Point Group S4



S4 E S4 C2 (S4)3
A 1 1 1 1
B 1 -1 1 -1
E* 1
1
i
-i
-1
-1
-i
i


Additional information

Number of symmetry elements h = 4
Number of classes, irreps n = 4
Number of real-valued irreducible representations n = 3
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E S4 C2 (S4)3



Genrate representation from irreducible representations


A B E*




Examples

1,3,5,7-Tetrachlorocyclooctatetraene Tetrachloroneopentane Tetraphenylmethane



Direct products of irreducible representations


Binary products
A B E*
A A
B BA
E* EE2A⊕2B

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E* A⊕2B2E3A⊕2B3E3A⊕4BMore



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 A 1 A
p (l=1) 3 Dipole B⊕E 3 B⊕E
d (l=2) 5 Quadrupole A⊕2B⊕E 6 2A⊕2B⊕E
f (l=3) 7 Octupole 2A⊕B⊕2E 10 2A⊕2B⊕3E
g (l=4) 9 Hexadecapole 3A⊕2B⊕2E 15 5A⊕4B⊕3E
h (l=5) 11 Dotricontapole 2A⊕3B⊕3E 21 4A⊕5B⊕6E
i (l=6) 13 Tetrahexacontapole 3A⊕4B⊕3E 28 8A⊕8B⊕6E
j (l=7) 15 Octacosahectapole 4A⊕3B⊕4E 36 8A⊕8B⊕10E
k (l=8) 17 256-pole 5A⊕4B⊕4E 45 13A⊕12B⊕10E
l (l=9) 19 512-pole 4A⊕5B⊕5E 55 12A⊕13B⊕15E
m (l=10) 21 1024-pole 5A⊕6B⊕5E 66 18A⊕18B⊕15E
n (l=11) 23 2048-pole 6A⊕5B⊕6E 78 18A⊕18B⊕21E
o (l=12) 25 4096-pole 7A⊕6B⊕6E 91 25A⊕24B⊕21E
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 S4
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E
D (L=2) 5 A⊕2B⊕E
F (L=3) 7 A⊕2B⊕2E
G (L=4) 9 3A⊕2B⊕2E
H (L=5) 11 3A⊕2B⊕3E
I (L=6) 13 3A⊕4B⊕3E


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