- Email: [email protected]
Closed-shell structures and the building game
Volume I4 1,number 6
CHEMICAL PHYSICS LETTERS
27 November 1987
CLOSED-SHELL STRUCTURES AND THE BUILDING GAME David J. WALES University Chemical Laboratories, Lensfield Road, Cambridge CB2 IE W, UK
Received 14 August 1987; in final form 2 1 September 1987
A simple model is developed to explore the statistics of dynamic assembly processes. The application of this buildinggameto the synthesis of close-boranes and carbon clusters is discussed. The theory may also be of interest in the study of transition metal clusters and biological structures.
1. Introduction The family of close-boranes, of general formula B,Hz- , have polyhedral structures with a BH unit at
each vertex and triangular faces (deltahedra). Every member of this series is known for n= 5 to n= 12, but none with n> 12 has been reported. Two theoretical studies have predicted that larger species with n = 13-24 should also be stable [ 1,2]. Furthermore, the electronic energy difference between any one of the large postulated structures and two known smaller species having the same total number of boron atoms is very large, so that the relative stability of the larger clusters with respect to a hypothetical decomposition into two smaller species appears to be high. The apparent non-existence of the close-boranes with more than twelve vertices is therefore puzzling. We suggest that the answer to this problem depends on kinetic, not thermodynamic, considerations. In 1970 Long [ 31 proposed a reaction pathway for the production of molecules with B3, Bq and BS skeletons from diborane, B2H6.The scheme basically involved the successive addition of BH, fragments to diborane with concomitant reductive elimination of hydrogen. Other authors [ 41 have also suggested the possibility of a stepwise pathway in the synthesis of B,,H:2. The latter experiment provides a number of important clues for the present study in the form of isolatable intermediates; for example the B11HG ion. In section 2 a general mechanistic scheme is considered which can account for the experimental observation that B,,H$ appears to be the largest 478
species on this sequential pathway. The wider utility of this theory is then illustrated in sections 3 and 4. Note that throughout this paper all coordination numbers refer to the connectivity of an atom within a skeletal framework, i.e. not including external bonds to hydrogen atoms.
2. The building game for close-boranes With the above observations in mind consider the following scheme. Starting from a triangle we add successive vertices so as to bridge external edges chosen at random. Each additional vertex produces another triangle which shares an edge with the fragment under construction. In addition, whenever a five-coordinate vertex is created we close up the structure around this atom and create another triangular face in the process. In doing this we neglect the hydrogen atoms completely and concentrate only on the boron skeleton. Hence it does not matter whether the boron species involved are anions, neutral molecules or even cations; we assume that the hydrogen atoms can look after themselves, for example in terms of reductive eliminations. This simple model, composed of the above two rules, is a game in the same sense as the “Game of Life” [ 51. It is possible to enumerate all the possibilities on the path from three vertices to twelve for this particularly simple scheme. The tree diagram is presented in fig. la, and the structures encountered on the pathway are illustrated in fig. lb.
0 009-2614/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 141,number 6
CHEMICAL PHYSICS LETTERS
a
3--4-
6a
9a
10a
6c
6b
9b
10b
1R VI
mib
Fig. I. (a) A tree diagram for the five-coordinate vertex building game; the labels, such as 6c, refer to the structures illustrated in (b) and the fractions give the probability that a particular pathway will he followed. Where no fraction is indicated there is only one distinct possibility. (b) Topologies of the various species encountered in (a).
Using the tree diagram it is easy to calculate the percentage of paths leading to the isosahedron when vertices are added stepwise to randomly selected external edges as described above. It is found that 99.96% of all such paths close after twelve vertices,
27 November 1987
so that the formation of any structures with more atoms by such a process is highly improbable. Furthermore, the pathway leading to structures which do not close after twelve vertices involves species which are probably electronically unfavourable. All the larger hypothetical close-boranes previously considered [ 1 ] have one or more six-coordinate vertices which only occur in this scheme for structures which follow from llb in lip. la. The higher coordination number was allowed to arise when closure around one live-coordinate vertex led to creation of a six-coordinate vertex. The rules of the above game involve closing up around five-connected vertices as they are formed, so it is not surprising that many pathways lead to the closed polyhedron whose vertices are all five-coordinate. If we had chosen to close up around four-coordinate vertices then a high percentage of pathways would have led to the octahedron. The usefulness of the new theory is immediately apparent; instead of asking whether the larger close-boranes are electronically stable we should investigate the rules of the above game. Using this insight the relevant question is why structures should close up around five-coordinate rather than four-coordinate vertices. Future theoretical studies may therefore be directed towards the solution of a more fruitful problem. Some qualitative explanations are advanced below. First we note that all the experimental evidence supports the above scheme; for example B3H9,B4H10 and B5H,, are all products of Long’s reaction scheme [ 31 and have the same boron skeletons as the unique three-, four- and five-vertex structures found in the building game. More significantly B,,H,, B, I Hi and a derivative of B,H, are all observed in a synthetic route [4] to B,,H:, . These have the same boron skeletons as the structures labelled lla, 10a and 9a in fig. 1 [6]. All pathways leading to the icosahedron must pass through lla in the above scheme, and many pass through 10a and 9a. Closer consideration of the BsHll species allows qualitative justification of the chosen rules. This molecules has a single four-coordinate vertex with two terminal bonds to hydrogen atoms. Consider the hypothetical process in which B5H,, closes up around this vertex to give B5Hg. This would require reductive elimination of H2 between the peripheral boron atoms which move to bonding distance. However, 479
Volume 141, number 6
CHEMICAL PHYSICS LETTERS
one of the B-H terminal bonds to the new apical atom must also break, since this atom must contribute three internal orbitals for skeletal bonding. Contrast this with the hypothetical case of closure around a fivecoordinate vertex. In this case the higher coordination number means that only one terminal hydrogen atom is likely to be bonded to the critical boron atom. Hence, closure around such a vertex would probably involve less disruption to the molecule. Steric factors may also be important; the distance between the peripheral terminally bonded hydrogen atoms before closure would be smaller for a five-coordinate vertex. The elimination of two of these hydrogen atoms may therefore relieve unfavourable steric repulsion, again suggesting that closure around a five-coordinate vertex may be more favourable. The change in boron-boron angles on closure to give a five-coordinate vertex would also be smaller than for a fourconnected vertex.
3. Application to large carbon clusters On vaporizing a graphite target with a laser Kroto et al. [ 71 discovered that a wide variety of large carbon clusters were formed. In particular they detected a remarkably stable fragment corresponding to a mass of sixty carbon atoms. On the basis of its non-reactivity and related evidence these workers proposed a closed, truncated-icosahedral geometry for the Ceo species, although some of their arguments have since been challenged [ 8 ] . Several theoretical investigations of this species have been performed ranging from simple Hiickel calculations to more extensive ab initio studies [ 91. The general conclusion of this theoretical work is that the icosahedral CbOmolecule may indeed be a stable molecule. However, Stone and Wales [ lo] have pointed out that various other CGO isomers with less symmetrical arrangements of the twelve pentagonal and six hexagonal rings may be nearly as stable. The application of the building game to the formation of closed shells of carbon atoms requires some modification of the rules used in section 2. Firstly we note that Kroto et al. observe the formation of many other species in their experiment corresponding to structures containing an even number of carbon atoms. The distribution of products depends strongly 480
27 November 1987
upon experimental conditions; the C,, peak predominates when cluster equilibration is allowed [ 71. Hence we would not necessarily expect to find a scheme in which most paths lead to closure after sixty atoms, in contrast with section 2. The following rules were chosen for the study of carbon clusters: (1) building blocks are allowed to add to the fragment under construction to give fiveor six-membered rings only and (2) two-coordinate atoms may link together to form five- or six-membered rings containing only three-coordinate vertices. The formation of four-membered rings is therefore forbidden; this seems a reasonable rule in view of the greater steric strain and antiaromatic character of four-membered conjugated rings. However, a larger ring may be formed along with a fiveor six-membered ring when the last bond is made to give a closed species. We also make the implicit assumption throughout these studies that the building process does not disrupt the bonding of the growing fragment. This is important because energy is sure to be released on bond-making and all the species are likely to be formed in excited states. However, the assumption that third-body collisions can drain away some of this excess energy is probably reasonable for the experiments under consideration. For this application the full schemes are too large to enumerate fully and computer programs were written to simulate the growth of the carbon clusters. Firstly a suitable pair of two-coordinate vertices is selected at random from the molecule under construction, and then a complementary pair of atoms is chosen on the building block to be added. The stepwise addition of building blocks and associated ring closures is then followed until the cluster contains sixty atoms. This process is repeated many times to give an average value for the number of closed paths, without regard for the detailed arrangement of the pentagonal and hexagonal faces. The addition of fragments to form five- and six-membered rings is illustrated in fig. 2a, and allowed ring closures are shown in fig. 2b. The choice of suitable building blocks is a crucial one. Some recent experimental work suggests that carbon cluster growth occurs from small species ( C1, Cz and C,) in the laser vapor&&ion of graphite [ 111, and from “large polyaromatic particles” in hydrocarbon flame experiments [ 121. To render the prob-
Volume 14 1,number 6
CHEMICAL PHYSICS LETTERS
27 November 1987
a
__-
Fig. 3. Building blocks considered for the carbon cluster schemes.
___ 88
fi
*
Fig. 2. (a) The addition of two Cl0 fragments to form tive-membered and six-membered rings. (b) Allowed ring closures in the carbon cluster building game.
lem as computationally simple as possible for this initial study we have first considered the fragments illustrated in fig. 3. The C6, Cl0 and C2,, species all consist of fused six-membered rings as found in graphite itself, while the C5 fragment also contains a five-membered ring and is probably a less likely candidate. Since two C6 rings cannot add together to give a five- or a six-membered ring no pathway involving this fragment alone can give any closed CsO according to the rules chosen. Using CIO,C, 5and C2,,
species alone 19,2 and 1Ohof pathways, respectively, lead to closed C6,,with twelve pentagonal and twenty hexagonal faces. These figures have been rounded to the nearest integer owing to the uncertainty entailed by the finite number of simulations used. The Cl0 fragment therefore gives by far the highest yield of closed CbO,although we should note that most of the closed molecules formed do not have icosahedral symmetry. When the game is allowed to continue to 120 atoms, corresponding to twelve Cl0 units, small amounts (of order 1% or less) of closed Cso, Cgo,Cloo and C, 1oare also found. In each case the number of simulations considered was 2000, which may be insufficient to detect some more improbable species. A scheme involving three Cl0 and five C6 fragments was also investigated. Two Cl0 and five C6 species were added to a Cl0 fragment in a random order according to the above rules; no closed C6,,molecules were found in 2000 simulations. A further study was carried out using the linear C3 molecule which may be a favourable fragmentation intermediate in the laser vaporisation experiment [ 111. In this case the assumed rules enabled live- and six-membered rings to form as above, but only additions giving no singly coordinate carbons atoms were allowed. Most simulations were found to lead 481
Volume 14 1,number 6
CHEMICAL PHYSICS LETTERS
to bispherical-type systems after sixty carbon atoms with two partly closed shells bridged by a few common live- and six-membered rings. Some interesting conclusions may be drawn from the above results. Firstly we have found that of the schemes tried so far the Cl0 fragment is by far the most efficient precursor of closed &,. Larger building blocks seem to be too inflexible to give a high yield of closed C6,,, whereas very small fragments lead to CbOspecies with regions which are locally almost closed linked together by smaller open areas. Of course, the restriction of using a single building block alone in each scheme may not be a good assumption. However, the use of a distribution of building blocks within a single game is much more expensive computationally, and it is not clear from the experiments what this distribution should be. The results also suggest that the formation of a closed CeOmolecule probably requires five or more collisions, and that if the closed-shell interpretation is correct then the C6,, mass peak may be due to a collision of isomers of similar stability. In summary, this preliminary application of the building game to carbon clusters has shown that closed sixty-atom species are indeed formed in an idealised growth process. The larger building blocks chosen may be more appropriate to the hydrocarbon flame experiments [ 121, but it is not impossible that they occur as intermediates under other conditions too. The appearance of prominent CbOmass peaks in various different experiments may indicate that some aspects of cluster growth are common to them. However, if all the even-numbered carbon clusters observed in, the mass spectra are closed shells, as previously proposed [ 71, then the models are clearly too biased towards closure at CbO.Observations such as this will be helpful for future applications with more complex rules, which may enable a more detailed description of the growth process to be extracted. Throughout the above treatment we have assumed that the ring closures occurring are electronically allowed processes from the point of view of orbital symmetry [ 131. Since the fragments may be electronically excited and both o and L interactions are involved this seems a reasonable postulate.
4. Further applications The scope for variation within the building game by modification of the rules and building blocks is very lare, and may represent a unifying scheme for the assembly of closed shells. For example, various authors have noted the structural similarities between simple plant and animal viruses and small RNA-containing bacteriophages [ 141. However, the assembly of virus capsids is a far more difficult problem than those discussed above since the building blocks involved are much more complicated and the relation between transcription, translation and assembly and the interaction with the host cell environment are not well understood. With this caveat established we will consider the construction of some icosahedral virus shells using the building game. Crick and Watson [ 151 first suggested that virus capsids might consist of a three-dimensional array of identical protein subunits. They argued that the viral genome is too small to contain sufficient information to code for a large number of different capsid proteins, and this view is now known to be substantially correct [ 161. Hence efficiency and economy are key factors in the consideration of the self-assembly of viruses. Caspar and Klug [ 171 showed that the number of structural possibilities available using only identical building blocks is actually very limited, and demonstrated how various shells with icosahedral point group symmetry might be formed. Electron microscopy, X-ray diffraction and, more recently, electron diffraction studies provide visualisations of virus shells, although considerable experimental effort is required. Two levels of organisation can be identified: the capsid may be viewed as an assembly of capsomers which often contain five or six protein subunits and are known as pentamers and hexamers respectively. Both the subunit-subunit and capsomer-capsomer bonds are non-covalent, but there is some experimental evidence that the capsomer-capsomer bonds are weaker [ 181. If we consider a game in which the building blocks are pentamers which can bind to five other pentamers then an equivalent scheme to that discussed for the close-boranes may be appropriate. In this case a closed icosahedral shell consisting of twelve pentamers would be produced by 99.96% of pathways. Of cource, we have neglected
482
27 November 1987
the possibility
of equili-
Volume 141,number 6
CHEMICAL PHYSICS LETTERS
Fig. 4. The 32-vertex omnicapped dodecahedron with twelve Bvecoordinate vertices and twenty six-coordinate vertices.
bration between structures and the interaction of the shell with the viral genome and core polypeptides, along with the other simplifications. It should also be noted that the angle at which the capsomers bind together probably introduces convexity into the growing shell [ 171 which will also favour the formation of closed structures. Nonetheless it is interesting to note that the random assembly of essentially pentavalent vertices in the building game almost always leads to the formation of a closed shell, and one might speculate on the possible evolutionary significance of this result. The admission of hexamers enables larger shells of icosahedral symmetry to be constructed [ 171. For example; a thirty-two-vertex structure (described by the labels [ 181 P= 3, f= 1, T=3) may be formed from twelve pentagons and twenty hexagons, as illustrated in tig. 4. The assembly of this structure may be simulated topologically with the following rules: (1) an edge connecting two hexamers may only bind a pentamer and (2) an edge connecting a pentamer and a hexamer may only bind another hexamer. In this case we close up around five-coordinate pentamers and six-coordinate hexamers. If we start with two hexamers and one pentamer at the vertices of a triangle, and edges are selected at random in the usual way, then it is found that 97% of paths lead to the closed shell of icosahedral symmetry (10000 simulations). Finally we consider some possible future appli-
27 November 1987
cations of the building game. It has been suggested that packing considerations may be important in determining the structures of transition metal complexes [ 191. In this case it would probably be more realistic to consider the stepwise bridging of faces to give fused tetrahedral structures, rather than the hollow shells discussed above. However, the potential energy surface for a transition metal cluster is probably “softer” than for the main-group boranes, so that rearrangement and equilibration of structures may be more important. These considerations are now being applied to the growth of some recently prepared microscopic colloidal metal particles [ 201. It is interesting to note the way in which some pathways leading to the icosahedron in the five-coordinate vertex scheme fold up spontaneously after the addition of a critical vertex to give a much more closed structure. Whether this observation has any physically interesting consequences remains to be seen. It is also possible that graph-theoretical analyses of the building game may lead to more compact representations of the various schemes we have considered above.
Acknowledgement DJW thanks the SERC for financial support and many friends for their encouragement of this project.
References [ 11L.D. Brown and W.N. Lipscomb, Inorg. Chem. 16 (1977) 2989; .I. Bicerano, D.S. Marynick and W.N. Lipscomb, Inorg. Chem. 17 (1978) 3443. [2] P.W. Fowler, Polyhedron 4 (1985) 2051. [ 31 L.H. Long, J. Inorg. Nucl. Chem. 32 (1970) 1097. [4] H.C. Miller, N.E. Miller and E.L. Muetterties, Inorg. Chem. 3 (1964) 1456. [ 51E.R. Berlekamp, J.H. Conway and R.K. Guy, Winning ways, Vol. 2 (Academic Press, New York, 1982) p. 8 19. [ 61 K. Wade, Electron deficient compounds (Nelson, London, 1971) p. 102. [7] H.W. Kroto, J.R. Heath, SC. O’Brien, R.F. Curl and R.E. Smalley, Nature 3 18 (1985) 162; Y. Liu, SC. O’Brien, Q. Zhang, J.R. Heath, EK. Tittel, R.F. Curl, H.W. Kroto and R.E. Smalley, Chem. Phys. Letters 126 (1986) 215.
483
Volume 141. number 6
CHEMICAL PHYSICS LETTERS
[ 81 D.M. Cox, D.J. Trevor, KC. Reichmann and A. Kaldar, J. Am. Chem. Sot. 108 (1986) 2457. [ 91 A.D.J. Haymet, Chem. Phys. Letters 122 (1985) 42 I; J. Am. Chem. Sot. 108 (1986) 319; R.C. Haddon, L.E. Brus and K. Raghavachari, Chem. Phys. Letters 12.5(1986) 459; R.L. Disch and J.M. Schulman, Chem. Phys. Letters 125 (1986) 465; P.W. Fowler, Chem. Phys. Letters 131 (I 986) 444; W.A. Schmalz, W.A. Seitz, D.J. Klein and GE. Hite, Chem. Phys. Letters 130 (1986) 203; H.P. Liithi and J. Almlijf, Chem. Phys. Letters 135 (1987) 357. [lo] A.J. Stone and D.J. Wales, Chem. Phys. Letters 128 (1986) 501. [ 111 R.D. Knight, R.A. Walch, SC. Foster, T.A. Miller, S.L. Mullen and A.G. Marshall, Chem. Phys. Letters 129 (1986) 331; M.Y. Hahn, EC. Honea, A.J. Paguia, K.E. Schriver, A.M. Camarena and R.L. Whetten, Chem. Phys. Letters 130 (1987) 12; A. Q’Keefe, M.M. Ross and A.P. Baronavski, Chem. Phys. Letters 130 (1987) 17. [ 121 P. Gerhardt, S. Liiffer and K.H. Homann, Chem. Phys. Letters 137 (1987) 306. [ 131T.L. Gilchrist and R.C. Starr, Organic chemical reactions and orbital symmetry (Cambridge Univ. Press, Cambridge, 1972) p. 38.
484
21 November 1987
[ 141 WC. Russell and W.D. Winters, Progr. Med. Virol 19 (1975) 1; S.J. Martin, The biochemistry of viruses (Cambridge Univ. Press, Cambridge, 1978). 151 F.H.C. Crickand J.D. Watson, Nature 177 (1956) 473. 161 M.Yanagida, J. Mol. Biol. 109 (1977) 515; J.M. Hogle, M. Chow and D.J. Filman, Sci. Am. 256 (1987) 28. 171 D.L.D. Caspar and A. Klug, Cold Spring Harbour Symp. Quant. Biol. 27 (1962) 1. 181 B.D. Davis, R. Dulbecco, H.N. Eisen and H.S. Ginsberg, Microbiology, 3rd Ed. (Harper and Row, London, 1980) p. 866; F.A. Eiserling and R.C. Dickson, Ann. Rev, B&hem. 41 (1972) 467. 191 B.F.G. Johnson and R.G. Woolley, J. Chem. Sot. Chem. Commun. (1987) 634. [20] D.G. Duff,A.C. Curtis,P.P. Edwards,D.A. Jefferson, B.F.G. Johnson and DE. Logan, J. Chem. Sot. Chem. Commun. (1987) 1264; D.G. Duff, A.C. Curtis,P.P. Edwards, D.A. Jefferson, B.F.G. Johnson, A.I. Kirkland and D.E. Logan, Angew. Chem. Intern. Ed. Engl. 26 (1987) 676.
CHEMICAL PHYSICS LETTERS
27 November 1987
CLOSED-SHELL STRUCTURES AND THE BUILDING GAME David J. WALES University Chemical Laboratories, Lensfield Road, Cambridge CB2 IE W, UK
Received 14 August 1987; in final form 2 1 September 1987
A simple model is developed to explore the statistics of dynamic assembly processes. The application of this buildinggameto the synthesis of close-boranes and carbon clusters is discussed. The theory may also be of interest in the study of transition metal clusters and biological structures.
1. Introduction The family of close-boranes, of general formula B,Hz- , have polyhedral structures with a BH unit at
each vertex and triangular faces (deltahedra). Every member of this series is known for n= 5 to n= 12, but none with n> 12 has been reported. Two theoretical studies have predicted that larger species with n = 13-24 should also be stable [ 1,2]. Furthermore, the electronic energy difference between any one of the large postulated structures and two known smaller species having the same total number of boron atoms is very large, so that the relative stability of the larger clusters with respect to a hypothetical decomposition into two smaller species appears to be high. The apparent non-existence of the close-boranes with more than twelve vertices is therefore puzzling. We suggest that the answer to this problem depends on kinetic, not thermodynamic, considerations. In 1970 Long [ 31 proposed a reaction pathway for the production of molecules with B3, Bq and BS skeletons from diborane, B2H6.The scheme basically involved the successive addition of BH, fragments to diborane with concomitant reductive elimination of hydrogen. Other authors [ 41 have also suggested the possibility of a stepwise pathway in the synthesis of B,,H:2. The latter experiment provides a number of important clues for the present study in the form of isolatable intermediates; for example the B11HG ion. In section 2 a general mechanistic scheme is considered which can account for the experimental observation that B,,H$ appears to be the largest 478
species on this sequential pathway. The wider utility of this theory is then illustrated in sections 3 and 4. Note that throughout this paper all coordination numbers refer to the connectivity of an atom within a skeletal framework, i.e. not including external bonds to hydrogen atoms.
2. The building game for close-boranes With the above observations in mind consider the following scheme. Starting from a triangle we add successive vertices so as to bridge external edges chosen at random. Each additional vertex produces another triangle which shares an edge with the fragment under construction. In addition, whenever a five-coordinate vertex is created we close up the structure around this atom and create another triangular face in the process. In doing this we neglect the hydrogen atoms completely and concentrate only on the boron skeleton. Hence it does not matter whether the boron species involved are anions, neutral molecules or even cations; we assume that the hydrogen atoms can look after themselves, for example in terms of reductive eliminations. This simple model, composed of the above two rules, is a game in the same sense as the “Game of Life” [ 51. It is possible to enumerate all the possibilities on the path from three vertices to twelve for this particularly simple scheme. The tree diagram is presented in fig. la, and the structures encountered on the pathway are illustrated in fig. lb.
0 009-2614/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 141,number 6
CHEMICAL PHYSICS LETTERS
a
3--4-
6a
9a
10a
6c
6b
9b
10b
1R VI
mib
Fig. I. (a) A tree diagram for the five-coordinate vertex building game; the labels, such as 6c, refer to the structures illustrated in (b) and the fractions give the probability that a particular pathway will he followed. Where no fraction is indicated there is only one distinct possibility. (b) Topologies of the various species encountered in (a).
Using the tree diagram it is easy to calculate the percentage of paths leading to the isosahedron when vertices are added stepwise to randomly selected external edges as described above. It is found that 99.96% of all such paths close after twelve vertices,
27 November 1987
so that the formation of any structures with more atoms by such a process is highly improbable. Furthermore, the pathway leading to structures which do not close after twelve vertices involves species which are probably electronically unfavourable. All the larger hypothetical close-boranes previously considered [ 1 ] have one or more six-coordinate vertices which only occur in this scheme for structures which follow from llb in lip. la. The higher coordination number was allowed to arise when closure around one live-coordinate vertex led to creation of a six-coordinate vertex. The rules of the above game involve closing up around five-connected vertices as they are formed, so it is not surprising that many pathways lead to the closed polyhedron whose vertices are all five-coordinate. If we had chosen to close up around four-coordinate vertices then a high percentage of pathways would have led to the octahedron. The usefulness of the new theory is immediately apparent; instead of asking whether the larger close-boranes are electronically stable we should investigate the rules of the above game. Using this insight the relevant question is why structures should close up around five-coordinate rather than four-coordinate vertices. Future theoretical studies may therefore be directed towards the solution of a more fruitful problem. Some qualitative explanations are advanced below. First we note that all the experimental evidence supports the above scheme; for example B3H9,B4H10 and B5H,, are all products of Long’s reaction scheme [ 31 and have the same boron skeletons as the unique three-, four- and five-vertex structures found in the building game. More significantly B,,H,, B, I Hi and a derivative of B,H, are all observed in a synthetic route [4] to B,,H:, . These have the same boron skeletons as the structures labelled lla, 10a and 9a in fig. 1 [6]. All pathways leading to the icosahedron must pass through lla in the above scheme, and many pass through 10a and 9a. Closer consideration of the BsHll species allows qualitative justification of the chosen rules. This molecules has a single four-coordinate vertex with two terminal bonds to hydrogen atoms. Consider the hypothetical process in which B5H,, closes up around this vertex to give B5Hg. This would require reductive elimination of H2 between the peripheral boron atoms which move to bonding distance. However, 479
Volume 141, number 6
CHEMICAL PHYSICS LETTERS
one of the B-H terminal bonds to the new apical atom must also break, since this atom must contribute three internal orbitals for skeletal bonding. Contrast this with the hypothetical case of closure around a fivecoordinate vertex. In this case the higher coordination number means that only one terminal hydrogen atom is likely to be bonded to the critical boron atom. Hence, closure around such a vertex would probably involve less disruption to the molecule. Steric factors may also be important; the distance between the peripheral terminally bonded hydrogen atoms before closure would be smaller for a five-coordinate vertex. The elimination of two of these hydrogen atoms may therefore relieve unfavourable steric repulsion, again suggesting that closure around a five-coordinate vertex may be more favourable. The change in boron-boron angles on closure to give a five-coordinate vertex would also be smaller than for a fourconnected vertex.
3. Application to large carbon clusters On vaporizing a graphite target with a laser Kroto et al. [ 71 discovered that a wide variety of large carbon clusters were formed. In particular they detected a remarkably stable fragment corresponding to a mass of sixty carbon atoms. On the basis of its non-reactivity and related evidence these workers proposed a closed, truncated-icosahedral geometry for the Ceo species, although some of their arguments have since been challenged [ 8 ] . Several theoretical investigations of this species have been performed ranging from simple Hiickel calculations to more extensive ab initio studies [ 91. The general conclusion of this theoretical work is that the icosahedral CbOmolecule may indeed be a stable molecule. However, Stone and Wales [ lo] have pointed out that various other CGO isomers with less symmetrical arrangements of the twelve pentagonal and six hexagonal rings may be nearly as stable. The application of the building game to the formation of closed shells of carbon atoms requires some modification of the rules used in section 2. Firstly we note that Kroto et al. observe the formation of many other species in their experiment corresponding to structures containing an even number of carbon atoms. The distribution of products depends strongly 480
27 November 1987
upon experimental conditions; the C,, peak predominates when cluster equilibration is allowed [ 71. Hence we would not necessarily expect to find a scheme in which most paths lead to closure after sixty atoms, in contrast with section 2. The following rules were chosen for the study of carbon clusters: (1) building blocks are allowed to add to the fragment under construction to give fiveor six-membered rings only and (2) two-coordinate atoms may link together to form five- or six-membered rings containing only three-coordinate vertices. The formation of four-membered rings is therefore forbidden; this seems a reasonable rule in view of the greater steric strain and antiaromatic character of four-membered conjugated rings. However, a larger ring may be formed along with a fiveor six-membered ring when the last bond is made to give a closed species. We also make the implicit assumption throughout these studies that the building process does not disrupt the bonding of the growing fragment. This is important because energy is sure to be released on bond-making and all the species are likely to be formed in excited states. However, the assumption that third-body collisions can drain away some of this excess energy is probably reasonable for the experiments under consideration. For this application the full schemes are too large to enumerate fully and computer programs were written to simulate the growth of the carbon clusters. Firstly a suitable pair of two-coordinate vertices is selected at random from the molecule under construction, and then a complementary pair of atoms is chosen on the building block to be added. The stepwise addition of building blocks and associated ring closures is then followed until the cluster contains sixty atoms. This process is repeated many times to give an average value for the number of closed paths, without regard for the detailed arrangement of the pentagonal and hexagonal faces. The addition of fragments to form five- and six-membered rings is illustrated in fig. 2a, and allowed ring closures are shown in fig. 2b. The choice of suitable building blocks is a crucial one. Some recent experimental work suggests that carbon cluster growth occurs from small species ( C1, Cz and C,) in the laser vapor&&ion of graphite [ 111, and from “large polyaromatic particles” in hydrocarbon flame experiments [ 121. To render the prob-
Volume 14 1,number 6
CHEMICAL PHYSICS LETTERS
27 November 1987
a
__-
Fig. 3. Building blocks considered for the carbon cluster schemes.
___ 88
fi
*
Fig. 2. (a) The addition of two Cl0 fragments to form tive-membered and six-membered rings. (b) Allowed ring closures in the carbon cluster building game.
lem as computationally simple as possible for this initial study we have first considered the fragments illustrated in fig. 3. The C6, Cl0 and C2,, species all consist of fused six-membered rings as found in graphite itself, while the C5 fragment also contains a five-membered ring and is probably a less likely candidate. Since two C6 rings cannot add together to give a five- or a six-membered ring no pathway involving this fragment alone can give any closed CsO according to the rules chosen. Using CIO,C, 5and C2,,
species alone 19,2 and 1Ohof pathways, respectively, lead to closed C6,,with twelve pentagonal and twenty hexagonal faces. These figures have been rounded to the nearest integer owing to the uncertainty entailed by the finite number of simulations used. The Cl0 fragment therefore gives by far the highest yield of closed CbO,although we should note that most of the closed molecules formed do not have icosahedral symmetry. When the game is allowed to continue to 120 atoms, corresponding to twelve Cl0 units, small amounts (of order 1% or less) of closed Cso, Cgo,Cloo and C, 1oare also found. In each case the number of simulations considered was 2000, which may be insufficient to detect some more improbable species. A scheme involving three Cl0 and five C6 fragments was also investigated. Two Cl0 and five C6 species were added to a Cl0 fragment in a random order according to the above rules; no closed C6,,molecules were found in 2000 simulations. A further study was carried out using the linear C3 molecule which may be a favourable fragmentation intermediate in the laser vaporisation experiment [ 111. In this case the assumed rules enabled live- and six-membered rings to form as above, but only additions giving no singly coordinate carbons atoms were allowed. Most simulations were found to lead 481
Volume 14 1,number 6
CHEMICAL PHYSICS LETTERS
to bispherical-type systems after sixty carbon atoms with two partly closed shells bridged by a few common live- and six-membered rings. Some interesting conclusions may be drawn from the above results. Firstly we have found that of the schemes tried so far the Cl0 fragment is by far the most efficient precursor of closed &,. Larger building blocks seem to be too inflexible to give a high yield of closed C6,,, whereas very small fragments lead to CbOspecies with regions which are locally almost closed linked together by smaller open areas. Of course, the restriction of using a single building block alone in each scheme may not be a good assumption. However, the use of a distribution of building blocks within a single game is much more expensive computationally, and it is not clear from the experiments what this distribution should be. The results also suggest that the formation of a closed CeOmolecule probably requires five or more collisions, and that if the closed-shell interpretation is correct then the C6,, mass peak may be due to a collision of isomers of similar stability. In summary, this preliminary application of the building game to carbon clusters has shown that closed sixty-atom species are indeed formed in an idealised growth process. The larger building blocks chosen may be more appropriate to the hydrocarbon flame experiments [ 121, but it is not impossible that they occur as intermediates under other conditions too. The appearance of prominent CbOmass peaks in various different experiments may indicate that some aspects of cluster growth are common to them. However, if all the even-numbered carbon clusters observed in, the mass spectra are closed shells, as previously proposed [ 71, then the models are clearly too biased towards closure at CbO.Observations such as this will be helpful for future applications with more complex rules, which may enable a more detailed description of the growth process to be extracted. Throughout the above treatment we have assumed that the ring closures occurring are electronically allowed processes from the point of view of orbital symmetry [ 131. Since the fragments may be electronically excited and both o and L interactions are involved this seems a reasonable postulate.
4. Further applications The scope for variation within the building game by modification of the rules and building blocks is very lare, and may represent a unifying scheme for the assembly of closed shells. For example, various authors have noted the structural similarities between simple plant and animal viruses and small RNA-containing bacteriophages [ 141. However, the assembly of virus capsids is a far more difficult problem than those discussed above since the building blocks involved are much more complicated and the relation between transcription, translation and assembly and the interaction with the host cell environment are not well understood. With this caveat established we will consider the construction of some icosahedral virus shells using the building game. Crick and Watson [ 151 first suggested that virus capsids might consist of a three-dimensional array of identical protein subunits. They argued that the viral genome is too small to contain sufficient information to code for a large number of different capsid proteins, and this view is now known to be substantially correct [ 161. Hence efficiency and economy are key factors in the consideration of the self-assembly of viruses. Caspar and Klug [ 171 showed that the number of structural possibilities available using only identical building blocks is actually very limited, and demonstrated how various shells with icosahedral point group symmetry might be formed. Electron microscopy, X-ray diffraction and, more recently, electron diffraction studies provide visualisations of virus shells, although considerable experimental effort is required. Two levels of organisation can be identified: the capsid may be viewed as an assembly of capsomers which often contain five or six protein subunits and are known as pentamers and hexamers respectively. Both the subunit-subunit and capsomer-capsomer bonds are non-covalent, but there is some experimental evidence that the capsomer-capsomer bonds are weaker [ 181. If we consider a game in which the building blocks are pentamers which can bind to five other pentamers then an equivalent scheme to that discussed for the close-boranes may be appropriate. In this case a closed icosahedral shell consisting of twelve pentamers would be produced by 99.96% of pathways. Of cource, we have neglected
482
27 November 1987
the possibility
of equili-
Volume 141,number 6
CHEMICAL PHYSICS LETTERS
Fig. 4. The 32-vertex omnicapped dodecahedron with twelve Bvecoordinate vertices and twenty six-coordinate vertices.
bration between structures and the interaction of the shell with the viral genome and core polypeptides, along with the other simplifications. It should also be noted that the angle at which the capsomers bind together probably introduces convexity into the growing shell [ 171 which will also favour the formation of closed structures. Nonetheless it is interesting to note that the random assembly of essentially pentavalent vertices in the building game almost always leads to the formation of a closed shell, and one might speculate on the possible evolutionary significance of this result. The admission of hexamers enables larger shells of icosahedral symmetry to be constructed [ 171. For example; a thirty-two-vertex structure (described by the labels [ 181 P= 3, f= 1, T=3) may be formed from twelve pentagons and twenty hexagons, as illustrated in tig. 4. The assembly of this structure may be simulated topologically with the following rules: (1) an edge connecting two hexamers may only bind a pentamer and (2) an edge connecting a pentamer and a hexamer may only bind another hexamer. In this case we close up around five-coordinate pentamers and six-coordinate hexamers. If we start with two hexamers and one pentamer at the vertices of a triangle, and edges are selected at random in the usual way, then it is found that 97% of paths lead to the closed shell of icosahedral symmetry (10000 simulations). Finally we consider some possible future appli-
27 November 1987
cations of the building game. It has been suggested that packing considerations may be important in determining the structures of transition metal complexes [ 191. In this case it would probably be more realistic to consider the stepwise bridging of faces to give fused tetrahedral structures, rather than the hollow shells discussed above. However, the potential energy surface for a transition metal cluster is probably “softer” than for the main-group boranes, so that rearrangement and equilibration of structures may be more important. These considerations are now being applied to the growth of some recently prepared microscopic colloidal metal particles [ 201. It is interesting to note the way in which some pathways leading to the icosahedron in the five-coordinate vertex scheme fold up spontaneously after the addition of a critical vertex to give a much more closed structure. Whether this observation has any physically interesting consequences remains to be seen. It is also possible that graph-theoretical analyses of the building game may lead to more compact representations of the various schemes we have considered above.
Acknowledgement DJW thanks the SERC for financial support and many friends for their encouragement of this project.
References [ 11L.D. Brown and W.N. Lipscomb, Inorg. Chem. 16 (1977) 2989; .I. Bicerano, D.S. Marynick and W.N. Lipscomb, Inorg. Chem. 17 (1978) 3443. [2] P.W. Fowler, Polyhedron 4 (1985) 2051. [ 31 L.H. Long, J. Inorg. Nucl. Chem. 32 (1970) 1097. [4] H.C. Miller, N.E. Miller and E.L. Muetterties, Inorg. Chem. 3 (1964) 1456. [ 51E.R. Berlekamp, J.H. Conway and R.K. Guy, Winning ways, Vol. 2 (Academic Press, New York, 1982) p. 8 19. [ 61 K. Wade, Electron deficient compounds (Nelson, London, 1971) p. 102. [7] H.W. Kroto, J.R. Heath, SC. O’Brien, R.F. Curl and R.E. Smalley, Nature 3 18 (1985) 162; Y. Liu, SC. O’Brien, Q. Zhang, J.R. Heath, EK. Tittel, R.F. Curl, H.W. Kroto and R.E. Smalley, Chem. Phys. Letters 126 (1986) 215.
483
Volume 141. number 6
CHEMICAL PHYSICS LETTERS
[ 81 D.M. Cox, D.J. Trevor, KC. Reichmann and A. Kaldar, J. Am. Chem. Sot. 108 (1986) 2457. [ 91 A.D.J. Haymet, Chem. Phys. Letters 122 (1985) 42 I; J. Am. Chem. Sot. 108 (1986) 319; R.C. Haddon, L.E. Brus and K. Raghavachari, Chem. Phys. Letters 12.5(1986) 459; R.L. Disch and J.M. Schulman, Chem. Phys. Letters 125 (1986) 465; P.W. Fowler, Chem. Phys. Letters 131 (I 986) 444; W.A. Schmalz, W.A. Seitz, D.J. Klein and GE. Hite, Chem. Phys. Letters 130 (1986) 203; H.P. Liithi and J. Almlijf, Chem. Phys. Letters 135 (1987) 357. [lo] A.J. Stone and D.J. Wales, Chem. Phys. Letters 128 (1986) 501. [ 111 R.D. Knight, R.A. Walch, SC. Foster, T.A. Miller, S.L. Mullen and A.G. Marshall, Chem. Phys. Letters 129 (1986) 331; M.Y. Hahn, EC. Honea, A.J. Paguia, K.E. Schriver, A.M. Camarena and R.L. Whetten, Chem. Phys. Letters 130 (1987) 12; A. Q’Keefe, M.M. Ross and A.P. Baronavski, Chem. Phys. Letters 130 (1987) 17. [ 121 P. Gerhardt, S. Liiffer and K.H. Homann, Chem. Phys. Letters 137 (1987) 306. [ 131T.L. Gilchrist and R.C. Starr, Organic chemical reactions and orbital symmetry (Cambridge Univ. Press, Cambridge, 1972) p. 38.
484
21 November 1987
[ 141 WC. Russell and W.D. Winters, Progr. Med. Virol 19 (1975) 1; S.J. Martin, The biochemistry of viruses (Cambridge Univ. Press, Cambridge, 1978). 151 F.H.C. Crickand J.D. Watson, Nature 177 (1956) 473. 161 M.Yanagida, J. Mol. Biol. 109 (1977) 515; J.M. Hogle, M. Chow and D.J. Filman, Sci. Am. 256 (1987) 28. 171 D.L.D. Caspar and A. Klug, Cold Spring Harbour Symp. Quant. Biol. 27 (1962) 1. 181 B.D. Davis, R. Dulbecco, H.N. Eisen and H.S. Ginsberg, Microbiology, 3rd Ed. (Harper and Row, London, 1980) p. 866; F.A. Eiserling and R.C. Dickson, Ann. Rev, B&hem. 41 (1972) 467. 191 B.F.G. Johnson and R.G. Woolley, J. Chem. Sot. Chem. Commun. (1987) 634. [20] D.G. Duff,A.C. Curtis,P.P. Edwards,D.A. Jefferson, B.F.G. Johnson and DE. Logan, J. Chem. Sot. Chem. Commun. (1987) 1264; D.G. Duff, A.C. Curtis,P.P. Edwards, D.A. Jefferson, B.F.G. Johnson, A.I. Kirkland and D.E. Logan, Angew. Chem. Intern. Ed. Engl. 26 (1987) 676.