C60 carbon cages

Volume 130. number 3

CHEMICAL PHYSICS LETTERS

3 October 1986

C, CARBON CAGES * T.G. SCHMALZ,

W.A. SEITZ,

Theoretical Chemical-Physrcs

D.J. KLEIN

Group, Department

and G.E. HITE

of Marine Sciences, Texas A&M Unrversity at Galveston,

Galveston, TX 77553, USA

Received 3 July 1986

Arrangements of carbon atoms in cage-like structures with no dangling bonds are considered as possible novel allotropic =___._ or _c caroon. ___L__ -1_._ _I%%-____. n __^__ l__..:__ ___r_:_ lavvlaolc: c_..___1_,_JCIUCLUI~~ _r_._r_.__*cmtial;~txls~~~s, _1_____*__1_*:__ ___ ICLGIIIL~I~U. l.Y__ll~~_l yuanrnarwe ~..__A~~_A~._ lorms rive uuIer:IBnL L~IJ cuges, navlrg U~,Lam art: resonance-theoretic calculations are made and compared to simpe Hiickel results. The favored structure is found to be the so-called Buckminsterfullerene structure.

1. Introduction

2. Choice of ChO structures

It has been observed [l] that laser vaporization of graphite in a high-pressure supersonic nozzle yields an especially stable C60 species. Further, Kroto et al. [l] suggest that this new species takes a “uniquely elegant” form corresponding to one of the Archimedian semiregular polyhedra, namely a truncated icosahedron with C atoms at each of the vertices and u-bonds along each edge. The remaining n-electrons, delocalized through resonance, presumably contribute to the high stability of the species. Roth _-Hiickel molecular-orbital 12-51 \___, L_-___ ___.____--._ ---- ____.__(MO) -> and -~~~ resonance-theory [S] calculations indicate that this proposed structure, dubbed [l] Buckminsterfullerene, has a sizable stabilizing resonance energy. But the question arises as to whether other C60 structures might exhibit comparable or even greater stability, and so be possible alternative candidate structures [4]. We address this question for all five structures of a “preferred” class of structures, as identified here, first via ..:--1,. ,,1,..1,+:,,, ,.-A cl.*, ..,L., ,...nr.*:+n+;.r.X bllllpl~ XIfi lvl” c;LLIcIlU(LLI”IIJ (111U CI‘IZU UJUIE)~UaluILILLICIY~ resonance theory.

The most stable structures should be those for which every carbon atom attains its tetravalency, preferably with little strain. This can be done [6] with a linear poly-yne chain gradually bent to close into a ring. Consideration of the u-strain implicates larger rings with little n-resonance energy. More significant resonance stabilization should occur with a planar framework bent upon itself to form a cage. To maximize stability there should be minimal curvature, for two reasons: fust, so that the u-skeleton achieves most nearly the $.hvhridizntinn ideal*)____ nnd sacnnrl __, ___--_____ _-__ “______, sn “_ that

* Research supported by the Robert A. Welch Foundation of Houston, Texas.

- _^^ ___ _I^ _ I_ ^_ _^ - -* U3.5U QElsevier (North-Holland Physics Publishing U UUY-X114/86/$

Science Pubiishers B.V. Division)

the overlap between the remaining neighbor n-orbitals is as large as possible. Thus, the cages we consider all correspond to convex polyhedra. The polyhedra represent the C-atom o-network and have three edges (u-bonds) incident at each vertex (carbon atom). Because of the well-known preference for hexagonal rings in order to achieve greatest aromatic stabilization, a maximum fraction of the rings (corre“^..^~L._ ..L-..11 I__ sp”UUlll~ L.. L” ~~1..L~~_.~~ p”ly1l~Ul”ll &._..^“\ lZtL%S,Sll”lllU “t: l__....__--1 r,exagor,w. IP II all the rings are hexagonal a planar (graphite) network results. The bending necessary for the formation of a polyhedral structure requires smaller rings [7]. Chemically the most favorable smaller-size rings are pentagonal. Thus we focus on three-valent convex polyhedra with only hexagonal and pentagonal faces. Via Euler’s theorem the number of pentagonal faces may readily 203

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Volume 130. number 3

be shown to be 12 and the number of hexagonal faces &J - 10, where u is the number of vertices. Hence for u = 60, the number of hexagonal faces is 20. NT_..A- .._.._L c_^__-^-l _^- L_ -_l_ *L-L -__-IYeXL,a IUU~I dI~UIIl~IIL IX11 DL: IEKLe LIlaL IIlUr‘S highly symmetric species are likely candidates for high stability. Since effective (screened) interactions are dominantly local, repetition of an especially stable local structure should lead to an overall global structure of especial stability. But such repetition, if feasible under the constraints of realization in Euclidean space, often leads to a system of high symmetry. Thence we restrict attention to cages with a “reasonable” amount of symmetry, which we take to include: icosahedral, octahedral, tetrahedral, and five- or higherfold axial symmetries. The originally suggested Buckminsterfullerene structure of Kroto et al. [l] is the only C6o structure with icosahedral symmetry. There are no structures possible with octahedral symmetry since there clearly is no way to realize a fourfold symmetry axis for a polyhedron of three-valent vertices with only five- and six-membered rings. Though tetrahedral symmetry cages are possible for other numbers of C atoms in the skeleton, we find none for C6u. The same rationale which excludes octahedral symmetry also excludes cages with axial symmetries of sevenfold or higher. The identification of C6u polyhedra with five- or six-fold axes of symmetry proceeds via identification of their “unit cells”. These unit cells are such that if repeated by rotation through 2n/n, n = 5 or 6, they yield complete C6u structures. We have found the five possible unit cells, whose planar representations are shown in fig. 1. The first two unit cells are to be re-

I

n

3 October 1986

peated six times, and the last three, five times. The fourth unit cell is that of Buckminsterfullerene, which has a fivefold axis (actually six fivefold axes). The first &I-_*_*___*____ --.I_?_%_ TT... -1 ..-A -,I, cu~~cspu~~us -..-..-“^-^A” *U~C GCU LULIBELSLIUCLUII:wnlcn naymer [4] termed “graphitene”.

3. Computational methods and estimates of resonance energies To obtain quantitative estimates of the stability of the selected polyhedral cages, we have carried out three types of calculations for the rr-resonance energy. The first scheme is via Htickel MO theory, with the total nelectron energy referenced against that of a corresponding set of (non-resonant) ethylenes. This calculation involves the usual matrix diagonalization, which can be aided through the use of symmetry to block the overall matrix. The remaining two computational schemes both utilize the various Kekule structures; i.e, the various covalent n-bonding patterns. The first of these two schemes, which has been found to be reasonable for benzenoid hydrocarbons composed solely of fused hexagons, assumes that the resonance energy is simply proportional to the logarithm of K, the number of Kekule structures [8]. The number of KekuM structures is obtained via a powerful transfer matrix technique which we have recently applied to a variety of related enumeration problems [9] of chemical interest. The last scheme, which has been found to be the most reliable for a ~.~~~_~ range of hvdrncnrhnna _. conirmated __..,-p-__-__,-_________“, including non-alternants, may be described either as

R w

Fig. 1. Unit cells for the fiie isomers of Ceo possessing cyclic symmetry. Cells I and II generate Cao upon sixfold repetition while III, IV (Buckminsterfullerene), and V generate C60 upon fivefold repetition.

Table 1 Kekuli and conjugated circuit counts for Gee isomers Structure

3 October 1986

CHEMICAL PHYSICS LETTERS

Volume 130. number 3

#(KEK)

#(e)

&3)

#W

I

12688

65968

17016

46416

II

12740

60680

31824

46056

III

9183

40140

10720

28220

IV

12500

83160

0

59760

V

16501

83000

88500

96380

Hemdon-Simpson resonance theory [lo] or as conjugated-circuit theory [ 111. In this approach the resonance energy is given as the ratio H/K with

(1) where the R, and Q, are oppositely signed parameters which decrease in magnitude (near geometrically) with increasing n, and where #th) is the sum over the number of conjugated 2m-circuits in the various Kekult? structures. (Here a conjugated 2m-circuit of a Kekule structure is a length-2m cycle with alternating single and double bonds in that structure.) These calculations too may be carried out via transfer matrix methods [!?I. The results for each of the five c60 structures, K and #th) with m G 5,are reported in table 1. (Here #4) is not reported, since for all our structures it is 0.) The computational schemes utilized here make

several approximations. They neglect (explicit) interactions with more highly excited configurations, and they neglect U--A couplings. In fact, the results in table 1 are purely graph theoretic in that they depend only on the system graph. Explicit calculation of resonance energies requires the modification of model parameters which have previously been determined for (planar) coniueated ____~_~____

hvdrocarhnn __, -_-_- -__-_

avntemn _ ,-------.

-Fnr -_ nnlvhdrnl r -_,__--_-

WP. “,”

terns, however, the deviation of these model parameters from their planar values may be obtained crudely as indicated below. First note that these parameters should be decreasing functions of 8, the angle of deviation of two nearneighbor n-orbitals away from being parallel. Since these parameters must be even functions of 19it is of relevance to estimate (132), the mean of the square of f3. m. . lnis mean, however, is very neariy the same for aii our polyhedra, and indeed should be so for rather general 60-atom cages that are simulated by convex polyhedra. This is so whenever the direction of the n-like orbital at site i is close to the outward direction making an equal angle with the incident polyhedron edges (corresponding to u-like bonds), since then standard trigonometric mlations imply that this angle deviates from $r radian by an amount = 6fi2, where 6 i is an assumed small “angle defect” around vertex i, i.e. Si is 2n radian minus the sum of the face angles at vertex i. Then

W2)=z l cs,, i

Table 2 n-electron resonance energies for the five cages studied Structure

I II III IV V

Hiickel LUMO-HOMO gap (IPI)

Resonance energies per site Hiickel (IPI)

Kekul&ount (J)

resonance-theory

conjugated circuits

(Herndon’s parameters) a)

(Randid’s parameters) b,

0.103

0.542

0.178

0.088

0.080

0

0.539

0.178

0.076

0.065

0 0.757 0

0.535 0.553 0.527

0.172 0.178 0.183

0.073 0.120 0.080

0.067 0.116 0.057

a) Rr = 0.841 eV, Qr = -0.26 eV, Rz = 0.336 eV, and 0 otherwise. b,

RI = 0.869 eV, Qr = -0.45 eV, Rz = 0.246 eV, and 0 otherwise. 205

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CHEMICAL PHYSICS LETTERS

but this latter sum is 4n, independent of the choice of the convex polyhedron [ 121. Thus all our polyhedra have similar mean curvatures and thus have similar overall u-strains. Further the mean angular correction factors to the n-electron coupling constants are the same for these different polyhedra. As a consequence of the arguments in the preceding narrtotanh the atahilitief nf r”-e‘-r-‘ “..” nvc?rall _ .--_-. relative ._._.*. 1 “...“y..~~” “_ the . .._ variml~ . ...-V..y CGocages are anticipated to be dominated by their rrelectron energies_.Thence in table 2 we report the nelectron resonance energies for the five cages studied. Three cages are predicted, via Htickel theory, to be open-shell species, and presumably then rather reactive. The other two (graphitene and Buckminsterfullerene) are closed-shell, and apparently stable as judged by what is believed to be the most reliable calculation, namely the conjugated-circuit estimate. Of these two the Buckminsterfullerene structure is clearly the more etnhla “CYVA”

.

4. Conclusion It is seen from table 2 that there is disagreement between the predictions of the three methods of computing resonance energy investigated here. The Hiickel theory and the Kekule count recognize very little difference in stability among the isomers. indeed, three of the cages have more Kekult? structures than the Buckminsterfullerene structure, but the patterns in which these additional Kekuld structures occur are evidently not very stabilizing. (This has long been realized as a possibility for non-benzenoid systems with rings of other than six atoms; the common such example is cyclobutadiene.) On the other hand, resonance theory, whether parameterized with the values of Hemdon [lo] or those of Randic [ 111, predicts Buckminsterfullerene to be significantly more stable than the other isomers. Evidently it is favorable to avoid a pair of abutting pentagons, since then destabilizing conjugated gcircuits (around this pair of pentagons) can occur. It is notable that Buckminsterfullerene is the unique C,, structure that satisfies both the criteria of section 2 and this additional condition that no two pentagons abut. Indeed this uniqueness statement remains valid even if the symmetry constraint of section 2 is discarded. There is also disagreement between the magnitude 206

3 October 1986

of the stabilization predicted from the different methods in table 2. As argued in the preceding section, all values in the table should be reduced by a factor related to (0 *>to account for deviations from planarity. When this is done [5], the stabilizations for Buckminsterfullerene predicted from Htickel theory and the Kekule count are still larger than those for benzene and close tn thncn nf n,.nnhita, l--h., ..~l.xa fm-.... ..anr\mnrra.n +ham..r b” UIVOY “A ~‘OpuLv. 11Lb .aluc ll”,,, ‘~,u”I‘OIIL&~ nLG”ry is already less than that of benzene even without correction, and is estimated in ref. [5] to be ~70% of benzene after correction. While this level of stabilization is not enormous, it is comparable to that of many other stable systems. In conclusion, we find that neither unmodified Hiickel theory nor counting Kekule structures - two methods which provide accurate resonance energy estimates for benzenoid systems - are likely to be useful in estimating the resonance energy of carbon cages ,.~,..w.oP ,.F +h, K.n,.rrnnns.,*~ X-~-I.-,.,. ,.C A%... -,.-l----A “GcI(IUJzi “I LIIG IIc;bBJJllly pmcmL;cj “1 II”c-lll~illl”~,vu rings. Herndon-Simpson resonance theory, however, predicts a stabilization for Buckminsterfullerene consistent with its being experimentally observable, and predicts clearly that the structure suggested by Kroto et al. [l] is the (thermodynamically) most favorable of all C,, candidate structures.

Note added Since submission of this paper several other calculations [ 131 of varying types have appeared on carbon cages of various sizes. Still the only previous C,, cages considered are the Buckminsterfullerene and “graphitene” structures.

References [l] H.W. Kroto, J.R. Heath, S.C. O‘Brien, R.F. Curl and R.E. Smalley, Nature 318 (1985) 162.

[ 21 D.A. Bochvar and G.E. Galpern, Dokl. Akad. Nauk SSSR 209 (1973) 610. [3] R.A. Davidson, Theoret. Chim. Acta 58 (1981) 193. [4] A.D.J. Haymet, J. Am. Chem. Sot. 108 (1986) 319. [5] D.J. Klein, T.G. Schmalz, W.A. Seitz andG.E. Hite, J. Am. Chem. Sot. 108 (1986) 1301. [6] K.S. Pitzer and E. Clementi, J. Am. Chem. Sot. 81(1959) 4477.

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[ 71 D’Arcy Thompson, Growth and form (Cambridge Univ. Press, Cambridge, 1943) ch. 9; A.F. Wells, The third dimension in chemistry (Oxford Univ. Press, Oxford, 1956) ch. 2; B. Grunbaum, Convex polytopes (Interscience, New York, 1967) ch. 13. 101 “C \1?tz7, ,,o”l-l\ .J7/,
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[lo] W.C. Herndon, J. Am. Chem. Sot. 95 (1973) 2404; W.C. Herndon and M.L. Ellzey Jr., J. Am. Chem. Sot. 96 (1974) 6631. [ 111 M. Randid, Tetrahedron 31(1975) 1477; M. Randid and N. Trinajsti;, J. Am. Chem. Sot. 106 (1984) 4428. r[A’, i 71 A.. R nevltter ~ntirlnnlm ~t~m~nti. Y’YUIIY, n+ I” YIY..Y...I.. 11”111.. 1.1. Yfri+rn \“_.a. ir;am ‘“d”,, sn u.1 originally unpublished manuscript, of which only (~05 sibly incomplete) hand-written notes made by W.G. Leibniz in 1676 have survived and been published In 1860. For the transcribed document, an extensive discussion of Descartes’ work, it’s relation to that of Euler (in circa 1750), and proofs of the various propositions enunciated see P.J. Federico, Descartes on polyhedra (Springer, Berlin, 1982). [ 131 M.D. Newton and R.E. Stanton, J. Am. Chem. Sot. 108 (1986) 2469; R.C. Haddon, L.E. Brus and K. Raghavachari, Chem. Phys. Letters 125 (1986) 459; R.L. Disch and J.M. Schulman, Chem. Phys. Letters 125 (1986) 465.

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