Point Group D
2
D
2
E
C
2
.
(z)
C
2
.
(y)
C
2
.
(x)
A
1
1
1
1
B
1
1
1
-1
-1
B
2
1
-1
1
-1
B
3
1
-1
-1
1
Additional information
Number of symmetry elements
h = 4
Number of classes, irreps
n = 4
Abelian group
yes
Optical Isomerism (Chirality)
yes
Polar
no
Parity
no
Reduce representation to irreducible representations
E
C
2
.
(z)
C
2
.
(y)
C
2
.
(x)
Genrate representation from irreducible representations
A
B
1
B
2
B
3
Examples
Biphenyl
Twistane
Direct products of irreducible representations
Binary products
⊙
A
B
1
B
2
B
3
A
A
B
1
B
1
A
B
2
B
2
B
3
A
B
3
B
3
B
2
B
1
A
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
A
p (l=1)
3
Dipole
B
1
⊕B
2
⊕B
3
3
B
1
⊕B
2
⊕B
3
d (l=2)
5
Quadrupole
2A⊕B
1
⊕B
2
⊕B
3
6
3A⊕B
1
⊕B
2
⊕B
3
f (l=3)
7
Octupole
A⊕2B
1
⊕2B
2
⊕2B
3
10
A⊕3B
1
⊕3B
2
⊕3B
3
g (l=4)
9
Hexadecapole
3A⊕2B
1
⊕2B
2
⊕2B
3
15
6A⊕3B
1
⊕3B
2
⊕3B
3
h (l=5)
11
Dotricontapole
2A⊕3B
1
⊕3B
2
⊕3B
3
21
3A⊕6B
1
⊕6B
2
⊕6B
3
i (l=6)
13
Tetrahexacontapole
4A⊕3B
1
⊕3B
2
⊕3B
3
28
10A⊕6B
1
⊕6B
2
⊕6B
3
j (l=7)
15
Octacosahectapole
3A⊕4B
1
⊕4B
2
⊕4B
3
36
6A⊕10B
1
⊕10B
2
⊕10B
3
k (l=8)
17
256-pole
5A⊕4B
1
⊕4B
2
⊕4B
3
45
15A⊕10B
1
⊕10B
2
⊕10B
3
l (l=9)
19
512-pole
4A⊕5B
1
⊕5B
2
⊕5B
3
55
10A⊕15B
1
⊕15B
2
⊕15B
3
m (l=10)
21
1024-pole
6A⊕5B
1
⊕5B
2
⊕5B
3
66
21A⊕15B
1
⊕15B
2
⊕15B
3
n (l=11)
23
2048-pole
5A⊕6B
1
⊕6B
2
⊕6B
3
78
15A⊕21B
1
⊕21B
2
⊕21B
3
o (l=12)
25
4096-pole
7A⊕6B
1
⊕6B
2
⊕6B
3
91
28A⊕21B
1
⊕21B
2
⊕21B
3
More
First nonvanshing multipole:
Quadrupole
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 D
2
L
2L+1
Term Splitting
S (L=0)
1
A
P (L=1)
3
B
1
⊕B
2
⊕B
3
D (L=2)
5
2A⊕B
1
⊕B
2
⊕B
3
F (L=3)
7
A⊕2B
1
⊕2B
2
⊕2B
3
G (L=4)
9
3A⊕2B
1
⊕2B
2
⊕2B
3
H (L=5)
11
2A⊕3B
1
⊕3B
2
⊕3B
3
I (L=6)
13
4A⊕3B
1
⊕3B
2
⊕3B
3
Last update November, 13
th
2023 by A. Gelessus,
Impressum
,
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