Point Group C19

C19 E C19 (C19)2 (C19)3 (C19)4 (C19)5 (C19)6 (C19)7 (C19)8 (C19)9 (C19)10 (C19)11 (C19)12 (C19)13 (C19)14 (C19)15 (C19)16 (C19)17 (C19)18
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E1* 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) 2cos(18π/19) 2cos(16π/19) 2cos(14π/19) 2cos(12π/19) 2cos(10π/19) 2cos(8π/19) 2cos(6π/19) 2cos(4π/19) 2cos(2π/19)
E2* 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) 2cos(2π/19) 2cos(6π/19) 2cos(10π/19) 2cos(14π/19) 2cos(18π/19) 2cos(16π/19) 2cos(12π/19) 2cos(8π/19) 2cos(4π/19)
E3* 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) 2cos(16π/19) 2cos(10π/19) 2cos(4π/19) 2cos(2π/19) 2cos(8π/19) 2cos(14π/19) 2cos(18π/19) 2cos(12π/19) 2cos(6π/19)
E4* 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) 2cos(4π/19) 2cos(12π/19) 2cos(18π/19) 2cos(10π/19) 2cos(2π/19) 2cos(6π/19) 2cos(14π/19) 2cos(16π/19) 2cos(8π/19)
E5* 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) 2cos(14π/19) 2cos(4π/19) 2cos(6π/19) 2cos(16π/19) 2cos(12π/19) 2cos(2π/19) 2cos(8π/19) 2cos(18π/19) 2cos(10π/19)
E6* 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) 2cos(6π/19) 2cos(18π/19) 2cos(8π/19) 2cos(4π/19) 2cos(16π/19) 2cos(10π/19) 2cos(2π/19) 2cos(14π/19) 2cos(12π/19)
E7* 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) 2cos(12π/19) 2cos(2π/19) 2cos(16π/19) 2cos(8π/19) 2cos(6π/19) 2cos(18π/19) 2cos(4π/19) 2cos(10π/19) 2cos(14π/19)
E8* 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) 2cos(8π/19) 2cos(14π/19) 2cos(2π/19) 2cos(18π/19) 2cos(4π/19) 2cos(12π/19) 2cos(10π/19) 2cos(6π/19) 2cos(16π/19)
E9* 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) 2cos(10π/19) 2cos(8π/19) 2cos(12π/19) 2cos(6π/19) 2cos(14π/19) 2cos(4π/19) 2cos(16π/19) 2cos(2π/19) 2cos(18π/19)

Additional information

Number of symmetry elements h = 19
Number of classes, irreps n = 19
Number of real-valued irreducible representations n = 10
Abelian group yes
Optical Isomerism (Chirality) yes
Polar yes
Parity no

Reduce representation to irreducible representations

E C19 (C19)2 (C19)3 (C19)4 (C19)5 (C19)6 (C19)7 (C19)8 (C19)9 (C19)10 (C19)11 (C19)12 (C19)13 (C19)14 (C19)15 (C19)16 (C19)17 (C19)18

Genrate representation from irreducible representations

A E1* E2* E3* E4* E5* E6* E7* E8* E9*

Direct products of irreducible representations

Binary products
A E1* E2* E3* E4* E5* E6* E7* E8* E9*
E1* E12A⊕E2
E2* E2E1⊕E32A⊕E4
E3* E3E2⊕E4E1⊕E52A⊕E6
E4* E4E3⊕E5E2⊕E6E1⊕E72A⊕E8
E5* E5E4⊕E6E3⊕E7E2⊕E8E1⊕E92A⊕E9
E6* E6E5⊕E7E4⊕E8E3⊕E9E2⊕E9E1⊕E82A⊕E7
E7* E7E6⊕E8E5⊕E9E4⊕E9E3⊕E8E2⊕E7E1⊕E62A⊕E5
E8* E8E7⊕E9E6⊕E9E5⊕E8E4⊕E7E3⊕E6E2⊕E5E1⊕E42A⊕E3
E9* E9E8⊕E9E7⊕E8E6⊕E7E5⊕E6E4⊕E5E3⊕E4E2⊕E3E1⊕E22A⊕E1

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⊕E4⊕E8E2⊕E6⊕E9A⊕E4⊕E7⊕E8More
E3* A⊕E6E3⊕E9A⊕E6⊕E7E3⊕E4⊕E9A⊕E1⊕E6⊕E7More
E4* A⊕E8E4⊕E7A⊕E3⊕E8E1⊕E4⊕E7A⊕E3⊕E5⊕E8More
E5* A⊕E9E4⊕E5A⊕E1⊕E9E4⊕E5⊕E6A⊕E1⊕E8⊕E9More
E6* A⊕E7E1⊕E6A⊕E5⊕E7E1⊕E6⊕E8A⊕E2⊕E5⊕E7More
E7* A⊕E5E2⊕E7A⊕E5⊕E9E2⊕E3⊕E7A⊕E4⊕E5⊕E9More
E8* A⊕E3E5⊕E8A⊕E3⊕E6E2⊕E5⊕E8A⊕E3⊕E6⊕E9More
E9* A⊕E1E8⊕E9A⊕E1⊕E2E7⊕E8⊕E9A⊕E1⊕E2⊕E3More

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 A⊕E1 3 A⊕E1
d (l=2) 5 Quadrupole A⊕E1⊕E2 6 2A⊕E1⊕E2
f (l=3) 7 Octupole A⊕E1⊕E2⊕E3 10 2A⊕2E1⊕E2⊕E3
g (l=4) 9 Hexadecapole A⊕E1⊕E2⊕E3⊕E4 15 3A⊕2E1⊕2E2⊕E3⊕E4
h (l=5) 11 Dotricontapole A⊕E1⊕E2⊕E3⊕E4⊕E5 21 3A⊕3E1⊕2E2⊕2E3⊕E4⊕E5
i (l=6) 13 Tetrahexacontapole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6 28 4A⊕3E1⊕3E2⊕2E3⊕2E4⊕E5⊕E6
j (l=7) 15 Octacosahectapole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7 36 4A⊕4E1⊕3E2⊕3E3⊕2E4⊕2E5⊕E6⊕E7
k (l=8) 17 256-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 45 5A⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7⊕E8
l (l=9) 19 512-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 55 5A⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕2E7⊕E8⊕E9
m (l=10) 21 1024-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕2E9 66 6A⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕3E6⊕2E7⊕2E8⊕2E9
n (l=11) 23 2048-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕2E8⊕2E9 78 6A⊕6E1⊕5E2⊕5E3⊕4E4⊕4E5⊕3E6⊕3E7⊕3E8⊕3E9
o (l=12) 25 4096-pole A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕2E7⊕2E8⊕2E9 91 7A⊕6E1⊕6E2⊕5E3⊕5E4⊕4E5⊕4E6⊕4E7⊕4E8⊕4E9

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 C19
L 2L+1 Term Splitting
S (L=0) 1 A
P (L=1) 3 A⊕E1
D (L=2) 5 A⊕E1⊕E2
F (L=3) 7 A⊕E1⊕E2⊕E3
G (L=4) 9 A⊕E1⊕E2⊕E3⊕E4
H (L=5) 11 A⊕E1⊕E2⊕E3⊕E4⊕E5
I (L=6) 13 A⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6

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