Point Group S16



S16 E S16 C8 (S16)3 C4 (S16)5 (C8)3 (S16)7 C2 (S16)9 (C8)5 (S16)11 (C4)3 (S16)13 (C8)7 (S16)15
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1* 2 2cos(π/8) 2cos(π/4) 2cos(3π/8) 0 -2cos(3π/8) -2cos(π/4) -2cos(π/8) -2 -2cos(π/8) -2cos(π/4) -2cos(3π/8) 0 2cos(3π/8) 2cos(π/4) 2cos(π/8)
E2* 2 2cos(π/4) 0 -2cos(π/4) -2 -2cos(π/4) 0 2cos(π/4) 2 2cos(π/4) 0 -2cos(π/4) -2 -2cos(π/4) 0 2cos(π/4)
E3* 2 2cos(3π/8) -2cos(π/4) -2cos(π/8) 0 2cos(π/8) 2cos(π/4) -2cos(3π/8) -2 -2cos(3π/8) 2cos(π/4) 2cos(π/8) 0 -2cos(π/8) -2cos(π/4) 2cos(3π/8)
E4* 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0
E5* 2 -2cos(3π/8) -2cos(π/4) 2cos(π/8) 0 -2cos(π/8) 2cos(π/4) 2cos(3π/8) -2 2cos(3π/8) 2cos(π/4) -2cos(π/8) 0 2cos(π/8) -2cos(π/4) -2cos(3π/8)
E6* 2 -2cos(π/4) 0 2cos(π/4) -2 2cos(π/4) 0 -2cos(π/4) 2 -2cos(π/4) 0 2cos(π/4) -2 2cos(π/4) 0 -2cos(π/4)
E7* 2 -2cos(π/8) 2cos(π/4) -2cos(3π/8) 0 2cos(3π/8) -2cos(π/4) 2cos(π/8) -2 2cos(π/8) -2cos(π/4) 2cos(3π/8) 0 -2cos(3π/8) 2cos(π/4) -2cos(π/8)


Additional information

Number of symmetry elements h = 16
Number of classes, irreps n = 16
Number of real-valued irreducible representations n = 9
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E S16 C8 (S16)3 C4 (S16)5 (C8)3 (S16)7 C2 (S16)9 (C8)5 (S16)11 (C4)3 (S16)13 (C8)7 (S16)15



Genrate representation from irreducible representations


A B E1* E2* E3* E4* E5* E6* E7*




Direct products of irreducible representations


Binary products
A B E1* E2* E3* E4* E5* E6* E7*
A A
B BA
E1* E1E72A⊕E2
E2* E2E6E1⊕E32A⊕E4
E3* E3E5E2⊕E4E1⊕E52A⊕E6
E4* E4E4E3⊕E5E2⊕E6E1⊕E72A⊕2B
E5* E5E3E4⊕E6E3⊕E72B⊕E2E1⊕E72A⊕E6
E6* E6E2E5⊕E72B⊕E4E3⊕E7E2⊕E6E1⊕E52A⊕E4
E7* E7E12B⊕E6E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕E32A⊕E2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1* A⊕E2E1⊕E3A⊕E2⊕E4E1⊕E3⊕E5A⊕E2⊕E4⊕E6More
E2* A⊕E4E2⊕E6A⊕2B⊕E4E2⊕2E6A⊕2B⊕2E4More
E3* A⊕E6E3⊕E7A⊕E4⊕E6E1⊕E3⊕E7A⊕E2⊕E4⊕E6More
E4* A⊕2B2E43A⊕2B3E43A⊕4BMore
E5* A⊕E6E1⊕E5A⊕E4⊕E6E1⊕E5⊕E7A⊕E2⊕E4⊕E6More
E6* A⊕E4E2⊕E6A⊕2B⊕E42E2⊕E6A⊕2B⊕2E4More
E7* A⊕E2E5⊕E7A⊕E2⊕E4E3⊕E5⊕E7A⊕E2⊕E4⊕E6More



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⊕E1 3 B⊕E1
d (l=2) 5 Quadrupole A⊕E2⊕E7 6 2A⊕E2⊕E7
f (l=3) 7 Octupole B⊕E1⊕E3⊕E6 10 2B⊕2E1⊕E3⊕E6
g (l=4) 9 Hexadecapole A⊕E2⊕E4⊕E5⊕E7 15 3A⊕2E2⊕E4⊕E5⊕2E7
h (l=5) 11 Dotricontapole B⊕E1⊕E3⊕E4⊕E5⊕E6 21 3B⊕3E1⊕2E3⊕E4⊕E5⊕2E6
i (l=6) 13 Tetrahexacontapole A⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 28 4A⊕3E2⊕E3⊕2E4⊕2E5⊕E6⊕3E7
j (l=7) 15 Octacosahectapole B⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4B⊕4E1⊕E2⊕3E3⊕2E4⊕2E5⊕3E6⊕E7
k (l=8) 17 256-pole A⊕2B⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 45 5A⊕2B⊕E1⊕4E2⊕2E3⊕3E4⊕3E5⊕2E6⊕4E7
l (l=9) 19 512-pole 2A⊕B⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕2E7 55 2A⊕5B⊕5E1⊕2E2⊕4E3⊕3E4⊕3E5⊕4E6⊕3E7
m (l=10) 21 1024-pole A⊕2B⊕2E1⊕E2⊕E3⊕E4⊕E5⊕2E6⊕E7 66 6A⊕4B⊕3E1⊕5E2⊕3E3⊕4E4⊕4E5⊕4E6⊕5E7
n (l=11) 23 2048-pole 2A⊕B⊕E1⊕2E2⊕E3⊕E4⊕2E5⊕E6⊕2E7 78 4A⊕6B⊕6E1⊕4E2⊕5E3⊕4E4⊕5E5⊕5E6⊕5E7
o (l=12) 25 4096-pole A⊕2B⊕2E1⊕E2⊕2E3⊕2E4⊕E5⊕2E6⊕E7 91 7A⊕6B⊕5E1⊕6E2⊕5E3⊕6E4⊕5E5⊕6E6⊕6E7
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 S16
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E7
D (L=2) 5 A⊕E2⊕E7
F (L=3) 7 A⊕E2⊕E5⊕E7
G (L=4) 9 A⊕E2⊕E4⊕E5⊕E7
H (L=5) 11 A⊕E2⊕E3⊕E4⊕E5⊕E7
I (L=6) 13 A⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7


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