Multipoles and Symmetry

Achim Gelessus, Walter Thiel, and Wolfgang Weber

Journal of Chemical Education 72, 505-508 (1995)

What is the first nonvanishing multipole of the icosahedral buckminsterfullerene C60? Are there general rules stating which multipoles are zero for a given point group? Answers to these and related questions are available from an exercise in group theory. The literature contains the principles of the connection between multipoles and symmetry, but their application to a wide range of multipoles and point groups seems to be lacking. We outline the necessary group-theoretical derivations and list explicit results for the 48 most common point groups. This information is relevant for the study of intermolecular interactions in which the electrostatic energy term can be expanded in multipole/multipole interactions.

Transformation Properties for Multipoles l=1 to l=4
dipole(p) quadrupole(d) octopole(f) hexadecapole(g)
l=1 l=2 l=3 l=4
C1 3A 5A 7A 9A
Cs 2A'+A'' 3A'+2A'' 4A'+3A'' 5A'+4A''
Ci 3Au 5Ag 7Au 9Ag
C2 A+2B 3A+2B 3A+4B 5A+4B
C3 A+E A+2E 3A+2E 3A+3E
C4 A+E A+2B+E A+2B+2E 3A+2B+2E
C5 A+E1 A+E1+E2 A+E1+2E2 A+2E1+2E2
C6 A+E1 A+E1+E2 A+2B+E1+E2 A+2B+E1+2E2
C7 A+E1 A+E1+E2 A+E1+E2+E3 A+E1+E2+2E3
C8 A+E1 A+E1+E2 A+E1+E2+E3 A+2B+E1+E2+E3
D2 B1+B2+B3 2A+B1+B2+B3 A+2B1+2B2+2B3 3A+2B1+2B2+2B3
D3 A2+E A1+2E A1+2A2+2E 2A1+A2+3E
D4 A2+E A1+B1+B2+E A2+B1+B2+2E 2A1+A2+B1+B2+2E
D5 A2+E1 A1+E1+E2 A2+E1+2E2 A1+2E1+2E2
D6 A2+E1 A1+E1+E2 A2+B1+B2+ E1+E2 A1+B1+B2+ E1+2E2
C2v A1+B1+B2 2A1+A2+B1+B2 2A1+A2+2B1+2B2 3A1+2A2+2B1+2B2
C3v A1+E A1+2E 2A1+A2+2E 2A1+A2+3E
C4v A1+E A1+B1+B2+E A1+B1+B2+2E 2A1+A2+B1+B2+ 2E
C5v A1+E1 A1+E1+E2 A1+E1+2E2 A1+2E1+2E2
C6v A1+E1 A1+E1+E2 A1+B1+B2+E1+ E2 A1+B1+B2+E1+ 2E2
C2h Au+2Bu 3Ag+2Bg 3Au+4Bu 5Ag+4Bg
C3h A''+E' A'+E'+E'' 2A'+A''+E'+E'' A'+2A''+2E'+E''
C4h Au+Eu Ag+2Bg+Eg Au+2Bu+2Eu 3Ag+2Bg+2Eg
C5h A''+E'1 A'+E'2+E''1 A''+E'1+E'2+E''2 A'+E'1+E'2+E''1+ E''2
C6h Au+E1u Ag+E1g+E2g Au+2Bu+E1u+E2u Ag+2Bg+E1g+2E2g
D2h B1u+B2u+B3u 2Ag+B1g+B2g+B3g Au+2B1u+2B2u+2B3u 3Ag+2B1g+2B2g+2B3g
D3h A''2+E' A'1+E'+E'' A'1+A'2+A''2+E'+E'' A'1+A''1+A''2+2E'+E''
D4h A2u+Eu A1g+B1g+B2g+Eg A2u+B1u+B2u+2Eu 2A1g+A2g+B1g+B2g+ 2Eg
D5h A''2+E'1 A'1+E'2+E''1 A''2+E'1+E'2+E''2 A'1+E'1+E'2+E''1+ E''2
D6h A2u+E1u A1g+E1g+E2g A2u+B1u+B2u+E1u+ E2u A1g+B1g+B2g+E1g+ 2E2g
D8h A2u+E1u A1g+E1g+E2g A2u+E1u+E2u+E3u A1g+B1g+B2g+E1g+ E2g+E3g
D2d B2+E A1+B1+B2+E A1+A2+B2+2E 2A1+A2+B1+B2+ 2E
D3d A2u+Eu A1g+2Eg A1u+2A2u+2Eu 2A1g+A2g+3Eg
D4d B2+E1 A1+E2+E3 B2+E1+E2+E3 A1+B1+B2+E1+ E2+E3
D5d A2u+E1u A1g+E1g+E2g A2u+E1u+2E2u A1g+2E1g+2E2g
D6d B2+E1 A1+E2+E5 B2+E1+E3+E4 A1+E2+E3+E4+ E5
S4 B+E A+2B+E 2A+B+2E 3A+2B+2E
S6 Au+Eu Ag+2Eg 3Au+2Eu 3Ag+3Eg
S8 B+E1 A+E2+E3 B+E1+E2+E3 A+2B+E1+E2+E3
T T E+T A+2T A+E+2T
Th Tu Eg+Tg Au+2Tu Ag+Eg+2Tg
Td T2 E+T2 A1+T1+T2 A1+E+T1+T2
O T1 E+T2 A2+T1+T2 A1+E+T1+T2
Oh T1u Eg+T2g A2u+T1u+T2u A1g+Eg+T1g+T2g
Cv ++ +++ +++ + +++ ++
Dh +u+ u +g+ g+ g +u+ u+ u+ u +g+ g+ g+ g+ g
I T1 H T2+G G+H
Ih T1u Hg T2u+Gu Gg+Hg

Transformation Properties for Multipoles l=5 to l=6
32-pole(h) 64-pole(i)
l=5 l=6
C1 11A 13A
Cs 6A'+5A'' 7A'+6A''
Ci 11Au 13Ag
C2 5A+6B 7A+6B
C3 3A+4E 5A+4E
C4 3A+2B+3E 3A+4B+3E
C5 3A+2E1+2E2 3A+3E1+2E2
C6 A+2B+2E1+2E2 3A+2B+2E1+2E2
C7 A+E1+2E2+2E3 A+2E1+2E2+2E3
C8 A+2B+E1+E2+2E3 A+2B+E1+2E2+2E3
D2 2A+3B1+3B2+3B3 4A+3B1+3B2+3B3
D3 A1+2A2+4E 3A1+2A2+4E
D4 A1+2A2+B1+B2+3E 2A1+A2+2B1+2B2+3E
D5 A1+2A2+2E1+2E2 2A1+A2+3E1+2E2
D6 A2+B1+B2+ 2E1+2E2 2A1+A2+B1+B2+ 2E1+2E2
C2v 3A1+2A2+3B1+3B2 4A1+3A2+3B1+3B2
C3v 2A1+A2+4E 3A1+2A2+4E
C4v 2A1+A2+B1+B2+ 3E 2A1+A2+2B1+2B2+ 3E
C5v 2A1+A2+2E1+2E2 2A1+A2+3E1+2E2
C6v A1+B1+B2+2E1+ 2E2 2A1+A2+B1+B2+ 2E1+2E2
C2h 5Au+6Bu 7Ag+6Bg
C3h 2A'+A''+2E'+2E'' 3A'+2A''+2E'+2E''
C4h 3Au+2Bu+3Eu 3Ag+4Bg+3Eg
C5h 2A'+A''+E'1+E'2+E''1+ E''2 A'+2A''+2E'1+E'2+E''1+ E''2
C6h Au+2Bu+2E1u+2E2u 3Ag+2Bg+2E1g+2E2g
D2h 2Au+3B1u+3B2u+3B3u 4Ag+3B1g+3B2g+3B3g
D3h A'1+A'2+A''2+2E'+2E'' 2A'1+A'2+A''1+A''2+ 2E'+2E''
D4h A1u+2A2u+B1u+B2u+ 3Eu 2A1g+A2g+2B1g+2B2g+ 3Eg
D5h 2A'1+A''2+E'1+E'2+ E''1+E''2 A'1+2A''2+2E'1+E'2+ E''1+E''2
D6h A2u+B1u+B2u+2E1u+ 2E2u 2A1g+A2g+B1g+B2g+ 2E1g+2E2g
D8h A2u+B1u+B2u+E1u+ E2u+2E3u A1g+B1g+B2g+E1g+ 2E2g+2E3g
D2d A1+A2+B1+2B2+ 3E 2A1+A2+2B1+2B2+ 3E
D3d A1u+2A2u+4Eu 3A1g+2A2g+4Eg
D4d A1+A2+B2+E1+ E2+2E3 A1+B1+B2+2E1+ 2E2+E3
D5d A1u+2A2u+2E1u+2E2u 2A1g+A2g+3E1g+2E2g
D6d B2+E1+E2+E3+ E4+E5 A1+B1+B2+E1+ E2+E3+E4+E5
S4 2A+3B+3E 3A+4B+3E
S6 3Au+4Eu 5Ag+4Eg
S8 2A+B+E1+E2+2E3 A+2B+2E1+2E2+E3
T E+3T 2A+E+3T
Th Eu+3Tu 2Ag+Eg+3Tg
Td E+T1+2T2 A1+A2+E+T1+2T2
O E+2T1+T2 A1+A2+E+T1+2T2
Oh Eu+2T1u+T2u A1g+A2g+Eg+T1g+ 2T2g
Cv +++ ++ +H +++ ++ +H+I
Dh +u+ u+ u+ u+ u+ Hu +g+ g+ g+ g+ g+ Hg+Ig
I T1+T2+H A+T1+G+H
Ih T1u+T2u+Hu A1g+T1g+Gg+Hg

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Last update November, 26th 2007 by A. Gelessus