Force Field for Copper Clusters and Nanoparticles

Transcription

1 Force Field for Copper Clusters and Nanoparticles CHENGGANG ZHOU, 1 JINPING WU, 1 LIANG CHEN, 2 YANG WANG, 2 HANSONG CHENG 3 ROBERT C. FORREY 4 1 Institute of Theoretical Chemistry and Computational Materials Science, China University of Geosciences, Wuhan , China 2 Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, Zhejiang , China 3 Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, PA Department of Physics, Penn State University, Berks Campus, Reading, PA Received 19 September 2008; Revised 12 December 2008; Accepted 13 December Published online in Wiley InterScience ( Abstract: An atomic force field for simulating copper clusters and nanoparticles is developed. More than 2000 cluster configurations of varying size and shape are used to constrain the parametrization of the copper force field. Binding energies for these training clusters were computed using density functional theory. Extensive testing shows that the copper force field is fast and reliable for near-equilibrium structures of clusters, ranging from only a few atoms to large nanoparticles that approach bulk structure. Nonequilibrium dissociation and compression structures that are included in the training set are also well described by the force field. Implications for molecular dynamics simulations and extensions to other metallic and covalent systems are discussed Wiley Periodicals, Inc. J Comput Chem 00: , 2009 Key words: potential energy function, force field, clusters, nanoparticles Introduction Metallic clusters and nanoparticles possess unique physicochemical properties and are widely used as catalysts in heterogeneous catalytic reactions. The evolution of structures and properties of such particles has been a subject of much interest in recent years because of the desire to control particle size and shape during synthesis. Formation energies and other size-dependent thermodynamic variables are generally difficult to measure, and reliance on simulations is often necessary for understanding mechanisms of growth and evaporation. 1 An atomic force field (FF) obtained as gradients of a potential energy function (PEF) is generally required in molecular dynamics simulations to describe the nuclear motion of a system of particles containing a large number of atoms. First principle electronic structure-based simulations are computationally challenging, and it is often convenient to parametrize an analytic PEF for use in the dynamical simulations. An analytic and transferable PEF parametrized by reliable theoretical or experimental data is essential for statistical-based methods using molecular dynamics and Monte Carlo simulations. Bulk metallic systems typically employ an embedded atom (EA) method, 2 12 which uses an energy functional that depends on the local electron density at a given atomic site. Parametrization of the PEF using bulk data, however, may lead to poor performance for small clusters and nanoparticles, as has been shown for the case of aluminum. 12 An accurate account of small cluster behavior may be important for metallic glasses that lack the long-range order of normal crystalline metals. Experiments and numerical simulations have shown that small clusters of atoms in metallic glassses may interconnect to form superclusters. 13, 14 Binary metallic alloys involving different chemistry and atomic size ratios have been studied to gain a clearer picture of the types of short- and medium-range order that occurs in these amorphous systems. 14 The role of atomic size ratio in the formation of copper metallic glass has been investigated 15 using a FF developed from an EA type of PEF. While these studies provided valuable insights, it is not clear what effect possible inaccuracies in small cluster and nanoparticle structures would have on the outcomes of the simulations. It would be interesting to perform similar studies using a FF that is known to be reliable for small clusters and nanoparticles. Theoretically, an accurate FF is required to understand the formation mechanisms of metal nanostructures, such as nanoparticles and nanowires, which usually contain a large number of metal atoms. Dynamically, the formation process may include a series of steps involving the nucleation of small clusters. These clusters, generated via sputtering or some other high-temperature process, are deposited on the surfaces of the nanostructures. Detailed understanding of the Correspondence to: H. Cheng or R. C. Forrey; chengh@ airproducts.com or rcf6@psu.edu 2009 Wiley Periodicals, Inc.

2 2 Zhou et al. Vol. 00, No. 00 growth of nanostructures would allow control of particle sizes and shapes and tailoring of specific properties for a variety of applications. Indeed, the ability to control such physical properties is critical in many practical processes. For example, the size effects of gold nanoparticles on catalytic properties have been demonstrated extensively in experiments. 16, 17 Computationally, a key step is to properly describe coalescences of nanostructures and small clusters that propel the growth. This requires that nanostructures and small clusters are described with accurate interatomic potential functions. One of the motivations of the present study is to develop a general PEF form that properly accounts for both covalent and metallic bonding interactions, which is essential for applications in heterogeneous catalysis. The development of reliable and transferable reactive FFs such as the commonly used ReaxFF 18 have allowed much progress to be made in understanding dynamical processes for a variety of systems The key to the success of these FFs is the use of extensive training sets of quantum mechanical calculations that not only include equilibrium cluster structures but also bond breaking and compression structures for all possible types of bonds and angles. The form of PEF must be flexible enough to account for important details of various local environments while at the same time allowing an adequate description of bond dissociation. The use of bond order functions together with attractive and repulsive pair potentials has proven to be a convenient and effective means for including local geometry dependencies explicitly into the PEF for covalent systems. 23, 24 This approach should also be applicable to metallic systems if the bond order terms are not too strongly dependent on coordination. Inclusion of bond order functions should allow a carefully parametrized FF to reproduce quantum mechanical calculations for small metal clusters with greater accuracy than would be expected by a bulk EA method. In this work, we follow this approach for copper clusters and develop a FF that should be reliable for clusters of arbitrary size and shape. In previous work, 25, 26 minimum energy structures and various metastable isomers were computed for copper clusters of up to 15 atoms using density functional theory (DFT). Here, we extend the DFT calculations to include additional cluster sizes and symmetries and also to include nonequilibrium configurations. The benchmark DFT calculations are then used as a training set for the development of an atomic FF for copper clusters and nanoparticles. This training set includes more than 2000 configurations. Although it is not practical to provide the details of each structure used in the training set, their coordinates and energies are available upon request. Form of the PEF The form of the PEF used in this work closely follows the ones used by Tersoff 23 and Brenner. 24 The binding energy is given in terms of attractive and repulsive pair potentials where b ij = (b ij + b ji )/2 with b ij discussed later. The atomic pair potentials are defined by V R (r) = V A (r) = D e (S e 1) exp { 2S e β(r r e ) } (2) D es e (S e 1) exp { 2/S e β(r r e ) } (3) where r e is the equilibrium distance, D e is the dissociation energy, β is an exponential length scale parameter, and S e is an additional adjustable parameter that equals 2 for a Morse potential. The first step in the FF development is the determination of this set of parameters for an isolated diatomic system. Adjustments may be made to these parameters to help fit clusters of larger size; however, the universal bonding behavior observed in binding energy curves for a wide variety of systems 27 suggests that such adjustments should be small. The influence of the local environment for an atomic pair is contained in the coefficient b ij. The form of the PEF given in eq. (1) differs slightly from that of Tersoff 23 and Brenner 24 which have separate coefficients for the repulsive and attractive pair potentials. Abell 28 suggested that the attractive potential coefficient represents a normalized bond order associated with sites i and j, while the coefficient of the repulsive potential is the net electron density associated primarily with site i. In practice, 23, 24 the coefficient of the repulsive potential is usually set to unity although some explicit forms have been proposed. Using a single coefficient that multiplies both the repulsive and attractive pair potentials offers less flexibility than the use of separate coefficients; however, we have found that this form is less prone to unphysical behavior in regions where the fit is not tightly constrained. The bond order function that we use is similar to that of Tersoff 23 and Brenner 24 and is given by b ij = f C (r ij ) 1 + f C (r ik )g(θ ijk ) exp{α(r ij r ik )} k =i,j The function f C is a smooth cutoff function that may be introduced for convenience in reducing the computational effort. The function is defined to be zero when r > r max and one when r < r min. In between, we use the form f C (r) = 1 2 [ ( )] π(r rmin ) 1 + cos r max r min ) and choose r min and r max according to the desired location and sharpness of the cutoff. The exponential term in eq. (4) is included in order to weaken the ij bond when r ij > r ik while leaving it essentially unchanged when r ij << r ik. Generally, we allow a bond-angle dependence of the form δ (4) (5) g(θ) = a 0 { 1 + c 2 0 /d 2 0 c2 0 /[ d (h 0 cos θ) 2]} (6) E = N N b ij [V R (r ij ) V A (r ij )] (1) i j>i where the parameters a 0, c 0, d 0, and h 0 are fit to the ab initio data for isolated triatomic molecules. A physical interpretation of these

3 Force Field for Copper Clusters and Nanoparticles 3 parameters is helpful in performing the fitting. The parameter h 0 should be the cosine of the angle for the configuration that yields the minimum energy. The parameters c 0 and d 0 determine the respective strength and sharpness of the angular dependence. The parameter a 0 gives added flexibility in reducing the ij bond strength relative to an isolated diatomic molecule. The bond-angle dependence is known to be important for covalent systems 23, 24 but may be unnecessary for large metallic clusters whose energies are determined primarily by coordination number. Because the earlier discussion follows so closely from methods that are designed for covalent bonding, the form of the PEF may also be expected to work well for small metallic clusters whose minimum energy structures depend on geometry. The main new feature of this PEF is the parametrization procedure described in the next section, which allows the pair potential parameters to depend on the local environment. The parameters typically increase with coordination number which weakens the bond strength and allows close-packed structures to form. This approach lessens the burden on the bond order function without compromising transferability and allows metallic and covalent bonds to be handled in a similar fashion. Parametrization As was found in the case of aluminum clusters, 12 we find that the dependence on bond angle is considerably weaker than the dependence on coordination number. Figure 1 shows the case of N = 3 isosceles triangles. The solid curves are energies computed without the use of the bond angle term, and the circles are the DFT results. The agreement is not too bad, although it is clear that the PEF is too attractive for equilateral triangles (in violation of the Jahn-Teller effect) and is not able to reproduce the minimum at 130. The dashed curves are energies computed using a bond angle term with h 0 = cos 130. There is modest improvement, and it is likely that an additional bond angle term could be introduced to handle the Jahn-Teller repulsion. In the following discussion, the bond angle dependence is small enough that we will ignore it by fixing g(θ) = a 0. Figure 1. Energy as a function of angle between two sides (r = Å) of an isosceles triangle. Solid circles are DFT data, solid curves are energies computed using a PEF without a bond angle term, and the dashed curves are energies computed using a PEF with a bond angle term centered about 130. We also find the universality 27 of metallic bonds for the various cluster sizes to be too approximate to obtain a set of parameters that is adequate for all cluster sizes. To overcome this difficulty, we have found it convenient to introduce parameters that depend on local and global coordination numbers N M i = l C (r ij ) 1 (7) N i = j=1 N g C (r ij ) 1 (8) j=1 where l C (r) and g C (r) are cutoff functions of the form (5) with (r min, r max ) = (2.7Å, 3.2Å) for l C and (10 Å, 15 Å) for g C. For comparison, the function f C uses (r min, r max ) = (5 Å, 7 Å) to Table 1. Parameters x(n i ) in Eq. (17). D e (N i ) r e (N i ) S e (N i ) β(n i ) a 0 (N i ) α(n i ) δ 0 (N i ) δ 1 (N i ) N i < N i < N i < N i < N i < N i < N i < N i < N i < N i < N i < N i < N i < N i < N i N i >

4 4 Zhou et al. Vol. 00, No. 00 Figure 2. Binding energies for N = 3 6 clusters. Solid circles are DFT data, the red curves are present FF1 results, and the blue curves are results using the Q-SC model. The lowest energy structures are twodimensional for each of these N. eliminate unnecessary computation of long-range interactions. The parameters of the pair potentials in eqs. (2) and (3) are generalized to allow a dependence on the atomic site indices i and j as follows: D eij = D e + D eij (9) r eij = r e + r eij (10) S eij = S e + S eij (11) β ij = β + β ij (12) where D e, r e, S e, and β are optimized diatomic parameters. We find these values to be D e = Å, r e = Å, S e = Å, and β = Å. The -shifts are defined as products of global and local functions where and D eij = G i (D e )L ij (13) r eij = G i (r e )L ij (14) S eij = G i (S e )L ij (15) β ij = G i (β)l ij (16) G i (x) = x(n i ) (17) L ij = Max{0, Min[8, M i + M j 2l C (r ij ) 2]} (18)

5 Force Field for Copper Clusters and Nanoparticles 5 Figure 3. Binding energies for N = 7, 8, 10, and 11 clusters. The solid circles are DFT data, the red curves are present FF1 results, and the blue curves are results using the Q-SC model. The lowest energy structures are three-dimensional for each of these N. [Color figure can be viewed in the online issue, which is available at The local function L ij provides modifications to the parameters of the pair potentials that arise from nearest neighbors and is independent of cluster size. The global function G i (x) depends on an effective cluster size that extends well beyond the nearest neighbors for a given atom i. This function is needed to provide physically reasonable break-up properties when smaller subclusters are pulled out of larger clusters of size N. The global parameters, which are given in Table 1, may vary significantly for small N i before settling to constant values for large N i. The parameters a 0, α, and δ in eq. (4) are generalized to a 0 (N i ), α(n i ) and δ ij = δ 0 (N i ) + δ 1 (N i )L ij which are also given in Table 1. In the discussion that follows, we will refer to the FF derived from eq. (1), which uses potentials (2) and (3) with the generalized parameter set described earlier as FF1. Results We have computed ab initio energies for many different cluster sizes of copper. 25, 26 These calculations include metastable isomers and serve as the starting point for the FF development. We have also performed many additional DFT calculations that were not previously reported in order to include structures that do not correspond to an energy minimum. As in our previous work, these new calculations were performed using DFT under the generalized gradients approximation with the Perdew-Wang exchange-correlation functional 29 as implemented in the DMOL 3 package. 30 A spin-polarization scheme was employed for all of the calculations. The valence electrons were described by a double numerical basis set augmented with polarization functions, and the core electrons were described by an effective

6 6 Zhou et al. Vol. 00, No. 00 Figure 4. Binding energies for N = 13, 14, 19, and 20 clusters. Solid circles are DFT data, the red curves are present FF1 results, and the blue curves are results using the Q-SC model. These clusters include icosahedral as well as triangular growth structures. [Color figure can be viewed in the online issue, which is available at potential. Many of the new structures that were added to the training set were obtained as intermediate steps in the optimization procedure, which leads to the stable or metastable structures reported previously. 26 Larger cluster sizes were also added to the training set following this same procedure, which generally includes a variety of initial starting structures to begin the energy optimization. Figures 2 6 compare binding energies computed by FF1 against the benchmark DFT results for a variety of cluster sizes and structures. Also shown are results of the quantum Sutton-Chen (Q-SC) model 9 which has a total energy of the form E = D i 1 2 j =i V(r ij ) cρ 1/2 i (19) ( ) 10 where V(r ij ) = α r ij and ρi = ( ) α 5. j =i r ij The Q-SC model is one of the most reliable of the EA methods. With the parameters D = mev, c = , and α = Å, it has been shown to provide accurate values for surface energies, vacancy energies, and stacking-fault energies. 15 The Q-SC potential is designed to reproduce an experimental cohesive energy of 3.5 ev in the bulk limit. Our DFT calculations, which use the same computational method employed for small clusters but with periodic boundary conditions, give a cohesive energy of 3.2 ev in the bulk limit. 25 For consistency, the FF1 energy is designed to approach our DFT result in the bulk limit. This gives a difference of 0.3 ev between the ab initio FF1 cohesive energy and the empirical Q-SC cohesive energy.

7 Force Field for Copper Clusters and Nanoparticles 7 Figure 5. Binding energies for N = 25, 55, 67, and 87 clusters. Solid circles are DFT data, the red curves are present FF1 results, and the blue curves are results using the Q-SC model. These clusters are primarily icosahedral structures. [Color figure can be viewed in the online issue, which is available at Figure 2 shows cluster sizes whose energy minima are 2- dimensional. 25 These cluster sizes also possess several metastable isomers that are 3-dimensional. 26 The training set for FF1 contains all of the structures denoted by the cluster index on the horizontal axis of each of the figures. The training set includes the various local and global minimum energy structures as well as intermediate non-equilibrium structures. This may be seen in the figures as the binding energy approaches a flat plateau as a function of cluster index. It is clear from Figure 2 that FF1 is doing a reasonably good job of fitting the DFT data. In contrast, the Q-SC model completely fails to reproduce the DFT curves for these small clusters. For the N = 7 11 clusters shown in Figure 3, FF1 again does a good job of fitting the DFT data. The Q-SC model is able to get the overall shape of the DFT curves for these cluster sizes but is significantly too attractive. All of the structures in Figure 3 are 3-dimensional and the minimum energy configuration follows a triangular growth path with increasing N. 26 Figure 4 includes icosahedral as well as triangular growth path structures. Again, FF1 reproduces the DFT data very well while the Q-SC results are too attractive. This trend continues for larger clusters. Figure 5 shows primarily icosahedral structures and Figure 6 shows fcc structures of various sizes and shapes. The FF1 and Q-SC results are very similar apart from an overall energy shift. It is interesting to study this energy shift for spherical fcc clusters as a function of cluster size. Figure 7 shows that the energy difference between the FF1 and Q-SC potential is about 0.2 ev per atom for small fcc-like clusters before increasing to 0.3 ev per atom as the bulk limit is approached.

8 8 Zhou et al. Vol. 00, No. 00 Figure 6. Binding energies for N = 32, 53, 88, and 172 clusters. Solid circles are DFT data, the red curves are present FF1 results, and the blue curves are results using the Q-SC model. These clusters are fcc structures. [Color figure can be viewed in the online issue, which is available at Because of the relative simplicity of the Q-SC model compared to FF1, it is tempting to rescale the Q-SC potential to see whether it is able to better reproduce the DFT data. Clearly, this cannot work for the small clusters shown in Figure 2. However, for the triangular growth clusters in Figure 3, an energy shift of ev per atom would put the curves in much better agreement. Likewise, an energy shift of ev per atom would move the Q-SC curves in Figure 4 into much better agreement with the FF1 and DFT curves. The problem with this approach is that a different energy shift is needed for different cluster sizes. Furthermore, within a given cluster size (e.g. N = 13), the energy shift for icosahedral structures is different than for the triangular growth clusters. This same difficulty occurred for all types of models and parametrizations that we studied, regardless of the form of the pair potentials that were used. Generalization of the potential parameters to include a dependence on the local environment appears to solve this problem. The danger with this approach is that extra care is needed to ensure that unphysical behavior does not arise in regions where the fits are not tightly constrained. This is particularly important for FF models that use many parameters to increase flexibility. For large clusters, the sensitivity of the potential parameters to the local environment is greatly reduced and the Q-SC potential works well. Figure 8 shows results for N = 25 using the old Q-SC potential, which has an overall energy scale of mev 15 and also a new Q-SC potential with a rescaled energy of 5.35 mev that gives the cohesive energy found in our bulk DFT calculations. 25 Also shown in the figure are the DFT and FF1 results for these structures. The rescaled Q-SC results show improved agreement with the DFT data but are still not as close as the FF1 energies. As N increases further, the rescaled Q-SC potential does an increasingly better job of

9 Force Field for Copper Clusters and Nanoparticles 9 Figure 7. Binding energies for spherical fcc clusters as a function of cluster size. Each cluster was constructed from unit cells with a lattice constant of Åwith no allowed relaxation for atoms near the surface. The blue curve is the present FF1 result and the red curve is that of the Q-SC model. [Color figure can be viewed in the online issue, which is available at reproducing the DFT data. Therefore, we have included an option within FF1 to use the rescaled Q-SC potential when N > 25. This reduces the number of summations and allows for faster evaluation of the energy and FF when the number of atoms is large. Using this option, we computed surfaces energies for the low-index surfaces (111), (100), and (110) to be 0.53 ev, 0.7 ev, and 1.06 ev per surface atom, respectively. As with cohesive energy, these surface energies are slightly lower than those reported previously, 31, 32 however, the anisotropy ratios are in excellent agreement with the broken-bond rule. 32 In the present study, in addition to benchmarking against DFT data, we tested FF1 by computing energy minima using all structures in the DFT training set as starting points for the optimization of the FF1 energy. The results for several different clusters sizes are shown in Figures 9 and 10. At the far right of each figure are structures that correspond to one atom being pulled far away from the others. These are included in order to demonstrate that FF1 provides the proper break-up energies. The red curves are the same results as in Figures 2 6, and the blue curves are the optimized FF1 energies. The minimum energies of FF1 in Figure 9 are very close to those of the DFT data and many of the metastable isomers found in the DFT calculations are also reproduced by FF1. The break-up structures are also well behaved, and the optimized energies reveal no unphysical behavior. The same is true for the clusters shown in Figure 10. The N = 14 case is particularly noteworthy in that the icosahedral structures at the far left of the figure all go to the same minimum energy icosahedral structure (cluster index 1) in agreement with the DFT calculations. However, the triangular growth structures with cluster indices between 30 and 50 yield energy minima upon optimization that are slightly less than that of the minimum energy icosahedral structure. The same thing happens for the N = 13 case, which may be seen on the N = 14 panel for cluster indices greater than 50. The icosahedral structures near cluster index 60 all yield the same minimum energy icosahedral structure upon optimization of the FF1 energy. Energy minima found from triangular growth structures with indices greater than 75 are again lower than the minimum energy icosahedral structure. The fluctuation in the optimized energy for these clusters shows that the potential energy surface derived from FF1 is considerably wrinkled in agreement with observations made previously. 26 Interpolations of our DFT data predicted 26 that the cross-over point from triangular to icosahedral growth occurs at N = 16. This is consistent with the present results of FF1. The N = 19 panel of Figure 10 shows that icosahedral structures with indices less than 30 all yield the same minimum energy icosahedral structure (cluster index 9) that was found by the DFT calculations. The same is true when triangular growth structures with indices between 30 and 40 are used as the starting point for the optimization. Clusters with indices greater than 40 correspond to an N = 18 icosahedral structure with an additional atom placed very far away. The N = 25 panel shows results that are similar to those of N = 19. Optimized FF1 energies for larger cluster sizes closely mirror the minimum energy structures of the DFT data shown in Figures 5 and 6. Additional tests were performed for break-up into subclusters with varying shapes and sizes. While small discontinuities in the FF are typical in regions where there is variation in the global coordination number N i, these forces are generally small and apply only to structures that are far from equilibrium. It is unlikely that such regions would influence dynamical simulations except at extremely low temperatures. Optimization of the FF1 energy showed that all break-up structures behaved as expected, and no unphysical behavior was found as the clusters approached equilibrium. Figure 8. Binding energy per atom for N = 25 clusters. As in previous figures, the solid circles are DFT data and the red curve is the FF1 result. The blue curves are results using a Q-SC model with an overall energy scale of mev for the old potential (solid curve) and 5.35 mev for the new potential (dashed curve) which is rescaled to give the cohesive energy obtained by DFT calculations in the bulk limit. 25 [Color figure can be viewed in the online issue, which is available at

10 10 Zhou et al. Vol. 00, No. 00 Figure 9. Binding energies for N = 4, 5, 7, and 8 clusters. The solid circles are DFT data, the red curves are present FF1 results, and the blue curves are the energy minima obtained by FF1 using structures denoted by the cluster index as the starting point for the optimization. [Color figure can be viewed in the online issue, which is available at Conclusions In this work, we have developed a PEF for copper that is highly flexible to account for the detailed local bonding environment of molecules and materials. The method builds on well-established techniques that are applicable to both covalent and metallic systems. We have parametrized the PEF for copper systems based on extensive DFT electronic structure calculations of copper clusters. The PEF was trained using more than 2000 configurations, with cluster sizes ranging from 2 to 4000 atoms. These included nonequilibrium dissociation and compression structures which are needed to describe bond breaking and formation. Therefore, we believe the resulting FF should be reliable for copper systems of arbitrary size and shape. Although the form of the PEF used in this work is comparable to other models, 23, 24 the usual pair potential parameters of the PEF are generalized to include the influence of the local environment. This lessens the burden on the bond-order coefficient b ij and allows the PEF to better fit the DFT data. The use of the global function G i (x) where x is any desired FF parameter allows the FF to be separately trained for each effective cluster size N i. As additional or improved benchmark data becomes available, the FF may be conveniently adjusted for the relevant size N i without the need for readjustment of parameters that depend on a different size. Also, different fitting functions may be used for different N i. For example, the similarity in the overall shape of the Q- SC and DFT results for large clusters suggest that the sum over k that is contained in the bond order function (4) is unnecessary for large N i. Removal of this summation provides a substantial increase in the efficiency of the FF code for large clusters without loss of accuracy for smaller clusters. Likewise, the angular part of the bond order

11 Force Field for Copper Clusters and Nanoparticles 11 Figure 10. Binding energies for N = 11, 14, 19, and 25 clusters. The solid circles are DFT data, the red curves are FF1 results, and the blue curves are the energy minima obtained by FF1 using structures denoted by the cluster index as the starting point for the optimization. [Color figure can be viewed in the online issue, which is available at function, which has been ignored in the present FF1 fitting, may be conveniently implemented to achieve better accuracy for small clusters, without causing a significant reduction in computational efficiency for large clusters. A numerical FF computed from the FF1 parametrization of the PEF (1) would have a greater accuracy than current standards and should be sufficiently fast for use in molecular dynamics simulations. For example, the FF could be used to determine standard Gibbs free energies of formation for copper clusters and nanoparticles as was done for aluminum. 1 The parametrization procedure described here could be extended to binary systems with the use of additional training sets that are comparable to the one used for the monatomic system. This would allow dynamical studies of copper alloys, such as copper zirconium metallic glasses 13 or other binary liquid metals, 33 and chemical reactions such as H 2 dissociative chemisorption. 34 An important issue that would need to be addressed before achieving this extension is how large the training set should be for the particular alloy system. A reasonable approach would be to develop the training set by systematically adding one non-cu atom at a time to the training set used in this work. Structures that are computed during the energy optimization could then be included in the training set for the binary FF. The pair potential parameters, which have already been generalized to include the influence of the local environment for a single atomic species, would need to be further generalized to allow different types of atoms at a particular site. The increasing size of both the training sets and the parameter sets will impose practical limits on this approach. Statistical measures such as those used in the alloy community 35 may be useful

12 12 Zhou et al. Vol. 00, No. 00 in establishing convergence of the FF with respect to the training set for large clusters. The decreasing sensitivity of the pair parameters to the local environment that occurs with increasing cluster size should help to reduce the parameter space and allow the form of the FF to undergo a smooth crossover to a simpler EA method, as was found here for the monatomic case. Acknowledgment The work conducted at CUG was supported by the National Natural Science Foundation of China for Youth (Grant No ). The work of RCF was supported by the National Science Foundation of the United States (Grant No. PHY ). References 1. Li, Z. H.; Bhatt, D.; Schultz, N. E.; Siepmann, J. I.; Truhlar, D. G. J Phys Chem C 2007, 111, (a) Daw, M. S.; Baskes, M. I. Phys Rev Lett 1983, 50, 1285; (b) Daw, M. S.; Baskes, M. I. Phys Rev B 1984, 29, Finnis, M. W.; Sinclair, J. E. Philos Mag A 1984, 50, Gupta, R. P. Phys Rev B 1985, 23, Tomanek, D.; Aligia, A. A.; Balseiro, C. A. Phys Rev B 1985, 32, Foiles, S. M.; Baskes, M. I.; Daw, M. S. Phys Rev B 1986, 33, Ercolessi, F.; Parrinello, M.; Tosatti, E. Philos Mag A 1988, 58, (a) Johnson, R. A. Phys Rev B 1988, 37, 3924; (b) Johnson, R. A. 1989, 39, 12554; (c) Johnson, R. A. 1990, 41, Sutton, A. P.; Chen, J. Philos Mag Lett 1990, 61, Baskes, M. I. Phys Rev B 1992, 46, Nishitani, S. R.; Ohgushi, S.; Inoue, Y.; Adachi, H. Mater Sci Eng A 2001, 309, (a) Jasper, A. W.; Staszewski, P.; Staszewska, G.; Schultz, N. E.; Truhlar, D. G. J Phys Chem B 2004, 108, 8996; (b) Jasper, A. W.; Schultz, N. E.; Truhlar, D. G. J Phys Chem B 2005, 109, Yavari, A. R. Nature 2006, 439, Sheng, H. W.; Luo, W. K.; Alamgir, F. M.; Bai, J. M.; Ma, E. Nature 2006, 439, Lee, H.-J.; Cagin, T.; Johnson, W. L.; Goddard, W. A. III. J Chem Phys 2003, 119, Hashmi, A. S. K. Chem Rev 2007, 107, Hashmi, A. S. K.; Frost, T. M.; Bats, J. W. J Am Chem Soc 2000, 122, van Duin, A. C. T.; Dasgupta, S.; Lorant, F.; Goddard, W. A. III. J Phys Chem A 2001, 105, Strachan, A.; van Duin, A. C. T.; Chakraborty, D.; Dasgupta, S.; Goddard, W. A. III. Phys Rev Lett 2003, 91, van Duin, A. C. T.; Strachan, A.; Stewman, S.; Zhang, Q.; Xu, X.; Goddard, W. A. III. J Phys Chem A 2003, 107, Zhang, Q.; Cagin, T.; van Duin, A. C. T.; Goddard, W. A. III. Qi, Y.; Hector, L. G. Phys Rev B 2004, 69, Nelson, K. D.; van Duin, A. C. T.; Oxgaard, J.; Deng, W.-Q.; Goddard, W. A. III. J Phys Chem A 2005, 109, Tersoff, J. Phys Rev B 1988, 37, Brenner, D. W. Phys Rev B 1990, 42, Guvelioglu, G. H.; Ma, P.; He, X.; Forrey, R. C.; Cheng, H. Phys Rev Lett 2005, 94, Guvelioglu, G. H.; Ma, P.; He, X.; Forrey, R. C.; Cheng, H. Phys Rev B 2006, 73, (a) Ferrante, J.; Smith, J. R.; Rose, J. H. Phys Rev Lett 1983, 50, 1385; (b) Rose, J. H.; Smith, J. R.; Ferrante, J. Phys Rev B 1983, 28, Abell, G. C. Phys Rev B 1985, 31, Perdew, J. P.; Yang, Y. Phys Rev B 1992, 45, DMOL 3, Accelrys Software, Inc., San Diego, Vitos, L.; Ruban, A. V.; Skriver, H. L.; Kollar, J. Surf Science 1998, 411, Galanakis, I.; Bihlmayer, G.; Bellini, V.; Papanikolaou, N.; Zeller, R.; Blugel, S.; Dederichs, P. H. Europhys Lett 2002, 58, Qi, Y.; Cagin, T.; Kimura, Y.; Goddard, W. A. III. Phys Rev B 1999, 59, Forrey, R. C.; Guvelioglu, G. H.; Ma, P.; He, X.; Cheng, H. Phys Rev B 2006, 73, d Avezac, M.; Zunger, A. Phys Rev B 2008, 78,