Point Group C
∞v
C
∞v
E
2C
∞
…
∞σ
v
A
1
Σ
+
1
1
1
1
A
2
Σ
-
-1
1
1
1
E
1
Π
0
1
1
1
E
2
Δ
0
1
1
1
E
3
Φ
0
1
1
1
…
…
0
1
1
1
E
n
0
1
1
1
Math View
Phys View
✅
Chem View
Additional information
Number of symmetry elements
h = ∞
Number of classes, irreps
n = ∞
Abelian group
no
Optical Isomerism (Chirality)
no
Polar
yes
Parity
no
Reduce representation to irreducible representations
E
2C
∞
…
∞σ
v
Genrate representation from irreducible representations
A
1
A
2
E
1
E
2
E
3
…
E
n
Examples
Hydrogen Chloride
Hydrogen Cyanide
Fluoroacetylene
Cyanoacetylene
Fluorodiacetylene
Direct products of irreducible representations
Binary products
⊙
A
1
A
2
E
1
E
2
E
3
…
E
n
A
1
A
2
E
1
E
2
E
3
…
E
n
Ternary Products
Quaternary Products
Spherical harmonics and Multipoles
Symmetric Powers of Γ
xyz
Spherical Harmonics Y
l
/ Multipole
Symmetric Power [Γ
l
(xyz)]
l
2l+1
Multipole
Symmetry
Rank
[Γ
l
(xyz)]
s (l=0)
1
Monopole
A
1
1
A
1
p (l=1)
3
Dipole
3
d (l=2)
5
Quadrupole
6
f (l=3)
7
Octupole
10
g (l=4)
9
Hexadecapole
15
h (l=5)
11
Dotricontapole
21
i (l=6)
13
Tetrahexacontapole
28
j (l=7)
15
Octacosahectapole
36
k (l=8)
17
256-pole
45
l (l=9)
19
512-pole
55
m (l=10)
21
1024-pole
66
n (l=11)
23
2048-pole
78
o (l=12)
25
4096-pole
91
More
First nonvanshing multipole:
Monopole
Further Reading
A. Gelessus, W. Thiel, W. Weber. J. Chem. Educ.
72
505 (1995)
Multipoles and symmetry
Ligand Field, d
n
term splitting
Term symbols for electronic configurations d
n
d
n
Term Symbols
d
1
= d
9
2
D
d
2
= d
8
1
S,
1
D,
1
G,
3
P,
3
F
d
3
= d
7
2
P,
2
D (2),
2
F,
2
G,
2
H,
4
P,
4
F
d
4
= d
6
1
S (2),
1
D (2),
1
F,
1
G (2),
1
I,
3
P (2),
3
D,
3
F (2),
3
G,
3
H,
5
D
d
5
2
S,
2
P,
2
D (3),
2
F (2),
2
G (2),
2
H,
2
I,
4
P,
4
D,
4
F,
4
G,
6
S
Term splitting in point group C
∞v
L
2L+1
Term Splitting
S (L=0)
1
A
1
P (L=1)
3
D (L=2)
5
F (L=3)
7
G (L=4)
9
H (L=5)
11
I (L=6)
13
Last update November, 13
th
2023 by A. Gelessus,
Impressum
,
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