Point Group D15h



D15h E 2C15 2(C15)2 2C5 2(C15)4 2C3 2(C5)2 2(C15)7 15C'2 σh 2S15 2(S15)13 2S5 2(S15)11 2S3 2(S5)3 2(S15)7 15σv
A'1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A'2 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 -1
E'1 2 2cos(2π/15) 2cos(4π/15) 2cos(2π/5) 2cos(8π/15) -1 2cos(4π/5) 2cos(14π/15) 0 2 2cos(2π/15) 2cos(4π/15) 2cos(2π/5) 2cos(8π/15) -1 2cos(4π/5) 2cos(14π/15) 0
E'2 2 2cos(4π/15) 2cos(8π/15) 2cos(4π/5) 2cos(14π/15) -1 2cos(2π/5) 2cos(2π/15) 0 2 2cos(4π/15) 2cos(8π/15) 2cos(4π/5) 2cos(14π/15) -1 2cos(2π/5) 2cos(2π/15) 0
E'3 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) 2 2cos(2π/5) 2cos(4π/5) 0 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) 2 2cos(2π/5) 2cos(4π/5) 0
E'4 2 2cos(8π/15) 2cos(14π/15) 2cos(2π/5) 2cos(2π/15) -1 2cos(4π/5) 2cos(4π/15) 0 2 2cos(8π/15) 2cos(14π/15) 2cos(2π/5) 2cos(2π/15) -1 2cos(4π/5) 2cos(4π/15) 0
E'5 2 -1 -1 2 -1 -1 2 -1 0 2 -1 -1 2 -1 -1 2 -1 0
E'6 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) 2 2cos(4π/5) 2cos(2π/5) 0 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) 2 2cos(4π/5) 2cos(2π/5) 0
E'7 2 2cos(14π/15) 2cos(2π/15) 2cos(4π/5) 2cos(4π/15) -1 2cos(2π/5) 2cos(8π/15) 0 2 2cos(14π/15) 2cos(2π/15) 2cos(4π/5) 2cos(4π/15) -1 2cos(2π/5) 2cos(8π/15) 0
A''1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1
A''2 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1
E''1 2 2cos(2π/15) 2cos(4π/15) 2cos(2π/5) 2cos(8π/15) -1 2cos(4π/5) 2cos(14π/15) 0 -2 -2cos(2π/15) -2cos(4π/15) -2cos(2π/5) -2cos(8π/15) 1 -2cos(4π/5) -2cos(14π/15) 0
E''2 2 2cos(4π/15) 2cos(8π/15) 2cos(4π/5) 2cos(14π/15) -1 2cos(2π/5) 2cos(2π/15) 0 -2 -2cos(4π/15) -2cos(8π/15) -2cos(4π/5) -2cos(14π/15) 1 -2cos(2π/5) -2cos(2π/15) 0
E''3 2 2cos(2π/5) 2cos(4π/5) 2cos(4π/5) 2cos(2π/5) 2 2cos(2π/5) 2cos(4π/5) 0 -2 -2cos(2π/5) -2cos(4π/5) -2cos(4π/5) -2cos(2π/5) -2 -2cos(2π/5) -2cos(4π/5) 0
E''4 2 2cos(8π/15) 2cos(14π/15) 2cos(2π/5) 2cos(2π/15) -1 2cos(4π/5) 2cos(4π/15) 0 -2 -2cos(8π/15) -2cos(14π/15) -2cos(2π/5) -2cos(2π/15) 1 -2cos(4π/5) -2cos(4π/15) 0
E''5 2 -1 -1 2 -1 -1 2 -1 0 -2 1 1 -2 1 1 -2 1 0
E''6 2 2cos(4π/5) 2cos(2π/5) 2cos(2π/5) 2cos(4π/5) 2 2cos(4π/5) 2cos(2π/5) 0 -2 -2cos(4π/5) -2cos(2π/5) -2cos(2π/5) -2cos(4π/5) -2 -2cos(4π/5) -2cos(2π/5) 0
E''7 2 2cos(14π/15) 2cos(2π/15) 2cos(4π/5) 2cos(4π/15) -1 2cos(2π/5) 2cos(8π/15) 0 -2 -2cos(14π/15) -2cos(2π/15) -2cos(4π/5) -2cos(4π/15) 1 -2cos(2π/5) -2cos(8π/15) 0


Additional information

Number of symmetry elements h = 60
Number of classes, irreps n = 18
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2C15 2(C15)2 2C5 2(C15)4 2C3 2(C5)2 2(C15)7 15C'2 σh 2S15 2(S15)13 2S5 2(S15)11 2S3 2(S5)3 2(S15)7 15σv



Genrate representation from irreducible representations


A'1 A'2 E'1 E'2 E'3 E'4 E'5 E'6 E'7 A''1 A''2 E''1 E''2 E''3 E''4 E''5 E''6 E''7




Direct products of irreducible representations


Binary products
A'1 A'2 E'1 E'2 E'3 E'4 E'5 E'6 E'7 A''1 A''2 E''1 E''2 E''3 E''4 E''5 E''6 E''7
A'1 A'1
A'2 A'2A'1
E'1 E'1E'1A'1⊕A'2⊕E'2
E'2 E'2E'2E'1⊕E'3A'1⊕A'2⊕E'4
E'3 E'3E'3E'2⊕E'4E'1⊕E'5A'1⊕A'2⊕E'6
E'4 E'4E'4E'3⊕E'5E'2⊕E'6E'1⊕E'7A'1⊕A'2⊕E'7
E'5 E'5E'5E'4⊕E'6E'3⊕E'7E'2⊕E'7E'1⊕E'6A'1⊕A'2⊕E'5
E'6 E'6E'6E'5⊕E'7E'4⊕E'7E'3⊕E'6E'2⊕E'5E'1⊕E'4A'1⊕A'2⊕E'3
E'7 E'7E'7E'6⊕E'7E'5⊕E'6E'4⊕E'5E'3⊕E'4E'2⊕E'3E'1⊕E'2A'1⊕A'2⊕E'1
A''1 A''1A''2E''1E''2E''3E''4E''5E''6E''7A'1
A''2 A''2A''1E''1E''2E''3E''4E''5E''6E''7A'2A'1
E''1 E''1E''1A''1⊕A''2⊕E''2E''1⊕E''3E''2⊕E''4E''3⊕E''5E''4⊕E''6E''5⊕E''7E''6⊕E''7E'1E'1A'1⊕A'2⊕E'2
E''2 E''2E''2E''1⊕E''3A''1⊕A''2⊕E''4E''1⊕E''5E''2⊕E''6E''3⊕E''7E''4⊕E''7E''5⊕E''6E'2E'2E'1⊕E'3A'1⊕A'2⊕E'4
E''3 E''3E''3E''2⊕E''4E''1⊕E''5A''1⊕A''2⊕E''6E''1⊕E''7E''2⊕E''7E''3⊕E''6E''4⊕E''5E'3E'3E'2⊕E'4E'1⊕E'5A'1⊕A'2⊕E'6
E''4 E''4E''4E''3⊕E''5E''2⊕E''6E''1⊕E''7A''1⊕A''2⊕E''7E''1⊕E''6E''2⊕E''5E''3⊕E''4E'4E'4E'3⊕E'5E'2⊕E'6E'1⊕E'7A'1⊕A'2⊕E'7
E''5 E''5E''5E''4⊕E''6E''3⊕E''7E''2⊕E''7E''1⊕E''6A''1⊕A''2⊕E''5E''1⊕E''4E''2⊕E''3E'5E'5E'4⊕E'6E'3⊕E'7E'2⊕E'7E'1⊕E'6A'1⊕A'2⊕E'5
E''6 E''6E''6E''5⊕E''7E''4⊕E''7E''3⊕E''6E''2⊕E''5E''1⊕E''4A''1⊕A''2⊕E''3E''1⊕E''2E'6E'6E'5⊕E'7E'4⊕E'7E'3⊕E'6E'2⊕E'5E'1⊕E'4A'1⊕A'2⊕E'3
E''7 E''7E''7E''6⊕E''7E''5⊕E''6E''4⊕E''5E''3⊕E''4E''2⊕E''3E''1⊕E''2A''1⊕A''2⊕E''1E'7E'7E'6⊕E'7E'5⊕E'6E'4⊕E'5E'3⊕E'4E'2⊕E'3E'1⊕E'2A'1⊕A'2⊕E'1

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E'1 A'1⊕E'2E'1⊕E'3A'1⊕E'2⊕E'4E'1⊕E'3⊕E'5A'1⊕E'2⊕E'4⊕E'6More
E'2 A'1⊕E'4E'2⊕E'6A'1⊕E'4⊕E'7E'2⊕E'5⊕E'6A'1⊕E'3⊕E'4⊕E'7More
E'3 A'1⊕E'6E'3⊕E'6A'1⊕E'3⊕E'6A'1⊕A'2⊕E'3⊕E'6A'1⊕2E'3⊕E'6More
E'4 A'1⊕E'7E'3⊕E'4A'1⊕E'1⊕E'7E'3⊕E'4⊕E'5A'1⊕E'1⊕E'6⊕E'7More
E'5 A'1⊕E'5A'1⊕A'2⊕E'5A'1⊕2E'5A'1⊕A'2⊕2E'52A'1⊕A'2⊕2E'5More
E'6 A'1⊕E'3E'3⊕E'6A'1⊕E'3⊕E'6A'1⊕A'2⊕E'3⊕E'6A'1⊕E'3⊕2E'6More
E'7 A'1⊕E'1E'6⊕E'7A'1⊕E'1⊕E'2E'5⊕E'6⊕E'7A'1⊕E'1⊕E'2⊕E'3More
E''1 A'1⊕E'2E''1⊕E''3A'1⊕E'2⊕E'4E''1⊕E''3⊕E''5A'1⊕E'2⊕E'4⊕E'6More
E''2 A'1⊕E'4E''2⊕E''6A'1⊕E'4⊕E'7E''2⊕E''5⊕E''6A'1⊕E'3⊕E'4⊕E'7More
E''3 A'1⊕E'6E''3⊕E''6A'1⊕E'3⊕E'6A''1⊕A''2⊕E''3⊕E''6A'1⊕2E'3⊕E'6More
E''4 A'1⊕E'7E''3⊕E''4A'1⊕E'1⊕E'7E''3⊕E''4⊕E''5A'1⊕E'1⊕E'6⊕E'7More
E''5 A'1⊕E'5A''1⊕A''2⊕E''5A'1⊕2E'5A''1⊕A''2⊕2E''52A'1⊕A'2⊕2E'5More
E''6 A'1⊕E'3E''3⊕E''6A'1⊕E'3⊕E'6A''1⊕A''2⊕E''3⊕E''6A'1⊕E'3⊕2E'6More
E''7 A'1⊕E'1E''6⊕E''7A'1⊕E'1⊕E'2E''5⊕E''6⊕E''7A'1⊕E'1⊕E'2⊕E'3More



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 1 A'1
p (l=1) 3 Dipole E'1⊕A''2 3 E'1⊕A''2
d (l=2) 5 Quadrupole A'1⊕E'2⊕E''1 6 2A'1⊕E'2⊕E''1
f (l=3) 7 Octupole E'1⊕E'3⊕A''2⊕E''2 10 2E'1⊕E'3⊕2A''2⊕E''2
g (l=4) 9 Hexadecapole A'1⊕E'2⊕E'4⊕E''1⊕E''3 15 3A'1⊕2E'2⊕E'4⊕2E''1⊕E''3
h (l=5) 11 Dotricontapole E'1⊕E'3⊕E'5⊕A''2⊕E''2⊕E''4 21 3E'1⊕2E'3⊕E'5⊕3A''2⊕2E''2⊕E''4
i (l=6) 13 Tetrahexacontapole A'1⊕E'2⊕E'4⊕E'6⊕E''1⊕E''3⊕E''5 28 4A'1⊕3E'2⊕2E'4⊕E'6⊕3E''1⊕2E''3⊕E''5
j (l=7) 15 Octacosahectapole E'1⊕E'3⊕E'5⊕E'7⊕A''2⊕E''2⊕E''4⊕E''6 36 4E'1⊕3E'3⊕2E'5⊕E'7⊕4A''2⊕3E''2⊕2E''4⊕E''6
k (l=8) 17 256-pole A'1⊕E'2⊕E'4⊕E'6⊕E'7⊕E''1⊕E''3⊕E''5⊕E''7 45 5A'1⊕4E'2⊕3E'4⊕2E'6⊕E'7⊕4E''1⊕3E''3⊕2E''5⊕E''7
l (l=9) 19 512-pole E'1⊕E'3⊕E'5⊕E'6⊕E'7⊕A''2⊕E''2⊕E''4⊕E''6⊕E''7 55 5E'1⊕4E'3⊕3E'5⊕E'6⊕2E'7⊕5A''2⊕4E''2⊕3E''4⊕2E''6⊕E''7
m (l=10) 21 1024-pole A'1⊕E'2⊕E'4⊕E'5⊕E'6⊕E'7⊕E''1⊕E''3⊕E''5⊕E''6⊕E''7 66 6A'1⊕5E'2⊕4E'4⊕E'5⊕3E'6⊕2E'7⊕5E''1⊕4E''3⊕3E''5⊕E''6⊕2E''7
n (l=11) 23 2048-pole E'1⊕E'3⊕E'4⊕E'5⊕E'6⊕E'7⊕A''2⊕E''2⊕E''4⊕E''5⊕E''6⊕E''7 78 6E'1⊕5E'3⊕E'4⊕4E'5⊕2E'6⊕3E'7⊕6A''2⊕5E''2⊕4E''4⊕E''5⊕3E''6⊕2E''7
o (l=12) 25 4096-pole A'1⊕E'2⊕E'3⊕E'4⊕E'5⊕E'6⊕E'7⊕E''1⊕E''3⊕E''4⊕E''5⊕E''6⊕E''7 91 7A'1⊕6E'2⊕E'3⊕5E'4⊕2E'5⊕4E'6⊕3E'7⊕6E''1⊕5E''3⊕E''4⊕4E''5⊕2E''6⊕3E''7
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 D15h
L 2L+1 Term Splitting
S (L=0) 1 A'1
P (L=1) 3 A'2⊕E''1
D (L=2) 5 A'1⊕E'2⊕E''1
F (L=3) 7 A'2⊕E'2⊕E''1⊕E''3
G (L=4) 9 A'1⊕E'2⊕E'4⊕E''1⊕E''3
H (L=5) 11 A'2⊕E'2⊕E'4⊕E''1⊕E''3⊕E''5
I (L=6) 13 A'1⊕E'2⊕E'4⊕E'6⊕E''1⊕E''3⊕E''5


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