Point Group D3d

Point Group D_3d

D_3d	E	2C₃	3C'₂	i	2S₆	3σ_d
A_1g	1	1	1	1	1	1
A_2g	1	1	-1	1	1	-1
E_g	2	-1	0	2	-1	0
A_1u	1	1	1	-1	-1	-1
A_2u	1	1	-1	-1	-1	1
E_u	2	-1	0	-2	1	0

Additional information

Number of symmetry elements	h = 12
Number of classes, irreps	n = 6
Abelian group	no
Optical Isomerism (Chirality)	no
Polar	no
Parity	yes

Reduce representation to irreducible representations

E	2C₃	3C'₂	i	2S₆	3σ_d

Genrate representation from irreducible representations

A_1g	A_2g	E_g	A_1u	A_2u	E_u

Examples

Sulfur S₆	Ethane (staggered)	Cyclohexane (chair)
Hexamethylbenzene

Direct products of irreducible representations

**Binary products**
⊙	A_1g	A_2g	E_g	A_1u	A_2u	E_u
A_1g	A_1g
A_2g	A_2g	A_1g
E_g	E_g	E_g	A_1g⊕A_2g⊕E_g
A_1u	A_1u	A_2u	E_u	A_1g
A_2u	A_2u	A_1u	E_u	A_2g	A_1g
E_u	E_u	E_u	A_1u⊕A_2u⊕E_u	E_g	E_g	A_1g⊕A_2g⊕E_g

Ternary Products
Quaternary Products

Symmetric powers [Γⁿ] of degenerate irreducible representations
Vibrational overtones

irrep	[Γ²]	[Γ³]	[Γ⁴]	[Γ⁵]	[Γ⁶]
E_g	A_1g⊕E_g	A_1g⊕A_2g⊕E_g	A_1g⊕2E_g	A_1g⊕A_2g⊕2E_g	2A_1g⊕A_2g⊕2E_g	More
E_u	A_1g⊕E_g	A_1u⊕A_2u⊕E_u	A_1g⊕2E_g	A_1u⊕A_2u⊕2E_u	2A_1g⊕A_2g⊕2E_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_1g	1	A_1g
p (l=1)	3	Dipole	A_2u⊕E_u	3	A_2u⊕E_u
d (l=2)	5	Quadrupole	A_1g⊕2E_g	6	2A_1g⊕2E_g
f (l=3)	7	Octupole	A_1u⊕2A_2u⊕2E_u	10	A_1u⊕3A_2u⊕3E_u
g (l=4)	9	Hexadecapole	2A_1g⊕A_2g⊕3E_g	15	4A_1g⊕A_2g⊕5E_g
h (l=5)	11	Dotricontapole	A_1u⊕2A_2u⊕4E_u	21	2A_1u⊕5A_2u⊕7E_u
i (l=6)	13	Tetrahexacontapole	3A_1g⊕2A_2g⊕4E_g	28	7A_1g⊕3A_2g⊕9E_g
j (l=7)	15	Octacosahectapole	2A_1u⊕3A_2u⊕5E_u	36	4A_1u⊕8A_2u⊕12E_u
k (l=8)	17	256-pole	3A_1g⊕2A_2g⊕6E_g	45	10A_1g⊕5A_2g⊕15E_g
l (l=9)	19	512-pole	3A_1u⊕4A_2u⊕6E_u	55	7A_1u⊕12A_2u⊕18E_u
m (l=10)	21	1024-pole	4A_1g⊕3A_2g⊕7E_g	66	14A_1g⊕8A_2g⊕22E_g
n (l=11)	23	2048-pole	3A_1u⊕4A_2u⊕8E_u	78	10A_1u⊕16A_2u⊕26E_u
o (l=12)	25	4096-pole	5A_1g⊕4A_2g⊕8E_g	91	19A_1g⊕12A_2g⊕30E_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 D_3d**
L	2L+1	Term Splitting
S (L=0)	1	A_1g
P (L=1)	3	A_2g⊕E_g
D (L=2)	5	A_1g⊕2E_g
F (L=3)	7	A_1g⊕2A_2g⊕2E_g
G (L=4)	9	2A_1g⊕A_2g⊕3E_g
H (L=5)	11	A_1g⊕2A_2g⊕4E_g
I (L=6)	13	3A_1g⊕2A_2g⊕4E_g

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