Point Group D16d



D16d E 2S32 2C16 2(S32)3 2C8 2(S32)5 2(C16)3 2(S32)7 2C4 2(S32)9 2(C16)5 2(S32)11 2(C8)3 2(S32)13 2(C16)7 2(S32)15 C2 16C'2 16σd
A1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1 2 2cos(π/16) 2cos(π/8) 2cos(3π/16) 2cos(π/4) 2cos(5π/16) 2cos(3π/8) 2cos(7π/16) 0 -2cos(7π/16) -2cos(3π/8) -2cos(5π/16) -2cos(π/4) -2cos(3π/16) -2cos(π/8) -2cos(π/16) -2 0 0
E2 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) 2 0 0
E3 2 2cos(3π/16) 2cos(3π/8) -2cos(7π/16) -2cos(π/4) -2cos(π/16) -2cos(π/8) -2cos(5π/16) 0 2cos(5π/16) 2cos(π/8) 2cos(π/16) 2cos(π/4) 2cos(7π/16) -2cos(3π/8) -2cos(3π/16) -2 0 0
E4 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) 2 0 0
E5 2 2cos(5π/16) -2cos(3π/8) -2cos(π/16) -2cos(π/4) 2cos(7π/16) 2cos(π/8) 2cos(3π/16) 0 -2cos(3π/16) -2cos(π/8) -2cos(7π/16) 2cos(π/4) 2cos(π/16) 2cos(3π/8) -2cos(5π/16) -2 0 0
E6 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) 2 0 0
E7 2 2cos(7π/16) -2cos(π/8) -2cos(5π/16) 2cos(π/4) 2cos(3π/16) -2cos(3π/8) -2cos(π/16) 0 2cos(π/16) 2cos(3π/8) -2cos(3π/16) -2cos(π/4) 2cos(5π/16) 2cos(π/8) -2cos(7π/16) -2 0 0
E8 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 0
E9 2 -2cos(7π/16) -2cos(π/8) 2cos(5π/16) 2cos(π/4) -2cos(3π/16) -2cos(3π/8) 2cos(π/16) 0 -2cos(π/16) 2cos(3π/8) 2cos(3π/16) -2cos(π/4) -2cos(5π/16) 2cos(π/8) 2cos(7π/16) -2 0 0
E10 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) 2 0 0
E11 2 -2cos(5π/16) -2cos(3π/8) 2cos(π/16) -2cos(π/4) -2cos(7π/16) 2cos(π/8) -2cos(3π/16) 0 2cos(3π/16) -2cos(π/8) 2cos(7π/16) 2cos(π/4) -2cos(π/16) 2cos(3π/8) 2cos(5π/16) -2 0 0
E12 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) 2 0 0
E13 2 -2cos(3π/16) 2cos(3π/8) 2cos(7π/16) -2cos(π/4) 2cos(π/16) -2cos(π/8) 2cos(5π/16) 0 -2cos(5π/16) 2cos(π/8) -2cos(π/16) 2cos(π/4) -2cos(7π/16) -2cos(3π/8) 2cos(3π/16) -2 0 0
E14 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) 2 0 0
E15 2 -2cos(π/16) 2cos(π/8) -2cos(3π/16) 2cos(π/4) -2cos(5π/16) 2cos(3π/8) -2cos(7π/16) 0 2cos(7π/16) -2cos(3π/8) 2cos(5π/16) -2cos(π/4) 2cos(3π/16) -2cos(π/8) 2cos(π/16) -2 0 0


Additional information

Number of symmetry elements h = 64
Number of classes, irreps n = 19
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2S32 2C16 2(S32)3 2C8 2(S32)5 2(C16)3 2(S32)7 2C4 2(S32)9 2(C16)5 2(S32)11 2(C8)3 2(S32)13 2(C16)7 2(S32)15 C2 16C'2 16σd



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15




Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E1 E1E1E15E15A1⊕A2⊕E2
E2 E2E2E14E14E1⊕E3A1⊕A2⊕E4
E3 E3E3E13E13E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E12E12E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕E8
E5 E5E5E11E11E4⊕E6E3⊕E7E2⊕E8E1⊕E9A1⊕A2⊕E10
E6 E6E6E10E10E5⊕E7E4⊕E8E3⊕E9E2⊕E10E1⊕E11A1⊕A2⊕E12
E7 E7E7E9E9E6⊕E8E5⊕E9E4⊕E10E3⊕E11E2⊕E12E1⊕E13A1⊕A2⊕E14
E8 E8E8E8E8E7⊕E9E6⊕E10E5⊕E11E4⊕E12E3⊕E13E2⊕E14E1⊕E15A1⊕A2⊕B1⊕B2
E9 E9E9E7E7E8⊕E10E7⊕E11E6⊕E12E5⊕E13E4⊕E14E3⊕E15B1⊕B2⊕E2E1⊕E15A1⊕A2⊕E14
E10 E10E10E6E6E9⊕E11E8⊕E12E7⊕E13E6⊕E14E5⊕E15B1⊕B2⊕E4E3⊕E15E2⊕E14E1⊕E13A1⊕A2⊕E12
E11 E11E11E5E5E10⊕E12E9⊕E13E8⊕E14E7⊕E15B1⊕B2⊕E6E5⊕E15E4⊕E14E3⊕E13E2⊕E12E1⊕E11A1⊕A2⊕E10
E12 E12E12E4E4E11⊕E13E10⊕E14E9⊕E15B1⊕B2⊕E8E7⊕E15E6⊕E14E5⊕E13E4⊕E12E3⊕E11E2⊕E10E1⊕E9A1⊕A2⊕E8
E13 E13E13E3E3E12⊕E14E11⊕E15B1⊕B2⊕E10E9⊕E15E8⊕E14E7⊕E13E6⊕E12E5⊕E11E4⊕E10E3⊕E9E2⊕E8E1⊕E7A1⊕A2⊕E6
E14 E14E14E2E2E13⊕E15B1⊕B2⊕E12E11⊕E15E10⊕E14E9⊕E13E8⊕E12E7⊕E11E6⊕E10E5⊕E9E4⊕E8E3⊕E7E2⊕E6E1⊕E5A1⊕A2⊕E4
E15 E15E15E1E1B1⊕B2⊕E14E13⊕E15E12⊕E14E11⊕E13E10⊕E12E9⊕E11E8⊕E10E7⊕E9E6⊕E8E5⊕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⊕E4⊕E8E2⊕E6⊕E10A1⊕E4⊕E8⊕E12More
E3 A1⊕E6E3⊕E9A1⊕E6⊕E12E3⊕E9⊕E15A1⊕E6⊕E12⊕E14More
E4 A1⊕E8E4⊕E12A1⊕B1⊕B2⊕E8E4⊕2E12A1⊕B1⊕B2⊕2E8More
E5 A1⊕E10E5⊕E15A1⊕E10⊕E12E5⊕E7⊕E15A1⊕E2⊕E10⊕E12More
E6 A1⊕E12E6⊕E14A1⊕E8⊕E12E2⊕E6⊕E14A1⊕E4⊕E8⊕E12More
E7 A1⊕E14E7⊕E11A1⊕E4⊕E14E3⊕E7⊕E11A1⊕E4⊕E10⊕E14More
E8 A1⊕B1⊕B22E82A1⊕A2⊕B1⊕B23E82A1⊕A2⊕2B1⊕2B2More
E9 A1⊕E14E5⊕E9A1⊕E4⊕E14E5⊕E9⊕E13A1⊕E4⊕E10⊕E14More
E10 A1⊕E12E2⊕E10A1⊕E8⊕E12E2⊕E10⊕E14A1⊕E4⊕E8⊕E12More
E11 A1⊕E10E1⊕E11A1⊕E10⊕E12E1⊕E9⊕E11A1⊕E2⊕E10⊕E12More
E12 A1⊕E8E4⊕E12A1⊕B1⊕B2⊕E82E4⊕E12A1⊕B1⊕B2⊕2E8More
E13 A1⊕E6E7⊕E13A1⊕E6⊕E12E1⊕E7⊕E13A1⊕E6⊕E12⊕E14More
E14 A1⊕E4E10⊕E14A1⊕E4⊕E8E6⊕E10⊕E14A1⊕E4⊕E8⊕E12More
E15 A1⊕E2E13⊕E15A1⊕E2⊕E4E11⊕E13⊕E15A1⊕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 B2⊕E1 3 B2⊕E1
d (l=2) 5 Quadrupole A1⊕E2⊕E15 6 2A1⊕E2⊕E15
f (l=3) 7 Octupole B2⊕E1⊕E3⊕E14 10 2B2⊕2E1⊕E3⊕E14
g (l=4) 9 Hexadecapole A1⊕E2⊕E4⊕E13⊕E15 15 3A1⊕2E2⊕E4⊕E13⊕2E15
h (l=5) 11 Dotricontapole B2⊕E1⊕E3⊕E5⊕E12⊕E14 21 3B2⊕3E1⊕2E3⊕E5⊕E12⊕2E14
i (l=6) 13 Tetrahexacontapole A1⊕E2⊕E4⊕E6⊕E11⊕E13⊕E15 28 4A1⊕3E2⊕2E4⊕E6⊕E11⊕2E13⊕3E15
j (l=7) 15 Octacosahectapole B2⊕E1⊕E3⊕E5⊕E7⊕E10⊕E12⊕E14 36 4B2⊕4E1⊕3E3⊕2E5⊕E7⊕E10⊕2E12⊕3E14
k (l=8) 17 256-pole A1⊕E2⊕E4⊕E6⊕E8⊕E9⊕E11⊕E13⊕E15 45 5A1⊕4E2⊕3E4⊕2E6⊕E8⊕E9⊕2E11⊕3E13⊕4E15
l (l=9) 19 512-pole B2⊕E1⊕E3⊕E5⊕E7⊕E8⊕E9⊕E10⊕E12⊕E14 55 5B2⊕5E1⊕4E3⊕3E5⊕2E7⊕E8⊕E9⊕2E10⊕3E12⊕4E14
m (l=10) 21 1024-pole A1⊕E2⊕E4⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E13⊕E15 66 6A1⊕5E2⊕4E4⊕3E6⊕E7⊕2E8⊕2E9⊕E10⊕3E11⊕4E13⊕5E15
n (l=11) 23 2048-pole B2⊕E1⊕E3⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12⊕E14 78 6B2⊕6E1⊕5E3⊕4E5⊕E6⊕3E7⊕2E8⊕2E9⊕3E10⊕E11⊕4E12⊕5E14
o (l=12) 25 4096-pole A1⊕E2⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9⊕E10⊕E11⊕E12⊕E13⊕E15 91 7A1⊕6E2⊕5E4⊕E5⊕4E6⊕2E7⊕3E8⊕3E9⊕2E10⊕4E11⊕E12⊕5E13⊕6E15
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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 D16d
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E15
D (L=2) 5 A1⊕E2⊕E15
F (L=3) 7 A2⊕E2⊕E13⊕E15
G (L=4) 9 A1⊕E2⊕E4⊕E13⊕E15
H (L=5) 11 A2⊕E2⊕E4⊕E11⊕E13⊕E15
I (L=6) 13 A1⊕E2⊕E4⊕E6⊕E11⊕E13⊕E15


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