Point Group C19v

C19v 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 19σv
A1 1 1 1 1 1 1 1 1 1 1 1
A2 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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0
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) 0

Additional information

Number of symmetry elements h = 38
Number of classes, irreps n = 11
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
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 19σv

Genrate representation from irreducible representations

A1 A2 E1 E2 E3 E4 E5 E6 E7 E8 E9

Direct products of irreducible representations

Binary products
A1 A2 E1 E2 E3 E4 E5 E6 E7 E8 E9
A1 A1
A2 A2A1
E1 E1E1A1⊕A2⊕E2
E2 E2E2E1⊕E3A1⊕A2⊕E4
E3 E3E3E2⊕E4E1⊕E5A1⊕A2⊕E6
E4 E4E4E3⊕E5E2⊕E6E1⊕E7A1⊕A2⊕E8
E5 E5E5E4⊕E6E3⊕E7E2⊕E8E1⊕E9A1⊕A2⊕E9
E6 E6E6E5⊕E7E4⊕E8E3⊕E9E2⊕E9E1⊕E8A1⊕A2⊕E7
E7 E7E7E6⊕E8E5⊕E9E4⊕E9E3⊕E8E2⊕E7E1⊕E6A1⊕A2⊕E5
E8 E8E8E7⊕E9E6⊕E9E5⊕E8E4⊕E7E3⊕E6E2⊕E5E1⊕E4A1⊕A2⊕E3
E9 E9E9E8⊕E9E7⊕E8E6⊕E7E5⊕E6E4⊕E5E3⊕E4E2⊕E3E1⊕E2A1⊕A2⊕E1

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⊕E9A1⊕E4⊕E7⊕E8More
E3 A1⊕E6E3⊕E9A1⊕E6⊕E7E3⊕E4⊕E9A1⊕E1⊕E6⊕E7More
E4 A1⊕E8E4⊕E7A1⊕E3⊕E8E1⊕E4⊕E7A1⊕E3⊕E5⊕E8More
E5 A1⊕E9E4⊕E5A1⊕E1⊕E9E4⊕E5⊕E6A1⊕E1⊕E8⊕E9More
E6 A1⊕E7E1⊕E6A1⊕E5⊕E7E1⊕E6⊕E8A1⊕E2⊕E5⊕E7More
E7 A1⊕E5E2⊕E7A1⊕E5⊕E9E2⊕E3⊕E7A1⊕E4⊕E5⊕E9More
E8 A1⊕E3E5⊕E8A1⊕E3⊕E6E2⊕E5⊕E8A1⊕E3⊕E6⊕E9More
E9 A1⊕E1E8⊕E9A1⊕E1⊕E2E7⊕E8⊕E9A1⊕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 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⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8 45 5A1⊕4E1⊕4E2⊕3E3⊕3E4⊕2E5⊕2E6⊕E7⊕E8
l (l=9) 19 512-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕E9 55 5A1⊕5E1⊕4E2⊕4E3⊕3E4⊕3E5⊕2E6⊕2E7⊕E8⊕E9
m (l=10) 21 1024-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕E8⊕2E9 66 6A1⊕5E1⊕5E2⊕4E3⊕4E4⊕3E5⊕3E6⊕2E7⊕2E8⊕2E9
n (l=11) 23 2048-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕E7⊕2E8⊕2E9 78 6A1⊕6E1⊕5E2⊕5E3⊕4E4⊕4E5⊕3E6⊕3E7⊕3E8⊕3E9
o (l=12) 25 4096-pole A1⊕E1⊕E2⊕E3⊕E4⊕E5⊕E6⊕2E7⊕2E8⊕2E9 91 7A1⊕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 C19v
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