Point Group D2h

Point Group D_2h

D_2h	E	C₂.(z)	C₂.(y)	C₂.(x)	i	σ.(xy)	σ.(xy)	σ.(xy)
A_1g	1	1	1	1	1	1	1	1
B_1g	1	1	-1	-1	1	1	-1	-1
B_2g	1	-1	1	-1	1	-1	1	-1
B_3g	1	-1	-1	1	1	-1	-1	1
A_1u	1	1	1	1	-1	-1	-1	-1
B_1u	1	1	-1	-1	-1	-1	1	1
B_2u	1	-1	1	-1	-1	1	-1	1
B_3u	1	-1	-1	1	-1	1	1	-1

Additional information

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

Reduce representation to irreducible representations

E	C₂.(z)	C₂.(y)	C₂.(x)	i	σ.(xy)	σ.(xy)	σ.(xy)

Genrate representation from irreducible representations

A_1g	B_1g	B_2g	B_3g	A_1u	B_1u	B_2u	B_3u

Examples

Ethylene	Diborane	Ethylenetetracarboxylic Acid Dianhydride
Tetrahydroxy-1,4-benzoquinone Biscarbonate	Naphthalene	Fulvalene
Tricyclo[3.1.1.1^(2,4)]octane	Anthracene	Pyrene
Tetracene	Pagodane	Dianthracene

Direct products of irreducible representations

**Binary products**
⊙	A_1g	B_1g	B_2g	B_3g	A_1u	B_1u	B_2u	B_3u
A_1g	A_1g
B_1g	B_1g	A_1g
B_2g	B_2g	B_3g	A_1g
B_3g	B_3g	B_2g	B_1g	A_1g
A_1u	A_1u	B_1u	B_2u	B_3u	A_1g
B_1u	B_1u	A_1u	B_3u	B_2u	B_1g	A_1g
B_2u	B_2u	B_3u	A_1u	B_1u	B_2g	B_3g	A_1g
B_3u	B_3u	B_2u	B_1u	A_1u	B_3g	B_2g	B_1g	A_1g

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_1g	1	A_1g
p (l=1)	3	Dipole	B_1u⊕B_2u⊕B_3u	3	B_1u⊕B_2u⊕B_3u
d (l=2)	5	Quadrupole	2A_1g⊕B_1g⊕B_2g⊕B_3g	6	3A_1g⊕B_1g⊕B_2g⊕B_3g
f (l=3)	7	Octupole	A_1u⊕2B_1u⊕2B_2u⊕2B_3u	10	A_1u⊕3B_1u⊕3B_2u⊕3B_3u
g (l=4)	9	Hexadecapole	3A_1g⊕2B_1g⊕2B_2g⊕2B_3g	15	6A_1g⊕3B_1g⊕3B_2g⊕3B_3g
h (l=5)	11	Dotricontapole	2A_1u⊕3B_1u⊕3B_2u⊕3B_3u	21	3A_1u⊕6B_1u⊕6B_2u⊕6B_3u
i (l=6)	13	Tetrahexacontapole	4A_1g⊕3B_1g⊕3B_2g⊕3B_3g	28	10A_1g⊕6B_1g⊕6B_2g⊕6B_3g
j (l=7)	15	Octacosahectapole	3A_1u⊕4B_1u⊕4B_2u⊕4B_3u	36	6A_1u⊕10B_1u⊕10B_2u⊕10B_3u
k (l=8)	17	256-pole	5A_1g⊕4B_1g⊕4B_2g⊕4B_3g	45	15A_1g⊕10B_1g⊕10B_2g⊕10B_3g
l (l=9)	19	512-pole	4A_1u⊕5B_1u⊕5B_2u⊕5B_3u	55	10A_1u⊕15B_1u⊕15B_2u⊕15B_3u
m (l=10)	21	1024-pole	6A_1g⊕5B_1g⊕5B_2g⊕5B_3g	66	21A_1g⊕15B_1g⊕15B_2g⊕15B_3g
n (l=11)	23	2048-pole	5A_1u⊕6B_1u⊕6B_2u⊕6B_3u	78	15A_1u⊕21B_1u⊕21B_2u⊕21B_3u
o (l=12)	25	4096-pole	7A_1g⊕6B_1g⊕6B_2g⊕6B_3g	91	28A_1g⊕21B_1g⊕21B_2g⊕21B_3g
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 D_2h**
L	2L+1	Term Splitting
S (L=0)	1	A_1g
P (L=1)	3	B_1g⊕B_2g⊕B_3g
D (L=2)	5	2A_1g⊕B_1g⊕B_2g⊕B_3g
F (L=3)	7	A_1g⊕2B_1g⊕2B_2g⊕2B_3g
G (L=4)	9	3A_1g⊕2B_1g⊕2B_2g⊕2B_3g
H (L=5)	11	2A_1g⊕3B_1g⊕3B_2g⊕3B_3g
I (L=6)	13	4A_1g⊕3B_1g⊕3B_2g⊕3B_3g

Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement