Point Group S30



ε=exp(2πi/15)
S30 E C15 (C15)2 C5 (C15)4 C3 (C5)2 (C15)7 (C15)8 (C5)3 (C3)2 (C15)11 (C5)4 (C15)13 (C15)14 i (S30)17 (S30)19 (S10)7 (S30)23 (S6)5 (S10)9 (S30)29 S30 S10 S6 (S30)7 (S10)3 (S30)11 (S30)13
Ag 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 1
E1g* 1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E2g* 1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
E3g* 1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
E4g* 1
1
ε4*
ε4*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε4*
ε4*
1
1
ε4*
ε4*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε4*
ε4*
E5g* 1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
E6g* 1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
E7g* 1
1
ε7*
ε7*
ε*
ε*
ε6*
ε6*
ε2*
ε2*
ε5*
ε5*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε5*
ε5*
ε2*
ε2*
ε6*
ε6*
ε*
ε*
ε7*
ε7*
1
1
ε7*
ε7*
ε*
ε*
ε6*
ε6*
ε2*
ε2*
ε5*
ε5*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε5*
ε5*
ε2*
ε2*
ε6*
ε6*
ε*
ε*
ε7*
ε7*
Au 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 -1
E1u* 1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
5*
5*
6*
6*
7*
7*
7*
7*
6*
6*
5*
5*
4*
4*
3*
3*
2*
2*
*
*
E2u* 1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
-1
-1
2*
2*
4*
4*
6*
6*
7*
7*
5*
5*
3*
3*
*
*
*
*
3*
3*
5*
5*
7*
7*
6*
6*
4*
4*
2*
2*
E3u* 1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
-1
-1
3*
3*
6*
6*
6*
6*
3*
3*
-1
-1
3*
3*
6*
6*
6*
6*
3*
3*
-1
-1
3*
3*
6*
6*
6*
6*
3*
3*
E4u* 1
1
ε4*
ε4*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε4*
ε4*
-1
-1
4*
4*
7*
7*
3*
3*
*
*
5*
5*
6*
6*
2*
2*
2*
2*
6*
6*
5*
5*
*
*
3*
3*
7*
7*
4*
4*
E5u* 1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
-1
-1
5*
5*
5*
5*
-1
-1
5*
5*
5*
5*
-1
-1
5*
5*
5*
5*
-1
-1
5*
5*
5*
5*
-1
-1
5*
5*
5*
5*
E6u* 1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
-1
-1
6*
6*
3*
3*
3*
3*
6*
6*
-1
-1
6*
6*
3*
3*
3*
3*
6*
6*
-1
-1
6*
6*
3*
3*
3*
3*
6*
6*
E7u* 1
1
ε7*
ε7*
ε*
ε*
ε6*
ε6*
ε2*
ε2*
ε5*
ε5*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε5*
ε5*
ε2*
ε2*
ε6*
ε6*
ε*
ε*
ε7*
ε7*
-1
-1
7*
7*
*
*
6*
6*
2*
2*
5*
5*
3*
3*
4*
4*
4*
4*
3*
3*
5*
5*
2*
2*
6*
6*
*
*
7*
7*


Additional information

Number of symmetry elements h = 30
Number of classes, irreps n = 30
Number of real-valued irreducible representations n = 16
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes


Reduce representation to irreducible representations


E C15 (C15)2 C5 (C15)4 C3 (C5)2 (C15)7 (C15)8 (C5)3 (C3)2 (C15)11 (C5)4 (C15)13 (C15)14 i (S30)17 (S30)19 (S10)7 (S30)23 (S6)5 (S10)9 (S30)29 S30 S10 S6 (S30)7 (S10)3 (S30)11 (S30)13



Genrate representation from irreducible representations


Ag E1g* E2g* E3g* E4g* E5g* E6g* E7g* Au E1u* E2u* E3u* E4u* E5u* E6u* E7u*




Direct products of irreducible representations


Binary products
Ag E1g* E2g* E3g* E4g* E5g* E6g* E7g* Au E1u* E2u* E3u* E4u* E5u* E6u* E7u*
Ag Ag
E1g* E1g2Ag⊕E2g
E2g* E2gE1g⊕E3g2Ag⊕E4g
E3g* E3gE2g⊕E4gE1g⊕E5g2Ag⊕E6g
E4g* E4gE3g⊕E5gE2g⊕E6gE1g⊕E7g2Ag⊕E7g
E5g* E5gE4g⊕E6gE3g⊕E7gE2g⊕E7gE1g⊕E6g2Ag⊕E5g
E6g* E6gE5g⊕E7gE4g⊕E7gE3g⊕E6gE2g⊕E5gE1g⊕E4g2Ag⊕E3g
E7g* E7gE6g⊕E7gE5g⊕E6gE4g⊕E5gE3g⊕E4gE2g⊕E3gE1g⊕E2g2Ag⊕E1g
Au AuE1uE2uE3uE4uE5uE6uE7uAg
E1u* E1u2Au⊕E2uE1u⊕E3uE2u⊕E4uE3u⊕E5uE4u⊕E6uE5u⊕E7uE6u⊕E7uE1g2Ag⊕E2g
E2u* E2uE1u⊕E3u2Au⊕E4uE1u⊕E5uE2u⊕E6uE3u⊕E7uE4u⊕E7uE5u⊕E6uE2gE1g⊕E3g2Ag⊕E4g
E3u* E3uE2u⊕E4uE1u⊕E5u2Au⊕E6uE1u⊕E7uE2u⊕E7uE3u⊕E6uE4u⊕E5uE3gE2g⊕E4gE1g⊕E5g2Ag⊕E6g
E4u* E4uE3u⊕E5uE2u⊕E6uE1u⊕E7u2Au⊕E7uE1u⊕E6uE2u⊕E5uE3u⊕E4uE4gE3g⊕E5gE2g⊕E6gE1g⊕E7g2Ag⊕E7g
E5u* E5uE4u⊕E6uE3u⊕E7uE2u⊕E7uE1u⊕E6u2Au⊕E5uE1u⊕E4uE2u⊕E3uE5gE4g⊕E6gE3g⊕E7gE2g⊕E7gE1g⊕E6g2Ag⊕E5g
E6u* E6uE5u⊕E7uE4u⊕E7uE3u⊕E6uE2u⊕E5uE1u⊕E4u2Au⊕E3uE1u⊕E2uE6gE5g⊕E7gE4g⊕E7gE3g⊕E6gE2g⊕E5gE1g⊕E4g2Ag⊕E3g
E7u* E7uE6u⊕E7uE5u⊕E6uE4u⊕E5uE3u⊕E4uE2u⊕E3uE1u⊕E2u2Au⊕E1uE7gE6g⊕E7gE5g⊕E6gE4g⊕E5gE3g⊕E4gE2g⊕E3gE1g⊕E2g2Ag⊕E1g

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1g* Ag⊕E2gE1g⊕E3gAg⊕E2g⊕E4gE1g⊕E3g⊕E5gAg⊕E2g⊕E4g⊕E6gMore
E2g* Ag⊕E4gE2g⊕E6gAg⊕E4g⊕E7gE2g⊕E5g⊕E6gAg⊕E3g⊕E4g⊕E7gMore
E3g* Ag⊕E6gE3g⊕E6gAg⊕E3g⊕E6g2Ag⊕E3g⊕E6gAg⊕2E3g⊕E6gMore
E4g* Ag⊕E7gE3g⊕E4gAg⊕E1g⊕E7gE3g⊕E4g⊕E5gAg⊕E1g⊕E6g⊕E7gMore
E5g* Ag⊕E5g2Ag⊕E5gAg⊕2E5g2Ag⊕2E5g3Ag⊕2E5gMore
E6g* Ag⊕E3gE3g⊕E6gAg⊕E3g⊕E6g2Ag⊕E3g⊕E6gAg⊕E3g⊕2E6gMore
E7g* Ag⊕E1gE6g⊕E7gAg⊕E1g⊕E2gE5g⊕E6g⊕E7gAg⊕E1g⊕E2g⊕E3gMore
E1u* Ag⊕E2gE1u⊕E3uAg⊕E2g⊕E4gE1u⊕E3u⊕E5uAg⊕E2g⊕E4g⊕E6gMore
E2u* Ag⊕E4gE2u⊕E6uAg⊕E4g⊕E7gE2u⊕E5u⊕E6uAg⊕E3g⊕E4g⊕E7gMore
E3u* Ag⊕E6gE3u⊕E6uAg⊕E3g⊕E6g2Au⊕E3u⊕E6uAg⊕2E3g⊕E6gMore
E4u* Ag⊕E7gE3u⊕E4uAg⊕E1g⊕E7gE3u⊕E4u⊕E5uAg⊕E1g⊕E6g⊕E7gMore
E5u* Ag⊕E5g2Au⊕E5uAg⊕2E5g2Au⊕2E5u3Ag⊕2E5gMore
E6u* Ag⊕E3gE3u⊕E6uAg⊕E3g⊕E6g2Au⊕E3u⊕E6uAg⊕E3g⊕2E6gMore
E7u* Ag⊕E1gE6u⊕E7uAg⊕E1g⊕E2gE5u⊕E6u⊕E7uAg⊕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 Ag 1 Ag
p (l=1) 3 Dipole Au⊕E1u 3 Au⊕E1u
d (l=2) 5 Quadrupole Ag⊕E1g⊕E2g 6 2Ag⊕E1g⊕E2g
f (l=3) 7 Octupole Au⊕E1u⊕E2u⊕E3u 10 2Au⊕2E1u⊕E2u⊕E3u
g (l=4) 9 Hexadecapole Ag⊕E1g⊕E2g⊕E3g⊕E4g 15 3Ag⊕2E1g⊕2E2g⊕E3g⊕E4g
h (l=5) 11 Dotricontapole Au⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u 21 3Au⊕3E1u⊕2E2u⊕2E3u⊕E4u⊕E5u
i (l=6) 13 Tetrahexacontapole Ag⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g 28 4Ag⊕3E1g⊕3E2g⊕2E3g⊕2E4g⊕E5g⊕E6g
j (l=7) 15 Octacosahectapole Au⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕E6u⊕E7u 36 4Au⊕4E1u⊕3E2u⊕3E3u⊕2E4u⊕2E5u⊕E6u⊕E7u
k (l=8) 17 256-pole Ag⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g⊕2E7g 45 5Ag⊕4E1g⊕4E2g⊕3E3g⊕3E4g⊕2E5g⊕2E6g⊕2E7g
l (l=9) 19 512-pole Au⊕E1u⊕E2u⊕E3u⊕E4u⊕E5u⊕2E6u⊕2E7u 55 5Au⊕5E1u⊕4E2u⊕4E3u⊕3E4u⊕3E5u⊕3E6u⊕3E7u
m (l=10) 21 1024-pole Ag⊕E1g⊕E2g⊕E3g⊕E4g⊕2E5g⊕2E6g⊕2E7g 66 6Ag⊕5E1g⊕5E2g⊕4E3g⊕4E4g⊕4E5g⊕4E6g⊕4E7g
n (l=11) 23 2048-pole Au⊕E1u⊕E2u⊕E3u⊕2E4u⊕2E5u⊕2E6u⊕2E7u 78 6Au⊕6E1u⊕5E2u⊕5E3u⊕5E4u⊕5E5u⊕5E6u⊕5E7u
o (l=12) 25 4096-pole Ag⊕E1g⊕E2g⊕2E3g⊕2E4g⊕2E5g⊕2E6g⊕2E7g 91 7Ag⊕6E1g⊕6E2g⊕6E3g⊕6E4g⊕6E5g⊕6E6g⊕6E7g
More

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 S30
L 2L+1 Term Splitting
S (L=0) 1 Ag
P (L=1) 3 Ag⊕E1g
D (L=2) 5 Ag⊕E1g⊕E2g
F (L=3) 7 Ag⊕E1g⊕E2g⊕E3g
G (L=4) 9 Ag⊕E1g⊕E2g⊕E3g⊕E4g
H (L=5) 11 Ag⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g
I (L=6) 13 Ag⊕E1g⊕E2g⊕E3g⊕E4g⊕E5g⊕E6g


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