Transition-State Searches in Metal Clusters by First-Principle Methods

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Transition-State Searches in Metal Clusters by First-Principle Methods Domingo Cruz-Olvera, Alejandra de la Trinidad Vasquez, Gerald Geudtner, José Manuel Vásquez-Pérez, Patrizia Calaminici,* and Andreas M. Köster* Departamento de Quı ́mica, CINVESTAV, Av. Instituto Politécnico Nacional 2508, AP 14-740, México D.F. 07000, México ABSTRACT: Elucidation of the chemical reactivity of metal clusters is often cumbersome due to the nonintuitive structures of the corresponding transition states. In this work, a hierarchical transition-state algorithm as implemented in the deMon2k code has been applied to locate transition states of small sodium clusters with 6−10 atoms. This algorithm combines the so-called double-ended interpolation method with the uphill trust region method. The minimum structures needed as input were obtained from Born−Oppenheimer molecular dynamics simulations. To connect the found transition states with the corresponding minimum structures, the intrinsic reaction coordinates were calculated. This work demonstrates how nonintuitive rearrangement mechanisms can be studied in metal clusters.

1. INTRODUCTION The study of metal clusters has intensively increased over the last decades, both from the experimental and theoretical points of view. Due to the fact that these systems are very important for the continuous development in the field of nanoscience and nanotechnology, the understanding of how their structure and energy properties change with cluster size has been the object of several investigations (see, for example, refs 1−38 and references therein). Generally, in cluster studies, the properties of interest are investigated considering the lowest minimum structure. However, the study of the full potential energy surface (PES) landscape is very important if one aims to understand how for a given finite system a chemical reaction may occur. A chemical reaction can be described in terms of an energy profile picture with which aspects of the mechanisms, structures and energies of the reaction are explained. A way to characterize a PES is by the location of its relevant critical points (CPs). In general, stationary points of a function that depends on N variables are reached when the gradient of the given function vanishes. For a multidimensional function such as the PES, several different kinds of stationary points usually exist. These are minima, maxima, and saddle stationary points. These CPs are characterized by the eigenvalue spectrum of the corresponding Hessian matrix. A minimum CP possesses only positive eigenvalues that correspond to positive curvatures, in all principal directions. A maximum possesses only negative eigenvalues that correspond to negative curvatures in all principal directions. Saddle points of different order can be present on a PES, too. A transition state is a saddle point of first order, that is, it possesses a negative curvature in one principle direction and positive curvatures in all other directions. From a chemical point of view, only minima and first-order saddle points are CPs of interest in a PES because they correspond to © XXXX American Chemical Society

stable isomers and to transition states, respectively. For a molecular system, the reaction coordinate is defined by the minimum-energy path on the PES that connects the energy minimum of the reactant with that of the product. For an elementary reaction step, this path has only one maximum, which is a first-order saddle point and named the transition state. For metal clusters, the PES of an even very simple system consisting of only a small number of atoms can possess a large number of local minimum structures. As a result, a complex network of rearrangement reactions exists in these systems. To unravel such networks, the systematic location of transition states is mandatory. Whereas the optimization of local minima is quite straightforward,39 the location of transition states is much more cumbersome due to their saddle point nature. Many different techniques for the search of transition states have been suggested (for an overview of the corresponding literature, see ref 40 and references therein). These algorithms work relatively well if intuitive starting structures are available, that is, structures that possess one negative eigenvalue in their Hessian matrix. Unfortunately, for metal clusters, such starting structures are most often not easy to find. Therefore, a more systematic approach is needed. To this end, a new hierarchical transition-state search algorithm40 has been recently developed in our laboratories. It combines the so-called double-ended interpolation method41 with the uphill trust region method.42,43 Special Issue: 25th Austin Symposium on Molecular Structure and Dynamics Received: June 19, 2014 Revised: July 21, 2014

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dx.doi.org/10.1021/jp506121f | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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The obtained minima were characterized by frequency analysis. The second derivatives were calculated by numerical differentiation (two-point finite difference) of the analytic energy gradients using a displacement of 0.001 au from the optimized geometry for all 3N coordinates. The harmonic frequencies were obtained by diagonalizing the mass-weighted Cartesian force constant matrix. As already discussed, the transition states were located with the hierarchical transition-state finder as implemented in the deMon2k code.40,48 The only input information were the minimum structures of the sodium clusters involved in the rearrangement. Starting from the obtained transition states, the IRC was calculated, using the approach from Gonzalez and Schlegel,61 either to find potential intermediates or to calculate the minimum-energy path that connects the two starting minimum structures. Both the IRC and the uphill trust region optimization in the hierarchical transition-state search were initialized by Hessian calculations in order to provide the correct eigenvalue spectrum.

Key to success is the formulation of the double-ended interpolation method analogue to a restricted step algorithm.40,44 This ensures robust and fast interpolations into the transition-state regions. Once an approximated transitionstate structure is located, the uphill trust region method is used to optimize the transition state. Thus, the transition-state search becomes free of intuitive starting structures. The only information needed are the initial reactant and product structures. The hierarchical transition-state search algorithm has so far been applied to small test reactions,40,44 Diels−Alder showcase reactions,40,44 Pd13 oxidation,45 and conformational rearrangements in glycerol.46 In this paper, we apply the hierarchical transition-state search algorithm for the first time to rearrangements reactions of pure metal clusters, in particular, selected sodium clusters. These systems are very challenging because neither the transition states nor the reaction coordinates are chemically intuitive. Therefore, it is usually not guaranteed that the two initial minima structures are indeed connected by an elementary rearrangement step free of intermediates. To this end, we performed intrinsic reaction coordinate (IRC) calculations starting from the located transition states. By the combination of the hierarchical transition-state search algorithm with the IRC calculations, reaction intermediates can be found. In this way, complex rearrangement networks can be completely unraveled. The paper is organized as follows. In the next section, the computational details are described. Results and discussion are presented in section 3. Finally, in the last section, the conclusions are summarized.

3. RESULTS AND DISCUSSION In this section, the results of selected transition-state searches in Nan clusters (n = 6−10) are discussed. Sodium clusters have been selected as prototypes of metal clusters for this type of investigation. The lowest minimum structures of Na6, Na7, Na8, Na9, and Na10 are depicted in Figure 1. All minimum structures are in their lowest multiplicity. The relative energies of the lowlying mimimum structures with respect to the found groundstate structure are given in kcal/mol in this figure. The lowest minimum structures found for Na6 are shown at the top of Figure 1. The ground-state structure is a pentagonal pyramid, and the first low-lying isomer is a planar structure,

2. COMPUTATIONAL DETAILS All calculations were performed using the linear combination of Gaussian-type orbital density functional theory (LCGTODFT) deMon2k program.47,48 The Coulomb energy was calculated by the variational fitting procedure proposed by Dunlap, Connolly, and Sabin.49,50 The auxiliary density was expanded in primitive Hermite Gaussian functions using the GEN-A2 auxiliary function set.51 The exchange−correlation functional was evaluated with this auxiliary density, that is, the auxiliary density functional theory (ADFT) method was used.52 The exchange−correlation energy and potential were numerically integrated on an adaptive grid.53 The grid accuracy was set to 10−5 au in all calculations. The structure optimization and the frequency analysis of the clusters were performed employing the gradient corrected exchange−correlation functional proposed by Perdew, Burke, and Ernzerhof (PBE)54 in combination with DFT optimized double-ζ valence plus polarization (DZVP) basis sets.55 In order to determine the geometry of the lowest ground state and of corresponding lowlying states of the studied clusters, a very extensive search was performed, and several dozens of initial configurations were studied. In this search, structures previously reported by other authors were included as well as many other initial structures, which were extracted from BOMD trajectories recorded at a temperature of 250 K. The BOMD trajectories were generated in the canonical ensemble employing a Nosé−Hoover chain thermostat56−59 with seven thermostats in the chain and a coupling frequency of 200 cm−1. For all structures, a full geometry optimization without any symmetry restriction was performed using a quasi-Newton optimization method in delocalized internal coordinates.60 The convergence was based on the Cartesian gradient and displacement vectors with a threshold of 10−4 and 10−3 au, respectively.

Figure 1. Lowest minima structures of the Nan (n = 6−10) clusters. The energy differences between the found ground-state structures (left) and the next-low-lying minima (right) are in kcal/mol. B



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Figure 2. Snapshots along the IRC for the Na6 rearrangement between the two lowest minimum structures of this cluster. The energy is given versus the IRC. The structure position is indicated by the black point on the energy path.


As can be seen from Figure 3 the IRC is initially very flat. Moreover, the transition state, characterized by an imaginary frequency of ω = 10i cm−1, is only 0.6 kcal/mol above the higher-energy isomer. As a result, the transition-state structure (top right in Figure 3) is very similar to the higher-energy isomer, and the IRC is very asymmetric. Such PES shapes are very challenging for transition-state finders, particularly if no initial structure guesses are available. The hierarchical transition-state finder worked flawlessly for this cluster rearrangement, underlying its robustness. The two lowest minimum structures that we found for the Na8 cluster are depicted in the third row of Figure 1. The ground-state structure is a slightly distorted snub disphenoid formed by a square-bipyramidal fragment with two extra Na atoms capping two triagonal faces (left structure in the third row of Figure 1). The first low-lying minimum structure is formed by five slightly distorted tetrahedral fragments (right structure in the third row of Figure 1) and lies 2.3 kcal/mol above the ground-state structure. The hierarchical transitionstate finder was applied with these two minimum structures as

which lies only 0.4 kcal/mol above the ground state (see Figure 1). By applying the hierarchical transition-state finder with these two minimum structures as input, a transition state characterized by an imaginary frequency of ω = 36i cm−1 is found. Figure 2 illustrates some snapshots of the IRC calculation, which allows one to follow the course of the structural rearrangement. The position of a structure along the IRC is indicated by the black point on the energy path. The transition-state structure is the third structure on the top right of Figure 2. The activation energy for this reaction (i.e., the energy difference between the low-lying minimum structure and the transition-state structure) is 0.8 kcal/mol, which is larger than the reaction energy of 0.4 kcal/mol (Figure 1). In the second row of Figure 1, the two lowest minimum structures found for the Na7 cluster are depicted. The groundstate structure is a pentagonal bipyramid followed by a lowlying minimum structure formed by four tetrahedral fragments, at 3.2 kcal/mol. In Figure 3, snapshots along the IRC calculations are presented in order to show the cluster rearrangement from the higher- to the lower-energy isomer. C



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Figure 5. Snapshots along the IRC for the Na8 ground-state rearrangement into its mirror image. The energy is given versus the IRC. The position of the structure is indicated by the black point on the energy path.

input. As a result, a transition state characterized by an imaginary frequency of ω = 24i cm−1 was found. Figure 4 illustrates some snapshots along the IRC for this cluster rearrangement reaction. The third structure on the right top of Figure 4 is the transition-state structure. As Figure 4 shows, the PES shape for this Na8 rearrangement is very similar to the situation in Na7. Again, the very small energy difference between the high-energy isomer and the transition state of only 0.2 kcal/mol challenges the transition-state search algorithm. As for the Na7 rearrangement, the hierarchical transition-state finder worked also flawlessly in the here-investigated Na8 rearrangement. The above-discussed rearrangement is, however, not the most common rearrangement in low-temperature Na8 molecular dynamics. Instead, rearrangements that transfer the minimum structure of Na8 into a mirrored image are typical for the low-temperature dynamics of this cluster. Also, these kinds of rearrangements can be studied with the hierarchical transition-state finder if the atoms of the reactant and product

structures are correctly labeled (otherwise, the initial alignment algorithm will superimpose the two identical structures). The corresponding IRC is depicted in Figure 5. At the starting point (first snapshot of Figure 5), the square-bipyramidal fragment of the ground-state structure is at the right side of the structure. Following the black point on the energy path, we can see how the structure deforms going uphill to the transition state (third snapshot of Figure 5). This transition state is characterized by an imaginary frequency of of ω = 22i cm−1. The energy difference between this transition state and the ground-state structure is 0.9 kcal/mol. Once the transition state is formed, the structure deforms then downhill to reach again the groundstate structure configuration at the end point of the energy path (last snapshot of Figure 5). At this last point, the squarebipyramidal fragment, characteristic of the ground-state structure, is now on the left side of the structure. The two lowest minimum structures for the Na9 cluster are depicted in the fourth row of Figure 1. The first low-lying D



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Figure 7. Snapshots along the IRC for the rearrangement between two higher-energy isomers of the Na9 cluster. The energy is given versus the IRC. The position of structure is indicated by the black point on the energy path.

minimum lies 0.7 kcal/mol above the ground-state structure. Figure 6 shows some snapshots along the IRC calculations for this cluster rearrangement. The transition state is characterized by an imaginary frequency of of ω = 33i cm−1. The calculated activation energy is 0.8 kcal/mol. We notice that minimum structures with peculiar conformations exist for this cluster. For example, a higher-lying isomer that possesses an almost planar minimum structure was found 13.4 kcal/mol above the groundstate structure. This is formed by a very flat pentagonal pyramid that is decorated by three Na atoms, all of them lying in the same plane (bottom right structure in Figure 7). In this way, a

hexagonal moiety is formed with a centered atom. The two remaining Na atoms are 3.45 Å from each other. The highestenergy minimum structure that we found for the Na9 cluster is a completely planar structure, which lies 15.0 kcal/mol above the ground-state structure (top left structure in Figure 7). It is formed by a planar hexamer with an extra Na atom in the center and two other Na atoms capping two successive Na−Na bonds of the hexamer. The distance between these two Na atoms is 6.27 Å. It is surprising to find almost planar or completely planar minimum structures for a sodium cluster containing nine atoms. For metal clusters, planar structures E



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Figure 9. Snapshots along the IRC for the Na10 rearrangement between a higher-energy planar isomer and the ground-state structure of this cluster. The energy is given versus the IRC. The structure position is indicated by the black point on the energy path.

structure is the third snapshot (top right in Figure 7). This transition state is characterized by an imaginary frequency of ω = 38i cm−1. The calculated activation energy is 1.0 kcal/mol. The inspection of the IRC calculations reveals that, as one would expect for this simple reaction, the transition state is formed by the deformation of the planar structure. The long distance of the terminating Na atoms becomes shorter, and one atom of the hexagonal fragment leaves planarity and starts to be the seed of the apex atom for the pentagonal moiety in the product structure. The two lowest minimum structures that we found for the Na10 cluster are depicted at the bottom of Figure 1. We notice that as the cluster size increases, it becomes more complicate to locate the ground-state stucture due to the increasing number of low-lying minima. In fact, the first low-lying minimum of Na10 lies only 0.2 kcal/mol above the found ground-state structure. The ground-state structure is built by two pentagonal

have been so far detected as characteristic structures for gold clusters. The transition from planar to three-dimensional structures in Au clusters occurs at 15 Au atoms, according to first-principles DFT calculations including spin−orbit coupling.62 To the best or our knowledge, these two minimum structures for the Na9 cluster are reported for the first time in the literature. These findings underline the fact that the PES landscape of metal clusters is very rich in minimum structures. In order to demonstrate that the hierarchical transition-state finder can also be applied to higher-energy isomers, we have searched the transition state between these two minima. The hierarchical transition-state finder was applied by considering these two structures as input. Snapshots along the IRC for this rearrangement are illustrated in Figure 7. The two extreme points depict the two minimum structures used as input for the hierarchical transition-state finder. The resulting transition-state F



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(4) Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer Series in Material Sciences; Springer-Verlag: Berlin, Germany, 1995. (5) Bonin, K. D.; Vollmer, M. Electric-Dipole Polarizabilities of Atoms, Molecules and Clusters; World Scientific: Singapore, 1997. (6) Knight, W. D.; Clemenger, K.; de Heer, W. A.; Saunders, W. A.; Chou, M. Y.; Cohen, M. L. Electronic Shell Structure and Abundances of Sodium Clusters. Phys. Rev. Lett. 1984, 52, 2141−2143. (7) Claridge, S. A.; Castleman, A. W., Jr.; Khanna, S. N.; Murray, C. B.; Sen, A.; Weiss, P. S. Cluster-Assembled Materials. ACS Nano 2009, 3, 244−255. (8) Khanna, S. N.; Castleman, A. W. Jr. Quantum Phenomena in Clusters and Nanostructures; Springer: Berlin, Germany, 2003. (9) Weltner, W.; Van Zee, R. J. Transition Metal Molecules. J. Annu. Rev. Phys. Chem. 1984, 35, 291−327. (10) Morse, M. D. Clusters of Transition-Metal Atoms. Chem. Rev. 1986, 86, 1049−1109. (11) Moskovits, M. Metal Clusters; Wiley: New York, 1986. (12) Duncan, M. A. Advances in Metal and Semiconductor Clusters; JAI Press: Greenwich, CT, 1993. (13) Salahub, D. R. Transition-Metal Atoms and Dimers. Adv. Chem. Phys. 1987, 69, 447−520. (14) Alonso, J. A. Electronic and Atomic Structure, and Magnetism of Transition-Metal Clusters. Chem. Rev. 2000, 100, 637−677. (15) Jug, K.; Zimmermann, B.; Calaminici, P.; Köster, A. M. Structure and Stability of Small Copper Clusters. J. Chem. Phys. 2002, 116, 4497−4507. (16) Calaminici, P.; Köster, A. M.; Russo, N.; Roy, P. N.; Carrington, T., Jr.; Salahub, D. R. V3: Structure and Vibrations from Density Functional Theory, Franck−Condon Factors, and the Pulsed-Field Ionization Zero-Electron-Kinetic Energy Spectrum. J. Chem. Phys. 2001, 114, 4036−4044. (17) Calaminici, P. Density Functional Calculations of Molecular Electric Properties in Iron Containing Systems. Chem. Phys. Lett. 2003, 374, 650−655. (18) Calaminici, P. Polarizability of Fen (n ≤ 4) Clusters: An AllElectron Density Functional Study. Chem. Phys. Lett. 2004, 387, 253− 257. (19) Lopez-Arvizu, G.; Calaminici, P. Assessment of Density Functional Theory Optimized Basis Sets for Gradient Corrected Functionals to Transition Metal Systems: The Case of Small Nin (n ≤ 5) Clusters. J. Chem. Phys. 2007, 126, 194102. (20) Calaminici, P. Is the Trend of the Polarizability per Atom for All Small 3d Transition Metal Clusters the Same? The Case of Nin (n ≤ 5) Clusters. J. Chem. Phys. 2008, 128, 164317. (21) Jellinek, J.; Garzon, I. L. Structural and Dynamical Properties of Transition Metal Clusters. Z. Phys. D 1991, 20, 239−242. (22) Lombardi, J. R.; Davis, B. Periodic Properties of Force Constants of Small Transition-Metal and Lanthanide Clusters. Chem. Rev. 2002, 102, 2431−2460. (23) Jackson, K. A. First-Principles Study of the Structural and Electronic Properties of Cu Clusters. Phys. Rev. B 1993, 47, 9715− 9722. (24) Gutsev, G. L.; Bauschlicher, C. W., Jr. Electron Affinities, Ionization Energies, and Fragmentation Energies of Fen Clusters (n = 2−6): A Density Functional Theory Study. J. Phys. Chem. A 2003, 107, 7013−7023. (25) Massobrio, C.; Pasquarello, A.; Car, R. Structural and Electronic Properties of Small Copper Clusters: A First Principles Study. Chem. Phys. Lett. 1995, 238, 215−221. (26) Green, S. M. E.; Alex, S.; Fleischer, N. L.; Millam, E. L.; Marcy, T. P.; Leopold, D. G. Negative Ion Photoelectron Spectroscopy of the Group 5 Metal Trimer Monoxides V3O, Nb3O, and Ta3O. J. Chem. Phys. 2001, 114, 2653−2668. (27) Vásquez-Pérez, J. M.; Gamboa-Martnez, G. U.; Köster, A. M.; Calaminici, P. The Discovery of Unexpected Isomers in Sodium Heptamers by Born−Oppenheimer Molecular Dynamics. J. Chem. Phys. 2009, 131, 124126.

bipyramidal moieties that are perpendicular to each other. The hierarchical transition-state finder was applied considering the two Na10 structures in Figure 1 as input. Snapshots along the IRC for this rearrangement are illustrated in Figure 8. The resulting transition-state structure is characterized by an imaginary frequency of ω = 33i cm−1 (structure of the third snapshot, top right in Figure 8). The calculated activation energy is 1.5 kcal/mol. As in the previous case of the Na9 cluster, also for the Na10 cluster we found a completely planar minimum structure. The hierarchical transition-state finder was applied considering this structure and the ground-state structure as input. Due to the shape of the two structures involved as the reactant and product, this reaction mechamism is not chemically intuitive. However, it is obvious that this rearrangement if free of intermediates, as our result shows. Snapshots along the corresponding IRC are illustrated in Figure 9. The activation energy is 1.5 kcal/mol, and an imaginary frequency of ω = 26i cm−1 characterizes the transition state.

4. CONCLUSIONS In this work, a hierarchical transition-state algorithm that combines the so-called double-ended interpolation method with the uphill trust region method was applied to search transition states of selected sodium clusters. These systems are prototypes of finite systems for which transition states are not chemically intuitive. Sodium clusters from 6 to 10 atoms have been investigated, and selected transition states have been analyzed. All found transition states have been confirmed by the IRC calculations to connect the reactant and product of the corresponding isomerization reaction. The successful application for the hierarchical transition-state finder for highly unsymmetrical reaction paths with small activation barriers underlines the robustness of this approach. Particular examples are the Na7 and Na8 cluster rearrangements. Again, it is important to note that no other information than the minimum structures of reactants and products are used in those transition-state optimizations. We also showed for Na8 that with the hierarchical transition-state finder, transition states between symmetry-equivalent minima can be found if reactant and product structures are correctly labeled. Similar studies for other metal clusters are currently underway in our laboratories.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (P.C.). *E-mail: [email protected] (A.M.K.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from CONACYT (Grants 130726 and 179409) is gratefully acknowledged. REFERENCES

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Transition-State Searches in Metal Clusters by First-Principle Methods

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