A'1	A'2	E'1	E'2	E'3	A''1	A''2	E''1	E''2	E''3

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Examples

Tropylium Cation	Cucurbit[7]uril

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Direct products of irreducible representations

Binary products

⊙	A'1	A'2	E'1	E'2	E'3	A''1	A''2	E''1	E''2	E''3
A'1	A'1
A'2	A'2	A'1
E'1	E'1	E'1	A'1⊕A'2⊕E'2
E'2	E'2	E'2	E'1⊕E'3	A'1⊕A'2⊕E'3
E'3	E'3	E'3	E'2⊕E'3	E'1⊕E'2	A'1⊕A'2⊕E'1
A''1	A''1	A''2	E''1	E''2	E''3	A'1
A''2	A''2	A''1	E''1	E''2	E''3	A'2	A'1
E''1	E''1	E''1	A''1⊕A''2⊕E''2	E''1⊕E''3	E''2⊕E''3	E'1	E'1	A'1⊕A'2⊕E'2
E''2	E''2	E''2	E''1⊕E''3	A''1⊕A''2⊕E''3	E''1⊕E''2	E'2	E'2	E'1⊕E'3	A'1⊕A'2⊕E'3
E''3	E''3	E''3	E''2⊕E''3	E''1⊕E''2	A''1⊕A''2⊕E''1	E'3	E'3	E'2⊕E'3	E'1⊕E'2	A'1⊕A'2⊕E'1

Ternary Products
Quaternary Products

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Symmetric powers [Γⁿ] of degenerate irreducible representations
Vibrational overtones

irrep	[Γ2]	[Γ3]	[Γ4]	[Γ5]	[Γ6]
E'1	A'1⊕E'2	E'1⊕E'3	A'1⊕E'2⊕E'3	E'1⊕E'2⊕E'3	A'1⊕E'1⊕E'2⊕E'3	More
E'2	A'1⊕E'3	E'1⊕E'2	A'1⊕E'1⊕E'3	E'1⊕E'2⊕E'3	A'1⊕E'1⊕E'2⊕E'3	More
E'3	A'1⊕E'1	E'2⊕E'3	A'1⊕E'1⊕E'2	E'1⊕E'2⊕E'3	A'1⊕E'1⊕E'2⊕E'3	More
E''1	A'1⊕E'2	E''1⊕E''3	A'1⊕E'2⊕E'3	E''1⊕E''2⊕E''3	A'1⊕E'1⊕E'2⊕E'3	More
E''2	A'1⊕E'3	E''1⊕E''2	A'1⊕E'1⊕E'3	E''1⊕E''2⊕E''3	A'1⊕E'1⊕E'2⊕E'3	More
E''3	A'1⊕E'1	E''2⊕E''3	A'1⊕E'1⊕E'2	E''1⊕E''2⊕E''3	A'1⊕E'1⊕E'2⊕E'3	More

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Spherical harmonics and Multipoles
Symmetric Powers of Γ_xyz

Spherical Harmonics Yl / Multipole				Symmetric Power [Γl(xyz)]
l	2l+1	Multipole	Symmetry	Rank	[Γl(xyz)]
s (l=0)	1	Monopole	A'1	1	A'1
p (l=1)	3	Dipole	E'1⊕A''2	3	E'1⊕A''2
d (l=2)	5	Quadrupole	A'1⊕E'2⊕E''1	6	2A'1⊕E'2⊕E''1
f (l=3)	7	Octupole	E'1⊕E'3⊕A''2⊕E''2	10	2E'1⊕E'3⊕2A''2⊕E''2
g (l=4)	9	Hexadecapole	A'1⊕E'2⊕E'3⊕E''1⊕E''3	15	3A'1⊕2E'2⊕E'3⊕2E''1⊕E''3
h (l=5)	11	Dotricontapole	E'1⊕E'2⊕E'3⊕A''2⊕E''2⊕E''3	21	3E'1⊕E'2⊕2E'3⊕3A''2⊕2E''2⊕E''3
i (l=6)	13	Tetrahexacontapole	A'1⊕E'1⊕E'2⊕E'3⊕E''1⊕E''2⊕E''3	28	4A'1⊕E'1⊕3E'2⊕2E'3⊕3E''1⊕E''2⊕2E''3
j (l=7)	15	Octacosahectapole	A'1⊕A'2⊕E'1⊕E'2⊕E'3⊕A''2⊕E''1⊕E''2⊕E''3	36	A'1⊕A'2⊕4E'1⊕2E'2⊕3E'3⊕4A''2⊕E''1⊕3E''2⊕2E''3
k (l=8)	17	256-pole	A'1⊕2E'1⊕E'2⊕E'3⊕A''1⊕A''2⊕E''1⊕E''2⊕E''3	45	5A'1⊕3E'1⊕4E'2⊕3E'3⊕A''1⊕A''2⊕4E''1⊕2E''2⊕3E''3
l (l=9)	19	512-pole	A'1⊕A'2⊕E'1⊕2E'2⊕E'3⊕A''2⊕2E''1⊕E''2⊕E''3	55	2A'1⊕2A'2⊕5E'1⊕4E'2⊕4E'3⊕5A''2⊕3E''1⊕4E''2⊕3E''3
m (l=10)	21	1024-pole	A'1⊕2E'1⊕E'2⊕2E'3⊕A''1⊕A''2⊕E''1⊕2E''2⊕E''3	66	6A'1⊕5E'1⊕5E'2⊕5E'3⊕2A''1⊕2A''2⊕5E''1⊕4E''2⊕4E''3
n (l=11)	23	2048-pole	A'1⊕A'2⊕E'1⊕2E'2⊕2E'3⊕A''2⊕2E''1⊕E''2⊕2E''3	78	3A'1⊕3A'2⊕6E'1⊕6E'2⊕6E'3⊕6A''2⊕5E''1⊕5E''2⊕5E''3
o (l=12)	25	4096-pole	A'1⊕2E'1⊕2E'2⊕2E'3⊕A''1⊕A''2⊕E''1⊕2E''2⊕2E''3	91	7A'1⊕7E'1⊕7E'2⊕7E'3⊕3A''1⊕3A''2⊕6E''1⊕6E''2⊕6E''3
More

First nonvanshing multipole: Quadrupole

Further Reading

• A. Gelessus, W. Thiel, W. Weber. J. Chem. Educ. 72 505 (1995)
  Multipoles and symmetry