Point Group D19h



D19h E 2C19 2(C19)2 2(C19)3 2(C19)4 2(C19)5 2(C19)6 2(C19)7 2(C19)8 2(C19)9 19C'2 σh 2S19 2(S19)17 2(S19)3 2(S19)15 2(S19)5 2(S19)13 2(S19)7 2(S19)11 2(S19)9 19σv
A'1 1 1 1 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 1 1 1 -1
E'1 2 2cos(2π/19) 2cos(4π/19) 2cos(6π/19) 2cos(8π/19) 2cos(10π/19) 2cos(12π/19) 2cos(14π/19) 2cos(16π/19) 2cos(18π/19) 0 2 2cos(2π/19) 2cos(4π/19) 2cos(6π/19) 2cos(8π/19) 2cos(10π/19) 2cos(12π/19) 2cos(14π/19) 2cos(16π/19) 2cos(18π/19) 0
E'2 2 2cos(4π/19) 2cos(8π/19) 2cos(12π/19) 2cos(16π/19) 2cos(18π/19) 2cos(14π/19) 2cos(10π/19) 2cos(6π/19) 2cos(2π/19) 0 2 2cos(4π/19) 2cos(8π/19) 2cos(12π/19) 2cos(16π/19) 2cos(18π/19) 2cos(14π/19) 2cos(10π/19) 2cos(6π/19) 2cos(2π/19) 0
E'3 2 2cos(6π/19) 2cos(12π/19) 2cos(18π/19) 2cos(14π/19) 2cos(8π/19) 2cos(2π/19) 2cos(4π/19) 2cos(10π/19) 2cos(16π/19) 0 2 2cos(6π/19) 2cos(12π/19) 2cos(18π/19) 2cos(14π/19) 2cos(8π/19) 2cos(2π/19) 2cos(4π/19) 2cos(10π/19) 2cos(16π/19) 0
E'4 2 2cos(8π/19) 2cos(16π/19) 2cos(14π/19) 2cos(6π/19) 2cos(2π/19) 2cos(10π/19) 2cos(18π/19) 2cos(12π/19) 2cos(4π/19) 0 2 2cos(8π/19) 2cos(16π/19) 2cos(14π/19) 2cos(6π/19) 2cos(2π/19) 2cos(10π/19) 2cos(18π/19) 2cos(12π/19) 2cos(4π/19) 0
E'5 2 2cos(10π/19) 2cos(18π/19) 2cos(8π/19) 2cos(2π/19) 2cos(12π/19) 2cos(16π/19) 2cos(6π/19) 2cos(4π/19) 2cos(14π/19) 0 2 2cos(10π/19) 2cos(18π/19) 2cos(8π/19) 2cos(2π/19) 2cos(12π/19) 2cos(16π/19) 2cos(6π/19) 2cos(4π/19) 2cos(14π/19) 0
E'6 2 2cos(12π/19) 2cos(14π/19) 2cos(2π/19) 2cos(10π/19) 2cos(16π/19) 2cos(4π/19) 2cos(8π/19) 2cos(18π/19) 2cos(6π/19) 0 2 2cos(12π/19) 2cos(14π/19) 2cos(2π/19) 2cos(10π/19) 2cos(16π/19) 2cos(4π/19) 2cos(8π/19) 2cos(18π/19) 2cos(6π/19) 0
E'7 2 2cos(14π/19) 2cos(10π/19) 2cos(4π/19) 2cos(18π/19) 2cos(6π/19) 2cos(8π/19) 2cos(16π/19) 2cos(2π/19) 2cos(12π/19) 0 2 2cos(14π/19) 2cos(10π/19) 2cos(4π/19) 2cos(18π/19) 2cos(6π/19) 2cos(8π/19) 2cos(16π/19) 2cos(2π/19) 2cos(12π/19) 0
E'8 2 2cos(16π/19) 2cos(6π/19) 2cos(10π/19) 2cos(12π/19) 2cos(4π/19) 2cos(18π/19) 2cos(2π/19) 2cos(14π/19) 2cos(8π/19) 0 2 2cos(16π/19) 2cos(6π/19) 2cos(10π/19) 2cos(12π/19) 2cos(4π/19) 2cos(18π/19) 2cos(2π/19) 2cos(14π/19) 2cos(8π/19) 0
E'9 2 2cos(18π/19) 2cos(2π/19) 2cos(16π/19) 2cos(4π/19) 2cos(14π/19) 2cos(6π/19) 2cos(12π/19) 2cos(8π/19) 2cos(10π/19) 0 2 2cos(18π/19) 2cos(2π/19) 2cos(16π/19) 2cos(4π/19) 2cos(14π/19) 2cos(6π/19) 2cos(12π/19) 2cos(8π/19) 2cos(10π/19) 0
A''1 1 1 1 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 -1 -1 -1 1
E''1 2 2cos(2π/19) 2cos(4π/19) 2cos(6π/19) 2cos(8π/19) 2cos(10π/19) 2cos(12π/19) 2cos(14π/19) 2cos(16π/19) 2cos(18π/19) 0 -2 -2cos(2π/19) -2cos(4π/19) -2cos(6π/19) -2cos(8π/19) -2cos(10π/19) -2cos(12π/19) -2cos(14π/19) -2cos(16π/19) -2cos(18π/19) 0
E''2 2 2cos(4π/19) 2cos(8π/19) 2cos(12π/19) 2cos(16π/19) 2cos(18π/19) 2cos(14π/19) 2cos(10π/19) 2cos(6π/19) 2cos(2π/19) 0 -2 -2cos(4π/19) -2cos(8π/19) -2cos(12π/19) -2cos(16π/19) -2cos(18π/19) -2cos(14π/19) -2cos(10π/19) -2cos(6π/19) -2cos(2π/19) 0
E''3 2 2cos(6π/19) 2cos(12π/19) 2cos(18π/19) 2cos(14π/19) 2cos(8π/19) 2cos(2π/19) 2cos(4π/19) 2cos(10π/19) 2cos(16π/19) 0 -2 -2cos(6π/19) -2cos(12π/19) -2cos(18π/19) -2cos(14π/19) -2cos(8π/19) -2cos(2π/19) -2cos(4π/19) -2cos(10π/19) -2cos(16π/19) 0
E''4 2 2cos(8π/19) 2cos(16π/19) 2cos(14π/19) 2cos(6π/19) 2cos(2π/19) 2cos(10π/19) 2cos(18π/19) 2cos(12π/19) 2cos(4π/19) 0 -2 -2cos(8π/19) -2cos(16π/19) -2cos(14π/19) -2cos(6π/19) -2cos(2π/19) -2cos(10π/19) -2cos(18π/19) -2cos(12π/19) -2cos(4π/19) 0
E''5 2 2cos(10π/19) 2cos(18π/19) 2cos(8π/19) 2cos(2π/19) 2cos(12π/19) 2cos(16π/19) 2cos(6π/19) 2cos(4π/19) 2cos(14π/19) 0 -2 -2cos(10π/19) -2cos(18π/19) -2cos(8π/19) -2cos(2π/19) -2cos(12π/19) -2cos(16π/19) -2cos(6π/19) -2cos(4π/19) -2cos(14π/19) 0
E''6 2 2cos(12π/19) 2cos(14π/19) 2cos(2π/19) 2cos(10π/19) 2cos(16π/19) 2cos(4π/19) 2cos(8π/19) 2cos(18π/19) 2cos(6π/19) 0 -2 -2cos(12π/19) -2cos(14π/19) -2cos(2π/19) -2cos(10π/19) -2cos(16π/19) -2cos(4π/19) -2cos(8π/19) -2cos(18π/19) -2cos(6π/19) 0
E''7 2 2cos(14π/19) 2cos(10π/19) 2cos(4π/19) 2cos(18π/19) 2cos(6π/19) 2cos(8π/19) 2cos(16π/19) 2cos(2π/19) 2cos(12π/19) 0 -2 -2cos(14π/19) -2cos(10π/19) -2cos(4π/19) -2cos(18π/19) -2cos(6π/19) -2cos(8π/19) -2cos(16π/19) -2cos(2π/19) -2cos(12π/19) 0
E''8 2 2cos(16π/19) 2cos(6π/19) 2cos(10π/19) 2cos(12π/19) 2cos(4π/19) 2cos(18π/19) 2cos(2π/19) 2cos(14π/19) 2cos(8π/19) 0 -2 -2cos(16π/19) -2cos(6π/19) -2cos(10π/19) -2cos(12π/19) -2cos(4π/19) -2cos(18π/19) -2cos(2π/19) -2cos(14π/19) -2cos(8π/19) 0
E''9 2 2cos(18π/19) 2cos(2π/19) 2cos(16π/19) 2cos(4π/19) 2cos(14π/19) 2cos(6π/19) 2cos(12π/19) 2cos(8π/19) 2cos(10π/19) 0 -2 -2cos(18π/19) -2cos(2π/19) -2cos(16π/19) -2cos(4π/19) -2cos(14π/19) -2cos(6π/19) -2cos(12π/19) -2cos(8π/19) -2cos(10π/19) 0


Additional information

Number of symmetry elements h = 76
Number of classes, irreps n = 22
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2C19 2(C19)2 2(C19)3 2(C19)4 2(C19)5 2(C19)6 2(C19)7 2(C19)8 2(C19)9 19C'2 σh 2S19 2(S19)17 2(S19)3 2(S19)15 2(S19)5 2(S19)13 2(S19)7 2(S19)11 2(S19)9 19σv



Genrate representation from irreducible representations


A'1 A'2 E'1 E'2 E'3 E'4 E'5 E'6 E'7 E'8 E'9 A''1 A''2 E''1 E''2 E''3 E''4 E''5 E''6 E''7 E''8 E''9




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 E'8 E'9 A''1 A''2 E''1 E''2 E''3 E''4 E''5 E''6 E''7 E''8 E''9
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'8
E'5 E'5E'5E'4⊕E'6E'3⊕E'7E'2⊕E'8E'1⊕E'9A'1⊕A'2⊕E'9
E'6 E'6E'6E'5⊕E'7E'4⊕E'8E'3⊕E'9E'2⊕E'9E'1⊕E'8A'1⊕A'2⊕E'7
E'7 E'7E'7E'6⊕E'8E'5⊕E'9E'4⊕E'9E'3⊕E'8E'2⊕E'7E'1⊕E'6A'1⊕A'2⊕E'5
E'8 E'8E'8E'7⊕E'9E'6⊕E'9E'5⊕E'8E'4⊕E'7E'3⊕E'6E'2⊕E'5E'1⊕E'4A'1⊕A'2⊕E'3
E'9 E'9E'9E'8⊕E'9E'7⊕E'8E'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''7E''8E''9A'1
A''2 A''2A''1E''1E''2E''3E''4E''5E''6E''7E''8E''9A'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''8E''7⊕E''9E''8⊕E''9E'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''8E''5⊕E''9E''6⊕E''9E''7⊕E''8E'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''8E''3⊕E''9E''4⊕E''9E''5⊕E''8E''6⊕E''7E'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''8E''1⊕E''9E''2⊕E''9E''3⊕E''8E''4⊕E''7E''5⊕E''6E'4E'4E'3⊕E'5E'2⊕E'6E'1⊕E'7A'1⊕A'2⊕E'8
E''5 E''5E''5E''4⊕E''6E''3⊕E''7E''2⊕E''8E''1⊕E''9A''1⊕A''2⊕E''9E''1⊕E''8E''2⊕E''7E''3⊕E''6E''4⊕E''5E'5E'5E'4⊕E'6E'3⊕E'7E'2⊕E'8E'1⊕E'9A'1⊕A'2⊕E'9
E''6 E''6E''6E''5⊕E''7E''4⊕E''8E''3⊕E''9E''2⊕E''9E''1⊕E''8A''1⊕A''2⊕E''7E''1⊕E''6E''2⊕E''5E''3⊕E''4E'6E'6E'5⊕E'7E'4⊕E'8E'3⊕E'9E'2⊕E'9E'1⊕E'8A'1⊕A'2⊕E'7
E''7 E''7E''7E''6⊕E''8E''5⊕E''9E''4⊕E''9E''3⊕E''8E''2⊕E''7E''1⊕E''6A''1⊕A''2⊕E''5E''1⊕E''4E''2⊕E''3E'7E'7E'6⊕E'8E'5⊕E'9E'4⊕E'9E'3⊕E'8E'2⊕E'7E'1⊕E'6A'1⊕A'2⊕E'5
E''8 E''8E''8E''7⊕E''9E''6⊕E''9E''5⊕E''8E''4⊕E''7E''3⊕E''6E''2⊕E''5E''1⊕E''4A''1⊕A''2⊕E''3E''1⊕E''2E'8E'8E'7⊕E'9E'6⊕E'9E'5⊕E'8E'4⊕E'7E'3⊕E'6E'2⊕E'5E'1⊕E'4A'1⊕A'2⊕E'3
E''9 E''9E''9E''8⊕E''9E''7⊕E''8E''6⊕E''7E''5⊕E''6E''4⊕E''5E''3⊕E''4E''2⊕E''3E''1⊕E''2A''1⊕A''2⊕E''1E'9E'9E'8⊕E'9E'7⊕E'8E'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'8E'2⊕E'6⊕E'9A'1⊕E'4⊕E'7⊕E'8More
E'3 A'1⊕E'6E'3⊕E'9A'1⊕E'6⊕E'7E'3⊕E'4⊕E'9A'1⊕E'1⊕E'6⊕E'7More
E'4 A'1⊕E'8E'4⊕E'7A'1⊕E'3⊕E'8E'1⊕E'4⊕E'7A'1⊕E'3⊕E'5⊕E'8More
E'5 A'1⊕E'9E'4⊕E'5A'1⊕E'1⊕E'9E'4⊕E'5⊕E'6A'1⊕E'1⊕E'8⊕E'9More
E'6 A'1⊕E'7E'1⊕E'6A'1⊕E'5⊕E'7E'1⊕E'6⊕E'8A'1⊕E'2⊕E'5⊕E'7More
E'7 A'1⊕E'5E'2⊕E'7A'1⊕E'5⊕E'9E'2⊕E'3⊕E'7A'1⊕E'4⊕E'5⊕E'9More
E'8 A'1⊕E'3E'5⊕E'8A'1⊕E'3⊕E'6E'2⊕E'5⊕E'8A'1⊕E'3⊕E'6⊕E'9More
E'9 A'1⊕E'1E'8⊕E'9A'1⊕E'1⊕E'2E'7⊕E'8⊕E'9A'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'8E''2⊕E''6⊕E''9A'1⊕E'4⊕E'7⊕E'8More
E''3 A'1⊕E'6E''3⊕E''9A'1⊕E'6⊕E'7E''3⊕E''4⊕E''9A'1⊕E'1⊕E'6⊕E'7More
E''4 A'1⊕E'8E''4⊕E''7A'1⊕E'3⊕E'8E''1⊕E''4⊕E''7A'1⊕E'3⊕E'5⊕E'8More
E''5 A'1⊕E'9E''4⊕E''5A'1⊕E'1⊕E'9E''4⊕E''5⊕E''6A'1⊕E'1⊕E'8⊕E'9More
E''6 A'1⊕E'7E''1⊕E''6A'1⊕E'5⊕E'7E''1⊕E''6⊕E''8A'1⊕E'2⊕E'5⊕E'7More
E''7 A'1⊕E'5E''2⊕E''7A'1⊕E'5⊕E'9E''2⊕E''3⊕E''7A'1⊕E'4⊕E'5⊕E'9More
E''8 A'1⊕E'3E''5⊕E''8A'1⊕E'3⊕E'6E''2⊕E''5⊕E''8A'1⊕E'3⊕E'6⊕E'9More
E''9 A'1⊕E'1E''8⊕E''9A'1⊕E'1⊕E'2E''7⊕E''8⊕E''9A'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'8⊕E''1⊕E''3⊕E''5⊕E''7 45 5A'1⊕4E'2⊕3E'4⊕2E'6⊕E'8⊕4E''1⊕3E''3⊕2E''5⊕E''7
l (l=9) 19 512-pole E'1⊕E'3⊕E'5⊕E'7⊕E'9⊕A''2⊕E''2⊕E''4⊕E''6⊕E''8 55 5E'1⊕4E'3⊕3E'5⊕2E'7⊕E'9⊕5A''2⊕4E''2⊕3E''4⊕2E''6⊕E''8
m (l=10) 21 1024-pole A'1⊕E'2⊕E'4⊕E'6⊕E'8⊕E'9⊕E''1⊕E''3⊕E''5⊕E''7⊕E''9 66 6A'1⊕5E'2⊕4E'4⊕3E'6⊕2E'8⊕E'9⊕5E''1⊕4E''3⊕3E''5⊕2E''7⊕E''9
n (l=11) 23 2048-pole E'1⊕E'3⊕E'5⊕E'7⊕E'8⊕E'9⊕A''2⊕E''2⊕E''4⊕E''6⊕E''8⊕E''9 78 6E'1⊕5E'3⊕4E'5⊕3E'7⊕E'8⊕2E'9⊕6A''2⊕5E''2⊕4E''4⊕3E''6⊕2E''8⊕E''9
o (l=12) 25 4096-pole A'1⊕E'2⊕E'4⊕E'6⊕E'7⊕E'8⊕E'9⊕E''1⊕E''3⊕E''5⊕E''7⊕E''8⊕E''9 91 7A'1⊕6E'2⊕5E'4⊕4E'6⊕E'7⊕3E'8⊕2E'9⊕6E''1⊕5E''3⊕4E''5⊕3E''7⊕E''8⊕2E''9
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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 D19h
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