J. CHEM. SOC. FARADAY TRANS., 1994, 90(8), 1061-1063 Clusters of C,, Molecules David J. Wales University Chemical Laboratories, Lensfield Road, Cambridge, UK CB2 1EW Clusters of 13, 19 and 55 c60 molecules are investigated by molecular dynamics simulation and in terms of their potential-energy surfaces. For (c60)55 nearly 3000 rearrangement pathways have been calculated and compared with a similar sample for a 55-atom cluster bound by the simpler Lennard-Jones potential. Striking differences are revealed between these systems, both in terms of the potential-energy surface and thermodynamic proper- ties. These results are due to the shorter range of the c60 intermolecular potential :for example, the rearrange- ment mechanisms of (c60)55 are generally much more localised that those of LJ5, .Often it is possible to obtain insight into bulk behaviour by considering the corresponding properties of finite clusters,' for which detailed and systematic calculations may be pos- sible. Here we consider clusters of 13, 19 and 55 c60 mol-ecules in the light of recent work by Hagen et d.and Cheng et al. which suggests that the bulk liquid phase of c60 has only a narrow range of stability2 or none at all.3 The model intermolecular potential employed is that of Girifal~o,~ which has a significantly shorter effective range than the atomic Lennard-Jones potential that is often used to provide a qual- itative description of inert gases.3 In fact, Ashcroft has used a system of rigid spheres as the starting point for his discussion of the unusual c60 phase diagram.' Reducing the range of a pairwise potential increases the number of minima on a cluster potential-energy surface (PES), decreases the probabil- ity of the system being associated with the global minimum, and leads to larger rearrangement barriers, making glassy or amorphous structure more The molecular dynamics calculations provide starting points for a separate energy minimisation procedure to locate a representative sample of local minima on the PES.High- energy trajectories were employed with average temperatures of between 400 and 500 K. At these energies it was necessary to place the clusters in a container to prevent evaporation; the evaporation rate would also be a useful diagnostic in the present context, but was not considered further in this study.A cluster PES may be viewed as a mountain range whose peaks and wells represent energy maxima and minima, respectively. Minimum-energy rearrangement pathways between two different minima are valley bottoms, and the transition state is the saddle point, or pass, between such valleys. Minima, transition states and rearrangement mecha- nisms in the present work were all calculated by eigenvector- followingg using an implementation that has been described before and applied to numerous cluster systems.lo This method enables transition states to be located by systemati- cally maximising the energy for one particular degree of freedom and simultaneously minimising in all the other direc- tions.Minima are located within the same framework by minimising for all degrees of freedom.? Analytic first and second derivatives of the energy were employed at every step, and the resulting energies and geometries are essentially exact for the model potential in question. In total 138, 1132 and 1947 different local minima were located for (c60)13, (c60)19 and (c60),,, respectively. The t Strictly speaking the pathways located by this method, namely minimising the energy using eigenvector-following after suitable dis-placements from the transition states, are not equivalent to minimum-energy pathways. However, they should be close enough for the present purposes. number of distinct minima actually increases much faster with the number of molecules than these numbers suggest; we assume that a representative sample has been obtained in each case.' For these sizes Lennard-Jones clusters (hereafter denoted LJ,) each exhibit a particularly stable, high-symmetry minimum.For 13 and 55 these are Mackay icosahedra12 with 1, point group symmetry (see Plate 1); for 19 the low-energy minimum is a D5, decahedron. The special stability of such high-symmetry structures for the Lennard- Jones potential, conferred by the completion of geometrical packing sequences, leads them to be known as 'magic numbers '. For (c60)13 and (c6O)lg only a few transition states and rearrangement pathways were calculated, with particular emphasis on the lowest-energy structures.For (c60)55 a sys- tematic survey was conducted, as results for this system are more likely to provide insight into bulk behaviour, and 2945 different transition states were located, all having C, sym- metry. A dataset of 3481 transition states has been analysed previously for LJ,, ,and all the corresponding rearrangement pathways have been characterised for both systems. Here we consider some of the key results. A selection of minima and transition states is given in Table 1. For (c60)13the icosahedron is the lowest-energy minimum found by quenching, and is probably the global minimum. However, there are two C, and C,minima which also have low energies. The former structure is linked to the 1, minimum by a facile rearrangement, and the C, minimum is linked to the C, by a mechanism that is almost barrierless.For LJ13 the 1, geometry has a uniquely low energy and must overcome large barriers in order to rearrange.13 The 0, cuboctahedron is a transition state for a degenerate rearrangement of the icosahedron, as it is for LJ,,. The D5, bicapped pentagonal prism is also a transition state, but mediates a degenerate rearrangement for a C, minimum in (c60)13 as opposed to the icosahedron in LJ13. The differ- ences are more pronounced between LJ19 and (c60),9, where the D,, global minimum of the former system corresponds to a saddle point of index 8 for (c60)19, i.e. it has eight normal modes with imaginary frequencies. Most low-energy minima and transition states of (c60)19have no symmetry elements at all.For (c60)55 both the cuboctahedron and icosahedron are minima and the former lies lower in energy. The two are linked by a transition state of Th symmetry (Plate 1) in a mechanism which is the realisation of a hypothetical process recognised by Mackay.' 2,14 The analogous mechanism has been found for LJ147 in an ongoing study of the rearrange- ments of model clusters with high symmetry. For LJ,,, the cuboctahedron is a transition state for a degenerate rearrangement of the icosahedron,' ' and cuboctahedral 1062 N energy 13 -38.19416413 13 -38.141 68081 13 -38.140 850 19 13 -38.140 842 04 13 -38.110192 58 13 -37.954 827 57 13 -37.944 162 11 13 -37.827 224 60 13 -37.175 586 27 19 -61.704 878 44 19 -61.703011 64 19 -59.1 88 687 96 55 -229.625 137 32 55 -229.491 394 04 55 -227.339 730 86 55 -223.873 227 74 55 -223.798 559 01 55 -223.791 83431 55 -219.767035 14 55 -218.484972 37 147 -700.581 584 72 147 -681.692 751 54 147 -667.527 597 90 J.CHEM. SOC. FARADAY TRANS., 1994,VOL. 90 index PG comments 0 0 'h c2 global minimum, Mackay icosahedron second lowest minimum 0 cs third lowest minimum 1 1 0 1 1 1 0 0 Cl c2 c5 c5 D5h Oh c2 cs links C2 and C, minima, w = 4.2ian-' links C2 and I, minima, w = 4.6ian-' links C, minimum to I,, w = 2.lian-' degenerate rearrangement of C, minimum, w = 1.6i cm-' degenerate rearrangement of minimum, w = 2.6i cm-' lowest minimum found for (c6,)19second lowest minimum 8 0 0 0 D5h Cl C, Oh decahedron analogous to global minimum for LJ,, lowest minimum found for (C60),, second lowest minimum found for (C,,),, cuboctahedron 0 0 'h c3v Mackay icosahedron C3, minimum 1 1 7 c3v Th D5h decahedron links C3, and I, minima, w = 3.3i cm-' mediates 0,to I, rearrangement, w = 2.9icm-' 0 Oh cuboctahedron 0 1 'h Th Mackay icosahedron mediates 0,to I,, rearrangement, w = 2.9icm-' The unit of energy is the pair well depth (3218.43K), N is the number of molecules in the cluster, the index is the number of imaginary normal-mode frequencies (0 for a minimum, 1 for a transition state), and PG is the point group. The unique imaginary frequency, w, is also given for the transition states. Lennard-Jones clusters do not become lower in energy than Mackay icosahedra of the same size until they are much larger.16 For LJ,, there is a D,, minimum,', while for (c60),,the corresponding structure is a saddle of index 7. However, there is a true C, minimum linked to the icosahe- dron uia a C, transition state.Neither the Oh nor the 1, minimum is found among the sample of 1947 minima obtained by quenching; two of the latter set belong to point group C,, and the rest are C,. Many significantly lower- energy minima exist for this cluster, and it is quite possible that the global minimum has not yet been located; the precise identity of the latter structure is probably unimpor- tant. These lower-energy disordered clusters also have larger mean vibrational frequencies than the 0,and 1, geometries.In previous studies where large numbers of minima have been located for model clusters, the most similar behaviour to (c60),, occurs for (H20)20, a 'frustrated' system." For LJ,, , in contrast to (c60),,and (H20)20, the global minimum is still located in quenches even at energies where the system is highly non-rigid. The tendency of c60 clusters to appear glassy or amorphous is fundamentally related to the form of the intermolecular potential. The latter governs the packing characteristics and the intolerance of strains which result for icosahedral geometries. The short-range potential means that local packing considerations are more important than the global correlations which would produce high-symmetry structures.These differences are also reflected in the distribu- tions of local minima with energy, where for LJ,, the low- energy range is dominated by structures based upon icosahedral order. More striking differences between the LJ,, and (c60)55 pathways are revealed in other properties. For example, Fig. l(a) and (b) show the normalised probability distributions for the smaller and larger barrier heights calcu- lated for all the rearrangement mechanisms. Clearly the (C60)5,cluster must generally overcome significantly greater barriers (relative to the pair well depth) in order to explore new regions of the PES. Fig. l(c) shows the probability dis- tributions for the cooperativity index,18 y, which is a measure of how many atoms are involved in a given rearrangement.Both LJ,, and (c&, show a peak for localised processes around y = 50, but the large peak for small y corresponding to cooperative mechanisms is entirely absent for (C,,),, . The short range of the potential for the latter cluster destroys the global correlations found for LJ,, . To relate the present results to the question of the bulk liquid stability we must consider two questions. First, how is the relatively unfavourable nature of the bulk liquid mani- fested in a microcanonical cluster simulation? Secondly, how are these characteristics of the cluster determined by the PES? As Hagen et al. point there will be no bulk liquid phase if the solidbiquid coexistence line lies above the liquid/ vapour critical point in temperature.This will occur if solid/ liquid coexistence is shifted to unusually high temperature, if liquid/vapour coexistence is shifted to unusually low tem- perature, or a combination of both factors. In the cluster simulation the presence of a container discriminates against the gas phase because it restricts the accessible volume. However, we should certainly be able to comment upon solidbiquid coexistence. A particular property of small systems is that the melting transition occurs over a finite range of temperature (or energy) for which solid- and liquid- like forms of the cluster coexist, and this leads to features in the microcanonical caloric curve, T(E),and the probability distribution of the short time averaged temperat~re.'*'~ The (c&, PES exhibits a complex glassy or amorphous struc- ture, with deep wells and high barriers (relative to the pair well depth).t The same is likely to be true in the bulk, and one might therefore argue that the solidbiquid coexistence region shifts to higher energy in the microcanonical ensemble because of the reduced entropy difference between the solid and liquid states.The present simulations of (c6& suggest t The nature of the surface may make ergodicity dificult to achieve in both cluster and bulk simulations. The true global minimum for (C&s with this potential is also unknown; the effect of starting molecular dynamics simulations from a higher-energy minimum would be to decrease the temperature for low-energy tra- jectories.5 .r(3 I 5 M rd Aa.r( v c eB9Y 0P2 (d 0 Y J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.6 1.2 0.8 0.4 0.0 012345 energy/pair well depth ?! 0.5-,/!, 0246810 energy/pair well depth 0 10 20 30 40 50 60 Y Fig. 1 Comparison of normalised probability distributions (obtained by binning and smoothing) between (C,,),, (-) and LJ,, (---). (a) Smaller and (b)larger barrier height sampled over all the different rearrangement mechanisms. The unit of energy for the horizontal axis is the appropriate pair well depth for both clusters. (c) Distribution for the dimensionless cooperativity index, y. The peaks around 50 are for relatively localised rearrangements where only a few atoms move significantly; the peak around 5 [absent for (c6o)ss] is for relatively delocalised processes, where most of the atoms in the cluster are involved.that the melting process is gradual (Fig. 2), as expected for a disordered solid,* and non-rigidity (as judged by the Linde- mann 6) sets in more slowly than for LJ,,, reaching essen- tially the same asymptotic value at higher total energy. The present results are also relevant to the formation of soot, in particular in the high-temperature synthesis of C6,. Deposition from the vapour must initially involve small clus- ters, and hence we would expect such soots to be amorphous. Note that the ‘magic numbers’ found by Martin et al.” for positively charged C,, clusters are actually similar to those found for Lennard-Jones clusters.Here the presence of the charge must modify the intermolecular potential significantly, introducing longer-range ordering and the recovery of special 0.34-1 I I 1 -7---40 50 60 70 80 energy relative to lowest minimum/pair well depth Fig. 2 Caloric curves for (C,,),, (---) and LJ,, (. *. * .) from molec- ular dynamics simulations. The temperature (reduced units) is calcu- lated by equipartition and the total energy is relative to the lowest known minimum. Each trajectory consisted of 2.5 x 10, time steps equivalent to about 25 ns for (C,,),, ; the error bars are one stan- dard deviation in height. Both curves were obtained by progressively raising the energy from the lowest minimum, with lo5 equilibration steps between production runs.Note the well defined S bend for LJSS . stability for structures with favourable geometrical packing arrangements. The author is a Royal Society Research Fellow. References 1 R. S. Berry, J. Chem. SOC.,Faraday Trans., 1990,84,2343. 2 A. Cheng, M.L. Klein and C.Caccamo, Phys. Rev. Lett., 1993, 71, 1200. 3 M. H. J. Hagen, E. J. Meijer, G. C. A. M. Mooij, D. Frenkel and H. N. W. Lekkerkerker, Nature (London), 1993,365,425. 4 L.A. Girifalco, J. Phys. Chem., 1992,%, 858. 5 N. W. Ashcroft, Nature (London), 1993,365,387. 6 P. A. Braier, R. S. Berry and D. J. Wales, J. Chem. Phys., 1990, 93,8145. 7 F. H. Stillinger and D.K. Stillinger, J. Chem. Phys., 1990,93, 6106. 8 J. Rose and R. S. Berry,J. Chem. Phys., 1993,98,3262. 9 J. Pancik, Collect. Czech. Chem. Commun., 1974,40,1112;C. J. Cerjan and W. H. Miller, J. Chem. Phys., 198 1,75,2800. 10 D. J. Wales, Mol. Phys., 1991, 74, 1; J. Chem. SOC., Faraday Trans., 1992,86,3505; 653; 1993,89,1305. 11 D. J. Wales, Mol. Phys., 1993,78, 151. 12 A. L. Mackay, Acta Crystallogr., 1962,15,916. 13 D. J. Wales and R. S. Berry, J. Chem. Phys., 1990,92,4283. 14 J. Farges, M.F. Feraudy, B. Raoult and G. Torchet, Acta. Crys- tallogr., Sect. A, 1982, 38,656. 15 J. Uppenbrink and D.J. Wales, J. Chem. SOC., Faraday Trans., 1991,87, 215. 16 R. W. Hasse, Phys. Lett. A, 1991,161, 130;B. Raoult, J. Farges, M. F. De Feraudy and G. Torchet, Philos. Mag. B, 1989,60, 881;J. A. Northby, J. Xie, D. L. Freeman and J. D. Doll, Z. Phys. D,1989,12,69. 17 D.J. Wales and I. Ohmine, J. Chem. Phys., 1993,98,7245. 18 F. H.Stillinger and T. A. Weber, Phys. Rev. A, 1983,28,2408. 19 D. J. Wales and R. S. Berry, Phys. Rev. Lett., submitted. 20 T. P. Martin, U. Naher, H. Schaber and U. Zimmerman, Phys. Rev. Lett., 1993,70, 3079. Paper 4/00718F; Received 8th February, 1994