.9532 -1.7137 0.5351 0.9368 -1.8953 1.8152 -0.7403 -0.7403 1.8152 -1.8953 0.9368 0.5351 -1.7137 1.9532 -1.1224 -0.3236 1.5922 -1.9883 1.2948 0.1083 -1.4520
E12* 2 -1.7137 0.9368 0.1083 -1.1224 1.8152 -1.9883 1.5922 -0.7403 -0.3236 1.2948 -1.8953 1.9532 -1.4520 0.5351 0.5351 -1.4520 1.9532 -1.8953 1.2948 -0.3236 -0.7403 1.5922 -1.9883 1.8152 -1.1224 0.1083 0.9368 -1.7137
E13* 2 -1.8953 1.5922 -1.1224 0.5351 0.1083 -0.7403 1.2948 -1.7137 1.9532 -1.9883 1.8152 -1.4520 0.9368 -0.3236 -0.3236 0.9368 -1.4520 1.8152 -1.9883 1.9532 -1.7137 1.2948 -0.7403 0.1083 0.5351 -1.1224 1.5922 -1.8953
E14* 2 -1.9883 1.9532 -1.8953 1.8152 -1.7137 1.5922 -1.4520 1.2948 -1.1224 0.9368 -0.7403 0.5351 -0.3236 0.1083 0.1083 -0.3236 0.5351 -0.7403 0.9368 -1.1224 1.2948 -1.4520 1.5922 -1.7137 1.8152 -1.8953 1.9532 -1.9883
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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 C29 (C29)2 (C29)3 (C29)4 (C29)5 (C29)6 (C29)7 (C29)8 (C29)9 (C29)10 (C29)11 (C29)12 (C29)13 (C29)14 (C29)15 (C29)16 (C29)17 (C29)18 (C29)19 (C29)20 (C29)21 (C29)22 (C29)23 (C29)24 (C29)25 (C29)26 (C29)27 (C29)28



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