Parametric Effects of the Potential Energy Function on the Geometrical

Article pubs.acs.org/JPCA

Parametric Effects of the Potential Energy Function on the Geometrical Features of Ternary Lennard-Jones Clusters Xia Wu,*,† Yan Sun,‡ Chunsheng Li,‡ and Wei Yang*,§ †

School of Chemistry and Chemical Engineering, Anqing Normal University, Anqing 246011, P. R. China College of Chemical Engineering, Hebei United University, Tangshan 063009, P. R. China § School of Physics and Electronic Engineering, Anqing Normal University, Anqing 246011, P. R. China ‡

ABSTRACT: The impact of parameters in potential function for describing atomic or molecular clusters is complex due to the complicated potential energy surface. Ternary Lennard-Jones (TLJ) AlBmCn clusters with two-body potential are investigated to study the effect of parameters. In the potential, the size parameter (σAA) of A atoms is fixed, and corresponding parameters of B and C atoms (relative to A atoms), i.e., σBB/σAA and σCC/σAA > 1.00, are used to control the atomic interaction among A, B, and C atoms in TLJ clusters. The minimum energy configurations of AlBmCn clusters with different species are optimized by adaptive immune optimization algorithm. Ternary cluster structures, bonds, and energies of the putative minima are studied. The results show that two different structures based on doubleicosahedra are found in 30-atom TLJ clusters. Furthermore, with increasing potential size parameters of B and C atoms, A atoms tend to be more compact for the increasing numbers of A−A bonds, but the short-range attractive part in TLJ clusters becomes weaker. To lower the potential energy, B and C atoms grow around the A atoms in pursuit of a compact configuration. The results are also approved in AlBmCn (l + m + n = 9−55) clusters and AlBmCn (l = 13, m + n = 42) clusters. tials,12 many-body potential, e.g., Gupta,13 embedded atom method (EAM),14 and bond-order potential such as Tersoff15 and Brenner.16 To investigate the strongly mixed rare gas clusters, various systems from binary to quinary mixtures of atom types were systematically analyzed by using LJ potentials fitting to high-end ab initio reference data.8 Furthermore, the impact of LJ, Pirani, and Hartree−Fock-dispersion individual damping (HFD-ID) potentials on the structure, bond length, and energy was investigated.17 It was found that although the three potentials were very similar near well depth, there existed differences at long and short ranges, and the number of the bonds that deviated from 1.00 might be the dominant factor determining the structure of a cluster. Moreover, for N-atom binary AlBm (l + m = N) LJ clusters, the stabilization of polytetrahedral structures18 was studied by structural phase diagram depending on atomic size N and the ratio parameter (σBB/σAA) of pure elemental B- and A-type LJ cluster sizes, and it was concluded that different structures, particularly those that are polytetrahedral, can be stabilized just through the two atom types in the cluster having different sizes.19 For Morse function, one parameter ρ0 controls the well “width” (range) of the potential, and larger ρ0 means more

1. INTRODUCTION The geometrical structure of ternary clusters has been the subject of much recent attention,1−3 which derives from the extensive studies of binary clusters in recent years. For binary and ternary clusters, potential applications in optical, electronic, magnetic, and catalytic fields provide an opportunity to investigate their structures because of size-dependent special properties. Furthermore, structural information of ternary clusters is of fundamental importance for their chemical and physical properties. For instance, in AlkTilNim (k + l + m = 4) clusters, most of the Al atoms were expected to displace themselves in the outermost part of the system, relatively few occupying the TiNi-rich inner part, by performing molecular dynamics (MD) simulations and density functional theory (DFT) calculations.4 A theoretical investigation of FexCoyNiz ternary clusters with x + y + z = 5 and 6 was performed with DFT, and the geometries, chemical order, local and total magnetic moments, binding energies, excess energies, and second difference in the energy were analyzed in the whole range of composition.5 The segregation phenomena of the Cu, Ag, and Au atoms in the Cu−Ag−Au trimetallic clusters were studied.6,7 Empirical potential energy functions (PEF) were widely applied for optimizing the stable structures and phase transitions of atomic clusters, including pairwise potential, e.g., Lennard-Jones (LJ),8,9 Morse,10,11 and Johnson poten© 2012 American Chemical Society

Received: April 18, 2012 Revised: July 18, 2012 Published: July 19, 2012 8218

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short-ranged interaction.10 Doye and Wales have shown how the ρ0 of Morse potential determines the structure of atomic clusters by the zero temperature phase diagram showing the variation of the lowest energy structure with atomic size N and parameter ρ0.10 Furthermore, the minimum structures of Morse clusters with size 161 ≤ N ≤ 240 as a function of ρ0 were located, and with the decreasing ρ0, the structures changed from disordered to icosahedral to decahedral to close-packed, but the size effect on structures became weaker.20 Then, a modified Morse (MM) model pair potential that unifies the pair interactions including the Morse and LJ potentials was designed.21,22 The meaning of parameter ρ in MM potential is the same as that of ρ0 in Morse potential, and another parameter φ is also used to control the long- and short-range interactions. The structural phase diagrams of MM cluster with sizes 11 ≤ N ≤ 30, and 38 were shown to have an overall view of how the minimum energy configurations depend on the two parameters.22 Furthermore, to predict geometrically more accurate and stable atomic clusters, the many-body potential, e.g., three-body potential, is necessary. For example, the energetically most favorable structure of a cluster was obtained by minimizing the total energy which comprised both two-body and three-body interactions,23,24 and it was found that threebody interactions play an important role in the equilibrium structure of small clusters. Optimization algorithms have been developed for the minimum energy configurations of monatomic clusters, such as genetic algorithm (GA),25 basin hopping method and its variants,26,27 heuristic algorithm combined with the surface and interior operators (HA-SIO),28 fast annealing evolutionary algorithm (FAEA),29 and dynamic lattice searching (DLS) methods.30 However, from monatomic clusters to binary and ternary clusters there is an increase in complexity due to the presence of two and three type components, which lead to the coexistence of geometrical isomers and “homotopic” isomers.31 Therefore, the modifications of GA,31 basin-hopping algorithm,32 adaptive immune optimization algorithm (AIOA),33 and an evolutionary algorithm (EA)34 were made for optimizing binary and ternary clusters. The focus of the current work is on how the structures of ternary clusters vary with variable parameters, which might be helpful for analyzing trimetallic clusters. Two-body LJ and Morse potentials represent such a system for the aim. In this study, the LJ potential is taken as a test system, because the mathematical form of the potential energy function is very simple, and it is relatively easy to apply most optimization methods. The putative minimum energy configurations of ternary LJ (TLJ) A10B10C10 clusters with different parameters, i.e., finite distance (LJ size) in TLJ potential, are located, and their bond numbers and average bond lengths are analyzed. Then the structures and energies of ternary AlBmCn (l + m + n = 9−55) clusters are investigated depending on atomic numbers and potential parameters. Furthermore, the stable structures of A13BmCn (m + n = 42) clusters are optimized, and the structural characteristic and the atomic distribution in the clusters are studied.

E = 4ε

∑ 1≤i1 correspond to short- and long-range part, respectively. Figure 4 shows the average lengths x̅ of A−A, A−B, B−B, A−C, C−C, and B−C bonds in A10B10C10 clusters with the variation of σBB/σAA = 1.02−1.30 and σCC/σAA = 1.02−1.30. At first, as σBB/σAA and σCC/σAA increase, along the vertical direction of σBB/σAA = σCC/ σAA, x̅ of A−A bonds gradually grows from 0.97 to 0.99. It means that the short-range part of A−A bond gets weaker. For the x̅ of B−B bonds, it can be seen that they vary from 1.03 to

3. RESULTS AND DISCUSSION 3.1. Optimized Ternary Lennard-Jones A 10B10C10 Clusters. The putative stable structures of A10B10C10 clusters with σBB/σAA and σCC/σAA both from 1.00 to 1.30 are investigated by AIOA, and the optimized structures are shown in Figure 2. From the figure, two types of the minimum

Figure 2. Structural distribution of two A10B10C10 clusters by diamond and square icons with the variation of σBB/σAA and σCC/σAA. A, B, and C atoms are represented by pink, green, and blue spheres, respectively.

energy configurations can be found, which are presented with diamonds and squares. The dominant structural motif is the configuration with squares, and the configurations with diamonds can only be found in the range of small σBB/σAA and σCC/σAA. For both configurations, the growth is based on the 19-atom double icosahedral (dIh19) geometries as shown in Figure 2, but their remaining 11 atoms demonstrate a different growth pattern on the outer-shell. Furthermore, the clusters with the same configuration even have some degree of distortion in many cases. It is because the variable σBB/σAA and σCC/σAA values determine the equilibrium distances (rij) of 8220

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Figure 3. Number of A−A, A−B, B−B, A−C, C−C, and B−C bonds in A10B10C10 clusters with σBB/σAA and σCC/σAA values ranging from 1.02 to 1.30. The labeled numbers stand for the numbers of bonds.

average lengths of B−B bonds in Figures 3 and 4, one can find that when the x̅ value is small ( σCC/σAA, which indicates that in this area with the increase of σBB/σAA, the interaction part of B−B bond changes from long-range to short-range. At the bottom-right corner (σBB/σAA < σCC/σAA), along the vertical direction of σBB/σAA = σCC/σAA line, x̅ values change from 1.03 to 1.00, and then to 0.97, except for x̅ = 1.02 at σBB/σAA = 1.02, 1.04, and σCC/σAA = 1.22−1.30. By analyzing the numbers and 8221

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Figure 4. Average lengths of A−A, A−B, B−B, A−C, C−C, and B−C bonds in A10B10C10 clusters with σBB/σAA and σCC/σAA values ranging from 1.02 to 1.30.

For A−B bonds in Figure 4, all x̅ values are lower than 1.00 except for the point at σBB/σAA = 1.18, σCC/σAA = 1.30. The short-range part is thus dominant. In the area of σBB/σAA > σCC/σAA, x̅ values of A−B bonds decrease to 0.9786 as σBB/σAA increases, which shows the short-range parts gradually get stronger. In the area σBB/σAA < σCC/σAA, x̅ values quickly decrease from 0.985 to 0.978 and then gradually grow until close to the well of the potential (1.00). In Figure 4, x̅ values of

A−B and A−C bonds exhibit symmetry. For B−C bonds, along the vertical direction of σBB/σAA = σCC/σAA, x̅ values decrease from 1.04 to 1.00 and then to 0.98. The potential function of B−C bonds changes from long-range to short-range. In conclusion, with the increasing σBB/σAA and σCC/σAA, to lower the potential energy, inner A atoms expand (x̅ values closer to 1.00) but with more packing motifs (larger numbers 8222

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is formed. From A8B8C7 to A18B18C18, clusters take the form based on the double-icosahedron. However, by comparing two clusters with different parameters, one can find that there exists difference for the atomic distribution. Apparently, A atoms in Figure 5b are much closer than those in Figure 5a. It is because σBB/σAA and σCC/ σAA values in Figure 5b are larger than those in Figure 5a, as discussed above. In Figure 5b, at A11B10C10 and A12B12C11, 11 and 12 A atoms occupy one icosahedron of the doubleicosahedral core. At A13B12C12, the icosahedral inner core is filled with 13 A atoms. Especially, in Figure 5a A19B18C18 cluster is a pancake structure, with A atoms located in the middle shell; however, in Figure 5b it has a 55-atom double-icosahedral motif, and its inner 19-atom double-icosahedral core is composed of A atoms. The second finite difference of the energy is a quantitative function indicating the stability of a cluster of a certain composition with respect to its neighbors. Figure 6 plots the

of bonds), and B and C atoms on the outer shell grow with a closer mode around the A atoms. 3.2. Analysis of AlBmCn (l + m + n = 9−55) Clusters with Different σ Values. AlBmCn (l + m + n = 9−55) clusters with σBB/σAA = 1.10, σCC/σAA = 1.20, and σBB/σAA = 1.20, σCC/ σAA = 1.40 are optimized by AIOA, and parts of their structures are drawn in Figure 5a,b, respectively. The proportion of l, m,

Figure 6. Second finite differences of the energies of the optimized AlBmCn (l + m + n = 55) clusters with σBB/σAA = 1.10, σCC/σAA = 1.20 and σBB/σAA = 1.20, σCC/σAA = 1.40.

second difference, Δ2E = EN+1 + EN−1 − 2EN, as a function of a N-atom AlBmCn (l + m + n = 9−55) clusters with σBB/σAA = 1.10, σCC/σAA = 1.20, and σBB/σAA = 1.20, σCC/σAA = 1.40. In the figure, both parametric clusters have the same positive peaks, which indicate particularly stable structures comparing to their neighbors. From the figure, 14 apparent positive peaks are found in the both curves, and they have relatively more stable structures. For example, 13-atom and 19-atom AlBmCn clusters have icosahedral and double-icosahedral configurations, respectively. 3.3. Stable Structures of A13BmCn (m + n = 42) with σBB/σAA = 1.05 and σCC/σAA = 1.10. The putative stable structures of 55-atom A13BmCn (m + n = 42) clusters are investigated, and several typical structures are shown in Figure 7. These 41 TLJ structures can be categorized into three classes, i.e., 6 ring-like structures linked by three face-sharing doubleicosahedra for m = 1, 6, 7, 9, 10, and 14, 3 6-fold pancake structures at m = 2−4, and 32 dominant Mackay icosahedra for m = 5, 8, 11−13, and 15−41. The three adjoining doubleicosahedral configuration was also found in Cu13AgnAu42‑n clusters previously.7 However, even for the same configuration of icosahedra, the distribution of 13 A atoms in A13BmCn (m + n = 42) clusters is different. With the increase of m values, the location of A atoms change from the surface (e.g., Al13B5C37 and Al13B13C29) to the interior (e.g., Al13B19C23 and A13B25C17).

Figure 5. Typical putative stable structures of AlBmCn (l + m + n = 9− 55) clusters with σBB/σAA = 1.10, σCC/σAA = 1.20 (a) and σBB/σAA = 1.20, σCC/σAA = 1.40 (b). A, B, and C atoms are represented by pink, green, and blue spheres, respectively.

and n is approximate to 1:1:1. From the figure, it can be seen that in the size range of 9−54, the configurations of both clusters are very similar. With the increase of sizes, clusters grow to become a complete icosahedron at A5B4C4. Then, the larger size clusters grow on the basis of the 13-atom icosahedron. At A7B6C6 cluster, a 19-atom double-icosahedron 8223

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Figure 7. Typical structures for A13BmCn (m + n = 42) clusters obtained by AIOA. A, B, and C atoms are represented by pink, green, and blue spheres, respectively. (2) Hungria, A. B.; Raja, R.; Adams, R. D.; Captain, B.; Thomas, J. M.; Midgley, P. A.; Golovko, V.; Johnson, B. F. G. Angew. Chem. 2006, 45, 4782−2785. (3) Fang, P. P.; Duan, S.; Lin, X. D.; Anema, J. R.; Li, J. F.; Buriez, O.; Ding, Y.; Fan, F. R.; Wu, D. Y.; Ren, B.; Wang, Z. L.; Amatore, C.; Tian, Z. Q. Chem. Sci. 2011, 2, 531−539. (4) Erkoc, S.; Oymak, H. J. Phys. Chem. B 2003, 107, 12118−12125. (5) Guzman-Ramirez, G.; Robles, J.; Vega, A.; Aguilera-Granja, F. J. Chem. Phys. 2011, 134, 054101. (6) Cheng, D. J.; Liu, X.; Cao, D. P.; Wang, W. C.; Huang, S. P. Nanotechnology 2007, 18, 475702. (7) Wu, X.; Wu, G. H.; Chen, Y. C.; Qiao, Y. Y. J. Phys. Chem. A 2011, 115, 13316−13323. (8) Dieterich, J. M.; Hartke, B. J. Comput. Chem. 2011, 32, 1377− 1385. (9) Lennard-Jones, J. E. Proc. Phys. Soc. 1931, 43, 461−482. (10) Doye, J. P. K.; Wales, D. J. J. Chem. Soc., Faraday Trans. 1997, 93, 4233−4243. (11) Morse, P. M. Phys. Rev. 1929, 34, 57−64. (12) Johnson, R. A.; Wilson, W. D. Defect calculations for Fcc and Bcc metals. Interatomic Potentials and Simulation of Lattice Defects; Plenum: New York, 1971; pp 301−305. (13) Gupta, R. P. Phys. Rev. B 1981, 23, 6265−6270. (14) Daw, M. S.; Baskes, M. I. Phys. Rev. B 1984, 29, 6443−6453. (15) Tersoff, J. Phys. Rev. Lett. 1984, 56, 632−635. (16) Brenner, D. W. Phys. Rev. B 1990, 42, 9458−9471. (17) Ma, Z. N.; Cai, W. S.; Shao, X. G. J. Comput. Chem. 2011, 32, 3075−3080. (18) Nelson, D. R.; Spaepen, F. Solid State Phys. 1989, 42, 1−90. (19) Doye, J. P. K.; Meyer, L. Phys. Rev. Lett. 2005, 95, 063401. (20) Feng, Y.; Cheng, L. J.; Liu, H. Y. J. Phys. Chem. A 2009, 113, 13651−13655. (21) Cheng, L. J.; Yang, J. L. J. Chem. Phys. 2007, 127, 124104. (22) Wu, J.; Cheng, L. J. J. Chem. Phys. 2011, 134, 194108. (23) Halicioglu, T.; White, P. J. J. Vac. Sci. Technol. 1980, 17, 1213− 1215. (24) Halicioglu, T; White, P. J. Surf. Sci. 1981, 106, 45−50. (25) Daven, D. M.; Tit, N.; Morris, J. R.; Ho, K. M. Chem. Phys. Lett. 1996, 256, 195−200. (26) Wales, D. J.; Doye, J. P. K. J. Phys. Chem. A 1997, 101, 5111− 5116. (27) Leary, R. H.; Doye, J. P. K. Phys. Rev. E 1999, 60, 6320−6322. (28) Takeuchi, H. J. Chem. Inf. Model. 2006, 46, 2066−2070. (29) Cai, W. S.; Shao, X. G. J. Comput. Chem. 2002, 23, 427−435. (30) Shao, X. G.; Cheng, L. J.; Cai, W. S. J. Comput. Chem. 2004, 25, 1693−1698. (31) Johnston, R. L. J. Chem. Soc., Dalton Trans. 2003, 22, 4193− 4207. (32) Kim, H. G.; Choi, S. K. J. Chem. Phys. 2008, 128, 144702.

In the size range m = 25−41, 13-atom A occupies the icosahedral inner core of the 55-atom Mackay icosahedra. By comparison of the bond numbers of three motifs, it is clear that three adjoining double-icosahedral structure and pancake structure have similar numbers (about 243), and the number is larger than that of Mackay icosahedron (234).

4. CONCLUSION The parametric impact for the potential functions in ternary Lennard-Jones (TLJ) AlBmCn clusters was developed for the experimental and theoretical studies of clusters. To simplify the problem, only the finite distance (σ) parameter for A, B, and C atoms is varied in the TLJ potential. Then, σBB/σAA (>1.00) and σCC/σAA (>1.00) are adopted to control the atomic interaction among A, B, and C atoms in AlBmCn clusters because σ is a TLJ size parameter, and in the calculation σAA is set as 1.00 for simplicity. Their stable structures were obtained by adaptive immune optimization algorithm. In A10B10C10 clusters, two different structures are found to be based on double-icosahedra. Then, the number of bonds and average lengths of bonds for the stable structures were studied. The results showed that as the separation distance of B and C atoms increases, A atoms tend to be closer, but the short-range attractive part in TLJ clusters became weaker. Thus, to lower the potential energy, B and C atoms grew around the A atoms in pursuit of much more compact configuration. Furthermore, the minimum energy configurations of AlBmCn (l + m + n = 9−55) clusters and AlBmCn (l = 13, m + n = 42) clusters were analyzed.

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AUTHOR INFORMATION

Corresponding Author

*X.W.: tel, +86-556-550-0090; fax, +86-556-550-0090; e-mail, [email protected]. W.Y.: e-mail, [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study is supported by National Natural Science Foundation of China (NNSFC) (Grant Nos. 21171008, 20901004, and 21001008). We thank X. G. Shao for a grant of AIOA program from Nankai University.

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