Point Group C16v



C16v E 2C16 2C8 2(C16)3 2C4 2(C16)5 2(C8)3 2(C16)7 C2 v d
A1 1 1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 1 -1
B2 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 0 0
E2 2 2cos(π/4) 0 -2cos(π/4) -2 -2cos(π/4) 0 2cos(π/4) 2 0 0
E3 2 2cos(3π/8) -2cos(π/4) -2cos(π/8) 0 2cos(π/8) 2cos(π/4) -2cos(3π/8) -2 0 0
E4 2 0 -2 0 2 0 -2 0 2 0 0
E5 2 -2cos(3π/8) -2cos(π/4) 2cos(π/8) 0 -2cos(π/8) 2cos(π/4) 2cos(3π/8) -2 0 0
E6 2 -2cos(π/4) 0 2cos(π/4) -2 2cos(π/4) 0 -2cos(π/4) 2 0 0
E7 2 -2cos(π/8) 2cos(π/4) -2cos(3π/8) 0 2cos(3π/8) -2cos(π/4) 2cos(π/8) -2 0 0


Additional information

Number of symmetry elements h = 32
Number of classes, irreps n = 11
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C16 2C8 2(C16)3 2C4 2(C16)5 2(C8)3 2(C16)7 C2 v d



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7




Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E7E7A1⊕A2⊕E2
E2 E2E2E6E6E1⊕E3A1⊕A2⊕E4
E3 E3E3E5E5E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E4E4E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕B1⊕B2
E5 E5E5E3E3E4⊕E6E3⊕E7B1⊕B2⊕E2E1⊕E7A1⊕A2⊕E6
E6 E6E6E2E2E5⊕E7B1⊕B2⊕E4E3⊕E7E2⊕E6E1⊕E5A1⊕A2⊕E4
E7 E7E7E1E1B1⊕B2⊕E6E5⊕E7E4⊕E6E3⊕E5E2⊕E4E1⊕E3A1⊕A2⊕E2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E1 A1⊕E2E1⊕E3A1⊕E2⊕E4E1⊕E3⊕E5A1⊕E2⊕E4⊕E6More
E2 A1⊕E4E2⊕E6A1⊕B1⊕B2⊕E4E2⊕2E6A1⊕B1⊕B2⊕2E4More
E3 A1⊕E6E3⊕E7A1⊕E4⊕E6E1⊕E3⊕E7A1⊕E2⊕E4⊕E6More
E4 A1⊕B1⊕B22E42A1⊕A2⊕B1⊕B23E42A1⊕A2⊕2B1⊕2B2More
E5 A1⊕E6E1⊕E5A1⊕E4⊕E6E1⊕E5⊕E7A1⊕E2⊕E4⊕E6More
E6 A1⊕E4E2⊕E6A1⊕B1⊕B2⊕E42E2⊕E6A1⊕B1⊕B2⊕2E4More
E7 A1⊕E2E5⊕E7A1⊕E2⊕E4E3⊕E5⊕E7A1⊕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 A1 1 A1
p (l=1) 3 Dipole A1⊕E1 3 A1⊕E1
d (l=2) 5 Quadrupole A1⊕E1⊕E2 6 2A1⊕E1⊕E2
f (l=3) 7 Octupole A1⊕E1⊕E2⊕E3 10 2A1⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A1⊕E1⊕E2⊕E3⊕E4 15 3A1⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A1⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A1⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6 28 4A1⊕3E1⊕3E2⊕2E3⊕2E4⊕E5⊕E6
j (l=7) 15 Octacosahectapole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4A1⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 45 5A1⊕B1⊕B2⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7
l (l=9) 19 512-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕2E7 55 5A1⊕B1⊕B2⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕3E7
m (l=10) 21 1024-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕E5⊕2E6⊕2E7 66 6A1⊕2B1⊕2B2⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕4E6⊕3E7
n (l=11) 23 2048-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕E4⊕2E5⊕2E6⊕2E7 78 6A1⊕2B1⊕2B2⊕6E1⊕5E2⊕5E3⊕4E4⊕5E5⊕4E6⊕5E7
o (l=12) 25 4096-pole A1⊕B1⊕B2⊕E1⊕E2⊕E3⊕2E4⊕2E5⊕2E6⊕2E7 91 7A1⊕3B1⊕3B2⊕6E1⊕6E2⊕5E3⊕6E4⊕5E5⊕6E6⊕5E7
More

First nonvanshing multipole: Dipole

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 C16v
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E1
D (L=2) 5 A1⊕E1⊕E2
F (L=3) 7 A2⊕E1⊕E2⊕E3
G (L=4) 9 A1⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A2⊕E1⊕E2⊕E3⊕E4⊕E5
I (L=6) 13 A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6


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