Point Group C4v



C4v E 2C4 C2 v d
A1 1 1 1 1 1
A2 1 1 1 -1 -1
B1 1 -1 1 1 -1
B2 1 -1 1 -1 1
E 2 0 -2 0 0


Additional information

Number of symmetry elements h = 8
Number of classes, irreps n = 5
Abelian group no
Optical Isomerism (Chirality) no
Polar yes
Parity no


Reduce representation to irreducible representations


E 2C4 C2 v d



Genrate representation from irreducible representations


A1 A2 B1 B2 E




Examples

Brominepentafluoride Sulfurchloridepentafluoride Pentaborane(9)



Direct products of irreducible representations


Binary products
A1 A2 B1 B2 E
A1 A1
A2 A2A1
B1 B1B2A1
B2 B2B1A2A1
E EEEEA1⊕A2⊕B1⊕B2

Ternary Products
Quaternary Products



Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep 2] 3] 4] 5] 6]
E A1⊕B1⊕B22E2A1⊕A2⊕B1⊕B23E2A1⊕A2⊕2B1⊕2B2More



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⊕E 3 A1⊕E
d (l=2) 5 Quadrupole A1⊕B1⊕B2⊕E 6 2A1⊕B1⊕B2⊕E
f (l=3) 7 Octupole A1⊕B1⊕B2⊕2E 10 2A1⊕B1⊕B2⊕3E
g (l=4) 9 Hexadecapole 2A1⊕A2⊕B1⊕B2⊕2E 15 4A1⊕A2⊕2B1⊕2B2⊕3E
h (l=5) 11 Dotricontapole 2A1⊕A2⊕B1⊕B2⊕3E 21 4A1⊕A2⊕2B1⊕2B2⊕6E
i (l=6) 13 Tetrahexacontapole 2A1⊕A2⊕2B1⊕2B2⊕3E 28 6A1⊕2A2⊕4B1⊕4B2⊕6E
j (l=7) 15 Octacosahectapole 2A1⊕A2⊕2B1⊕2B2⊕4E 36 6A1⊕2A2⊕4B1⊕4B2⊕10E
k (l=8) 17 256-pole 3A1⊕2A2⊕2B1⊕2B2⊕4E 45 9A1⊕4A2⊕6B1⊕6B2⊕10E
l (l=9) 19 512-pole 3A1⊕2A2⊕2B1⊕2B2⊕5E 55 9A1⊕4A2⊕6B1⊕6B2⊕15E
m (l=10) 21 1024-pole 3A1⊕2A2⊕3B1⊕3B2⊕5E 66 12A1⊕6A2⊕9B1⊕9B2⊕15E
n (l=11) 23 2048-pole 3A1⊕2A2⊕3B1⊕3B2⊕6E 78 12A1⊕6A2⊕9B1⊕9B2⊕21E
o (l=12) 25 4096-pole 4A1⊕3A2⊕3B1⊕3B2⊕6E 91 16A1⊕9A2⊕12B1⊕12B2⊕21E
More

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 C4v
L 2L+1 Term Splitting
S (L=0) 1 A1
P (L=1) 3 A2⊕E
D (L=2) 5 A1⊕B1⊕B2⊕E
F (L=3) 7 A2⊕B1⊕B2⊕2E
G (L=4) 9 2A1⊕A2⊕B1⊕B2⊕2E
H (L=5) 11 A1⊕2A2⊕B1⊕B2⊕3E
I (L=6) 13 2A1⊕A2⊕2B1⊕2B2⊕3E


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