Point Group D15d



D15d E 2C15 2(C15)2 2C5 2(C15)4 2C3 2(C5)2 2(C15)7 15C'2 i 2(S30)13 2(S30)11 2(S10)3 2(S30)7 2S6 2S10 2S30 15σd
A1g 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2g 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 -1
E1g 2 2cos(2π/15) 2cos(4π/15) 2cos(2π/5) 2cos(8π/15) -1 2cos(4π/5) 2cos(14π/15) 0 2 2cos(2π/15) 2cos(4π/15) 2cos(2π/5) 2cos(8π/15) -1 2cos(4π/5) 2cos(14π/15) 0
E2g 2 2cos(4π/15) 2cos(8π/15) 2cos(4π/5) 2cos(14π/15) -1 2cos(2π/5) 2cos(2π/15) 0 2 2cos(4π/15) 2cos(8π/15) 2cos(4π/5) 2cos(14π/15) -1 2cos(2π/5) 2cos(2π/15) 0
E3g 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) 2 2cos(2π/5) 2cos(4π/5) 0 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) 2 2cos(2π/5) 2cos(4π/5) 0
E4g 2 2cos(8π/15) 2cos(14π/15) 2cos(2π/5) 2cos(2π/15) -1 2cos(4π/5) 2cos(4π/15) 0 2 2cos(8π/15) 2cos(14π/15) 2cos(2π/5) 2cos(2π/15) -1 2cos(4π/5) 2cos(4π/15) 0
E5g 2 -1 -1 2 -1 -1 2 -1 0 2 -1 -1 2 -1 -1 2 -1 0
E6g 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) 2 2cos(4π/5) 2cos(2π/5) 0 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) 2 2cos(4π/5) 2cos(2π/5) 0
E7g 2 2cos(14π/15) 2cos(2π/15) 2cos(4π/5) 2cos(4π/15) -1 2cos(2π/5) 2cos(8π/15) 0 2 2cos(14π/15) 2cos(2π/15) 2cos(4π/5) 2cos(4π/15) -1 2cos(2π/5) 2cos(8π/15) 0
A1u 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1
A2u 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1
E1u 2 2cos(2π/15) 2cos(4π/15) 2cos(2π/5) 2cos(8π/15) -1 2cos(4π/5) 2cos(14π/15) 0 -2 -2cos(2π/15) -2cos(4π/15) -2cos(2π/5) -2cos(8π/15) 1 -2cos(4π/5) -2cos(14π/15) 0
E2u 2 2cos(4π/15) 2cos(8π/15) 2cos(4π/5) 2cos(14π/15) -1 2cos(2π/5) 2cos(2π/15) 0 -2 -2cos(4π/15) -2cos(8π/15) -2cos(4π/5) -2cos(14π/15) 1 -2cos(2π/5) -2cos(2π/15) 0
E3u 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) 2 2cos(2π/5) 2cos(4π/5) 0 -2 -2cos(2π/5) -2cos(4π/5) -2cos(4π/5) -2cos(2π/5) -2 -2cos(2π/5) -2cos(4π/5) 0
E4u 2 2cos(8π/15) 2cos(14π/15) 2cos(2π/5) 2cos(2π/15) -1 2cos(4π/5) 2cos(4π/15) 0 -2 -2cos(8π/15) -2cos(14π/15) -2cos(2π/5) -2cos(2π/15) 1 -2cos(4π/5) -2cos(4π/15) 0
E5u 2 -1 -1 2 -1 -1 2 -1 0 -2 1 1 -2 1 1 -2 1 0
E6u 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) 2 2cos(4π/5) 2cos(2π/5) 0 -2 -2cos(4π/5) -2cos(2π/5) -2cos(2π/5) -2cos(4π/5) -2 -2cos(4π/5) -2cos(2π/5) 0
E7u 2 2cos(14π/15) 2cos(2π/15) 2cos(4π/5) 2cos(4π/15) -1 2cos(2π/5) 2cos(8π/15) 0 -2 -2cos(14π/15) -2cos(2π/15) -2cos(4π/5) -2cos(4π/15) 1 -2cos(2π/5) -2cos(8π/15) 0


Additional information

Number of symmetry elements h = 60
Number of classes, irreps n = 18
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E 2C15 2(C15)2 2C5 2(C15)4 2C3 2(C5)2 2(C15)7 15C'2 i 2(S30)13 2(S30)11 2(S10)3 2(S30)7 2S6 2S10 2S30 15σd



Genrate representation from irreducible representations


A1g A2g E1g E2g E3g E4g E5g E6g E7g A1u A2u E1u E2u E3u E4u E5u E6u E7u




Direct products of irreducible representations


Binary products
A1g A2g E1g E2g E3g E4g E5g E6g E7g A1u A2u E1u E2u E3u E4u E5u E6u E7u
A1g A1g
A2g A2gA1g
E1g E1gE1gA1g⊕A2g⊕E2g
E2g E2gE2gE1g⊕E3gA1g⊕A2g⊕E4g
E3g E3gE3gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕E6g
E4g E4gE4gE3g⊕E5gE2g⊕E6gE1g⊕E7gA1g⊕A2g⊕E7g
E5g E5gE5gE4g⊕E6gE3g⊕E7gE2g⊕E7gE1g⊕E6gA1g⊕A2g⊕E5g
E6g E6gE6gE5g⊕E7gE4g⊕E7gE3g⊕E6gE2g⊕E5gE1g⊕E4gA1g⊕A2g⊕E3g
E7g E7gE7gE6g⊕E7gE5g⊕E6gE4g⊕E5gE3g⊕E4gE2g⊕E3gE1g⊕E2gA1g⊕A2g⊕E1g
A1u A1uA2uE1uE2uE3uE4uE5uE6uE7uA1g
A2u A2uA1uE1uE2uE3uE4uE5uE6uE7uA2gA1g
E1u E1uE1uA1u⊕A2u⊕E2uE1u⊕E3uE2u⊕E4uE3u⊕E5uE4u⊕E6uE5u⊕E7uE6u⊕E7uE1gE1gA1g⊕A2g⊕E2g
E2u E2uE2uE1u⊕E3uA1u⊕A2u⊕E4uE1u⊕E5uE2u⊕E6uE3u⊕E7uE4u⊕E7uE5u⊕E6uE2gE2gE1g⊕E3gA1g⊕A2g⊕E4g
E3u E3uE3uE2u⊕E4uE1u⊕E5uA1u⊕A2u⊕E6uE1u⊕E7uE2u⊕E7uE3u⊕E6uE4u⊕E5uE3gE3gE2g⊕E4gE1g⊕E5gA1g⊕A2g⊕E6g
E4u E4uE4uE3u⊕E5uE2u⊕E6uE1u⊕E7uA1u⊕A2u⊕E7uE1u⊕E6uE2u⊕E5uE3u⊕E4uE4gE4gE3g⊕E5gE2g⊕E6gE1g⊕E7gA1g⊕A2g⊕E7g
E5u E5uE5uE4u⊕E6uE3u⊕E7uE2u⊕E7uE1u⊕E6uA1u⊕A2u⊕E5uE1u⊕E4uE2u⊕E3uE5gE5gE4g⊕E6gE3g⊕E7gE2g⊕E7gE1g⊕E6gA1g⊕A2g⊕E5g
E6u E6uE6uE5u⊕E7uE4u⊕E7uE3u⊕E6uE2u⊕E5uE1u⊕E4uA1u⊕A2u⊕E3uE1u⊕E2uE6gE6gE5g⊕E7gE4g⊕E7gE3g⊕E6gE2g⊕E5gE1g⊕E4gA1g⊕A2g⊕E3g
E7u E7uE7uE6u⊕E7uE5u⊕E6uE4u⊕E5uE3u⊕E4uE2u⊕E3uE1u⊕E2uA1u⊕A2u⊕E1uE7gE7gE6g⊕E7gE5g⊕E6gE4g⊕E5gE3g⊕E4gE2g⊕E3gE1g⊕E2gA1g⊕A2g⊕E1g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g A1g⊕E2gE1g⊕E3gA1g⊕E2g⊕E4gE1g⊕E3g⊕E5gA1g⊕E2g⊕E4g⊕E6gMore
E2g A1g⊕E4gE2g⊕E6gA1g⊕E4g⊕E7gE2g⊕E5g⊕E6gA1g⊕E3g⊕E4g⊕E7gMore
E3g A1g⊕E6gE3g⊕E6gA1g⊕E3g⊕E6gA1g⊕A2g⊕E3g⊕E6gA1g⊕2E3g⊕E6gMore
E4g A1g⊕E7gE3g⊕E4gA1g⊕E1g⊕E7gE3g⊕E4g⊕E5gA1g⊕E1g⊕E6g⊕E7gMore
E5g A1g⊕E5gA1g⊕A2g⊕E5gA1g⊕2E5gA1g⊕A2g⊕2E5g2A1g⊕A2g⊕2E5gMore
E6g A1g⊕E3gE3g⊕E6gA1g⊕E3g⊕E6gA1g⊕A2g⊕E3g⊕E6gA1g⊕E3g⊕2E6gMore
E7g A1g⊕E1gE6g⊕E7gA1g⊕E1g⊕E2gE5g⊕E6g⊕E7gA1g⊕E1g⊕E2g⊕E3gMore
E1u A1g⊕E2gE1u⊕E3uA1g⊕E2g⊕E4gE1u⊕E3u⊕E5uA1g⊕E2g⊕E4g⊕E6gMore
E2u A1g⊕E4gE2u⊕E6uA1g⊕E4g⊕E7gE2u⊕E5u⊕E6uA1g⊕E3g⊕E4g⊕E7gMore
E3u A1g⊕E6gE3u⊕E6uA1g⊕E3g⊕E6gA1u⊕A2u⊕E3u⊕E6uA1g⊕2E3g⊕E6gMore
E4u A1g⊕E7gE3u⊕E4uA1g⊕E1g⊕E7gE3u⊕E4u⊕E5uA1g⊕E1g⊕E6g⊕E7gMore
E5u A1g⊕E5gA1u⊕A2u⊕E5uA1g⊕2E5gA1u⊕A2u⊕2E5u2A1g⊕A2g⊕2E5gMore
E6u A1g⊕E3gE3u⊕E6uA1g⊕E3g⊕E6gA1u⊕A2u⊕E3u⊕E6uA1g⊕E3g⊕2E6gMore
E7u A1g⊕E1gE6u⊕E7uA1g⊕E1g⊕E2gE5u⊕E6u⊕E7uA1g⊕E1g⊕E2g⊕E3gMore



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 A1g 1 A1g
p (l=1) 3 Dipole A2u⊕E1u 3 A2u⊕E1u
d (l=2) 5 Quadrupole A1g⊕E1g⊕E2g 6 2A1g⊕E1g⊕E2g
f (l=3) 7 Octupole A2u⊕E1u⊕E2u⊕E3u 10 2A2u⊕2E1u⊕E2u⊕E3u
g (l=4) 9 Hexadecapole A1g⊕E1g⊕E2g⊕E3g⊕E4g 15 3A1g⊕2E1g⊕2E2g⊕E3g⊕E4g
h (l=5) 11 Dotricontapole A2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u 21 3A2u⊕3E1u⊕2E2u⊕2E3u⊕E4u⊕E5u
i (l=6) 13 Tetrahexacontapole A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g 28 4A1g⊕3E1g⊕3E2g⊕2E3g⊕2E4g⊕E5g⊕E6g
j (l=7) 15 Octacosahectapole A2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕E6u⊕E7u 36 4A2u⊕4E1u⊕3E2u⊕3E3u⊕2E4u⊕2E5u⊕E6u⊕E7u
k (l=8) 17 256-pole A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g⊕2E7g 45 5A1g⊕4E1g⊕4E2g⊕3E3g⊕3E4g⊕2E5g⊕2E6g⊕2E7g
l (l=9) 19 512-pole A2u⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕2E6u⊕2E7u 55 5A2u⊕5E1u⊕4E2u⊕4E3u⊕3E4u⊕3E5u⊕3E6u⊕3E7u
m (l=10) 21 1024-pole A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕2E5g⊕2E6g⊕2E7g 66 6A1g⊕5E1g⊕5E2g⊕4E3g⊕4E4g⊕4E5g⊕4E6g⊕4E7g
n (l=11) 23 2048-pole A2u⊕E1u⊕E2u⊕E3u⊕2E4u⊕2E5u⊕2E6u⊕2E7u 78 6A2u⊕6E1u⊕5E2u⊕5E3u⊕5E4u⊕5E5u⊕5E6u⊕5E7u
o (l=12) 25 4096-pole A1g⊕E1g⊕E2g⊕2E3g⊕2E4g⊕2E5g⊕2E6g⊕2E7g 91 7A1g⊕6E1g⊕6E2g⊕6E3g⊕6E4g⊕6E5g⊕6E6g⊕6E7g
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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 D15d
L 2L+1 Term Splitting
S (L=0) 1 A1g
P (L=1) 3 A2g⊕E1g
D (L=2) 5 A1g⊕E1g⊕E2g
F (L=3) 7 A2g⊕E1g⊕E2g⊕E3g
G (L=4) 9 A1g⊕E1g⊕E2g⊕E3g⊕E4g
H (L=5) 11 A2g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g
I (L=6) 13 A1g⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g


Last update August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement