Point Group C4h

Point Group C_4h

C_4h	E	2C₄	C₂	i	2S₄	σ_h
A_g	1	1	1	1	1	1
B_g	1	-1	1	1	-1	1
E_g*	2	0	-2	2	0	-2
A_u	1	1	1	-1	-1	-1
B_u	1	-1	1	-1	1	-1
E_u*	2	0	-2	-2	0	2

Additional information

Number of symmetry elements	h = 8
Number of classes, irreps	n = 8
Number of real-valued irreducible representations	n = 6
Abelian group	yes
Optical Isomerism (Chirality)	no
Polar	no
Parity	yes

Reduce representation to irreducible representations

E	2C₄	C₂	i	2S₄	σ_h

Genrate representation from irreducible representations

A_g	B_g	E_g*	A_u	B_u	E_u*

Direct products of irreducible representations

**Binary products**
⊙	A_g	B_g	E_g*	A_u	B_u	E_u*
A_g	A_g
B_g	B_g	A_g
E_g*	E_g	E_g	2A_g⊕2B_g
A_u	A_u	B_u	E_u	A_g
B_u	B_u	A_u	E_u	B_g	A_g
E_u*	E_u	E_u	2A_u⊕2B_u	E_g	E_g	2A_g⊕2B_g

Ternary Products
Quaternary Products

Symmetric powers [Γⁿ] of degenerate irreducible representations
Vibrational overtones

irrep	[Γ²]	[Γ³]	[Γ⁴]	[Γ⁵]	[Γ⁶]
E_g*	A_g⊕2B_g	2E_g	3A_g⊕2B_g	3E_g	3A_g⊕4B_g	More
E_u*	A_g⊕2B_g	2E_u	3A_g⊕2B_g	3E_u	3A_g⊕4B_g	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_g	1	A_g
p (l=1)	3	Dipole	A_u⊕E_u	3	A_u⊕E_u
d (l=2)	5	Quadrupole	A_g⊕2B_g⊕E_g	6	2A_g⊕2B_g⊕E_g
f (l=3)	7	Octupole	A_u⊕2B_u⊕2E_u	10	2A_u⊕2B_u⊕3E_u
g (l=4)	9	Hexadecapole	3A_g⊕2B_g⊕2E_g	15	5A_g⊕4B_g⊕3E_g
h (l=5)	11	Dotricontapole	3A_u⊕2B_u⊕3E_u	21	5A_u⊕4B_u⊕6E_u
i (l=6)	13	Tetrahexacontapole	3A_g⊕4B_g⊕3E_g	28	8A_g⊕8B_g⊕6E_g
j (l=7)	15	Octacosahectapole	3A_u⊕4B_u⊕4E_u	36	8A_u⊕8B_u⊕10E_u
k (l=8)	17	256-pole	5A_g⊕4B_g⊕4E_g	45	13A_g⊕12B_g⊕10E_g
l (l=9)	19	512-pole	5A_u⊕4B_u⊕5E_u	55	13A_u⊕12B_u⊕15E_u
m (l=10)	21	1024-pole	5A_g⊕6B_g⊕5E_g	66	18A_g⊕18B_g⊕15E_g
n (l=11)	23	2048-pole	5A_u⊕6B_u⊕6E_u	78	18A_u⊕18B_u⊕21E_u
o (l=12)	25	4096-pole	7A_g⊕6B_g⊕6E_g	91	25A_g⊕24B_g⊕21E_g
More

First nonvanshing multipole: Quadrupole

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 C_4h**
L	2L+1	Term Splitting
S (L=0)	1	A_g
P (L=1)	3	A_g⊕E_g
D (L=2)	5	A_g⊕2B_g⊕E_g
F (L=3)	7	A_g⊕2B_g⊕2E_g
G (L=4)	9	3A_g⊕2B_g⊕2E_g
H (L=5)	11	3A_g⊕2B_g⊕3E_g
I (L=6)	13	3A_g⊕4B_g⊕3E_g

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