Ab initio study of rearrangements between C 60 fullerenes

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1 Chemical Physics Letters 374 (2003) Ab initio study of rearrangements between C 60 fullerenes Yuko Kumeda, David J. Wales * The University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, UK Received 27 March 2003; in final form 27 March 2003 Abstract Rearrangement mechanisms between different C 60 fullerenes are characterised for all the local minima and transition states up to five steps away from the buckminsterfullerene global minimum. The electronic structure is treated using plane-wave density-functional theory, and combined with hybrid eigenvector-following techniques to obtain accurately converged transition states. Our results basically confirm the picture deduced in a previous study that employed tightbinding theory: the low energy region of the potential energy surface leads to efficient relaxation to buckminsterfullerene if the system has sufficient energy to overcome the high downhill barriers. Ó 2003 Elsevier Science B.V. All rights reserved. 1. Introduction The formation mechanism of icosahedral buckminsterfullerene (BF) from a carbon vapour containing single atoms, dimers and trimers is still not fully resolved, and several different schemes have been suggested [1 3]. The Ôpentagon roadõ is based on the growth of a nautilus-like shell, or ÔicospiralÕ [4 6], and a number of possible intermediates have been characterised theoretically [7]. The Ôfullerene roadõ instead involves graphite-like sheets, which close up when the cluster contains around thirty carbon atoms [8]. Subsequent growth occurs via insertion of C 2 fragments [9] until a favourable structure such as C 60 or C 70 is reached. Analysis of the fullerene road using a Ôbuilding gameõ approach [10] suggests that it can * Corresponding author. Fax: address: dw34@cam.ac.uk (D.J. Wales). explain the experimental abundance of C 60 or C 70 if the driving force is simply energy minimisation [11]. Mechanisms based on ring stacking [12,13] and cycloadditions [14 16] have also been described, and drift-tube ion mobility experiments provide evidence that cycloaddition of rings of carbon atoms with subsequent 1,2-carbon shifts is important for fullerene formation [14 17]. Around 20 ev of energy would probably be liberated in such paths [18]. A key ingredient required in many of these schemes is the ability of fullerenes to rearrange into different isomers with the same number of pentagonal and hexagonal faces. In fact, for C 60 itself there are 1812 different fullerenes (excluding permutation-inversion isomers) [19]. The most likely rearrangement mechanism involves a ÔpyracyleneÕ or ÔStone-WalesÕ (SW) rearrangement [20], where two carbon atoms effectively rotate through 90 about the midpoint of the bond that connects them. In this transformation a pair of hexagons /03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi: /s (03)

2 126 Y. Kumeda, D.J. Wales / Chemical Physics Letters 374 (2003) and a pair of pentagons switch places, and hence the pentagons can migrate around the surface of the fullerene shell. Austin et al. [21] have analysed all the possible connections between C 60 fullerenes on the basis of the SW rearrangement, considered as a topological operation. The resulting ÔSW mapõ reveals that BF is connected to 1709 of the possible C 60 fullerene isomers through one or more SW rearrangements. Hence, 94% of all C 60 fullerenes can rearrange to BF within thirty SW transformations, or less. The map can be used to determine sets of minima that each require a certain minimum number of SW rearrangements to reach BF [21], and these sets have been termed ÔSW stacksõ. BF itself is connected to only one other fullerene structure (excluding permutation-inversion isomers), which has C 2v symmetry and two pairs of adjacent pentagons. Any annealing process involving SW rearrangements must therefore pass through this isomer before reaching BF. Transition states and pathways have previously been characterised for all the geometrically distinct SW rearrangements up to seven transformations from BF [22], i.e. stack seven. In total 197 minima and 547 transition states were obtained in this study, which used a tight-binding potential. The results enabled the low-lying part of the C 60 potential energy surface to be visualised using a disconnectivity graph [23,22], and the annealing dynamics were further investigated using a master equation approach [22]. More recently, Bettinger et al. [24] have investigated the pathway between BF and the nextlowest fullerene in SW stack two using a hybrid Hartree Fock density functional theory (DFT), including gradient approximations and basis sets up to polarised triple-f quality. The present report also uses DFT, and presents geometry optimisations and pathway calculations for all the SW transition states and fullerene minima up to stack five. For the SW rearrangement of BF itself we examine the mechanism and energy barriers obtained for several different density functionals. These results are also compared with previous work, and agree well with the study of Bettinger et al. [24]. For the rearrangements between the higher energy stacks we compare our DFT results with the tight-binding calculations in our previous report. Although some of the mechanistic details change, the resulting disconnectivity graphs are in very good agreement with our previous results. The present work therefore strengthens our interpretation of the annealing dynamics in terms of the underlying potential energy surface [22]. 2. Methods All the energies and gradients employed in the present work for geometry optimisations were calculated using the plane-wave DFT program CPMD [25]. Results were obtained for every local minimum and transition state using both the local density approximation (LDA) and the BLYP functional, which consists of Becke nonlocal exchange [26] and Lee Yang Parr correlation [27] terms. For the SW rearrangement of BF itself, where a wider variety of other results are available in the literature, we also considered the PBE [28] and OLYP [29] functionals. Kleinman Bylander pseudopotentials [30] were employed throughout with an orbital convergence of 10 6 a.u. and a maximum nonlocal pseudopotential quantum number l max ¼ 1. The plane-wave cutoff was chosen at 40 Ry for the SW rearrangement of BF, unless stated otherwise, and a value of 30 Ry was employed for the all other geometry optimisations and pathway calculations. Periodic boundary conditions were applied using a face-centred-cubic supercell with rhombic dodecahedral symmetry. Following initial convergence tests the supercell dimension was fixed at a value that corresponds to a side length of 35 a.u. for a cubic cell containing four supercells. All the calculations were geometrically unrestricted, i.e. there are no frozen atoms. A modified version of NocedalÕs LBFGS algorithm [31] was used for all the minimisations, including the variational eigenvector calculation employed in the transition state searches [32,33]. The line search was removed and a dynamic step scaling scheme was employed, which proved to be more efficient in previous work. All stationary points were converged to a root-mean-square gradient of a.u. (about 0:001 ev A 1 ), beyond which numerical imprecisions in the energy

3 Y. Kumeda, D.J. Wales / Chemical Physics Letters 374 (2003) and gradient become troublesome. The diagonal elements of the inverse Hessian were initially set to 0.1 for all the LBFGS minimisations. Transition states were located using the hybrid eigenvector-following (EF) approach described in previous accounts [32,34], and the parameters employed were similar to those used in our study of defect migration in crystalline silicon [33]. In particular, a maximum of ten iterations were permitted in the variational calculation of the smallest Hessian eigenvalue and the corresponding eigenvector, and no tangent space minimisations were performed until the smallest non-zero eigenvalue became negative and itself converged in two iterations or less. There is no need to converge the variationally determined eigenvector accurately at the beginning of a search, and since the eigenvector from the last cycle is used as the starting point for subsequent variational calculations after the first step, a small number of iterations can be employed. The variational calculations were deemed to be converged when the root-meansquare gradient obtained for the eigenvalue [32] fell below a.u., or 0:15 ev A 3. If the eigenvalue calculation converged in two steps or less then up to ten LBFGS energy minimisation steps were performed in the tangent space. The uphill eigenvector-following step along the variationally determined eigenvector was set to 0.16 A until the corresponding eigenvalue became negative, after which the formulation for the step described in previous work was used [32,34]. The maximum eigenvector-following step size in this scheme is adjusted dynamically using a trust radius approach with a trust ratio of 0.05 [34]. To prevent contamination of the required eigenvector by modes corresponding to overall translation and rotation a projection scheme was applied, as in previous work [32,34]. For each transition state the corresponding pathway was characterised using LBFGS energy minimisation following small displacements of 0.05 A parallel and antiparallel to the unique Hessian eigenvector corresponding to the negative Hessian eigenvalue in the transition state. The resulting pathways should be a good enough approximation to the true steepest-descent paths for the present purposes. In our previous study we characterised all the geometrically distinct SW rearrangements up to and including stack seven using a tight-binding potential [22]. The resulting transition states were employed as the starting geometries for the new calculations, although every stationary point and pathway was reconverged. We have analysed all the local minima and transition states up to and including stack five in the present work, which includes a total of 66 minima and 160 transition states. 3. Results and discussion We first considered the SW rearrangement of BF itself in some detail, since a number of previous calculations are available for comparison in this case. In particular, Bettinger et al. located two transition states for this process, both corresponding to concerted, single step mechanisms with similar barriers. The first transition state was described as a diradical with C 2 symmetry, and the second as an asymmetric structure with a stretched triple bond [24]. Both pathways are classified as degenerate rearrangements between permutational isomers [34]; the symmetric pathway corresponds to a symmetric degenerate rearrangement, where a new C 2 symmetry operation appears at the transition state, while the other pathway corresponds to an asymmetric degenerate rearrangement, where the two steepest-descent paths are not related by a symmetry operation [34]. In the present work we located the same two transition states when the LDA functional was used with a cutoff at 30 Ry, and uphill barriers of 6.30 and 6.64 ev were obtained for the symmetric and asymmetric paths, respectively (Table 1). In the asymmetric transition state one of the atoms in the C C unit that ÔrotatesÕ moves quite a long way out of the surface (Fig. 1). In fact a cutoff of 30 Ry for carbon is not quite enough for the results to converge to the planewave basis set limit. We therefore repeated the calculations with a cutoff of 40 Ry to converge the energies of the stationary points and the barrier heights more accurately. For this cutoff the asymmetric transition state could no longer be located with the LDA functional. To investigate

4 128 Y. Kumeda, D.J. Wales / Chemical Physics Letters 374 (2003) Table 1 Downhill and uphill barriers (in ev) for the rearrangement between buckminsterfullerene and the next-lowest fullerene minimum with C 2v symmetry Theory Cutoff Symmetric Asymmetric DF-TB [22] 1.18/5.17 LDA / /6.64 LDA /6.91 BLYP /6.40 OLYP /7.09 PBE /6.77 B3LYP/3-21G [24] /7.35 B3LYP/6-311G [24] 1.68/7.27 PBE/3-21G [24] /6.99 /6.98 PBE/6-31G [24] 1.55/6.91 Results for both symmetric (point group C 2 ) and asymmetric transition states are included, if they were found, at various levels of theory. The plane-wave cutoff is in rydberg. The results correspond to the present work unless otherwise indicated. this phenomenon further we therefore repeated the transition state searches with three other functionals, namely BLYP [26,27], OLYP [29] and PBE [28]. In each case we were only able to locate the symmetric transition state, and the resulting barrier heights are collected in Table 1. The transition state obtained in our previous tight-binding calculations corresponds to the asymmetric path. The disappearance of the asymmetric transition state for the LDA functional when the basis set is improved seems to agree quite well with the results obtained by Bettinger et al. [24]. They found that for a B3LYP functional the asymmetric transition state could be located for a relatively modest 3-21G basis set. However, for larger basis sets corresponding to 6-311G quality or better, only the symmetric transition state could be found [24]. Such basis set effects are probably indicative of a rather flat potential energy surface. For the BLYP, OLYP and PBE functionals the results obtained with cutoffs of 30 and 40 Ry are very similar, and for PBE they agree very well with the previous calculations employing Gaussian basis sets [24]. We therefore used a cutoff of 30 Ry in all the LDA and BLYP calculations for the higher energy transition states. The results for the OLYP functional are the closest to those obtained with Gaussian basis sets and the B3LYP functional [24]. For higher energy transition states it is harder to classify the geometry as ÔsymmetricÕ or ÔasymmetricÕ in character, since the point group symmetry is C 1 in either case. However, the transition states corresponding to rearrangements between stacks two and three generally appear to have asymmetric character for the LDA functional, and symmetric character for the BLYP functional, when a plane-wave cutoff of 30 Ry is used. We have not attempted to classify the transition states for rearrangements between higher energy stacks. To interpret the low-lying part of the C 60 fullerene potential energy surface we again employ disconnectivity graphs, following Becker and Karplus [35]. These graphs have now been applied to a wide variety of systems [23,34], including C 60, where a pattern corresponding to high barriers with an underlying potential energy gradient leading to buckminsterfullerene was revealed [22]. In brief, the disconnectivity graph construction requires the energies of a connected set of local minima and the transition states that link them. Then for any given total energy, E, the local minima can be divided into disjoint sets, or ÔsuperbasinsÕ [35], where any two minima in the same set can be interconverted without passing through a transition state of energy greater than E. However, rearrangements between minima in different sets involve one or more transition states that exceed this threshold. When E lies above the energy of the global minimum, but below the next-lowest minimum, there is a single superbasin containing just the global minimum. When E lies above the two lowest minima, but below any transition states, there are two superbasins, which merge together when E is high enough for them to be connected directly or indirectly by a pathway involving one or more transition states. The superbasin analysis is performed at a discrete set of energies, and each superbasin is represented by a point on the horizontal axis. Lines are then drawn to connect these points if they correspond to the same superbasin, or to superbasins that merge together at the higher energy node. At the low energy end of the vertical energy scale nodes are terminated when they reach the value corresponding to the appropriate local minimum. However, displacements on the horizontal axis are

5 Y. Kumeda, D.J. Wales / Chemical Physics Letters 374 (2003) Fig. 1. Top and side views of the asymmetric (top) and symmetric (bottom) pathways connecting buckminsterfullerene to the C 2v isomer in SW stack two. The middle panel is the transition state and the end panels are the two minima in each case. The asymmetric path was calculated using the LDA functional with a cutoff of 30 Ry and the symmetric path corresponds to the BLYP functional and a cutoff of 40 Ry. arbitrary, and can be chosen to give the clearest representation. Connected graphs such as these, which contain no cycles, are known as ÔtreesÕ, and several different patterns have been identified in previous work [23]. Although additional information is generally used to calculate thermodynamic or dynamic properties [34], the appearance of the graph alone can provide useful insight into these properties without further calculation. For example, multiple potential energy funnels correspond to competing low energy structures, and generally result in a separation of time scales in the relaxation dynamics [34]. In contrast, single funnels correspond to systems that relax to their global minima efficiently. The graph obtained in our previous study using a tight-binding potential, and reproduced for comparison in Fig. 2, exhibits long branches compared to the energy separation between successive

6 130 Y. Kumeda, D.J. Wales / Chemical Physics Letters 374 (2003) local minima, and was classified as a Ôweeping willowõ pattern [23]. The corresponding graphs obtained for the present plane-wave DFT calculations with the LYP and BLYP functionals are shown in Fig. 3, where the local minima have been ordered on the horizontal axis in the same sequence as for Fig. 2. The overall appearance of the three graphs is very similar, and confirms our previous conclusion that relaxation to buckminsterfullerene should be quite efficient if the system has sufficient energy to overcome the large downhill barriers [22,23]. This result is also consistent with the successful annealing observed by Xu and Scuseria in their constant energy simulations [36]. The range of energies spanned in the LYP tree is about 9% greater than for BLYP, and 30% greater than for the tight-binding model. Reordering the local minima also introduces two branch crossings in the BLYP tree, and three for the LDA. However, the pattern of connections previously predicted by the tight-binding model is otherwise remarkably good. Fig. 2. Disconnectivity graph including the lowest 66 fullerene minima of C 60 calculated using a tight-binding approach [22]. The structures of six minima are indicated, including buckminsterfullerene and the next-lowest structure with C 2v symmetry. 4. Conclusion Local minima, transition states and pathways have been recalculated for C 60 fullerenes up to SW stack five, i.e. including all the local minima that can be interconverted to buckminsterfullerene by five SW rearrangements or less. In the present Fig. 3. Disconnectivity graphs for minima and transition states in the five lowest SW stacks of C 60 calculated using a plane-wave implementation of density functional theory. The graphs on the left and right correspond to the LDA and BLYP functionals, respectively, and the minima have been ordered in the same sequence as for the tight-binding results in Fig. 2. The vertical energy scales are the same and the energy zero has been shifted to buckminsterfullerene in both cases. The same six structures are marked as for Fig. 2.

7 Y. Kumeda, D.J. Wales / Chemical Physics Letters 374 (2003) study we combined plane-wave DFT evaluations of the energy with hybrid eigenvector-following techniques to converge the transition states, and we compared a number of different functionals for the SW rearrangement of buckminsterfullerene itself. Both asymmetric and symmetric transition states were identified, as in the study of Bettinger et al. [24], but the asymmetric transition state was only found for the LDA functional and a lower plane-wave cutoff of 30 Ry. The asymmetric transition state was also the one previously located using a tight-binding potential [22]. The disconnectivity graphs constructed using both the LDA and BLYP functionals are similar in appearance, and also resemble quite closely the graph obtained using a tight-binding model in previous work. In particular, they have long branches compared to the energy difference between successive minima, corresponding to high downhill barriers. These barriers must be overcome for the system to relax to buckminsterfullerene. Hence these more accurate calculations confirm the picture that emerged from the tightbinding study, namely that the low-lying part of the C 60 potential energy surface acts as an efficient funnel leading to buckminsterfullerene, provided the system has enough energy to overcome the large downhill barriers. Acknowledgements Y.K. gratefully acknowledges a Research Fellowship from the Japanese Society for the Promotion of Science. This research was also supported by an allocation of computing resources on the SGI2800 machine at the Institute of Statistical Mathematics, Tokyo. References [1] H.W. Kroto, J.R. Heath, S.C. OÕBrien, R.F. Curl, R.E. Smalley, Nature 318 (1985) 162. [2] J.R. Heath, S.C. OÕBrien, Q. Zhang, Y. Liu, R.F. Curl, H.W. Kroto, F.K. Tillel, R.E. Smalley, J. Am. Chem. Soc. 107 (1985) [3] R.F. Curl, Philos. Trans. R. Soc. Lond. A 343 (1993) 19. [4] J.R. Heath, S.C. OÕBrien, R.F. Curl, H.W. Kroto, R.E. Smalley, Comment. Condens. Matter Phys. 13 (1987) 119. [5] H.W. Kroto, Science 242 (1988) [6] R.E. Smalley, Acc. Chem. Res. 25 (1992) 98. [7] K.R. Bates, G.E. Scuseria, J. Phys. Chem. A 101 (1997) [8] J.R. Heath, in: G.S. Hammond, V.J. Kuck (Eds.), Fullerenes Synthesis, Properties, and Chemistry of Large Carbon Clusters, ACS Symposium Series, vol. 481, American Chemical Society, Washington, DC, 1992, p. 1. [9] M. Endo, H.W. Kroto, J. Phys. Chem. 96 (1992) 220. [10] D.J. Wales, Chem. Phys. Lett. 141 (1987) 478. [11] D.E. Manolopoulos, P.W. Fowler, in: W. Andreoni (Ed.), The Far Reaching Impact of the Discovery of C 60, NATO ASI Series, Series E, vol. 316, NATO, Kluwer Academic Publishers, Dordrecht, Netherlands, 1996, p. 51. [12] T. Wakabayashi, Y. Achiba, Chem. Phys. Lett. 190 (1992) 465. [13] T. Wakabayashi, H. Shiromaru, K. Kikuchi, Y. Achiba, Chem. Phys. Lett. 201 (1993) 470. [14] G. von Helden, N.G. Gotts, M.T. Bowers, Nature 363 (1993) 60. [15] G. von Helden, M.-T. Hsu, N. Gotts, M.T. Bowers, J. Phys. Chem. 97 (1993) [16] J.M. Hunter, J.L. Fye, M.F. Jarrold, J. Chem. Phys. 99 (1993) [17] J.M. Hunter, M.F. Jarrold, J. Am. Chem. Soc. 117 (1995) [18] D.L. Strout, G.E. Scuseria, J. Phys. Chem. 100 (1996) [19] D.E. Manolopoulos, J.C. May, S.E. Down, Chem. Phys. Lett. 181 (1991) 105. [20] A.J. Stone, D.J. Wales, Chem. Phys. Lett. 128 (1986) 501. [21] S.J. Austin, P.W. Fowler, D.E. Manolopoulos, F. Zerbetto, Chem. Phys. Lett. 235 (1995) 146. [22] T.R. Walsh, D.J. Wales, J. Chem. Phys. 109 (1998) [23] D.J. Wales, M.A. Miller, T.R. Walsh, Nature 394 (1998) 758. [24] H.F. Bettinger, B.I. Yakobsen, G.E. Scuseria, J. Am. Chem. Soc. 125 (2003) [25] CPMD V3.5 Copyright IBM Corp , Copyright MPI fuer Festkoerperforschung Stuttgart [26] A.D. Becke, Phys. Rev. A 38 (1988) [27] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [28] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) [29] N.C. Handy, A.J. Cohen, J. Chem. Phys. 116 (2002) [30] L. Kleinman, D.M. Bylander, Phys. Rev. Lett. 48 (1982) [31] D. Liu, J. Nocedal, Math. Prog. 45 (1989) 503. [32] L.J. Munro, D.J. Wales, Phys. Rev. B 59 (1999) [33] Y. Kumeda, L.J. Munro, D.J. Wales, Chem. Phys. Lett. 341 (2001) 185. [34] D.J. Wales, J.P.K. Doye, M.A. Miller, P.N. Mortenson, T.R. Walsh, Adv. Chem. Phys. 115 (2000) 1. [35] O.M. Becker, M. Karplus, J. Chem. Phys. 106 (1997) [36] C.H. Xu, G.E. Scuseria, Phys. Rev. Lett. 72 (1994) 669.