Generalizations of the Stone-Wales rearrangement for cage compounds, including fullerenes

THEO CHEM Journal of Molecular Structure (Theochem~ 363 (1996) 291-301

Generalizations of the Stone-Wales rearrangement compounds, including fuflerenes

for cage

Alexandru T. Balaban*, Thomas G. Schmalz, Hongyao Zhu, Douglas J. Klein* I’k~us A&MUniwrsif,s. Gal~,e.~lon TX ?7553-1675. USA Received 5 September 1995: accepted 10 November 1995

Abstract Generalizations of the Stone-Wales (pyracylene) rearrangement are proposed for fullerenes and other cage compounds. These involve various num~rs of pairs of hexagons fused via a zig-zag sequence of edges, ending in two faces which are polygons of any ring size. This generalized rearrangement may lead either to antomerization, or more often to rearrangements changing one isomer of a cage into another. Applications to fullerenes with S- and 6-membered rings are discussed, as well as other types of cages with larger and smaller rings. Kewwds; Buckminsterfui~erene; rearrangement

Fuilerene

rearrangement:

1. Introduction The carbon cages C,, of which buckminsterfuilerene is the most famous, can have various numbers n of carbon atoms. For each n value, one or many isomeric structures are possible. In agreement with Euler’s theorem, a cage (or a polyhedral cubic graph, i.e. a planar S-connected graph in which every point is connected to three adjacent points) consisting of m-gons must obey the following reIationship:

where ,f;, is the number of ~-membered faces. If one allows only pentagons and hexagons as faces * Corresponding author. ’ Permanent address: Poly~~ic University, Organic Chemistry Department. 77207 Bucharest, Roumania.

Pyracylene

rearrangement;

Rearrangement

class:

Stone-Wales

of “proper fullerenes”, such as in the known and well-studied C& and CT0 compounds [l-S], the above relationship reduces to the necessity of having 12 pentagons and any number of hexagons. The numbers of possible carbon cages (proper fullerenes) C, (n = 20 + ~2fk) with .f; = 12 pentagons and .f6= (rr/2) - 10 = 0 through 20 hexagons are presented in Table 1. Of these, only a few of those which obey the “non-abutting pentagon rule” (9,101, i.e. “isolated pentagon cages”, have so far been experimentally identified. Two condensed Smembered rings have an antiaromati~ &ring periphery, and are expected to destabilize the cage [11-141. Table 2 presents the partition of the 1812 isomers of proper fullerenes Cm according to the types AD of their carbon atoms as in Fig. 1; this is equivalent to partitioning these isomers according to the numbers p of pairs of abutting pentagons (i.e. pairs

0166-1280~96~~1~.00 6 1996 Elsevier Science B.V. All rights reserved SSDI 0 166- 1280(95)04448-5

A.T. Balaban et al./Journal

292

Table 1 Counts for the carbon cages C, (proper vertices; all havefS = 12 No. of isomers

,I

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

fullerenes)

of MolecularStructure

with 20 to 60

No. of isomers with q = 0

0

I

0

I 2 3 4 5 6 7 8 9 IO 11 12 I3 14 15 16 17 18 19 20

0

0

1

0

(Theochem)

Table 2 Partition A = q is B is the hexagon: 2B = 60

363 (1996) 291-301

of the 1812 C6s isomers in terms of A and B, where the number of points common to three pentagons, and number of points common to two pentagons and one A(5j) = q; B(5*.6) = 2p - 3q; C(5,6*) = 60 - 3A- 4p + 3q; D(63) = 2A + B = 2p - y 12

A0

3

4

5

6

7

8

9

IO

B

0

0

I

0

2

_

0

3 4 5 6 7 8 9 10 11

0

2 3 6 6 15 17 40 45 89 116 199 271 437 580 924 1205 1812

0

2

I 5 5 16 20 43 62 123 174 300 442 704

of pentagons sharing one edge), and q of triplets of pentagons sharing a common vertex. If only the isolated pentagon cages are selected, then we obtain p = q = A = B = 0, i.e. atom types A and B are excluded. The smallest such case is buckminsterfullerene C6e, and the next one is CT0 [12]. The numbers of isolated pentagon cages (proper fullerenes) C, with n = 60 - 96 are presented in Table 3 [15-171.

2. The pyracylene rearrangement of fullerenes with 5 and &membered rings, and its generalization Two types of theoretical transformations

A Fig. 1. The four possible environments

for C,

B

I 3 5 17 81 215

I4 I5 I6 17 18 19 20 21 22 23 24

210

I1

54

25

131

IO

116 132

2

1 _

I

I6 3

31 28

23

_

5

54

I

2 7

42

214

145

2

I1

I47

12 13

4

6 39

2 6

1 7

1

isomers have received attention in the literature: the “leap-frog” conversion [l&22] of C, into Cjn, and the Stone-Wales pyracylene isomerization (C, + C,) involving a net displacement of two bonds [23]. One usually depicts this reaction as in Fig. 2. Often this is identified as a 4-electron pyracylene rearrangement, which would be thermally forbidden in a standard view [24], though since the bond order in C, is higher than unity, more than four electrons are actually involved.

C of a vertex in a fullerene

D having

5- and 6-membered

rings

A. T. Balahan et al./Journal Table 3 Counts for preferable PI

Counts

60

I

cages (isolated 70

1

pentagon

of Molecular

fullerenes

C,)

Structure

(Theochem)

293

363 (19961 291-301

72

74

76

78

X0

82

84

86

88

90

92

94

96

Sum

I

1

2

5

7

9

24

19

35

46

86

134

187

558

@-@+@

Fig. 2. The Stone-Wales pyracylene rearrangement. In pyracylene this is an automerization, but when this is a subgraph in a fullerene, the result is usually an isomerization.

Theoretical calculations in fact indicate that the barrier to the Stone-Wales rearrangement is quite high if the atoms are constrained to remain near the surface of the fullerene [13,25,26]. However Scuseria and co-workers [25,26] have found that significantly lower barriers are encountered if an sp3-sp intermediate is formed in which the sp hybridized carbon atom is allowed to rise above the fullerene surface. In this paper we do not propose to study the detailed physical path by which a rearrangement might occur, but wish instead to focus on rearrangement mechanisms, as formal relationships between one fullerene structure and another. It is quite likely that all of the

concerted “graphical” rearrangements discussed below actually have lower energy pathways, analogous to that found for the Stone-Wales rearrangement itself. In papers discussing the mechanism of fullerene formation, Curl described how one may extend the pyracylene isomerization to systems containing more than four rings in the subgraph undergoing rearrangement [27]. Moreover he noted that in the same sense that the pyracylene rearrangement involves four electrons, his generalization could be viewed to involve six, which in one view would make it thermally allowed [24]. However, this rearrangement involves long migration paths for two of the atoms, which may again lead to a large barrier. Several other graphical rearrangement patterns for fullerenes have been indicated by Murry et al. [26]. In the present paper we introduce further generalizations of the Stone-Wales isomerization. We discuss first the case where two pentagons are separated by several pairs of condensed hexagons as shown in Fig. 3. In the structures with broken

-1

Fig. 3. The generalized

pyracylene

rearrangement

/\ @b -1, \ \/

for systems involving

5- and 6-membered

rings

294

A.T. Bulaban et al./Journal of Molecular Structure (Theochem)

363 (1996) 291-301

-= /\-\ Q (63 I’

\ I’ \

-

--

1;

I

1

r”

I

I’

\ /

\

-

Fig. 4. Upper row: stepwise approach to the generalized Stone-Wales rearrangement; middle row: two different ways to proceed stepwise when the zig-zag trail is longer; lower row: a different mode by which the process from the upper row of Fig. 3 can be imagined, involving two shorter zig-zags instead of a long zig-zag of bonds

lines of Figs. 2 and 3 we preserve for convenience both the outer geometry and the electronic structure, but of course they must change during the isomerization (right-hand structures in Fig. 3). The rearrangement shown in Fig. 3 leads to the same molecular graph as that proposed by Curl [27] if the atoms are all equivalent; but if the carbon atoms are distinguished, e.g. by isotopic labelling, then the connectivity pattern is found to be different. Thus these two processes correspond to distinct physical mechanisms. Four ideas must be stressed. (i) This generalization to any number of rows of hexagons separating the two pentagons cannot be obtained by stepwise application of the pyracyclene rearrangement (unless rings of sizes other than 5 and 6 are allowed between subsequent steps). (ii) Similarly to the pyracylene rearrangement (Fig. 2), this generaliza-

tion as depicted in Fig. 3 is equivalent to an apparent degenerate isomerization (automerization) [28] in which the initial and final valence structures are the same, but the atoms in various positions are different. (It should be recalled that for detecting rapid reversible automerizations NMR techniques are adequate as in the case of bullvalene or semibullvalene; for slower automerizations, isotopic labeling is the usual detection method, as in the case of the pyrolytic automerization of naphthalene and other aromatics, or of the catalytic automerization of phenanthrene [29]). (iii) However, when the structural fragments depicted in Figs. 2 and 3 are part of fullerene cages, the rearrangement is no longer degenerate, and the process is a true isomerization. (iv) A primary focus of the present generalization is to construct many possible isomers of fullerenes starting from a given structure,

A.T. Balaban et al./Journal of Molecular Structure (Theochem)

363 (1996) 291-301

/\ @03 /\ @Q ‘- \-

c

= 0 I\

”

I

\

i-1

-

\

295

I

*

-

-__ \ \-:

-

I

\ /

Fig. 5. Application of the generalized pyracylene rearrangement an isomerization (non-degenerate rearrangement).

besides probing into the mechanism of fullerene formation. In its essence, the generalization described in the present paper consists of: (i) conserving a conjugated string (bold lines in Figs. 2 and 3) of 2k carbon atoms inside an outer periphery formed from 2k - 2 peri-condensed hexagons and two pentagons on opposite sides at the ends of the string, but (ii) changing in parallel all connections of the 2k - 2 inner points to the carbons on the periphery. Then the pyracylene rearrangement (Fig. 2) would be the particular case for k = 2, whereas Fig. 3 represents the next cases for k = 3 and 4. Two important restrictions apply. First, the inner string of 2k atoms must have a strict zig-zag geometry: this is equivalent, in a cage system, to choosing alternatively left/right directions at each branching point starting at one end of the string. Second, the two next-to-marginal edges of the inner string of 2k

to systems with 5- and 6-membered

rings: (A), (B) automerizations;

(C)

vertices must belong to each of the B-membered rings at the ends of this string. Considered as a concerted process the generalized pyracylene rearrangement involves transition states with multiple Hiickel 4n cycles. However, this process may be imagined to take place stepwise, as in Fig. 4, provided that we are ready to admit ‘I-membered rings on a temporary basis. With longer zig-zag trails, as shown in the middle row of Fig. 4, a stepwise process may start at different locations. As an alternative to conserving the long zig-zag trail of bonds, one can consider the rearrangement suggested by Curl [27] which conserves shorter cross-zig-zags, such as indicated in the bottom row of Fig. 4. The resulting graph via this process is isomorphic to that resulting via that of Fig. 3. However, different atoms are permuted to different positions. Moreover, the process of Fig. 3 breaks

296

A.T. Bulaban et ~l.lJo~rn~~o~~oiecl~~ar StrMeture f Theoche~~ 363 (1996) 291-301

Fig. 6. Application of the generalized pyracylene rearrangement to the least stable isomer of buckminsterful~erene.

and reforms more a-bonds (globally) than that of Fig. 4, though it can be argued that in Fig. 4 less “damage” is done to the r-network during the rearrangement. That is, if in the transition state for short zig-zags (the bottom row of Fig. 4) the bonds buckle up and down (so as to keep the conserved a-bonds from over-shortening), then the r-bonding resonance integrals diminish and compromise the n-bonding there. The work of Murry et al. 1261 also considers rearrangements involving changes between the initial and final patterns of Fig. 3. Further generalizations are possible, which allow any even number of j-membered rings to be involved in the generalized pyracylene rearrangement. As shown in Fig. 5(A), what is required is that the conjugated inner string of 2k carbon atoms with the zig-zag geometry must have at each of its ends one j-membered and one 6- membered ring, with mutually opposite geometries: the remaining

ring sizes flanking this inner string may be 5- or 6membered, because in this situation the rearrangement will preserve these rings, and will not introduce 4- or 7-membered rings. Examples are shown in Fig. SA for a non-degenerate rearrangement. It was shown [30] that in the photoionization and fragmentation of Cm, in addition to CZ fragments, longer C, fragments may also be involved. The present paper shows that by analogy with the StoneWales pyracylene rearrangement (which involves the disconnection and the reconnection in a different way of an internal C2 fragment) one can conceive of generalized isomerizations involving a longer CZk internal atomic chain with a zig-zag geometry.

3. Applications of the generalized pyracylene isomerization The least stable of the 1812 C6,, isomers (with

B

Fig. 7. Three-fold application of the generalized pyracylene rearrangement: in (A) there are two belts of six Smembered rings, and in (B) there are six pairs of abutting pentagons.

A.T. Balaban et al./Journal of Molecular Structure (Theochem)

291

363 (1996) 291-301

-

Fig. 8. Three-fold application of the generalized pyracylene rearrangement: rings, but in (B) only isolated pentagons and type B groups of pentagons.

in (A) there are three type A groups

Fig. 9. The fullerene with 60 carbon atoms and two 4-membered rings with point group symmetry [37], and its conversion into a fullerene with only 5- and 6-membered rings.

$: Fig. IO. The generalized also involved, provided

of abutting

C,, described

Smembered

by Gao and Herndon

I

,’ I’ I’ I’

@

rearrangement, trading off two 4-gons and two 6-gons for four 5-gons; any numbers that there is a continuous zig-zag geometry of the middle heavy line.

of layers of 6-gons may be

298

A.T. Balaban et al./Journal of Molecular Structure (Theochem)

363 (1996) 291-301

Fig. 11. Three conversions of the truncated octahedron: (A) into a cage with two hexagons and a belt of 12 type B condensed (abutting) pentagons; (B) changing two 4-gons and two 6-gons into four Sgons; (C) changing one 4-gon and one 6-gon into a 3-gon, a 5-gon and two 7-gons.

maximal p, q, A and D values) has two strained caps of six pentagons (two halves of a regular pentagonal dodecahedron) connected by a belt of 20 hexagons. It is depicted as a Schlegel diagram in Fig. 6(A). On applying one generalized pyracylene isomerization with k = 10 it is converted into the isomer shown in Fig. 6(B). A different C& isomer, which also posseses two antipodal separate aggregates of six pentagons, is presented in Fig. 7(A), along with the result of the generalized isomerization (Fig. 7B, with six pairs of

abutting pentagons); in both cases, only type B condensation is present. In Fig. 8(A), a ChOisomer with three groups of four condensed pentagons of type A is shown to be converted by the generalized pyracylene rearrangement into an isomer with three type B condensed triplets of pentagons and three isolated pentagons (Fig. 8B). Babic and Trinajstic [31] divided fullerene isomers with up to 70 carbon atoms into “pyracylene rearrangement” classes; they showed that 31 from

A.T. Balaban et al/Journal of Molecular Structure (Theochem)

363 (1996) 291-301

299

processes are examined, but here they are not discussed in detail. However, we note that Babic et al. performed rearrangements on the duals of the molecular graphs, and these do not distinguish between different permutations of atoms in the products. Consequently they do not differentiate between the rearrangements proposed here and those of the type proposed by Curl [27], i.e. those exemplified in the upper row of Fig. 3 and the lower row of Fig. 4. It has been argued that although the pyracylene rearrangement cannot explain interconversions of CT8 isomers after they are formed [33], such rearrangements may operate in annealing or fragmentations [25,26,34] and in the opening of “windows” allowing the incorporation of endohedral foreign atoms [35]. In addition to metallic atoms or ions, 3He atoms were shown to penetrate fullerenes allowing the determination of ring currents in ChO and CT0 [36]. By analogy, the present generalized pyracylene rearrangement allows the opening of larger “windows” or the expulsion of larger fragments.

4. Generalization Fig. 12. Another type of generalization for the Stone-Wales rearrangement, involving four abutting polygons with any ring sizes (a-, b-, c- and d-membered rings), as seen in the first row. For some particular values of these ring sizes, illustrated in the third row, ring sizes are kept, hut their order (according to the sense of rotation marked by the curved arrow) is changed. The second and fourth rows illustrate cases where they become pairwise equal.

the 1812 CeO isomers do not possess an arrangement of pentagons that would allow a StoneWales rearrangement to be performed. We investigated all of these 31 isomers, and proved that by means of the generalized rearrangement described in the present paper, all 31 isomers are able to be converted into other isomers that can then undergo normal Stone-Wales rearrangements. This suggests the possibility of using generalized pyracylene rearrangements as an efficient method for systematically generating all isomers of a given C, starting from any one structure as a seed. Just recently, Babic et al. [32] have submitted for publication a manuscript in which such

of the pyracylene rearrangement of fullerenes with any ring size

Eq. (1) applies regardless of ring sizes in a polyhedral cage. Two groups of authors [37,38] calculated ring strain in fullerenes with 4-membered rings (these no longer being “proper” fullerenes), and found that the strain in some cases may be lower than in related, proper fullerenes. According to Eq. (2), one 4-membered ring can replace in a cage two 5-membered rings, as demonstrated in Fig. 9, where two 6-gons, one 5-gon and one 4-gon are interconverted into one 6-gon and three 5-gons. Similarly, Fig. 10 shows the interconversion between a subgraph with four 5-gons and a system with two 6-gons and two 4-gons. As in cases discussed earlier for proper fullerenes, the numbers of layers of 6-gons between the two rings with different sizes can be increased arbitrarily; this is shown by an example in Fig. 11 which again is a trade-off between a subgraph with four 5-gons and a system with two 6-gons and two 4-gons.

300

A.T. Balaban et al./Journal qf Molecular Structure (Theochem)

Fig. 13. A, different type of generalized in&ease/decrease by 2.

Stone-Wales

rearrangement

Fullerenes with ring sizes larger than six possess negative Gaussian curvature, which needs to be compensated for by smaller ring sizes. Allowing such larger (and smaller) rings, there are even further possibilities. Another way to generalize the Stone-Wales process for interconverting an internally cubic subgraph with four abutting faces is illustrated in Fig. 12: if these faces are a-, b-, c- and d-membered rings before the rearrangement, then after the interconversion they will pairwise (across the shifting bond) increase and decrease their size by one vertex, respectively. Thus, pyrene will be converted by this process into an azuleno-azulene. Interestingly, depending on the initial topology of the assembly of four polygons with ring sizes progressively increasing by one, this process will convert such a system either into the same sequence rotated in the opposite sense, or into an assembly maintaining two adjacent polygons and changing by one the sizes of two other ones at the expense of each other. Notably the zipper mechanism in Fig. 4 involves 7-membered rings at intermediate (transition) stages. For a general grouping of six polygons as in Fig. 13 one can also imagine a rearrangement of three bonds so as to cause three polygons to increase and three alternate polygons to decrease their ring sizes by 2, as indicated in Fig. 13. Woodward-Hoffman-type correlation diagrams for such generalized Stone-Wales rearrangements could offer additional insight into possible barriers. Rather generally from a graph-theoretical viewpoint one may devise rearrangement processes by first cutting out a (small) fragment; second, rotating it so that broken bonds match again; and third, reconnecting dangling bonds at new positions around the periphery. If some bonds are reformed

involving

363 (1996) 291-301

three bonds and six polygons

of any ring sizes: their sizes

between atoms that were previously bonded, then these bonds can be imagined to be retained in the actual mode of rearrangement. In this light, the Stone-Wales rearrangement (corresponding to cutting out a 2-site fragment) is the simplest possibility (involving the smallest possible fragment, since cutting out only a l-site fragment does nothing). The fact that graphically these rearrangements can all be pictured as “planar” rotations of the excised fragment does not restrict the physical motion of the atoms to lie near the surface of the fullerene, as long as the appropriate reconnections are eventually made. Then the physically more realistic rearrangement mechanisms considered by Scuseria and co-workers [25,26,34,35], which involve dangling bonds and larger holes, can also be categorized in this way.

5. Conclusion It was shown that a theoretical derivation of constitutionally isomeric cage compounds is possible via a generalized pyracylene rearrangement: in the 3-connected graph formed from 5and 6-membered rings, the numbers fs and f6 of each type of ring are conserved, but their mutual positions are changed. Moreover, this generalized Stone-Wales rearrangement can be generalized even further, being applicable to cage systems with any ring sizes.

Acknowledgements Discussions acknowledged.

with

Dr.

D. Babic

are gratefully

A.T. B&ban

et al./Jotrrnal of Molecular Structure (Theochem)

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