Point Group S14

S14 E 2C7 2(C7)2 2(C7)3 i 2(S14)5 2(S14)3 2S14
Ag 1 1 1 1 1 1 1 1
E1g* 2 2cos(2π/7) 2cos(4π/7) 2cos(6π/7) 2 2cos(2π/7) 2cos(4π/7) 2cos(6π/7)
E2g* 2 2cos(4π/7) 2cos(6π/7) 2cos(2π/7) 2 2cos(4π/7) 2cos(6π/7) 2cos(2π/7)
E3g* 2 2cos(6π/7) 2cos(2π/7) 2cos(4π/7) 2 2cos(6π/7) 2cos(2π/7) 2cos(4π/7)
Au 1 1 1 1 -1 -1 -1 -1
E1u* 2 2cos(2π/7) 2cos(4π/7) 2cos(6π/7) -2 -2cos(2π/7) -2cos(4π/7) -2cos(6π/7)
E2u* 2 2cos(4π/7) 2cos(6π/7) 2cos(2π/7) -2 -2cos(4π/7) -2cos(6π/7) -2cos(2π/7)
E3u* 2 2cos(6π/7) 2cos(2π/7) 2cos(4π/7) -2 -2cos(6π/7) -2cos(2π/7) -2cos(4π/7)

Additional information

Number of symmetry elements h = 14
Number of classes, irreps n = 14
Number of real-valued irreducible representations n = 8
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity yes

Reduce representation to irreducible representations

E 2C7 2(C7)2 2(C7)3 i 2(S14)5 2(S14)3 2S14

Genrate representation from irreducible representations

Ag E1g* E2g* E3g* Au E1u* E2u* E3u*

Direct products of irreducible representations

Binary products
Ag E1g* E2g* E3g* Au E1u* E2u* E3u*
Ag Ag
E1g* E1g2Ag⊕E2g
E2g* E2gE1g⊕E3g2Ag⊕E3g
E3g* E3gE2g⊕E3gE1g⊕E2g2Ag⊕E1g
Au AuE1uE2uE3uAg
E1u* E1u2Au⊕E2uE1u⊕E3uE2u⊕E3uE1g2Ag⊕E2g
E2u* E2uE1u⊕E3u2Au⊕E3uE1u⊕E2uE2gE1g⊕E3g2Ag⊕E3g
E3u* E3uE2u⊕E3uE1u⊕E2u2Au⊕E1uE3gE2g⊕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⊕E3gE1g⊕E2g⊕E3gAg⊕E1g⊕E2g⊕E3gMore
E2g* Ag⊕E3gE1g⊕E2gAg⊕E1g⊕E3gE1g⊕E2g⊕E3gAg⊕E1g⊕E2g⊕E3gMore
E3g* Ag⊕E1gE2g⊕E3gAg⊕E1g⊕E2gE1g⊕E2g⊕E3gAg⊕E1g⊕E2g⊕E3gMore
E1u* Ag⊕E2gE1u⊕E3uAg⊕E2g⊕E3gE1u⊕E2u⊕E3uAg⊕E1g⊕E2g⊕E3gMore
E2u* Ag⊕E3gE1u⊕E2uAg⊕E1g⊕E3gE1u⊕E2u⊕E3uAg⊕E1g⊕E2g⊕E3gMore
E3u* Ag⊕E1gE2u⊕E3uAg⊕E1g⊕E2gE1u⊕E2u⊕E3uAg⊕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⊕2E3g 15 3Ag⊕2E1g⊕2E2g⊕2E3g
h (l=5) 11 Dotricontapole Au⊕E1u⊕2E2u⊕2E3u 21 3Au⊕3E1u⊕3E2u⊕3E3u
i (l=6) 13 Tetrahexacontapole Ag⊕2E1g⊕2E2g⊕2E3g 28 4Ag⊕4E1g⊕4E2g⊕4E3g
j (l=7) 15 Octacosahectapole 3Au⊕2E1u⊕2E2u⊕2E3u 36 6Au⊕5E1u⊕5E2u⊕5E3u
k (l=8) 17 256-pole 3Ag⊕3E1g⊕2E2g⊕2E3g 45 7Ag⊕7E1g⊕6E2g⊕6E3g
l (l=9) 19 512-pole 3Au⊕3E1u⊕3E2u⊕2E3u 55 9Au⊕8E1u⊕8E2u⊕7E3u
m (l=10) 21 1024-pole 3Ag⊕3E1g⊕3E2g⊕3E3g 66 10Ag⊕10E1g⊕9E2g⊕9E3g
n (l=11) 23 2048-pole 3Au⊕3E1u⊕3E2u⊕4E3u 78 12Au⊕11E1u⊕11E2u⊕11E3u
o (l=12) 25 4096-pole 3Ag⊕3E1g⊕4E2g⊕4E3g 91 13Ag⊕13E1g⊕13E2g⊕13E3g

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 S14
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⊕2E3g
H (L=5) 11 Ag⊕E1g⊕2E2g⊕2E3g
I (L=6) 13 Ag⊕2E1g⊕2E2g⊕2E3g

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