Indian Journal of ChemistryVol. 31 MB, May 1992, pp. F51-F55

Simple structural and bonding representations of fullerenesA Rathna & J Chandrasekhar*

Department of Organic Chemistry, Indian Institute of Science, Bangalore 560 012

Received 20 February 1992

Bonding in buckminsterfullerene, C60, can be described in terms of a unique canonical representa-tion in which all six membered rings have a benzenoid Kekule structure while the pentagons are allmade of exclusively single bonds, The corresponding valence bond structure reflects the full symmetryof the molecule and is consistent with the observed bond length variations. Computational support forthe bonding description is provided using localized MO's obtained at the MNDO level. The require-ment of benzenoid structures for all the' hexagons can be used as a criterion of stability of fullereneswhich complements the pentagon isolation rule. A convenient two-dimensional representation of thefullerene structures incorporating the above bonding description is suggested, especially for use in me-chanistic discussions.

The first report} of the mass spectral characterisa-tion of a sixty carbon cluster along with the re-markable structural proposal derived from thegeodesic dome of Buckminster Fuller (1) immedi-ately attracted the attention of numerous theoreti-cal chemists", The intriguing question concerningthe nature of bonding in the truncated icosahedralframework was the focus of many of these stud-ies. Simple valence rules can be satisfied if all thecarbon atoms of C60 are considered to be uni-formly Sp2 hybridized, with 60 p orbitals of pseu-do Jt symmetry participating in a three-dimen-sional aromatic interaction. Effectively, 60 elec-trons are delocalized over the surface of thespheroidal molecule. In fact, the free electronmodel! reproduced the essential features of theelectronic structure of the molecule, viz., thesymmetry and the associated degeneracy of all themolecular orbitals. Huckel calculations confirmedthe pattern", while more sophisticated methodslike the INDOS, PRDD06, MND02.7 AMI8, andab initio=? procedures provided additional de-tails. These calculations led to reliable structuraland spectral parameters, in particular those asso-ciated with vibrational, electronic and photoelec-tron spectra. Many of these insights were ob-tained even before the recent explosion of activityinvolving the fullerenes following the developmentof simpler procedures for their synthesis and ex-perimental study!",

While the theoretical results mentioned abovehave been quite useful, a simpler representationof the structure and bonding of fullerenes is prob-ably of greater value, especially for understandingtheir reactivity patterns. The formation of newbonds and the reorganisation of existing bonds inthese compounds following a chemical reaction

are rather difficult to visualize at present withinthe delocalized bonding models. For example, themechanistic details of a reaction such as the Birchreduction of C60 appear complicated, while thecorresponding information for a typical aromaticmolecule is textbook material. In this paper, weconsider a simple approach based on localizedmolecular orbitals for understanding the structure,stability and reactivity offullerenes. We discuss prev-ious studies employing related concepts and also pro-vide additional details.

Computational detailsThe standard MOPAC package" was imple-

mented on a CYBER 992 -computer and en-hanced to handle molecules containing more than60 non-hydrogen atoms. Using this program, thegeometry of C60 was fully optimized imposing Ihsymmetry at the MNDO level. The wave func-tions were successively transformed until a set oflocalized molecular orbitals was obtained 12. Thedefault criteria for localization built in the pro-gram was used without modification. Convergenceoccurred in 48 cycles.

A modified version of Jorgensen's program 13, in-stalled on the VAX 8810 computer, was employ-ed to plot the contours of the MNDO localizedwave functions. The PERGRA program 1\ deve-loped for a PC, was also used to pictorially repre-sent the relative coefficients of the AO's in thevarious types of localized molecular orbitals.

Results and DiscussionA convenient starting point for understanding

the bonding in C60 is provided by the experimen-tal and computed structural parameters of thismolecule (Table 1). Many of the more quantitative

F52 INDIAN JCHEM,SEC.A&B,MAY 1992

theoretical methods clearly predict the presenceof bond alternancy in C60. The bond shared bytwo hexagons (6-6) is relatively short, while thecommon bond between a hexagon and a pentagon(6-5) is long. NMR studies" and the more recentneutron", Xvray'? and electron diffraction" re-sults also confirm that there are effectively twotypes of bond lengths in C60. A range of distancesis observed in the solid state, due to intermolecu-lar interactions 16, 17. However, the general patternfollows the calculated data, which refer to the gasphase.

The bond length variations suggest the presenceof alternating single and double bonds in C60. Thecomputed bond orders reflect the changes inlengths. A linear muffin-tin orbitals (LMTO) cal-culation gave bond orders of 1.54 and 1.41, re-spectively for the 6-6 and the 6-5 bonds!". Thecorresponding values computed by us at theMNDO level (1.51 and 1.10) show a larger differ-ence. This is also reflected in the greater degreeof bond alternancy obtained at this theoretical le-vel (Table 1).

In addition to bond orders, another criterioncan be used to qualify the differences in the na-ture of bonds in a molecule. The individual mo-lecular orbitals can be transformed into localizedorbitals leaving the total wave function un-changed. Although the nature of the transformedMO's sometimes depend on the localizationprocedure employed, especially in polycyclic aro-matic compounds and three-dimensional polyhe-dral boron hydrides"; the results are quite in-structive. A previous study using the Boys method

Table I-Calculated and experimental bond lengths (A) in C60

Method: r, r2 Reference

6-6 6-5Calculated

MNDO: 1.400 1.474 2,7AMI: 1.385 1.464 8PRDDO: 1.360 1.436 6LMTO: 1.410 1.440 19HF/STO-3G: 1.376 1.463 8a,9aHF/DZ: 1.368 1.451 9bHF/DZP: 1.372 1.453 9cHF/TZP: 1.370 1.448 9cMP2: 1.405 1.445 9d

ExperimentalSolid State NMR : 1.40 1.45 15Single crystal 1.340-1.391 1.378-1.561 17X-ray diffraction: av.1.355 av.l.467Electron 1.401 1.458 18diffraction:Neutron powder 1.366-1.412 1.420-1.487 16diffraction' : av.1.391 av.1.455

/

of maxirnising the sum of squares of orbital cen-troids at the PRDDO level showed that the MO'sof C60 can be localized as efficiently as in ben-zene", The calculations led to three types of local-ized bonds in C60, viz., a well defined 6-5 0 bondand a pair of 1: bonds. The latter correspond to abent bond representation of the 6-6 double bond.The pyramidality at carbon makes the two 1:bonds non-equivalent. The nature of hybridizationof the three unique bonds were compared withthe corresponding data obtained for benzene atthe same theoretical level",

The localized bonds obtained in the presentstudy using MNDO wave functions correspond toan even simpler bonding description. Again, threeunique types of localized MO's were obtained.The orbital contour diagrams (Fig. 1) clearly in-dicate that one of them is the 6- 5 0 bond foundin the earlier study. However, the 6-6 doublebond appears as a 0 bond and a pseudo :Jt bond,rather than as a pair of 1: bonds. An alternativepictorial representation of these orbitals whichreflects the relative atomic orbital coefficients and

(a)

c

(b) c

~-gy~c \c

Fig. I-Contours of the localized orbitals of C60• Only a smallfragment of the molecule is shown for clarity [(a) 6-5 0 bond;

(b) 6-6 o bond; (c) 6-6 11 bond]

RATIINA et al.: STRUCTURAL & BONDING REPRESENTATIONS OF FULLERENES

hybridization (Fig. 2) is also instructive. The ext-ent of localization and the nature of orbital distor-tions are clearly seen. For example, the Jt bond isnot completely localized, as evident from thesmall lobes on the adjacent atom orbitals. The Jt

bond is also splayed outwards on the exterior ofthe spheroidal molecule, reflecting the pyramidal-it)' at the carbon atoms. The orbital contours in-dicate significant differences in the distortion ofthe 0 and rr MO's. The former has a greaterdensity on the outside, while the reverse is true orthe rt bond. Such orbital effects have been asso-ciated with interesting chemical consequences inmany organic systems". In the case of C60, en-dohedral interactions are expected to be quite dif-ferent in magnitude compared to interactions ofthe Jt cloud with reagents on the exterior.

The above analysis clearly implies that buck-minsterfuUerene consists of 60 single and 60 dou-ble bonds. The pentagons are constructed of ex-clusively single bonds. On the. other hand, the

Fig. 2-An alternative view emphasizing the relative LCAO coe-fficients oflocalized orbitals ofC80• Only a small fragment of themolecule is shown for clarity, [(a) 6-5 (J bond; (b) 6-6 (J bond; (c)

6-6nbondl

/

F53

hexagons are made of alternating single and dou-ble bonds. In effect, the electronic structure ofCoo is well represented by a single canonicalstructure, which remarkably reflects the icosahe-dral symmetry of the molecule. In contrast, reson-ance structures such as the Kekule forms for ben-zene do not have the full symmetry of the mole-cule. Multiple related structures have to be in-voked to restore the molecular symmetry. Forbuckminsterfullerene, alternative resonance formsprobably do not contribute as much as the uniquehighly symmetric pairing scheme which parallelsthe localized molecular orbital description.

The localized bonding description has beenused to advantage earlierI6.17.22.For example, pen-tagon faces are readily recognised as being relat-ively electron deficient, while the hexagons areelectron rich. Intermolecular interactions, in solidfullerene'P-" and perhaps also in a donor-acceptorcomplex'", are determined by these electron dens-ity variations.

The valence bond description also enables theextension of previous rules of stability of polycyc-lic conjugated compounds to the fuUerene family.For example, the known antiaromatic character ofpentalene has been invoked by Taylor'? to suggestthat structures with shared pentagons are un-stable. This factor can be used to rationalize the'pentagon isolation rule' and the uniqueness of theC60 structure. Buckminsterfullerene represents thesmallest closed structure with exclusively five andsix membered rings in which no two pentagonsare adjacent. Extending the argument, it has beenproposed that structures with a double bond in apentagon face are destabilized. This proposal wasused to rationalize the relative stabilities of sever-al of the larger fullerens, C, (n >60).

We prefer an alternative interpretation of thefactors contributing to the relative stabilities offuUerenes. The reference system, graphite. is ofcourse aromatic with its planar sheets of hexa-gons. The structure can be folded to yield a closedsurface by introducing 12 pentagons. In order tohave maximum stability for the cluster, it is essen-tial that all the hexagons retain benzenoid charac-ter. Alternative canonical structures like the qui-nanoid forms do not lead to stabilization. That is,the spin pairing scheme as in the Kekule struc-ture 2a is more stable than those in 2b-2e. Thesmallest closed structure made of pentagons andhexagons in which every double bond is optimallypart of a benzenoid structure such as 2a is thefuUerene, C60. Such an ideal bonding situation isnot possible in the larger clusters, say C70 or Cgo.

F54 INDIANJCHEM,SEC.A&B,MAY 1992

2d 2•.

In the DSh form of C70, for any given resonancestructure, at least five of the hexagons would notretain their Kekule-type benzenoid spin pairing.Hence, the presence of additional hexagons in thiscluster relative to C60 does not lead to correspon-dingly greater stability. The same reasoning ap-plies to the larger clusters as well. Ten hexagonsof the C80 cluster (assuming a five-fold symmetricstructure) would lose benzenoid character.

In our revised proposal for rationalizing the rel-ative stabilities of fullerenes, the focus has beenshifted from the pentagons to the hexagons. Rath-er than suggesting that double bonds avoid pen-tagons", we insist that hexagons should adoptbenzenoid structures for stability. The latter is awell known empirical rule for understanding thestabilities of polycyclic aromatic molecules. It isinteresting that it can be used for three-dimen-sional clusters as well. However, as emphasizedby earlier workers+", conjugation contributes on-ly partly to the stabilization of fullerenes. Straineffects associated with the 0 skeleton, pyramidal-ity at the carbon, and the varying magnitude of 3toverlap must all be taken into account for a com-plete understanding of the stability of these com-pounds. In the gas phase, entropic and other fac-tors have also been suggested to be important'".

The localized bonds of C60 can be incorporatedin an exploded view of the molecule, employedearlier by Saunders". The procedure of drawingsuch a structure (Fig. 3) is similar to that used oc-casionally for dodecahedrane, C2oHzo. A pentagonis first drawn, followed by five bonds from eachof the vertices. In the case of dodecahedrane, ad-ditional 10 atoms are drawn staggered with re-spect to the first set and all the bonds are suitablyconnected to complete the structure. On the otherhand, for pictorially representing fullerene, a setof 5 bonds (two-carbon units) is drawn, parallel tothe edges of the base pentagon. Connecting thenearest atoms would lead to corannulene, with apentagon surrounded by 5 hexagons. Another setof 10 atoms (5 bonds) is drawn, staggered withrespect to the previous set of bonds and mainta-

Fig. 3-Projected two-dimensional representation of C60

ining five-fold symmetry. The nearest bondsare connected and the process is repeatedup to Cso. The five bonds (10 atoms) of the finalset are drawn radially. The nearest atoms are con-nected and finally the outermost five atoms arelinked as a pentagon. The localized bonds, alter-nating single and double bonds, can then be easilyinserted by ensuring that no double bonds areplaced in any pentagon. For C70 or CgO'the sameprocedure can be employed, with additional setsof 10 carbon units inserted beyond Cso beforedrawing the outermost ten atoms. The structureof C70 drawn following this procedure with a rep-resentative canonical structure is shown in Fig. 4.

The projected views of course do not reflectthe spheroidal shape of the fullerenes. In particu-lar, the above representations (Figs 3 and 4) em-phasize only the five-fold symmetry of the mole-cules. However, it is a straight forward exercise toredraw the structures with three-fold or two-foldsymmetry being highlighted. These may be appro-priate in some discussions, e.g., structures of thetrianion and the dianion of buckministerfullereneor some partially hydrogenated forms.

The exploded representations have several ad-vantages. All the atoms, bonds, pentagons andhexagons are visible in these structures. Thus, thekey structural features can be recognised withoutthe use of three-dimensional models. The peda-gogical value hardly needs to be emphasized.More importantly, it would be possible to consid-er and evaluate bond reorganizations which mayaccompany chemical reactions involving fullerenesusing these simplified structures. We recommendthe routine use of these representations, including

RAlHNA et al: SlRUcruRAL & BONDING REPRESENTATIONS OF FUU..ERENES F55

Fig. 4-Projected two-dimensional representation of C70

the alternating single and double bonds, for me-chanistic discussions.

AcknowledgementWe thank the Supercomputer Education and

Research Centre, IISc, for help and cooperation.Discussions with P Balaram and S Chandrasekar-an were valuable.

References1 Smalley R, Curl E, Heath W, O'Brien J & Kroto J, Na-

ture, 123 (1985) 143.2 For a comprehensive list of theoretical studies and

MNDO results on numerous fulJerenes, see: Thiel W &Bakowies D, JAm chem Soc, 113 (1991) 3704.

3 Takahashi A & Ozaki M, Chem Phys u« 127 (1986)242.

4 (a) Hess B A& Schaad LJ,J org Chern, 51 (1986) 3902.(b) Haddon R C, Brus L E & Raghavachari K, Chem physLett; 125 (1986) 459.(c) Haddon R C, Brus L E & Raghavachari K, Chem physLett, I3l (1986) 165.(d) Randic M, Nikolic S & Trinajstic N, Croat chim Acta,60 (1987) 595.(e) Schmalz T G, Seizt W A, Klein D J & Hite G E, JAm chem Soc, 110 (1988) 1113.

5 Shibuya T I & Yoshitani M, Chem phys u« 137 (1987)13.

6 Marynick D S & Estreicher S, Chem phys Lett, 132(1986) 383.

7 (a) Newton M D & Stanton R E, J Am chem Soc, 108(1986) 2469.(b) McKee M L & Herndon W C, J Mol Struct (Thelrchemt; 153 (1987) 75.

8 (a) Schulman J M, Disch R L, Miller M A & Peck R C,ChemphysLett, 141 (1987)45.(b) Rudzinski J M, Slanina Z, Togasi M, Osawa E & Iizu-kaT, Thermochim Acta; 125(1988) 155.

9 (a)Disch R L & Schulman J M, Chem phys Lett, 125(1986)465.(b) Liithi H P & .AlmlofJ Chem phys Lett, 135 (1987)357.(c) Scuseria G E, Chem phys Lett; 176 (1991) 423.(d) Haeser M, Almlof J& Scuseria G E, Chem phys Lett, 181(1991) 487.

10 Kratschmer W, Lamb L D, Fostiropoulos K & Huffman DR, Nature, 347 (1990) 354.

11 Stewart J J P, J Computer-aided molecular design, 4(1990) 1.

12 Perkins P G & Stewart J J P, J chem Sac Faraday Trans-Z,78 (1982) 285.

13 Jorgensen W L & Salem L, The organic chemist s book oforbitals (Academic Press, New York) 1973.

14 For typical applications of this program, see: Sustmann R&SickingU, Chem Ber, 120(1987) 1323, 1471, 1653.

15 (a) Yannoni C S, Johnson R D, Meijer G, Bethune D S &SalemJ R, J phys Chern, 95 (1991) 9.(b) TyckoR, J phys Chern, 95 (1991) 518.

16 David W I F, Ibberson R M, Matthewsman J C, PrassidesK, Dennis T J S, Hare J P, Kroto H W, Taylor R & Wal-ton D R M, Nature, 353 (1991) 147.

17 liu S, Lu Y-J, Kappers M M & Ibers J A, Science, 254(1991)408.

18 Hedberg K, Hedberg L, Bethune, Brown C A, Dorn H C,Johnson RD & De Varies M, Science, 254 (1991)410.

19 Satpathy S, Cbem phys Len; 130 (1986) 545.20 (a) Dixon P, Kleiner D A & lipscomb W N, JAm chem

Soc, 100 (1978) 5681.(b) lipscomb W N, Science, 196 (1977) 1047.

21 Gleiter R & Paquette L A, Acc Chem Res, 16 (1983)328.

22 Taylor R, Tetrahedron Lett, 32 (1991) 3731.23 Ermer 0, HelvchimActa, 74(1991) 1339.24 Saunders M, Science, 253 (1991) 330.