Multipoles and Symmetry Achim Gelessus, Walter Thiel, and Wolfgang Weber Organisch-chemisches Institut, Universitat Zurich, CH-8057 Zurich
What is the first nonvanishing multipole of the icosahedral buckminsterfullerene Cso? Are there general rules stating which multipoles are zero for a given point group? Answers to these and related questions are available from a n exercise in group theory. The literature contains the principles of the connection between multipoles and symmetry (1, 21, hut 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 (3).This information is relevant for the study of intermolecular interactions ( 4 4 ) in which the electrostatic energy term can he expanded in multipole/multipole interactions.
sentation r,, respectively (xl(R)= x(R) in the usual texb book notation (3)). The application of this formula is shown for the point group C3"below, where R' (first diagram) and the characters rdR1. (second diamam) . - . for each of the three different symmetry classes are given for the first five product representations Ti (1 = 0-4).These Droduct re~resentationsare generally redicible to lower &mension& irreps with the use of the standard reduction formula (3, 11).
,..
Derivation and Results The multipole moment Ql, of the total charge distribution p(?) is defined (1,2, 7) a s
where Yl, denotes a spherical harmonic with quantum numbers 1 and m. The multipole Ql consists of all the (21 + 1) multipole moments Qlm,so the existence of a multipole Q, requires a t least one nonvanishing multipole moment Qi,. As i s known from standard textbooks a n integral can he nonzero only if the integrand is totally symmetric (3, 8). Because the total charge distribution p(r) and the factor # transform according to the totally symmetric irreducible representation (irrep), the symmetry of the integrand is completely determined by t h e spherical harmonic. The investigation of the transformation properties of a multipole is therefore equivalent to the investigation of the transformation properties of the corresponding set of spherical harmonics. Determining Transformation Properties
The transformation properties of the dipole and quadrupole moments can easily be determined by inspection of ordinary character tables. They usually list the coordinates and the auadratic forms of coordinates to indicate the rcprcscntat~onarcording to which they transform. The transtbrmatim pn~pcniesof higher multipolcs Q, 11> 1 can he ohtainedefro& the directsymmetrized product of 1 dipoles. The dimension of this direct symmetrized product is equal to the number of distinguishable terms formed by 1 coordinates lx, y, zl which, for example, is 6 in the case of 1 = 2 (because xy = yx, xz = zx, and yz = zy). A general recursion formula for the direct symmetrized product of a three-dimensional representation such a s the dipole has been derived (8-10)
whereRbnd X ~ Rdenote I the 1-fold application of the symmetry operation R and the character of the product repre-
Decomposing Reducible Representations The reducible representations T2, r3,... can be decomposed into series of multipoles with either even or odd 1 values. This follows from the parity properties of the multipoles. Because the dipole is of ungerade parity, its lth symmetrized power can be expanded into a series that contains all multi oles up to the 2' pole that are of the same parity . . as the 2 pole (gerade for even 1, ungerade for odd I). Hcnre the 6-dimensional repn:sentarion I > consists of thc monopole and the (~uadrupolcboth gcrade, I = '2,. and the 10-dimensional refiesent-ation r3ofthe dipole and the octopole (both ungerade, 1 = 3). In order to obtain the irreps of a multipole Q1 from the irreps of the lower multipoles must be subtracted. For the determination of the hexadecapole in the example of the C3"point group, two A, irreps (one for the monopole and one for the quadrupole) and two E irreps (for the quadrupole only) must be subtracted from r4.Therefore the set of the nine hexadecapole moments transforms a s 2A, + A z + 3E.
P
Occurrence of Multiples Acomplete list of the transformation properties up to 64poles of all 48 point groups given in Cotton's character tables (3)is presented in Tables 1 and 2. This compilation extends similar lists that contain only the number of the totally symmetric irreps (I,2,121 or cover only the crystallographically important point groups (13).Inspection of the tables immediately identifies all nonvanishing totally Volume 72 Number 6 June 1995
505
Table 1. Transformation Properties for Multipoles I = 1 to I = 3
symmetric multipole moments. Beyond this explicit information, there a r e some general rules about t h e occurrence of multipoles i n molecules of c e r t a i n point groups t h a t a r e now derived. 1. The monopole Qo always transforms as the totally symmetric irrep because the spherical harmonic Y m is a constant factor. Therefore all molecules can be charged. 2. For parity reasons, multipoles Qi with odd 1 values cannot occur in molecules with a center of inversion because t h e odd4 spherical harmonics Yi, transform antisymmetrically under an inversion. 3. If a molecule has a dipole, then all other multipoles occur. This can he shown most easily by a closer inspection of the spherical harmonics. A molecule can always be oriented so that the dipole coincides with the z-axis. For this orientation there are only symmetry operations under which t h e spherical harmonic Y l o transforms symmetrically (i.e., rotations about the raxis and mirror planes including the zaxis). All the functions Ylo(1> 1)have the same transformation properties as Yio. Therefore a t least the multipole moment Ylooccurs for all multipoles QI. 4. There is only one nonzero component of the quadrupole in all symmetric tops (molecules with exactly one high-order rotation axis C,, n > 2, for example, henzene). By convention, the z axis is the C, axis and transforms as A or 5, whereas the degenerate x and y coordinates transform asE. Because the square of any irrep contains t h e totally symmetric representation exactly once, the direct symmetrized product of the dipole contains exactly two totally symmetric representations
dipole (p) I= 1
T xT=lA
3A 2A'+A
5A 3A'
Ci
3-4" A+2B A+E A+E
5.40 3A+2B A+2E A+2B+E A + El + E2 A+&+& A+E,+Ez A + El + E2 ZA+BI+BZ+B~ A1+2E AI+BI+Bz+E A,+E,+Ez A, + El + Ez 2A, + A2 + B j + Bz A1+2E Al+B,+B2+E A1 + E, + E2 A i + EI + Ez 3Ag + 2Bg
7Au 3A + 4 8 3A + 2E A +2B+2E A + E r +2E2 A+ZB+El+Ez A E, + €2 + €3 A + El E2 + €3 A + 2 & +ZB2++1?83 A1 + 2 A z + 2 E Az+B1+82+2E A2 + El + 2Ez Az+B~+Bz+EI+Ez 2A7 + A z + Z B ? +2Bz 2Al+Az+2E AI+BI+Bz+~E Al+El+2Ez A1 + BI + 82 + EI + EZ 3A" + 48"
A'+ E ' + E"
2A'+A+E'+EZ' Au + 28" + 2E" A" + El' + E2' + 4''
C2 C3
C4
Ag + Elg+ Ezg
4A'
+
+ 3A"
+
2Ag + B I + ~ BZO+ 839 At' + E' + E"
A. + 2Bu + Elu+ EZU Au + 2BfU+ 2Bzu + 2B3" 2.41+'Az' + A," +Az"+E'+E"
AZU+ Eu A2" + El'
A79 + Blg + Bzg + Eg A,'+ E2'+ El"
Azu + BIU + BZU + 2.5" A2" + El' + Ez' + E2"
Am + Em Azu + El" Bz+E
A1g + fig + Ezg Aig + Elg+ Ezg Al+Bl+Bz+E
A2u + Eu 8 2 + El A2" + El" 82 + E, B+E
Afg + 2Eg A, + E z + E3
A2" + B l " + B2" + El" + E ~ u AZU+ EIU+ Ezu + E m Af+Az+Bz+ZE ,A ,, + 2A2" + 2Eu BZ+EI+EZ+E~ Azu + Elu+ 2E2u 5 2 + EI + E3 + E4 2A+B+2E
+ nu
Dooh
Alg + hg + €20 A,+Ez+E5 A+2B+E
G+llg+Ag
I
TI
H
~+Il,+Au+0, T2 + G
ih
TI"
Ho
Tzu + Gu
+ ..., H x H = l A + ...
Journal of Chemical Education
7A
+ 2A"
AS + 2Bg + Eg A' + Ez' + El''
which must he the monopole. Hence, the first nanvanishingmultipole oftetrahedral molecules such as methane is t h e aetopole. Octahedral molecules like sulfur hexafluoride do not even have an actopale due to parity (see 21, so their first nonvanishing multipale is the hexadecapole.
506
octopole (f) I= 3
h Cs
( A o r B , E ) x ( A o r B , E ) = M +... One of t h e totally symmetric representations is the monopale of the molecule; the other one is the single nonzero quadrupole moment (Qzounder the above axis conventions). 5. If all multipole moments Qi, of a given multipole Qi (1 > 0)together span an irreducible basis, this multipole vanishes. This representation m u s t ~ b edegenerate because there is more than one component far any 1 > 0, and thus it cannot he totally symmetric. This explains most easily the absence of dipoles in t h e cuhic paint groups and of quadrupoles in the icosahedral groups. 6. No quadrupole occurs in cubic point groups, and not even a hexadeeapale in icosahedral paint groups. In these eases all comoonents of the d i ~ o l and e the cluadmpole belong to one degenerate irrep ( T andH), so the direct symmetrized product farmed from them contains the totally symmetric representation exactly once
quadrupole (d) I= 2
7. The 64-pole occurs in all molecules regardless of their symmetry This is the first nonvanishing multipole for the most symmetrical molecules possible that belong to the icosahedral group Zh. The 64-pale is of even parity and thus occurs for all molecules that have a nonvanishing dipole or quadrupole (symmetric tops). For the seven eubie point groups, the explicit reduction confirms the oceurrenee of the 64-pole.
Table 2. Transformation Propertiesfor Multipoles I = 4 to I= 6
hexadecapole (g)
5A'
+ 4A"
9.49 5A + 4B 3A + 3E 3A+2B+2E A + 2Ei + 2Ez A+ZB+Ei+2Ez A+E,+Ez+2E3 A+2B+Et+Ez+E3 3A + 281 + 2Bz + 2B3 2A, + A z + 3 E 2A,+Az+Bf+B2+2E Ai +2Ei +2E2 Ai+Bi+Bz+E,+2Ez 3Ai+2Az+ZBi+Z& 2Ai+A2+3E 2Ai+Az+Bi+Bz+2E A, + 2Ei + 2E2 Ai+B,+Bz+Ei+2Ez 5Ag + 4Bg A ' + 2 A " + 2 E ' + E" 3Ag + 2Bg + 2Eg A'+ E l ' + W + El"+& Ag + 2Bg + Eig + 2Ezg 3Ag + 2 4 + 2B29 + 2B3g Al'+Al"+ Az"+ 2 E ' + E" 2Aig + Asg + Big + Bzg + 2Eg A~'+EI'+&'+EI"+E~" Aig + Big + Bzg + Eig + 2Ezg Aig + Big + Bzg + Eig + E a + E* 2Ai+Az+Bi+Bz+2E 2Aig + Azg + 3Eg A,+B,+Bz+E,+Ez+O Aig + 2Eig + 2 E a Ai+Ez+E3+Ed+Es 3A+28+2E 3Ag + 3Eg A+2B+b+Ez+6 A+E+2T A, + Eg + 2Tg Ai+E+T,+Tz Ai+E+T,+Tz Aig + Eg + Tig + Tzg
F+n+a+o+r g+ng+ag+og+rg G+H Go + HO
.
.,
32-oole (h) 1=5
64- ole lil
11A 6A' + 5A" 1IA" 5A + 6B 3A + 4E 3A+2B+3E ~A+~EI+~Ez A+2B+2Ei+2Ez A+Ei+2Ez+2E3 A + Z B + E , +E2+2E3 2A + 381 + 382 + 383 A, +2A2+4E A~+~AZ+BY+BZ+~E 2A1+A2+2Ei+2Ez Az+Bi+Bz+2Ei+2Ez 3A1+ 2A2 + 38, + 382 2Ai+Az+4E 2Ai+Az+Bi+Bz+3E 2Ai +Az+2Ei +2Ez Ai+Bi+Bz+ZEj+2Ez 5Au + 6Bu 2A' +A" + 2E' + 2E" 3Au + 2B" + 31% 2A' + A"+ El'+ E i + El"+ E2" A" + 2B" + 2Ei" + 2Ezu 2A" + 3Biu + 382" + 3B3u A,'+&+ A2"+ 2 E ' + 2 E " Am + 2Am + Biu + Bzu + 3Eu 2A1' +A$' + El' + E2' + El" + E2" AZ"
+ Bl" + B2u + 2.5" + 2 5 "
&u
+ Biu + BZU+ Eiu + E2u + 2E3u
Ai+Az+Bi+ZB2+3E Aiu+2Azu+4Eu Ai+Az+&+E,+Ez+ZE3 A," + 2Azo + 2Eiu + 2Ezu BZ+E~+EZ+E~+E~+E~ 2A+3B+3E 3Au + 4Eu 2A+B+Ei+Ez+ZE3 E+3T Eu + 3Tu E + Tf+2T2 E + 2T,+Tz Eu + 2T7u+Tzu l?+II+A+Q+r+H z+IIu+Au+@u+ru+Hu Ti + T2 + H Ti" + Tzu + H"
Discussion
The classical electrostatic interaction between two molecules carrying multipoles Ql, and Qlb falls off as R".-la1 with increasing distance R . For the interaction of two buckminsterfullerene molecules with Ih symmetry, the first nonvanishing contribution to the electrostatic energy is the 64-polel64-pole term. This results in an extremely
steep and short-range potential with a distance dependence of R-13.Thus, the electrostatic contribution to the interaction potential can be neglected to a first approximation. The long-range interaction potential of such highly svmmetrical molecules will be dominated bv the dis-oersion that shows an ~4 dependence asathe leading term (induced dipole-induced dipole interaction). These considerations are qualitative. A quantitative asVolume 72 Number 6 June 1995
507
sessment of the different interaction terms require quantum-chemical calculations. I t should be emphasized that the applicability of Tables 1 and 2 is not restricted to multipole analyses. Other posare the 'plitting of a set of atomic tals or the splitting of degenerate electronic states of a n atom in an environment of lower symmetry (ligand field theory). Literature Cited 1. Ruekingham. A. D. In Inlermolpculor Inhrnclions: From Diofomics lo Biopo?vmris: Pullmnnn B.. Ed.: W h y : New York, 1978: pp 1-67.
508
Journal of Chemical Education
2. Buekhgham,A. D. InPhysienl Chemistry ~ ~ A d v o n c 7kolise; ed Ey"ng, H.; Jost W.: Henderson. D., Eds.; Academic: New York, 1970; Vol 4, C h a w ? 8. 3. cotfan. E A. ~ h ~ ~ i o~f ~ m ~ u pTi~ P O ~ V3,1~d ~ d~wiley: ; i N ~~ YO=*. W~ 1990. t i 4. stone,^. ~ . ; ~ nS.~L. eJ.,phys c h p m 1 ~ 8 8 , ~ ~ 2 . 3 3 2 5 3 3 3 5 .
::~'~~;~:~:~:~u;i~;,Eyt:z$j;:t7-142,
7. Panofsky W K H.; Phillips, M. Classical Electricity ond Magnetism, 2nd ed.: Addim"-wedey: &ad!ng. MA. 1962. 8. c a l i f a m S. ~ i b m t i o n dstates: wiley: ~ e w ~ o r 1976; k . p 201. 9. Boyle, L. L. Ini.J Quantum Chem 1972.6.725-746. 10. Planelies, J.: Zieouieh-Wilson. C. M. Int. J Qanniilm Chem. 1993.47.319323, 11, nnkham. M.: Gmvp Theory a n d Quantum Mechanics: McGraw-Hill: New York, 1964. 12. Bhagauantam. S.; Suryanarayana. DActa Cryst. 1949.2.21-26. 13. Korter, G. E:Dimmock. J. 0.; Wheeler R. G.:Statz, H. Pmperties of the Thirly~Tmo Point Groups; MIT Press: Camb"dge, MA. 1960.
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