Point Group C
3h
C
3h
E
2C
3
σ
h
2S
3
A'
1
1
1
1
E'
*
2
-1
2
-1
A''
1
1
-1
-1
E''
*
2
-1
-2
1
Math View
Phys View
Chem View
✅
Additional information
Number of symmetry elements
h = 6
Number of classes, irreps
n = 6
Number of real-valued irreducible representations
n = 4
Abelian group
yes
Optical Isomerism (Chirality)
no
Polar
no
Parity
no
Reduce representation to irreducible representations
E
2C
3
σ
h
2S
3
Genrate representation from irreducible representations
A'
E'
*
A''
E''
*
Examples
Boric Acid
1,3,5-Trihydroxybenzene
Tricyclo[3.3.3.0
(1,5)
]undecane
Bicyclo[3.3.3]undecane
Direct products of irreducible representations
Binary products
⊙
A'
E'
*
A''
E''
*
A'
A'
E'
*
E'
2A'⊕E'
A''
A''
E''
A'
E''
*
E''
2A''⊕E''
E'
2A'⊕E'
Ternary Products
Quaternary Products
Symmetric powers [Γ
n
] of degenerate irreducible representations
Vibrational overtones
irrep
[Γ
2
]
[Γ
3
]
[Γ
4
]
[Γ
5
]
[Γ
6
]
E'
*
A'⊕E'
2A'⊕E'
A'⊕2E'
2A'⊕2E'
3A'⊕2E'
More
E''
*
A'⊕E'
2A''⊕E''
A'⊕2E'
2A''⊕2E''
3A'⊕2E'
More
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
E'⊕A''
3
E'⊕A''
d (l=2)
5
Quadrupole
A'⊕E'⊕E''
6
2A'⊕E'⊕E''
f (l=3)
7
Octupole
2A'⊕E'⊕A''⊕E''
10
2A'⊕2E'⊕2A''⊕E''
g (l=4)
9
Hexadecapole
A'⊕2E'⊕2A''⊕E''
15
3A'⊕3E'⊕2A''⊕2E''
h (l=5)
11
Dotricontapole
2A'⊕2E'⊕A''⊕2E''
21
4A'⊕4E'⊕3A''⊕3E''
i (l=6)
13
Tetrahexacontapole
3A'⊕2E'⊕2A''⊕2E''
28
6A'⊕5E'⊕4A''⊕4E''
j (l=7)
15
Octacosahectapole
2A'⊕3E'⊕3A''⊕2E''
36
6A'⊕7E'⊕6A''⊕5E''
k (l=8)
17
256-pole
3A'⊕3E'⊕2A''⊕3E''
45
9A'⊕8E'⊕6A''⊕7E''
l (l=9)
19
512-pole
4A'⊕3E'⊕3A''⊕3E''
55
10A'⊕10E'⊕9A''⊕8E''
m (l=10)
21
1024-pole
3A'⊕4E'⊕4A''⊕3E''
66
12A'⊕12E'⊕10A''⊕10E''
n (l=11)
23
2048-pole
4A'⊕4E'⊕3A''⊕4E''
78
14A'⊕14E'⊕12A''⊕12E''
o (l=12)
25
4096-pole
5A'⊕4E'⊕4A''⊕4E''
91
17A'⊕16E'⊕14A''⊕14E''
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 C
3h
L
2L+1
Term Splitting
S (L=0)
1
A'
P (L=1)
3
A'⊕E''
D (L=2)
5
A'⊕E'⊕E''
F (L=3)
7
A'⊕E'⊕2A''⊕E''
G (L=4)
9
A'⊕2E'⊕2A''⊕E''
H (L=5)
11
A'⊕2E'⊕2A''⊕2E''
I (L=6)
13
3A'⊕2E'⊕2A''⊕2E''
Last update November, 13
th
2023 by A. Gelessus,
Impressum
,
Datenschutzerklärung/DataPrivacyStatement