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Isomerisations of the fullerenes
Vol. 30. No. 8, pp. 1235-1250. I” Great Bntain.
1992 Copyright
ISOMERISATIONS
0008.6223/92 $5.00 + .OO Cc, I992 Pergamon Press Ltd.
OF THE FULLERENES
PATRICKW.FOWLER,'DAVIDE.MANOLOPOULOS,~~~~ROBERTP.RYAN'.* ‘Department of Chemistry, University of Exeter, Exeter EX4 4QD, England ‘Department of Chemistry, University of Nottingham, Nottingham NC7 2RD. England (Accepted 28 .4pril1992) Abstract-The Stone-Wales (pyracylene) rearrangement is a hypothetical mechanism for interconversion of fullerene isomers, accounting for mobility of the pentagons on the surfaces of these clusters. Two versions of the transformation, involving patches of 4 and 12 rings, respectively, can be defined. Energetic and group-theoretical aspects ofthe transformation are discussed, symmetry-based selection rules devised, and complete isomerisation maps presented for small fullerenes (C20to Cd,,)and isolated-pentagon isomers (CT8to Css). Fullerenes that self-racemise under this transformation include D2 C2*. Limitations on the usefulness of this mechanism as a means of rationalising experimental isomer distributions are briefly discussed. Key Words-Carbon,
fullerene, isomerisation, chirality, symmetry
1. INTRODUCTION
The fast-growing family of fullerenes now includes a range of chiral and achiral molecules in addition to the CbOarchetype and its CT0satellite. Chromatography of soluble products of graphite evaporation yields Go, C,,,, C,,, G8, CXZ,C,,, G, Gr . . 1l-61,SOme in several isomeric forms. This new experimental evidence presents an opportunity to test the qualitative theoretical approaches[7-9[ developed in the years since the first observation of&[ lo]. Three main targets for a systematic theory of fullerenes are: enumeration of possible isomers, classification of their electronic structures, and estimation of steric factors in overall stability. As argued elsewhere[ 1 I], the first two problems are solved by the spiral algorithm[9] and the leapfrog/cylinder magic-number rules [ 12,131, respectively. Some progress has been made on the steric problem, but more remains to be done. In the present paper we concentrate on another factor that may help to decide the experimental isomer distributions and stable magic numbers (i.e., the likelihood of interconversion between isomers of a given fullerene). It is now possible to compute a complete solution to the mapping of C, interconversions for any given mechanism, and here we study the most promising candidate mechanism: the StoneWales[ 141 (or pyracylene[4]) transformation. A recent suggestion links the experimental isomer distribution to thermodynamic equilibrium within closed groups of Stone-Wales interconverting isomers[4]. lsomerisation maps for particular fullerenes (C,,, C,,, C&) have been published[4,15t, 161, but our aim here is to provide a general discussion of the mathematical and chemical characteristics of the transformation. Such a discussion gives a basis for prediction of experimentally isolable isomers within any particular *Permanent address: Department of Chemistry, St. Patrick’s College, Maynooth, Co. Kildare, Ireland. tin Fig. 2 of this paper the transformation between isomers 9 and 13 is to be deleted.
model of electronic and steric stability. Maps tbr other higher and lower fullerenes of interest are also presented. 2. THE STONE-WALES TRANSFORMATION The pyracylene transformation was first proposed in a paper by Stone and Wales on possible nonicosahedral isomers of CGO[ 141, and was taken up by Coulombeau and Rassat[ 17,181 in a further exploration of low-symmetry C,, isomers. Although there is no experimental evidence for stability of any other C,, isomer, and characterisation of bulk fullerite has confirmed the primacy of Zh&,, the transformation itself gives a general hypothetical mechanism for interconversion between fullerene isomers and may explain features of the observed isomer distributions for higher carbon cages. To perform th’e transformation, we first search the fullerene surface for a patch (I) where two pentagons and two hexagons meet in a pyracylene/pyracene pattern. If such a patch is found, we twist the central bond (the Stone-Wales or SW bond), breaking two neighbouring bonds and making two more, thereby swapping pentagons and hexagons and leading to a “rotated” patch within the same 12-atom perimeter. The product is still a fullerene (in general different from the starting isomer) and any single transformation of this kind is clearly reversible. The “sense” of the twist is immaterial and so the transformation links in a well-defined way two isomers of a given fullerene C, (or one isomer to its enantiomer. or conceivably one isomer of low symmetry to itself). There are steric[7] and electronic[ 18.19,20] grounds for believing that isolated-pentagon isomers are more stable than fullerenes with abutting pentagons. This intuition is now often described as a rulethe isolated-pentagon rule (IPR). Certainly it appears to be obeyed by all fullerenes isolated and characterised so far. It is useful, therefore, to consider a restricted version of the Stone-Wales transformation
1235
P. W. FOWLERet ul
1236
I in which both initial and final molecules have isolated pentagons[4,15]. This isomerisation, if geometrically possible, should have a more favourable activation energy. It involves an active patch of I2 rings in a 20-atom perimeter (II). The transfo~ation proceeds via an anti-aromatic transition state and is therefore thermally forbidden[ 141 under the Woodward-Hoffman rules for concerted reactions. Figure 1 shows the (approximately independent) transformation of CTand ?r orbitals as the isome~~tion converts “reactant” to “product” fullerene. A crude model for the energetic-s of the process can be constructed by applying the purely topological Hiickel theory at all stages along the reaction coordinate. The four active u neighbours of the central SW bond are changing from fully formed (order 1) to broken links (order 0) or vice versa. The total a bond order is thus roughly conserved, but the adjacency matrix from which the pattern of a molecule orbitals is derived has changing entries. If we define a reaction coordinate r by linear interpolation of adjacency matrices and calculate the
4xsp2
4xsp2
4XPT
total 7cenergy as a function E,(r) we have a crude estimate of the activation energy. The model is, of course, ove~implified in that it throws all the energy change into the rearrangement of the H system and ignores the steric cost of distorting, breaking and reforming bonds within the a framework, but the derived reaction profile is qualitatively reasonable. Figure 2 shows three reaction profiles computed in this way. Figure 2a iflustrates the variation of x energy for & CzB,the smallest fullerene cage to support a Stone-Wales transformation. Under rotation of a small (type I) patch, the molecule simply converts between (isoenergetic) left- and right-handed forms. The transition state has C,, symmetry (Fig. 3)-in the topological model at r = 0 there are four half-formed bonds symmet~cally disposed about the active SW link, and so it is possible for this point on the reaction coordinate to have symmetry elements that are absent in both reactant and product. In general, C,, symmetry would be expected to be achieved by the transition state whenever a self-racemisation occurs via a C, SW patch. The reaction profile for Cz8is symmetrical and shows a significant activation energy of -0.3 1 I p I (or just over half of the delocalisation energy per atom), most (95%) of which arises from the shift in energy of an orbital that is anti-symmetric under the preserved C, operation {antibonding along the SW bond) and therefore has a nodal plane containing the SW bond in the transition state. An example of a transformation between isolatedand adjacent-pentagon isomers is given in Fig. 2b. Icosahedral C,, (with 12 isolated pentagons) is con-
4XP*
4XPz
Fig. 1. Orbital changes during the course ofthe Stone-Wales transformation (reactant - transition state - product). The changesin the Q(top row) and x (bottom row) systemsare inde~ndent (o~hogonal) in firstorder. The symmetric reaction coordinate r is defined in terms of Hiickel resonance integrals Pi/: 8~ = 034 = CM1 - W, P23 = 841 = (WI + r)P.
lsomerisations of the fullerenes
1237
41.62
41.92
09
92.6.
r
93.2 M r
(4
120.5
1
Fig. 2. A topological model for the reaction profile of the Stone-Wales transformation for (a) racemisation of LIZCzXt(b) intr~uction of one pentagon adjacency into r, Cm to give CIUCm, and (c) interconversion of isolated oentaeon isomers of CT,. The labelling of the four linked isomers of C,, follows the interconversion mad ores&ted later in this paper. The total Htickel ?r energy is plotted in units of 8. In (a) and (b) the reaction coordinate r goes from - I to + 1; in (c) there are three consecutive reaction coordinates of this type.
vetted to a C,, isomer (with 8 single pentagons and 2 pairs) by twisting any one ofits SW bonds. Reactant and product have different K energies and the profile is shifted slightly to the product side. The model predicts an activation energy of -0.33 / 8 / for the reverse reaction (i.e., the removal of the two pentagon adjacencies from the C,, isomer). Finally, Fig. 2c shows a profile for isolate-to-isolate transformation within the family of isomers of CT8. The pathway connects four isomers of CT8 by
consecutive transformations of type-II SW patches. The simple Hi.ickel calculations on which this graph is based show that a conversion from least to most stable isomer of the family is accompanied by a roughly constant (5 10%) activation energy of 0.33 I@1for each step. Other empirical and semi-empirical models give different energy orderings. No reliable estimate of activation energy is yet available for comparison with this topological model, but we note that fullerenes are formed under violent
1238
P. W. FOWLERet al.
Fig. 3. Transition state for the Stone-Wales transformation of CZs.& reactant (one enantiomer), CZUtransition state, and D2 product (the opposite enantiomer) are shown.
conditions and both ingestion and extrusion mechanisms for cage growth/decay presuppose some mobility of pentagons on the cluster surface[21,22]. In the absence of a thermally allowed, low-energy alternative to the Stone-Wales mechanism, we take it here as the starting point for an investigation of the mathematically possible interconversions amongst C, isomers. 3. SYMMETRY ASPECIS Fullerenes belong to a limited subset of the molecular point symmetry groups[23]. Examination of the structural components (3-connected atoms, 4-connetted edges, hexagonal and pentagonal faces) shows that a fullerene skeleton must belong to one of 28 point groups: I,, I, Th, Td, T, D,, DA Dndr(n = 2,3,5,6), G Gir, G, &,+ (n = 2,3), C,, Ci, G. An edge of the polyhedron has at best C,, site symmetry and so a Stone-Wales bond must lie at a site of C,,, C,, C, or C, symmetry. An important characteristic of the transformation (in both general and restricted forms) is that it preserves the site symmetry of the active Stone- Wales bond. The overall molecular symmetry may (and usually does) change, but the subset of symmetry elements joining the midpoint of the active SW bond to the “centre” of the cluster (i.e., to the “point” of the point group) must be common to both initial and final fullerene. This preservation of site symmetry is the basis of the selection rule for single-step transformations. The SW patches (I or II) of a given fullerene fall into equivalent sets or “orbits”[24], with all members characterised by a common site symmetry. The size of the orbit is equal to the ratio of orders of the molecular point group and the site group, and so for a SW bond the possible number of patches within a set is h (site group C,), h/2 (site group C, or C.,) or h/4 (site group C,,), where h is the order of the molecular point group. The number of orbits gives the maximum number of “exits” from a given fullerene (i.e., the number of distinct neighbours on the isomerisation map, not counting enantiomers). The qualification “maximum” is needed here because a fullerene may conceivably transform into itself or its own enantiomer. Possible orbits for all fullerene groups have been tabulated[24,19]. In the notation of [24] the symbol
&,(I,,), for example, denotes a set of 30 equivalent positions with C,, site symmetry (order = 4) in a molecule belonging to the I,, group (order = 120). A necessary but not sufficient condition for an SW transformation to connect two fullerene isomers is that the point groups of each should have at least one SW orbit site symmetry in common. Thus, Ih - C, is forbidden because though C, is a subgroup if I,, the only possible orbit of SW patches in I,, has C,, site symmetry (see below) and no orbit of this symmetry can be present in C,. It is often but not invariably the case that the two fullerene groups are related as group and subgroup; a counterexample exists for two isomers of C,, Dzd- D1,,,for example. Table 1 shows the possible orbits of SW bonds in the fullerene symmetry groups. The first criterion in the construction of Table 1 was the identification of orbits with site symmetry C,, or one of its subgroups. However, not all such orbits are possible sets of SW bonds, since the maximum number of simultaneous SW patches is limited by the fixed number of pentagons ( 12) in a fullerene. A given pentagon can be a component of at most 5 type-1 patches and each patch involves 2 pentagons, so the maximum number of SW bonds in a fullerene is 30. This is realised uniquely in ZhCm where all SW patches are equivalent and span O,,(Z,,). Not only is icosahedral C, the only fullerene with 30 SW bonds, it is the only icosahedral (I or Z,,)fullerene with any SW bonds at all, since for C, no face is hexagonal, and for all icosah~ral cages larger than C, the pentagons are more than one bond apart. A corollary is that no fullerene of I symmetry takes part in a StoneWales transformation. Similar reasoning shows that the maximum number of simultaneous large patches (type II) to be just 12. (Proof: each pentagon within II may belong to at most 2 large patches, each patch involves 2 pentagons and there are exactly 12 pentagons available.) The maximum is realised, for example, in the D6)6h isomer of C,,[25] and its derivatives Cs4+,2pproduced by insertion of p equatorial belts of 6 hexagons[ 131. Even where a group has a I2-orbit ofsuitable site symmetry it may be impossible to achieve 12 large patches; tetrahedral groups are cases in point. The groups T,,, Td, and Teach have SW orbits of order 6 and 12. On a pseudosphe~cal cluster the 6-orbit can occur at most once, and so it is not possible to reach 12 patches by taking two copies of this orbit. The 1Zorbit itself has the topology of a (twisted) truncated tetrahedron and the crowding at the base of each frustrum is such that only small patches may coexist there (Fig. 4), with the result that a maximum of 6 large patches is possible within these groups. The construction of Table 1 takes these and related considerations into account, distinguishing orbits of SW bonds that may lie in patches of either type from those that are confined to small patches. Although the SW transformation preserves site symmetry, it does not in general conserve the number
1239
Isomerisations of the fullcrenes Table I. Catalogue of equivalent sets of Stone-Wales point groups
bonds within the fullerene
Site group c:zu
Molecular group
-
a
021,
0,
-
4.m
9s:
0,
Oh Q?v,02. 03 05
-
Ob,O6
02 02:
03 05 0.5,
c',
-
02s.
CT
c2
ob
02 -
Oh Oh 0%
-
03 O2h 03 axy, %h,
ax:,
olOhr
010"
0,2C
0,2v,
0 4d 0 6d 0 IOd 0 12d
04 0, a0 012 02
Ob
-
O,Zd
034
08 012 a0 0t4
-
-
-
04v:
06~
0, 4 02 02 03 04 06 OlO OIZ 04 Oh 04 Ob 08 012 OYO
0*
04 Oh 012 @4
4:
oy,
-
-
Orbits are grouped according to the site symmetry, and named within the conventions of [24]. A star denotes an orbit that is possible for small patches only. The group I is not listed, as no fullerene of this topological symmetry has SW bonds
of SW bonds. The active bond remains embedded in a Stone-Wales patch, but twisting about this bond may entail making or breaking of up to eight neighbou~ng patch patterns (as illustrated in Fig. 5). If the active bond is itself within a large patch, then up to four SW motifs may be made or broken by its rotation,
~~\
___--
_
/ _
Fig. 4. Arrangement of I2 SW patches on a tetrahedral fullerene. The 12 bond centres (marked by black circles) lie at the vertices of a truncated tetrahedron. Crowding on each triangular face (inset) implies that all 12 are patches of type I (smail) and not type II (large),
4. CHIRALITY AND THE STONE-WALES TRANSFORMATION
Most fullerenes have both chiraI and achiral isomers and the Stone-Wales interconversion maps will generally incfude transformations between these types. It is instructive to consider the general relation-
ship of the SW transformation to chirality. For A B there are three distinct possibilities: (a) A and B are both achiral, (b) one of A and B is chiral, or(c) A and B are both chiral. In case (a) the active SW bond may have any of the four allowed site symmet~es, but in cases (b) and (c) only the rotational groups (Cl or C,) are possible. Case (b) is considered first. When achiral and chiral isomers A and B are linked by a single SW transformation there is an equal probability of producing left- and ~ght-hands forms of B starting from A. The SW patches feading to the two enantiomers are equivalent by symmetry, that is to say they form an orbit of the point group of A, but they differ in the handedness of their local environments. Examination of specific cases (Fig. 6) reveals the general rufewithin an orbit of patches in A character&d by a common site symmetry, two patches lead to the same
1240
P. W. FOWLER et al.
Fig. 5. Changes in the number of Stone-Wales patches can accompany twisting of a single SW bond. All eight bonds marked with an “‘a” lose their Stone-Wales status upon rotation of the central bond.
enantiomer of B if they are exchanged by a proper operation, but opposite enantiomers of B if they are exchanged by an improper operation. When B is also achiral (case (a)), all patches in the orbit lead to the same product as B is superimposable on its mirror image. Both examples shown in Fig. 6 are of patches with C, site symmetry, which span the regular orbit[24] of the molecular point group. In this orbit every pair of patches is swapped by just one operation and since half of the operations of an achiral group are proper and haIf improper, the orbit splits into two equal disjoint sets, one for each handedness of the product. If we associate signs i- and - with enantiome~ of the product, then the pattern of signs transforms as the antisymmet~c repre~ntation I’, of the molecular point group (T, has character + I under proper and - 1 under improper operations). Similar remarks apply to the “half-orbit” of the molecular group
(a)
(4
Fig. 6. Transformation of an achiral to a chiral fullerene. Orbits of SW patches for the conversions(a) Go CM (isomer 8) - Cr CG (isomer 6, L and R forms), (b) CJ, Cx8 (isomer 16) - C, Cjg (isomer 14, L and R forms). The labels L and R are arbitrarily assigned to the two enantiomers of the product. Within the orbit of SW sites, all patches producing a given enantiomer are related to proper operations ((a) C,. (b) C?. Cf about an axis ~~ndicular to the ulane of thk ‘paper), and swap with patches of opposite effect under improper operations (u reflections ~~ndicular to the paper}.
spanned by patches with C, site symmetry. The permutation representation of the C,- and C,-orbits of an achiral group contains distinct copies of both Tt and To (the totally symmetric representation), Case (c) is the conversion of a chiral isomer A to a chiral product B. All patches of an orbit in A in this case are exchanged amongst themselves by proper rotations alone and so all lead to the same enantiomer of B. There are three subcases, depending on whether A is identical with B, enantiome~c to B or simply different from B. A chiral-to”chira1 SW tmnsfo~ation may thus produce a null result, self-mcemisation (e.g., twisting of one of the two C, patches in L&Czs), or conversion between fixed enantiome~ of A and B (e.g., the map 1 & - 2 C, +-+5 Z.&for C,,[ 151 where two mirror-image triads exist and interconvert one enantiomereach of 1,2, and 5 without racemisation). These considerations have direct consequences for the prospects of resolution of optically active fullerenes. If an isomerisation map includes a single achiral structure, then resolution of enantiomers of even the most stable isomer on that map is possible only under conditions where the SW transformation is “frozen out.” On the other hand, if the map contains only chiral isomers and has no case of self-racemisation either explicit or implicit in a cycle, then the freezing out of the SW transformation is not a prerequisite for resolution 5. STONE-WALES
T~NSFORMATiONS LEAPFROGS
AND
The leapfrog fullerenes form a class of isolatedpentagon polyhedra of particular theoretical interest, They have the formula C, (n = 60 + 6k, k f I) and are obtained by omni~pping a fuflerene C,,, and taking the dual ofthe resulting deltahedron. They are the only fullerenes with closed electronic shells and ant~bonding LUMOs within simple Hiickel theory. An equivalent definition can be made in terms of locaiised bonding[26]: every leapfrog fullerene has a unique Kekule structure that is found by placing 5 double bonds exo to any one pentagon and filling in the single and double bonds over the rest of the structure as forced by valency considerations. Every pentagon in this Kekule structure has single bonds along its sides and all exo bonds double. The SW bonds, if any, on a leapfrog fullerene are therefore formally double. The special Kekuli: structure ofa leapfrog fullerene has the property of achieving the maximum possible number of benzenoid hexagonal faces (n/3X261. As Fig. 7 shows, the Stone-Wales transformation on a leapfrog fullerene disrupts this special Kekute structure. It follows that no two leapfrog fullerene isomers of C, may be connected by a single SW step, thougb they may be connected by a sequence of such steps. Figure 8 shows the genesis and fate of SW patches under the leapfrog operation, showing that every patch in a leapfrog comes from a pair of pentagons
Isomerisations of the fullerenes
Fig. 7. Stone-Wales transformation and the leapfrog construction. (a) In the special Kekuli structure of the leapfrog fullerene every pentagon has 5 cm double bonds. Bond orders of contacts across the boundary of the 4&g patch are indicated by broken lines. (b) On rotation of the SW bond within the patch a mismatch is produced at the edges, and the transformed fullerene is therefore no longer a leapfrog isomer.
sharing an edge in the parent, and every large patch in the leapfrog from an “anti-patch” where a bond common to two pentagons joins atoms in two hexagons. The figure also shows that one repetition of the leapfrog operation is sufficient to destroy all SW patches. Double-Ieapfrog fullerenes are thus disconnected from all other isomers on an interconversion map, but as the first double-leapfrog not already disqualified by icosahedral symmetry is Czla this fact is unlikely to have chemical relevance in the near future. The smallest leapfrog fullerene with a large SW patch, and therefore an allowed isome~~tion to another isolated-pentagon fullerene, is a Dlk isomer of C,,[27,4]. 6. GENERATION OF ISOMERISATION MAPS It is a finite task to determine all possible SW tmnsfo~ations between isomers C,. For each rz it is relatively straightforward to catalogue all possible fullerene isomers using the spiral algorithm[9]. The symmetry, spectroscopic signature and approximate (topological) atomic coordinates are all available from the adjacency matrix( 191 which is itself con-
1241
structible from a I D sequence of 12 pentagonal and (%n - 10) hexagonal faces in the spiral that encodes each fullerene isomer. For a given isomer, searching of the face adjacency patterns can be used to find SW bonds (in patches of type I or II or both), site symmetries can then be assigned and the effect of the transformation ascertained by diagonalisation of the transformed adjacency matrix. A map of allowed isomerisations can thus be constructed computationally. One informal indication of the completeness of the spiral algorithm is that this Stone-Wales procedure has never yet generated an isomer outside the starting set. Handling of chiral SW transformations of the type discussed earlier requires some care as the coordinates generated from the eigenvectors of the adjacency matrix are of arbitrary handedness[ 191. The results of a survey using these programs will now be discussed in two parts. First, we look at the results for small (n I 40) fullerenes with abutting pentagons. Secondly, the results for the restricted transformation in higher fullerenes are presented. The isomer counts for fullerene polyhedra Czo to Cm appear as part of a larger tabulation in[ 191. Combining the spiral algorithm with new programs to identify and perform the SW tmnsformation, we have mapped isomerisations for this range ofclusters. A classification of the patches is given in Table 2 and maps are shown in Figs. 9- 12. Several general properties of the transformation can be illustrated by reference to the maps. Their most obvious feature is that they are incompletely connected: the isomers of C, fall into families and are not all interconvertible by repeated transformation. An analogous “factorisation” is found for many rearrangement mechanisms in chemistry. For example, the diamond-square-diamond borane rearrangement is the dual of the Stone-Wales transformation when the four atoms involved are appropriately connected. A concerted multiple DSD rearrangement has been postulated as a mechanism converting ( I ,2) to (1,7) C,B,0H,2 via a cuboctahedral transition state[28]. The third isomer, the ( 1,12) form, cannot be reached by this mechanism from either of the
Fig. 8. Generation of a Stone-Wales patch (I) by leapfrogging, and destruction by double leapfrogging Leapfrogging a pre-existing SW patch destroys it but may create new patches on its periphery.
1242
P. W.
F~wLER
ef al.
Table 2. Catalogue of fullerene isomers CzOto C& Isomer
c20l(fh) c24
1@6a)
c26
1@3h)
c28
l(DZ)
c28
2( r,)
Go
W5hj
c30
2(C2”)
c30
3(Czv)
c32
l(c2)
c32
WA)
c32
3(4df
c32 4(c2) c32 5(&h) c32 6(03) c34
l(C2)
c34
2(G)
c34
3(G)
C34 4(Cd C34 5(G)
CM 6fC3J f3b
lfc2)
C36 W2) c36
3(c,)
c36 4(c,) c36 5(4) c36 c36
6(f)2& 7(C))
c36 a C36 9(G) C36 WC2) c36
f j(c2)
c36
12(c2)
c36
13@3h)
C36 W&d) c36
15ff.h)
c38
l(c2)
c3s
2@3h)
c38
3(C, )
C384(CJ C38
S(G)
c38
6(C2) 7(cI)
c38
c38 8(c)
)
C38 9(4) C38
WG)
c38
ll(ci)
C38 12(G”)
13(CZ) 14(C,) C38 15tCz”)
c38 c3*
c38
16(c3”)
C38 17(G) &I
1@5d)
Go
2(C2)
C40 WA) C404(Cl) c‘la
5(G)
Go
I
7(CJ Go 8(G”) G
C40 9(G) c40
MCI)
Go
11(G)
Go
12(Ct)
13(G) Go 14(CJ c40 ls(C2) Go
Ring spiral 555555555555 55555655655555 555556565655555 5555565665655555 5556565655555556 55555566666SSSSSS 55555656665655555 55556656655555556 555556566665655555 555556656665565555 555556665665SS6555 555566566555655655 555566566556556555 555656565565656555 5555565666665655555 5555566666655555556 555566566556565655s 5555665666555655565 555656566556565565s 5556565665566556555 55555656666665655555 55555665666656655555 55555666666555655655 55555666666556556555 5555665665566566SSSS 55556656656566565555 55556656665565655655 55556656665566556555 55556666665555555566 55565656655665656555 55565656656566556555 55565666655655555656 55566566655565556565 55656656556555656565 5S656665556555566565 555556566666665655555 555556665666566655555 555556666665565656555 555556666666555655565 SS55665666556656565SS 555566566656565655655 555566566656566556555 555566666655556556655 555566666655655565565 555656566565665656555 555656566566656555556 555656566656656555655 555656666556565566555 555656666565655556556 555656666655655555456 555666666555555555666 556566565565565665655 5555556666666666555555 5555565666666665655555 5555566S66666666555555 5555566666655665665555 5555566666656566565555 5555566666665565655655 5555566666665566556555 5555566666665655655565 5555665666556665665555 5555665666565665656555 5555665666566566556555 5555665666566656555556 5555665666656656555655 5555666666555656566555 5555666666556555656565
SW bonds
Orbits 0
0
2 0 5 3 2 65 6 4 0 3 2 6 3 6 5 3 4 2 6 4 : 5 8 7 2 7 7 9 6 6 2 0 5 4 7 : 7 6 9 4 0 9 6 4 6 8 0 (: 3 : 3 4 8 5 : ; 6
x 1x
c,
0 1 x
c2u
1 x
c,,
1 x
c,
1 x
CT,,
1 x
c,
2 x CI
1 x c,, 1 x c, 1 x c2 2 x 0 1x 1x 3 x 1x 3 x 1x 1x 2 x 1x 6 x 2 x 0 I
c, c, c, c, c,, 1 x c, c, c2,2 x c, c, c,, 1 x C, C, C, C, .
x c,
1
x cz
5 x c, 2 x c,, 3 x c, 1 x C,“, 1 x c,, 1 x c, 1 x c, 1 x c-2,3 x c, 1 x c2,3 x c, I x C2”, 1 x c, 1 x c,, 1 x C,
1x
c2,
I x
c,
0 5x 4 x 7x 1X 3x I x 1x 1x 4 x 0
c, c, c, C,, 2 x c, c,
c,
Cr c,, 4 x c, C,
1 x c,, 4 x c, 6 X 1x 1x 4 x 0 1x 0 3x 1x 3 x 1x
c, c,, 3 x c, c, c,, 1 x c,
1x
c2,
4
C, c, c, c, c,
x c,
5 x 1x 5 x 2x
C! c; c, c,
1 x c,, 4 x c, 3 x c, (Continued)
1243
Isomerisations of the fullerenes Table 2. (Continued) Isomer
Ring spiral
SW bonds
GJ 16(C2) Go 17(C,) Go WC2)
5555666666556556556556 5555666666556556565655 5555666666556655665555 5555666666565556555656 5556565656666665555556 5556565665665665656555 5556565665666565655655 5556565665666566556555 5556565665666656555556 5556566665566555665565 5556566665656555656565 5556566665656556565655 5556566665656655665555 5556566665665555655665 5556566665665556555656 5556566665665655655565 5556566666556556556655 5556566666556655656555 5556566666566555565556 5565665655656566656555 5565665656556565665655 5565665656556656565556 5565665656565656565565 5565665656565656655655 5566656655655655656565
1 7
Cm
WC2)
Go
WC,“)
c40
21(C2)
c40
WC,)
Go
WC21
Cm
MC<)
c4025(c,j Go WC,) C4027(C2) c40
2WCJ
29(C2) 3WC3) Go 31(G) Cd032(b) c4o 33(&h) Go 34(C,) c40 c40
c40
35(c2)
GO
WC2)
&I
37(c2,)
c40
38(02)
39(&d) Go 40( Td)
c-40
4 7 3 7 6 3 8 7 8 6 7 11 9 I1 2 4 4 5 5 6 10 10 12
Orbits 1 x c2,3 x c, I x c, 2 x c, I x c,, 3 x c,
I x c, I x C&3 x c, 6 x C, 1 x Cl, I x 2 x c,, 3 x I x c,, 3 x 8 x C, 2 x c,, 2 x 3 x c,, 2 x I x c,, 5 x 3 x c, I x c,, 5 x 1 x c, 1 x c, 4 x c, ! x C&2 x 1 x c,, 2 x 1 x c,, I x I x c,, 4 x
c, c, c, c, c, c, c,
c, c,
c, c‘,
I x c, 1 x C.T
Each spectrally distinct isomer is represented by a ring spiral[9]. For each isomer the total number of Stone-Wales bonds is listed, along with its decomposition into orbits of the point group. Thus, for example, isomer 9 of C3, has 7 SW bonds, 4 in a set of C, site symmetry, 2 in a set of C, site symmetry and 1 in a set with the full C2, symmetry. The interconversions are mapped out later in the present paper.
Table 3. Interconversions Reactant isomer
2(C2) 4(C,) 5(C,) I 7(CJ g(C2”)
I lO(C,) I l(C2)
12(C,) 13(C,) 14(C,) 15(C2) 16(C2) 17(C,)
1S(C2) l%Cd 2o(cd 21(C2) WC,) 23(c2)
24(c) 25(C2) 26(C,) 27(C2) 2g(C,) 29(C2) 3O(C3) 3 I(C.J 32@2) 33@2h)
of the 40 isomers of CM Product isomers
5(C,) 4(CA5(C,),lO(CI) 2(C,),4(C,),7(CM(C,) g(C,),lXC,),2l(C,) 5(C,),l2(C,) 6(C,) 5(C,),lO(C,),l2(C,),l4(C,) 4(C,),9(C,), lO(C,),2 l(C,),22(C,) 23(G)
1244
P. W. FOWLER et al. Table 3. (Continued) Product isomers
Reactant isomer
lS(C,),25tC,),36(C,),37(C,) 15(C,),35(C,),36(Cz) 26(C,),35(c,),35(C2)
34(Cl) 35(C2) 36(G) 37(CZ”)
38(&f 39(&)
28(C,)MG) 26(C,),27(Cz),3I(G) 28(G)
40( Td)
3 l(G)
Isomers 1and 3 have no SW bond and therefore no allowed conversion. For each isomer in the numbering scheme of Table 2 and Fig. 12, this table lists products of twisting each distinct type of SW bond.
other two, and the three isomers are thus split into two disjoint families on the transformation map. In this case it is easy to see the reason for the selection rule: the concerted transformation preserves inversion partners and so cannot exchange isomers with two (B,C) antipodal pairs for one with a single (C,C)
c
C
C
(b)
42
,: 1
2 2
3
Fig. 9. Stone-Wales interconversion maps for isomers of C30 to Cj4. (a) CJOforms a single family, (b) C32 forms two families (isomer 5 is disconnected from the rest), (c) CM forms two families of three isomers each. The conventions used here apply to all other maps in the present paper: each isomer is labelled by its place in the list of ring spirals given in Table 2 or 4, enantiomers are not distinguished, the site symmetry of the active bond is indicated on the arrow. Spectrally neutral transformations (i.e., those where the product is enantiomeric with, or identical to, the reactant) are not shown but are listed in the captions. Such transformations occur for CJ2 (isomer 4) and Cj4 (isomers 2,4, and 5).
pair. In the case of the SW transformation the constant factor linking families is presumably also geometrical, though less easy to see. Examples of single isomers, self-racemising fullerenes, chiral pathways, and closed loops of transformations are all present in these maps for small fullerenes and show what may be expected in the superficially more complex maps for higher fullerenes. The only fullerene for which the unrestricted Stone-Wales transformation has received attention in the past has been CM. A full isomerisation map for the t 8 12 spectrally distinct isomers of this fullerene would be a daunting prospect, and instead the present programs have been used only to check the first few steps away from the canonical icosahedral isomer. Coulombeau and Rassat studied the introduction of Stone-Wales “defects” into the C,, structure, confining attention to transformations of those SW bonds
5
Fig. 10. The Stone-Wales interconversion map for C16. Conventions as in Fig. 9. Only isomer 5 is disconnected from the others. Spectrally neutral transformations occur for Cj6 (isomers 3. 7, and 12).
Isomerisations of the fullerenes
1245
Fig. I 1. The Stone-Wales interconversion map for C18. Conventions as in Fig. 9. Isomers 2 and I2 are disconnected from the family that incfudes the other 15 isomers. Spectrally neutral transformations occur for CIx (isomers 3.4, 7. 10, 13, and 17). IO and I3 also convert via two C, patches.
Fig. I?. The 40 fullerene isomers of C,,. Taken with Table 3. !hjs allows the construction of the interconversion map. Spectrally neutral transformations occur for C4ii (isomers 4. IO. 19, and 35).
present in the original structure] 17.181. tifting this restriction we find the cascade, with only one “ I-defect” (C1,) isomer. 6 “2-defect” isomers and IX “3defect” forms. It is a general feature of the isomerisation maps that high-symmet~ isomers with their few, large orbits tend to lie on the ends of branches whereas low-symmetry isomers are often multiple junction points on the map. It is also notable that. if in the formation process any nonicosahedral isomer of ChO is ever produced, the overwhelming dominance of I), CM)indicates that all material must funnel through the C21,bottleneck on the interconversion map if the number of carbon atoms per molecule remains constant. Of more chemical significance are the isolatedpentagon isomers ofthe higher fullerenes. It is not yet clear what determines the isomer distribution in every case, but electronic and sterio factors are often finely balanced. CT6 adopts a closed-shell chiral 1): structure rather than the geodesic but open-shell tetrahedral cage[29,3]; CJ8 occurs as several isomers[27,4], no one of which has a properly closed shell in simple Huckel theory. Isolation of pentagons would appear to be necessary for stability, and the rcstricted Stone-Wales transformation. linking only isolated-pentagon fullerenes, is plausibly of lower energy than the general transformation (see Fig. 2). The smallest fullerencs to support the larger type II patches are isomers of C,,, and a map of Cqp isomer-
1246
P. W. FowLER~~ C
C
C, -
2v c)
(0) 4
2"
ct
2
3
1
(bf
c
c* Cc
I
L"
-
2
4
7
Fig. 13. Stone-Wales transformations between isolatedpentagon isomers of (a) CT8(two families), and (b) CsO(five families).
14
Fig. 14. Stone-Wales t~nsfo~atio~s
al
isations was presented in [4] asa rationalisation ofthe experimental observation of two isomers of this cluster. The map splits into two: a linked group of four isomers and an unconnected single isomer. There are difficulties with interpretation-the isomer predicted to be most stable in [4] was not observed at all, though it may be the predominant isomer in independent experiments performed in Tokyo[30], and in any case one might expect a thermodynamic equilibrium between all four isomers on the larger branch of the map. Maps for isoIated-~ntagon isomers C,, to C,, are presented in Figs. 13- 15. A classification of their SW patches is given in Table 4. One interesting point that has been emphasised in our recent treatment of CB2[161 is that this fullerene appears to be the only one for which all isolated-pentagon isomers form a single SW map. For CsOwe see the isolation ofthe icosahedral isomer from all the others. For Cs4 (map plotted in [ 1.51)the two leapfrog isomers belong to distinct families, as do the three leapfrog isomers of &, (map not shown here), whereas for G6 2 of the 6 leapfrog isomers are found in one family. The partition of isolated-pentagon isomers into distinct families for CT8is a consequence of disallow-
7
8
between isolated-~ntagon isomers of Cs6(three families).
Isomer&ions
Fig. 15. Stone-Wales transformations
between isolated-~ntagon
Table 4. Catalogue of isolate-~ntagon Isomer
G36ll(C,) c86
12(c,)
1247
of the fullerenes
isomers of CR8(six famiIies).
fullerene isomers CT8to Csa
Ring spiral 56666656565656666665666656656665656565666 56666656565656666666666566566565666565656 56666656565665666666656665656566656666565 56666656565666566566656656666666566656565 56666656566565666666566665656566656666565 566666565656565666666666666565656565666665 566666565656566665666666566665656565666665 566666565656656666666566665665666565656566 566666565656656666666656666565665656565666 566666565665656666656666656566656566665665 566666566565656666566666656~666~6566665665 566666656565656566666666666565656565666665 5666665656565666666666656566565656665666665 5666665656565666666666656656565656656666665 5666665656566566666665666566565665665666665 5666665656566566666665666656565665656666665 5666665656656566666566666566566565665666665 5666665656656566666566666656566565656666665 5666665656656566666566666656466565656565666 5666665656665666666565666565656666566666565 5666665665656566656666666566656565665666665 566666565656566666666665666566565666565666566 566666565656566666666665666566656666565656566 566666565656566666666666566656566566656666565 564666565656656666666566646~66~66566565666566 566666565656656666666566666566566656565665666 566666565656656666666656666566565666565666566 566666565656666666666565656656656666656566566 566666565656666666666565665656656666566566566 566666565656666666666656565656665666666656565 566666565665656666656666665666665656656566665 566666565665656666656666666566656656565665666 566666565665656666656666666566665656565656666
Orbits
MII) 0
3 3 3 3 0 0 ; 3 5 0 2 4 4 4 4 5 3 3 6 : 0 3 3 3 3 3 0 6 5 5
0 I
x cl?,. I x c;
I x c;,,,1 x C’, I x c;, I x c-2, 0 0 I x c, 0 I x c;,.,1 x c; 1 x C’?, 0 I x C‘I
2 2 2 2
x x x x
1x Ix 1x 1x 2x Ix 0 1x
C’, c’, c‘r c:,
c,, 2 x c’, c‘, (; c,,I x f', c‘, c, c-2. 1 x C,
3 x L’,
1 x c,. 1 x (‘, 3 x c,
I x c',. I x C‘, 0 I x c’,, 1 x c', 5 x C, 5 x c’, (C‘ontinwdj
1248
P. W. Fowmt et al. Table 4. (Continue) Ring spiral
Isomer 1
C86 13(G) c86 14(c,) C86 1 %Gf c86
l’%Cr)
C86 mG) c86
I%c3)
Gf’
WD3)
G8
W2)
Gs
2(Cd
Ces
3(G) 4(c,)
c88
Csa 5(C2J Gs
@Cd
c8S 7( cd cas 8(G) Gs
%a
css
WCZ”)
c86
ll(cd
C88 WC,) p
wj 88
f
css
WCI)
Gs
WC,)
Gs
ma
CSS l@Cif css
19( CJ
CSS 2o(cZt CSS =fc,) css
WG”)
23(c,) G8 -WC?) egg
C88 25(c21 2
;$j
28($$ Cg8 29(u CT::
CSS M(cil
31(G) Cea 3wx CS, 33( C,t Css 34( 73 G?
C88 3W2)
Orbits
n(II)
566666565665656666656666666566665656566566665 S664665656656S6666665666666566566656565665666 566666565665656666666666666565656565656666665 566666565665666666656665656566666566665656656 566666565665666666656665656656665666656566566 566666565665666666656665665665666566565665666 566666566565656666566666666566~5656566566665 5666665656565666656666666666666656566565656566 5666665656565666666666656666565666656566665665 5666665656565666666666665666566566666566656565 5666665656565666666666666665665665656565665666 566666565656656666666S666656666666566565656665 5666665656566566666665666665666666565665656665 5666665656566566666666566666565666656566665665 5666665656566566666666666665666565656565656666 5666665656566566666666666665666565656566566665 5666665656566666666665656565666666666656565656 5666665656566666666665656566566666666566565656 5666665656566666666665656656566666656666565665 5666665656566666666665656656566666665666565656 ~~66665656566666666665665656566666566666565665 5666665656S66666666666565656566565666665666566 5666665656566666666666565656566656666665656566 5666665656566666666666565656656565666656666566 5666665656566666666666565656656656666656665665 5666665656566666666666566565656656665666665665 5666665656566666666666566566566565656656566666 5666665656566666666666566566566656656656565666 5666665656566666666666566656566656656566565666 5666665656566666666666566656656565656565666666 5666665656656566666566666656666665666565656566 56666656566565666665666666656666566S6665665656 5666665656656566666566666665666665656665656656 5666665656656566666566666665666665665665656566 5666665656656566666656666665666666565665656665 S6666656566S6566666666666665656565665666665656 5666665656656566666666666665656566565666656656 5666665656654566666666666665656656565665666665 5666665656656566666666666665656656565666566656 5666665656656666666656656566566666666566565656 5666665656656666666666656656665656566565656666 S666665665656566665666666665666665656665656656
6 4 : 6 3 6 0 :, 0 5 4 2 2 4 5 : 4 3 5 3 6 2 3 4 6 4 5 4 4 4 3 4 5
6 X C, 2 x c,
1 x c,, 2 X c, 1 x c,, 2 x C, 3x 1x 1x 0 1x
c, c, c, c,
: 1 x Cz,, 1 x c, 4 x c,
1 x c, 2 x C, 2 x C,, I x c,
1 x c,,, 1 x c, 5 x c, 5 x c, 4 x C$
1 x c,, I x C) 5 x CI
1 x c,, I x c, 2 x C,, 2 x c, 2 x c,
I x c,, 1 x ci 1 x c, 4 x C,
I x c,, 1 x c, 2 x c,, I x c, I x c,, 2 x c, 2 x Ci 4 x c, 2 x c,
1 x c-2,1 x CI 4 x c, 5 x C,
1
1 x c,
4
2 x c, 1 x c, 1 x c,, 2 x c, 0
;: 4
1 x c,
Each spectrally distinct isomer is represented by a ring spiral[9]. A list of spirals for Cs4 is given in [ 191 and is not repeated here. For each isomer the total number n(H) of type II Stone-Wales bonds is listed, along with their decom~sition into orbits of the point group. The ~nterconversions are mapped for CT8 to Css in the present paper, with the exceptions of C82 and C 84r for which maps are available else-
where[ 15,161.
ing small-patch SW transformations. The -t13isomer of C,, has no type-11 patch but 18 type-1 patches, and can be converted to the Dshisomer by three consecutive small-patch rotations and thence to any other isolated-~ntagon isomer by rotation of newly formed larger patches (Fig. 16). The role ofthe restricted SW transformation in rationalising isomer distributions is an intrinsically limited one. Clearly, it cannot be a helpful concept in the range of n below 78, because no such transformations are possible there. It will also become progressively less discriminating in the high n range, since most isolated-pentagon isomers of the very large higher fullerenes will tend to have large separations between their pentagons and hence, no SW patch. The rationalisation proposed in [4] has thus a
naturally finite range of usefulness. It is interesting to speculate on whether all or nearly all isolated-pentagon isomers of the super-high fuilerenes will eventually be found, or whether other more complicated rea~angements of pentagons on their surfaces will act to equilibrate the isomers and bring the number of isolable species down to a manageable level. Given the enormous number of isolated-pentagon isomers for n = 120 and above[ 191, the absence of superStone-Wales or other rea~ngements would make the chemist’s life a difficult one. Prediction of the isomer distribution even within the Stone-Wales regime depends upon assumptions about the relative stability and activation energies across the isomerisation map. At one extreme, under the assumption of freely converting isolated-penta-
I249
lsomerisations of the fullerenes
fullerene isomers and to list all their interconversions. Computer algorithms capable of carrying out both of these jobs have been implemented and described here. An atlas of all spectroscopic signatures and of all isomer families, complete to any desired carbon number n is now within reach. The next step in this task of classification is the generalisation from the two-dimensional maps for constant n to the threedimensional ladders connecting fullerenes C, and accessible by a combination of isoc “i-2. Cni4, merisation and CZ ingestion/extrusion. Programs for the systematic treatment of this step are under development. Fig. 16. Conversion between families of isolated-pentagon isomers. The D3 isomer of CT8is in a family by itself and has no type II patch. Successive twisting of the marked small patches leads to a L&hisomer and hence to the second family of isolated-pentagon isomers.
,tcknow/rl~~:fnLi,s-R.P.R. thanks the Irish-American Partnership and AGB Scientific Ltd. for financial support. Help from Robin Batten (Exeter) with computer graphics is gratefully acknowledged.
gon isomers of equal stability, it can be shown that the equilibrium mole fraction of each isomer in any
given map is inversely proportional to the order of its molecular point group. (Proof: Consider a SW transformation A - B in which the conserved site group of the transforming patch has order OS, the molecular point group of A has order O,,, and the molecular point group of B has order 0,. The number of SW patches in A which give B on transformation is n, .B = 0 JO,. and the number of patches in B which give A is n I. B = OB/Os. The equilibrium concentrations of A and B are related by microscopic reversibility, so that K, = [B],./[A],. = k,/k-,, where k, and km, are the total forward and reverse reaction rate constants. Since LIH for the transformation is assumed to be zero, these total rate constants are simply k, = n ,+k and k , = n ,_,k, where k is the generic rate constant of a single A - B transformation. Hence, 0,/O,, as claimed. The re[W[4,. = n,-B/n,-H= sulting distribution can be interpreted as entropic, the molar statistical entropy ofeach isomer X in the map being given by S, = R In W\- with W,- being proportional to l/O,.) Under the opposite assumption of significant differences in stability between rapidly interconverting isomers, a distribution governed by enthalpy (rather than entropy) differences is to be expected. In particular, if one isomer is very much more stable than any other. this thermodynamic distribution may further simplify to one observable isomer per map. Experimental results for C,R do not decide conclusively between the two extreme pictures ([4.30], see discussion in [ 161).
REFERENCES I. W. Kratschmer, L. D. Lamb. K. Fostiropoulos, and D. R. Huffman, Nature 347, 354 (1990). 2. R. Tavlor, J. P. Hare, A. K. Abdul-Sala, and H. W. Kroto; J. Chem. SW. Chem. Commun. I423 ( 1990). 3. R. Ettl. 1. Chao. F. Diederich. and R. L. Whetten. Naturc353, 149 (1991). 4. F. Diederich, R. L. Whetten, C. Thilgen. R. Ettl. I. Chao, and M. Alvarez, Science 254, I768 ( I99 I ). 5. F. Diederich, R. Ettl. Y. Rubin, R. L. Whetten. R. Beck. M. Alvarez, S. Anz, D. Sensharma, F. Wudl. K. C. Khemani, and A. Koch, Science 252. 548 ( I99 I ). 6. K. Kikuchi. N. Nakahara. T. Wakabavasi. M. Honda, H. Matsumiya. T. Moriwaki, S. Suzuki, H. Shiromaru, K. Saito, K. Yamauchi. I. Ikemoto, and Y. Achiba. Chum. Phys. Lett. 188, 177 (1992). I. H. W. Kroto, Nu:ature329, 529 (1987). 8. P. W. Fowler, Chem. Phys. Left. 131,444 (1986). 9. D. E. Manolopoulos, J. C. May, and S. E. Down, Chem. Phys. Lelt. 181. 105 (1991). 10. H. W. Kroto, J. R. Heath, J. C. O’Brien. R. F. Curl, and R. E. Smalley, Nuture318, 162 (1985). I I. P. W. Fowler and D. E. Manolopoulos, Nuturc355. 162 (1985). 12. P. W. Fowler and J. I. Steer. J. Chem. Sot. C‘hem. C’ommun. 1403 (1987). 13. P. W. Fowler, J. Chem. Sot. Furuduy 86.2073 (1990). 14. A. J. Stone and D. J. Wales, Chem. Pllys. Len. 128.501 (1986). IS. P. W. Fowler. D. E. Manolopoulos. and R. P. Ryan, .1. Chem. Sot. Chem.
Commun.
408 (1992).
16. D. E. Manolopoulos,
P. W. Fowler, and R. P. Ryan. /. Chem. Sot. Furudq~, 88, I225 ( 1992). 17. C. Coulombeau and A. Rassat, J Chim Phy~. 88. I73 (1991). IS. C. Coulombeau and A. Rassat. J. C/M. P/qx 88. 665 (1991).
19. D. E. Manolopoulos and P. W. Fowler, J. C‘hcm. PhJ:r.. 96.7603
( 1992).
7. CONCLUSIONS
20. T. G. Schmalz, W. A. Seitz, D. J. Klein, and G. E. Hite.
The focus of this paper has been mainly mathematical and taxonomic. We have been discussing the systematics of a particular mechanism of isomerisation, because we believe that this is an important preliminary to detailed modelling. In order to test ideas about fullerene stability and isomerisation it is necessary, at least in principle, to be able to generate all
21. S. C. O’Brien, J. R. Heath, R. F. Curl. and R. E. Smalley, J Chem. Phys. 88, 220 ( 1988). 22. M. Endo and H. W. Kroto. to be published. 23. A list of 36 possible groups is given in P. W. Fowler, J. E. Cremona, and J. I. Steer, Theor. Chim. Actu 73, I (1988). More detailed considerations (P. W. Fowler, D. E. Manolopoulos, C. M. Quinn, D. B. Redmond, and R. P. Ryan. to be published) show that the symmetry
Chem.
P~QJS.Left. 130, 203 (1986).
1250
24. 25. 26. 27.
P. W.
FOWLER
criterion can be tightened because groups Szn, C,, C,,t, C,, (n = 56) cannot be real&d as fulierenes. P. W. Fowler and C. M. Quinn, Theor. Chim. Acta 70, 333 (1986). P. W. Fowler, J. Chem. Sot. Faraday87, 1945 (1991). P. W. Fowler, J. Chem. Sot. Perkin 2, 145 (1992). P. W. Fowler, D. E. Manolopouios. and R. C. Batten, J. Chem. Sot. Faraday 87,3 103 ( 199 1).
et
al.
28. See for example, F. A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry, 3rd Ed. Wiley, New York (1972). 29. D. E. Manolopoulos, J. Chem. Sot. Faraday 87,2861 (1991). 30. K. Kikuchi, N. Nakahara, T. Wakabayasi, S. Suzuki, II. Shiroma~, Y. Miyake, K. Saito, 1. Ikemoto, M. Kianosbo, and Y. Achiba, Nature 357, 142 (1992).