Point Group C15h



C15h E 2C15 2(C15)2 2C5 2(C15)4 2C3 2(C5)2 2(C15)7 σh 2S15 2(S15)13 2S5 2(S15)11 2S3 2(S5)3 2(S15)7
A' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E'1* 2 1.8271 1.3383 0.6180 -0.2091 -1 -1.6180 -1.9563 2 1.8271 1.3383 0.6180 -0.2091 -1 -1.6180 -1.9563
E'2* 2 1.3383 -0.2091 -1.6180 -1.9563 -1 0.6180 1.8271 2 1.3383 -0.2091 -1.6180 -1.9563 -1 0.6180 1.8271
E'3* 2 0.6180 -1.6180 -1.6180 0.6180 2 0.6180 -1.6180 2 0.6180 -1.6180 -1.6180 0.6180 2 0.6180 -1.6180
E'4* 2 -0.2091 -1.9563 0.6180 1.8271 -1 -1.6180 1.3383 2 -0.2091 -1.9563 0.6180 1.8271 -1 -1.6180 1.3383
E'5* 2 -1 -1 2 -1 -1 2 -1 2 -1 -1 2 -1 -1 2 -1
E'6* 2 -1.6180 0.6180 0.6180 -1.6180 2 -1.6180 0.6180 2 -1.6180 0.6180 0.6180 -1.6180 2 -1.6180 0.6180
E'7* 2 -1.9563 1.8271 -1.6180 1.3383 -1 0.6180 -0.2091 2 -1.9563 1.8271 -1.6180 1.3383 -1 0.6180 -0.2091
A'' 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1
E''1* 2 1.8271 1.3383 0.6180 -0.2091 -1 -1.6180 -1.9563 -2 -1.8271 -1.3383 -0.6180 0.2091 1 1.6180 1.9563
E''2* 2 1.3383 -0.2091 -1.6180 -1.9563 -1 0.6180 1.8271 -2 -1.3383 0.2091 1.6180 1.9563 1 -0.6180 -1.8271
E''3* 2 0.6180 -1.6180 -1.6180 0.6180 2 0.6180 -1.6180 -2 -0.6180 1.6180 1.6180 -0.6180 -2 -0.6180 1.6180
E''4* 2 -0.2091 -1.9563 0.6180 1.8271 -1 -1.6180 1.3383 -2 0.2091 1.9563 -0.6180 -1.8271 1 1.6180 -1.3383
E''5* 2 -1 -1 2 -1 -1 2 -1 -2 1 1 -2 1 1 -2 1
E''6* 2 -1.6180 0.6180 0.6180 -1.6180 2 -1.6180 0.6180 -2 1.6180 -0.6180 -0.6180 1.6180 -2 1.6180 -0.6180
E''7* 2 -1.9563 1.8271 -1.6180 1.3383 -1 0.6180 -0.2091 -2 1.9563 -1.8271 1.6180 -1.3383 1 -0.6180 0.2091
Math View
Phys View A
Phys View B
Chem View A
Chem View B
✅


Additional information

Number of symmetry elements h = 30
Number of classes, irreps n = 30
Number of real-valued irreducible representations n = 16
Abelian group yes
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2C15 2(C15)2 2C5 2(C15)4 2C3 2(C5)2 2(C15)7 σh 2S15 2(S15)13 2S5 2(S15)11 2S3 2(S5)3 2(S15)7



Genrate representation from irreducible representations


A' E'1* E'2* E'3* E'4* E'5* E'6* E'7* A'' E''1* E''2* E''3* E''4* E''5* E''6* E''7*



Direct products of irreducible representations


Binary products
⊙ A' E'1* E'2* E'3* E'4* E'5* E'6* E'7* A'' E''1* E''2* E''3* E''4* E''5* E''6* E''7*
A' A'
E'1* E'12A'⊕E'2
E'2* E'2E'1⊕E'32A'⊕E'4
E'3* E'3E'2⊕E'4E'1⊕E'52A'⊕E'6
E'4* E'4E'3⊕E'5E'2⊕E'6E'1⊕E'72A'⊕E'7
E'5* E'5E'4⊕E'6E'3⊕E'7E'2⊕E'7E'1⊕E'62A'⊕E'5
E'6* E'6E'5⊕E'7E'4⊕E'7E'3⊕E'6E'2⊕E'5E'1⊕E'42A'⊕E'3
E'7* E'7E'6⊕E'7E'5⊕E'6E'4⊕E'5E'3⊕E'4E'2⊕E'3E'1⊕E'22A'⊕E'1
A'' A''E''1E''2E''3E''4E''5E''6E''7A'
E''1* E''12A''⊕E''2E''1⊕E''3E''2⊕E''4E''3⊕E''5E''4⊕E''6E''5⊕E''7E''6⊕E''7E'12A'⊕E'2
E''2* E''2E''1⊕E''32A''⊕E''4E''1⊕E''5E''2⊕E''6E''3⊕E''7E''4⊕E''7E''5⊕E''6E'2E'1⊕E'32A'⊕E'4
E''3* E''3E''2⊕E''4E''1⊕E''52A''⊕E''6E''1⊕E''7E''2⊕E''7E''3⊕E''6E''4⊕E''5E'3E'2⊕E'4E'1⊕E'52A'⊕E'6
E''4* E''4E''3⊕E''5E''2⊕E''6E''1⊕E''72A''⊕E''7E''1⊕E''6E''2⊕E''5E''3⊕E''4E'4E'3⊕E'5E'2⊕E'6E'1⊕E'72A'⊕E'7
E''5* E''5E''4⊕E''6E''3⊕E''7E''2⊕E''7E''1⊕E''62A''⊕E''5E''1⊕E''4E''2⊕E''3E'5E'4⊕E'6E'3⊕E'7E'2⊕E'7E'1⊕E'62A'⊕E'5
E''6* E''6E''5⊕E''7E''4⊕E''7E''3⊕E''6E''2⊕E''5E''1⊕E''42A''⊕E''3E''1⊕E''2E'6E'5⊕E'7E'4⊕E'7E'3⊕E'6E'2⊕E'5E'1⊕E'42A'⊕E'3
E''7* E''7E''6⊕E''7E''5⊕E''6E''4⊕E''5E''3⊕E''4E''2⊕E''3E''1⊕E''22A''⊕E''1E'7E'6⊕E'7E'5⊕E'6E'4⊕E'5E'3⊕E'4E'2⊕E'3E'1⊕E'22A'⊕E'1

Ternary Products
Quaternary Products


Symmetric powers [Γn] of degenerate irreducible representations
Vibrational overtones


irrep [Γ2] [Γ3] [Γ4] [Γ5] [Γ6]
E'1* A'⊕E'2E'1⊕E'3A'⊕E'2⊕E'4E'1⊕E'3⊕E'5A'⊕E'2⊕E'4⊕E'6More
E'2* A'⊕E'4E'2⊕E'6A'⊕E'4⊕E'7E'2⊕E'5⊕E'6A'⊕E'3⊕E'4⊕E'7More
E'3* A'⊕E'6E'3⊕E'6A'⊕E'3⊕E'62A'⊕E'3⊕E'6A'⊕2E'3⊕E'6More
E'4* A'⊕E'7E'3⊕E'4A'⊕E'1⊕E'7E'3⊕E'4⊕E'5A'⊕E'1⊕E'6⊕E'7More
E'5* A'⊕E'52A'⊕E'5A'⊕2E'52A'⊕2E'53A'⊕2E'5More
E'6* A'⊕E'3E'3⊕E'6A'⊕E'3⊕E'62A'⊕E'3⊕E'6A'⊕E'3⊕2E'6More
E'7* A'⊕E'1E'6⊕E'7A'⊕E'1⊕E'2E'5⊕E'6⊕E'7A'⊕E'1⊕E'2⊕E'3More
E''1* A'⊕E'2E''1⊕E''3A'⊕E'2⊕E'4E''1⊕E''3⊕E''5A'⊕E'2⊕E'4⊕E'6More
E''2* A'⊕E'4E''2⊕E''6A'⊕E'4⊕E'7E''2⊕E''5⊕E''6A'⊕E'3⊕E'4⊕E'7More
E''3* A'⊕E'6E''3⊕E''6A'⊕E'3⊕E'62A''⊕E''3⊕E''6A'⊕2E'3⊕E'6More
E''4* A'⊕E'7E''3⊕E''4A'⊕E'1⊕E'7E''3⊕E''4⊕E''5A'⊕E'1⊕E'6⊕E'7More
E''5* A'⊕E'52A''⊕E''5A'⊕2E'52A''⊕2E''53A'⊕2E'5More
E''6* A'⊕E'3E''3⊕E''6A'⊕E'3⊕E'62A''⊕E''3⊕E''6A'⊕E'3⊕2E'6More
E''7* A'⊕E'1E''6⊕E''7A'⊕E'1⊕E'2E''5⊕E''6⊕E''7A'⊕E'1⊕E'2⊕E'3More


Spherical harmonics and Multipoles
Symmetric Powers of Γxyz


Spherical Harmonics Yl / Multipole Symmetric Power [Γl(xyz)]
l 2l+1 Multipole Symmetry Rank [Γl(xyz)]
s (l=0) 1 Monopole A' 1 A'
p (l=1) 3 Dipole E'1⊕A'' 3 E'1⊕A''
d (l=2) 5 Quadrupole A'⊕E'2⊕E''1 6 2A'⊕E'2⊕E''1
f (l=3) 7 Octupole E'1⊕E'3⊕A''⊕E''2 10 2E'1⊕E'3⊕2A''⊕E''2
g (l=4) 9 Hexadecapole A'⊕E'2⊕E'4⊕E''1⊕E''3 15 3A'⊕2E'2⊕E'4⊕2E''1⊕E''3
h (l=5) 11 Dotricontapole E'1⊕E'3⊕E'5⊕A''⊕E''2⊕E''4 21 3E'1⊕2E'3⊕E'5⊕3A''⊕2E''2⊕E''4
i (l=6) 13 Tetrahexacontapole A'⊕E'2⊕E'4⊕E'6⊕E''1⊕E''3⊕E''5 28 4A'⊕3E'2⊕2E'4⊕E'6⊕3E''1⊕2E''3⊕E''5
j (l=7) 15 Octacosahectapole E'1⊕E'3⊕E'5⊕E'7⊕A''⊕E''2⊕E''4⊕E''6 36 4E'1⊕3E'3⊕2E'5⊕E'7⊕4A''⊕3E''2⊕2E''4⊕E''6
k (l=8) 17 256-pole A'⊕E'2⊕E'4⊕E'6⊕E'7⊕E''1⊕E''3⊕E''5⊕E''7 45 5A'⊕4E'2⊕3E'4⊕2E'6⊕E'7⊕4E''1⊕3E''3⊕2E''5⊕E''7
l (l=9) 19 512-pole E'1⊕E'3⊕E'5⊕E'6⊕E'7⊕A''⊕E''2⊕E''4⊕E''6⊕E''7 55 5E'1⊕4E'3⊕3E'5⊕E'6⊕2E'7⊕5A''⊕4E''2⊕3E''4⊕2E''6⊕E''7
m (l=10) 21 1024-pole A'⊕E'2⊕E'4⊕E'5⊕E'6⊕E'7⊕E''1⊕E''3⊕E''5⊕E''6⊕E''7 66 6A'⊕5E'2⊕4E'4⊕E'5⊕3E'6⊕2E'7⊕5E''1⊕4E''3⊕3E''5⊕E''6⊕2E''7
n (l=11) 23 2048-pole E'1⊕E'3⊕E'4⊕E'5⊕E'6⊕E'7⊕A''⊕E''2⊕E''4⊕E''5⊕E''6⊕E''7 78 6E'1⊕5E'3⊕E'4⊕4E'5⊕2E'6⊕3E'7⊕6A''⊕5E''2⊕4E''4⊕E''5⊕3E''6⊕2E''7
o (l=12) 25 4096-pole A'⊕E'2⊕E'3⊕E'4⊕E'5⊕E'6⊕E'7⊕E''1⊕E''3⊕E''4⊕E''5⊕E''6⊕E''7 91 7A'⊕6E'2⊕E'3⊕5E'4⊕2E'5⊕4E'6⊕3E'7⊕6E''1⊕5E''3⊕E''4⊕4E''5⊕2E''6⊕3E''7
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 C15h
L 2L+1 Term Splitting
S (L=0) 1 A'
P (L=1) 3 A'⊕E''1
D (L=2) 5 A'⊕E'2⊕E''1
F (L=3) 7 A'⊕E'2⊕E''1⊕E''3
G (L=4) 9 A'⊕E'2⊕E'4⊕E''1⊕E''3
H (L=5) 11 A'⊕E'2⊕E'4⊕E''1⊕E''3⊕E''5
I (L=6) 13 A'⊕E'2⊕E'4⊕E'6⊕E''1⊕E''3⊕E''5

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