Unbiased Determination of Structural and Electronic Properties of

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J. Phys. Chem. C 2007, 111, 12528-12535

Unbiased Determination of Structural and Electronic Properties of Gold Clusters with up to 58 Atoms Yi Dong* and Michael Springborg† Physical and Theoretical Chemistry, UniVersity of Saarland, 66123 Saarbru¨cken, Germany ReceiVed: February 9, 2007; In Final Form: June 6, 2007

Isolated neutral AuN clusters are studied using a parametrized density-functional tight-binding method combined with genetic algorithms for all N from 2 up to 58. Various descriptors are used in analyzing the results, including stability, shape, and similarity functions, as well as radial distances of the atoms and the orbital energies, all as functions of N. Also, dissociation patterns and the symmetry of the clusters are analyzed. By comparing with results of jellium calculations and with those of earlier embedded-atom studies, it is demonstrated that, for gold clusters, electronic effects are very important, leading to a partial suppression of the occurrence of magic numbers, as well as to a low-symmetry partly compact clusters. Also, shell-like structures are found. It is suggested that the ability of gold to form strong binding in low-coordinated systems is the reason for the occurrence of planar and shell-like structures.

I. Introduction It is well-known that materials’ properties depend sensitively on their structure and composition, and actually most technological products are to a large extent a consequence of our capability to exploit this fact, although a precise prediction of the materials’ properties for a given system has not yet become possible. During the last quarter of a century, another approach for controlling and varying materials’ properties has been intensively studied and also partly exploited in practical applications. Thus, when the materials’ dimensions are reduced to the nanometer range, their properties change markedly from those of their macroscopic counterparts. One class of such systems is provided by clusters that typically contain between some 10s and some 100 000s of atoms and most often only a few types of atoms (with, however, the possible exception of surfactants that saturate dangling bonds on the surface of the clusters). Quantum-size effects combined with the fact that the number of surface atoms relative to the total number of atoms is far from vanishing may be held responsible for the unique, size-dependent properties of those materials, and they have, accordingly, been the subject of many experimental and theoretical studies (see, e.g., ref 1). The precise determination of the relationship between size and property is, however, not easy to determine for clusters. In experimental studies, the clusters are rarely isolated, but instead, they often interact with some other medium like a solvent or a supporting surface, they possess surfactants, or their precise size is only approximately known. On the other hand, theoretical studies most often deal with isolated clusters of a well-defined size and often without ligands for which it is overwhelmingly complicated to determine the structure. Without any further information, the identification of the structure with lowest total energy for a cluster of N atoms requires searching in a geometric space of 3N - 6 dimensions, which for any but the smallest values of N are hardly possible. Therefore, theoretical studies of the properties of clusters have * Corresponding author. E-mail: [email protected]. † E-mail: [email protected].

to incorporate one or more approximations: (i) only small clusters are studied, (ii) only few, selected structures are studied, (iii) empirical potentials that depend on only the interatomic distances are employed; that is, the total energy for a given structure is approximated through

Etot ) Etot({rij})

(1)

(with rij being the distance between atoms i and j) and electronic degrees of freedom are ignored, or (iv) parametrized methods that include electronic degrees of freedom are used. In combination with parameter-free electronic-structure methods, approximations i and ii have been applied for several different types of metal clusters, whereby accurate information on the structure of the smallest clusters with up to 15-20 atoms has been obtained. Also, selected structures and sizes of larger clusters with up to 75 atoms have been studied with such methods. On the other hand, unbiased structure-optimization methods in combination with approximation iii have been applied to metal clusters with well above 100 atoms, although it is unclear to which extent the neglect of orbital interactions seriously influence the results. Approach iv does therefore constitute an intermediate approach that can be used in addressing, qualitatively and quantitatively, the importance of the inclusion of orbital interactions. However, it is still computationally considerably heavier than approach iii and has, therefore, only been applied in some few cases in combination with unbiased structure optimization for a larger set of larger clusters. It is the purpose of the present work to present results of such a study on gold clusters; that is, here, we shall report results of a study on AuN clusters that explicitly include a description of the electronic degrees of freedom and for which we have optimized the structure for every single N up to 58 using an unbiased approach. Gold clusters constitute a special case within clusters. They have attracted much attention over the last 20 years, partly because of applications in, for example, catalysis, biology, and nanotechnology2-4 and partly because gold clusters can be a useful model system for theoretical studies. On the other hand,

10.1021/jp071120x CCC: $37.00 © 2007 American Chemical Society Published on Web 08/03/2007

Properties of Gold Clusters with Up to 58 Atoms it has turned out to be particularly difficult to determine the structures of these clusters. When attempting to perform an unbiased structure optimization of a larger range of cluster sizes, parameter-free electronic-structure methods cannot be applied because of the large computational needs; instead, more or less accurate approximate methods have to be applied. Then, it has turned out that, in particular, for gold clusters, the resulting structures depend very sensitively on the applied method (see, e.g., ref 1), which is to a much lesser extent the case for clusters of most other elements. Also, for gold clusters, the above-mentioned computational limitations have put restrictions on the theoretical studies that have been performed so far. For smaller clusters (with up to ∼20 atoms), various electronic-structure methods have been applied, in part, with the special emphasis of identifying the size at which the clusters change from two- to three-dimensional (2D, 3D),5-17 whereby more or less unbiased methods were applied in the optimization of the structure. It has been found (see, e.g., ref 14) that the size for which the structures change from two- to three-dimensional depends not only on the charge of the clusters but also on the computational details. Ultimately, this suggests that there is a complicated competition between different types of interactions, including relativistic effects, the range of the potential for the interatomic interactions, the localization of the electrons, and so forth, that favor different types of structures. Parameter-free electronic-structure calculations for larger clusters exist only for few, selected sizes for which, in addition, only special specifically chosen structures were considered (see, e.g., refs 14 and 18-22). Here, special emphasis has been put on understanding the occurrence of low-symmetry amorphous structures which, like the occurrence of smaller planar clusters, has been interpreted in terms of relativistic effects and range of the interatomic interactions (see, e.g., refs 18 and 23). Alternatively, unbiased studies of the structures of gold clusters for larger N are all based on more or less empirical interaction potentials, like the embedded-atom, the Sutton-Chen, the Murrell-Mottram, and the Gupta potential (see, e.g., refs 1,23, and 24) that most often are of the form of eq 1; that is, electronic degrees of freedom are ignored, and the total energy is written as a function of the interatomic distances. As a consequence of this, the above-mentioned 2D-3D transition occurs for very low values of N (i.e., N < 10). Moreover, these studies support the consensus that the precise form of the interatomic potential can be held responsible for the occurrence of low-symmetry structures for Au clusters (see, e.g., refs 23 and 32). However, for the smaller clusters, these studies tend to produce more compact, closely packed structures than those found in the studies that include electronic degrees of freedom. Most likely, the total-energy expression of eq 1 does favor close packing and high coordination. Furthermore, Wilson and Johnston25 have reported a result of a study for which also threebody interactions were explicitly included in the approximate descriptions of the total energy as a function of the structure; also, in this study, electronic degrees of freedom were neglected, and they find structures characterized by close packing and high coordination. Thus, electronic degrees of freedom need to be included in order to obtain a correct description of the structures of the smallest clusters of gold. Although there exists such studies for clusters with up to 15-20 atoms as well as for selected larger clusters, no systematic study of all cluster sizes with up to several 10s of atoms has been presented. It is the purpose of

J. Phys. Chem. C, Vol. 111, No. 34, 2007 12529 this work to present results of a study that explicitly includes a description of the electronic degrees of freedom. Using a parametrized density-functional method in combination with genetic algorithms for an unbiased structure determination, we have optimized the structures for AuN clusters with every N up to 58. By optimizing the structure of AuN clusters for a larger, continuous sequence of N values, we are able to identify trends. This is emphasized by analyzing the results by means of special tools that we have developed explicitly for this purpose. Our computational approach shall be described in section II, and the results will be described in section III. We shall put special emphasis on analyzing our findings through some few key descriptors and comparing them with those of two extreme situations: the results of the above-mentioned empirical calculations that do not include electronic degrees of freedom and those of jellium calculations that exclude structural degrees of freedom. Finally, section IV summarizes our findings. II. Computational Method In order to calculate the electronic properties and total energy of a given structure, we have used the parametrized densityfunctional tight-binding (DFTB) method of Seifert et al.35-37 For the determination of the structure of the lowest total energy, we have combined this method with genetic algorithms. The DFTB method is based on the density functional theory of Hohenberg and Kohn in the formulation of Kohn and Sham. Then, the Kohn-Sham orbitals Ψi of the system of interest are expanded in terms of atom-centered localized basis functions χm,

Ψi(b) r )

cimχm(b r -B Rj) ∑ m

(2)

The Kohn-Sham single-particle operator is approximated as

r ) ˆt + hˆ ) ˆt + Veff(b)

∑j Vj0(|br -RBj|)

(3)

with ˆt being the kinetic-energy operator and Veff(r b) being the effective Kohn-Sham potential which is approximated as a simple superposition of the potentials of the neutral atoms. Furthermore, we assume that the matrix elements 〈χm|V0j |χn〉 vanish unless at least one of the two basis functions is centered at atom j. Finally, all remaining matrix elements are calculated accurately. From the Kohn-Sham eigenvalues of the isolated atoms and of the system of interest, we calculate the total energy (relative to that of the isolated atoms) according to occ

EB =

1

jm + ∑ Ujj′ (|R Bj -R Bj′|) ∑i i - ∑j ∑ 2 j*j′ m

(4)

(with m being an atom index and j being an orbital index). Here, Ujk is a short-range pair potential between atoms j and k that is so adjusted that results from parameter-free density-functional calculations on two-atomic systems as a function of the interatomic distance are accurately reproduced. Finally, in this study, only the 5d and 6s electrons were explicitly included in the calculations, whereas the others were treated within a frozen-core approximation. Moreover, scalarrelativistic effects (but not spin-orbit couplings) are included in the calculations. Since the approach is based on extrapolating results from the two-atomic systems to larger systems, we tested it by calculating the lattice constant of crystalline Au. We found 7.73 au, which

12530 J. Phys. Chem. C, Vol. 111, No. 34, 2007 is in excellent agreement with the experimental value of 7.71 au.38 As a further test, we studied the planar Au6 cluster with the D3h symmetry. This has three corner atoms forming an equilateral triangle and has three additional side atoms displaced slightly away from the center of the triangle. Remacle and Kryachko13 have reported bond lengths between the side atoms of 5.03-5.14 au and bond lengths between the corner and the side atoms of 5.31-5.48 au, where the different values result from different computational approaches. We find values of 4.77 and 4.94 which is somewhat smaller than those above. For a tetrahedral Au20, we find the three shortest bond lengths to be around 4.77-4.94 au, whereas the original study39 on this system reported values of 5.06-5.35 au. In total, this suggests that our approach leads to structures that are too compact, whereas we believe, in agreement with the statement of Koskinen et al.40 who compared the performance of a DFTB method with that of a parameter-free density-functional method on small Au clusters, that the two methods give geometrically and energetically very similar results. In optimizing the structures of gold clusters, we have used a method based on genetic algorithms that we had used earlier in optimizing the structures of (HAlO)N clusters.41,42 The genetic algorithms are based on the principles of natural evolution, are therefore also called evolutionary algorithms,43,44 and have been found to provide an efficient tool for global geometry optimizations. Our version of the genetic algorithms is as follows. Suppose that we have optimized the structure of the cluster with N - 1 atoms. From this structure, we construct a so-called generation consisting of M independent clusters for the N-atom system by randomly adding one Au atom and letting each of these M structures relax to its nearest total-energy minima. Subsequently, a new set of clusters is constructed by cutting each of the original ones randomly into two parts that are interchanged (under the constraint that no atom should become too close to any other atom or too far from all the other ones) and, afterward, is allowed to relax. Out of the total set of 2M structures, the M ones of the lowest total energy are kept as the next generation. This procedure is repeated until the lowest total energy is unchanged for a large number of generations. In addition to these calculations, we also considered the spherical jellium model. Here, it was assumed that all but the 11 5d and 6s valence electrons per gold atom as well as the nuclei were smeared out to a spherical medium (jellium) with a constant density inside which the valence electrons were moving. The density of the jellium was taken to be the same as in crystalline Au, and a local-density approximation within density-functional theory was assumed valid for the valence electrons. The resulting one-dimensional, single-particle equations were solved numerically and self-consistently. Finally, throughout the paper, we shall use atomic units as the length unit and electronvolts as the energy unit. Moreover, brief accounts of our results for clusters with N e 20 have been presented elsewhere.45,46 III. Results and Discussion From the DFTB calculations, we extract the total energy as a function of size of the cluster, EB(N). In Figure 1, we show EB(N)/N as a function of N for the globally optimized structures together with the value of the icosahedral Au13 cluster (for which the total energy could be lowered upon a local symmetrybreaking relaxation, which is seen as the two triangles in Figure 1, with the lower one corresponding to the lower-symmetry structure) and that of the tetragonal Au 20 cluster (for which the total energy was only marginally higher than that of the

Dong and Springborg

Figure 1. Variation in the total energy per atom (relative to that of the isolated atom) for the optimized AuN clusters (solid curve) together with those of planar structures (dark circles) and those of icosahedral Au13 and tetragonal Au20 clusters (dark triangles).

globally optimized structure). In the figure, we also show results for planar clusters with N e15. The figure shows that, for clusters with up to 6 atoms, the structure of the lowest total energy is planar. For larger clusters, truly three-dimensional objects are found (see below), but for clusters with up to 15 atoms, planar structures lie very close in energy to those found in the unbiased search. We cannot exclude that removing inaccuracies in our approach would change the relative ordering of those. On the other hand, Koskinen et al.40 found also that the DFTB method predicts a transition from 2D to 3D structures for a relatively low value of N. As also discussed by others (cf. the discussion in Introduction), the precise size at which the 2D-3D transition occurs depends very sensitively on the computational details. Finally, the experimentally observed tetragonal Au20 cluster39 is indeed a structure of a very low total energy. In Figure 1, we see that the total energy per atom is an overall decreasing function of N until N = 20, after which value, the function instead oscillates around values within roughly 10% of that of the infinite crystal. When approximating the interatomic interactions so that the electronic degrees of freedom are not directly included, structure optimizations often tend to produce structures that are characterized by closed packing whereby as many atoms as possible obtain a high coordination. In that case, the total energy per atom is overall decaying as a function of cluster size (see, e.g., refs 27 and 28). However, already the existence of planar gold clusters suggests that such a description is inaccurate for gold. Therefore, the structure of gold clusters is determined not only by a condition of minimizing the surface area but also by electronic interactions between nearest neighbors which are important, meaning ultimately that the total energy per atom will not be a simple, slowly decaying function of N but will possess a rather different functional behavior, as seen in the figure. As we shall see below, these properties of gold clusters manifest themselves at many different places. In order to identify particularly stable clusters, we consider the stability function

∆2EB(N) ) EB(N + 1) + EB(N - 1) - 2EB(N)

(5)

that has maxima (minima) for particularly (un)stable structures. This function is shown in Figure 2. On the basis of the abovementioned change in EB(N) for N = 20, the stability function is much more smooth for N < 20 than for N > 20. For N e 20, our results confirm those of Wang et al.11 and of Li et al.17 who predicted a clear even-odd oscillatory pattern; that is, clusters with even N were more stable than those with odd N. On the other hand, the most pronounced maxima, at N ) 24, 33, 40, 42, 51, and 54, do not follow this even-odd pattern and are only marginally in agreement with the results of the

Properties of Gold Clusters with Up to 58 Atoms

J. Phys. Chem. C, Vol. 111, No. 34, 2007 12531

Figure 3. Three energy gaps, Eg (solid curve) for even N as well as Eg,1 (dashed curve) and Eg,2 (dash-dotted curve) for odd N, as a function of N.

TABLE 1: Point Groups (PG) of the Optimized AuN Clusters from the DFTB Calculations as Function of N N PG N PG N PG N PG N PG N PG N PG N PG Figure 2. Stability function which has local maxima (minima) for particularly (un)stable structures. The upper panel shows the results from the DFTB calculations, and the lower panel shows those from the jellium calculations.

embedded-atom calculations27,28 that do not explicitly include electronic degrees of freedom. Since our structure-optimization approach, that is, the genetic algorithms, operate with a smaller pool of structures, the calculations may result in different, optimized structures that in some cases (at least when N is not too large) may correspond to the energetic next-higher isomers of these cluster sizes. Thus, for N ) 8, we find a second isomer 0.222 eV above the first one, and for N ) 12, we find a second isomer only 0.005 eV above the first one. For N ) 11, we find even two different structures, 0.062 and 0.072 eV, respectively, above the energetic lowest-lying one. These numbers are considerably smaller than those quoted in the most recent, parameter-free study on small gold clusters, suggesting that the limited structure search of Li et al.17 has not been sufficiently exhaustive. In order to analyze the origin of the particularly stable clusters (i.e., magic numbers), we first consider the effects of electronic degrees of freedom. In Figure 2, we also show the stability function as obtained from the jellium calculations. It is clear that this stability function takes somewhat larger positive or negative values than that of the DFTB calculations, suggesting that the fact that the DFTB calculations also allow for structural relaxations leads to an overall damping of the stability of the clusters as a function of their size. Moreover, the two curves show only a marginal agreement. In fact, embedded-atom calculations27,28 lead to even smaller absolute values of the stability, suggesting that stability is dictated by a complicated interplay between structural and electronic degrees of freedom. Whether the jellium model is adequate can be studied, for example, by looking at the orbital energies. When plotting these for the jellium model as a function of cluster size (not shown), one obtains a broad range of roughly evenly spaced levels above approximately -30 eV with the Fermi level at around -4 eV. The levels show no particular structure. In contrast, the DFTB calculations lead to a narrower, deep-lying band (below -10 eV) that is formed by the 5d functions. Thus, around the Fermi level, orbitals formed mainly by the 6s functions are found. This could suggest that in the jellium calculations one should treat not 11 but only 1 valence electron per atom. This would, however, lead to magic numbers at 2, 8, 18, 20, 34, and 58, which hardly is in agreement with the results of the DFTB calculations. Because of the low symmetry of the

3 4 5 6 7 8 9

C2V D2h Cs D3d D5h D2d C2V

10 11 12 13 14 15 16

C4d C2V C5V D2h C2 D2d D2

17 18 19 20 21 22 23

C1 C5V D5h C2V C1 C1 C1

24 25 26 27 28 29 30

C1 C1 C1 C1 C1 C1 C1

31 32 33 34 35 36 37

C1 C1 D2 C1 C1 C1 C1

38 39 40 41 42 43 44

C1 C1 C1 C1 C1 C1 C1

45 46 47 48 49 50 51

C1 C1 C1 C1 C1 C1 C1

52 53 54 55 56 57 58

C1 C1 C1 C1 C1 C1 C1

structures of the clusters, the orbitals have low degeneracies in the DFTB calculations leading to many different orbital energies. A remarkable exception is N ) 33 that has only a few different orbital energies and, in addition, is particularly stable (see Figure 2). Before leaving the discussion of electronic effects, we present one result that indicates that the electronic degrees of freedom have some impact on the stability of the clusters. In the DFTB calculations, we do not allow for a spin-polarization. Therefore, the gap Eg between the highest occupied and the lowest unoccupied orbital (HOMO and LUMO) vanishes for odd N. However, for odd N, we may consider two other gaps: one between the single-occupied orbital and the lowest completely empty orbital, Eg,1; and one between the highest completely filled orbital and the single-occupied one, Eg,2. In Figure 3, we show Eg for even N and Eg,1 and Eg,2 for odd N. The maxima for N around 6, 8, 18, and 34 (Eg) and 9, 19, 23, 33, and 35 (Eg,1) correlate fairly well with some of the maxima in Figure 2. These results are only marginally in agreement with those of Wang et al.11 who, however, found that many of the clusters in the range 2 e N e 20 were planar, which most likely is the case in that size range, pointing to some limitations of the present approach. As illustrated in Table 1, where the point groups of the optimized structures from the DFTB calculations are listed, the optimized structures of the AuN clusters possess a low symmetry. A similar tendency is also found in embedded-atom calculations28 where electronic degrees of freedom are not included directly, indicating that the occurrence of the low symmetry is not only an electronic effect like a Jahn-Teller distortion. In Figure 4, we show our optimized structures for N ) 3-20 and for N ) 33 and 55. Upon visualization, we find that the structures for N ) 4-8 are similar to those found by Wang et al.11 in their parameter-free density-functional study, whereas for N ) 3 they find a linear structure just 0.04 eV below the triangular structure. In the more recent density-functional study, Li et al.17 found planar structures to be those of the global minima for N e14, whereas structures like those of Figure 4 correspond to higher-lying isomers. Furthermore, for N g10, all studies agree that the 3D structures, whether being those of

12532 J. Phys. Chem. C, Vol. 111, No. 34, 2007

Dong and Springborg

Figure 4. Structures of the AuN clusters from the global structure optimization. From left to right, N goes from 3-7, 8-12, and 13-17 in the first, second, and third row, respectively, and equals 18, 19, 20, 33, and 55 in the fourth row.

the global minima or higher-energetic ones, all are lowsymmetric, making a direct comparison difficult. In order to obtain additional information on the structure, we first determine the center for each cluster,

B R0 )

1 N

N

B Ri ∑ i)1

(6)

with B Ri being the position of the ith atom, and then we determine the radial distance for each atom,

(7)

Figure 5. Radial distances as a function of cluster size, i.e., each small line represents (at least) one atom with that radial distance. The curve shows the radius of the spherical jellium with a density as in the crystal.

First, we analyze the moments of inertia of the cluster. From those, it is possible to distinguish between spherical, lens-shaped, and cigar-shaped clusters. It turned out that no cluster, except for a single atom, has an overall spherical shape, in agreement with the results of Table 1. However, the differences from a spherical shape are particularly low for N ) 8, 9, 18, 25, and 33. Some of these values correspond to particularly stable clusters, for which the stability accordingly may be related to a roughly spherical structure, whereas other values occur for clusters with just some few extra atoms on the surface of a compact core. This is confirmed by Figure 5 that shows the radial distances for the different AuN clusters. For N up to around 10, all atoms have a relatively large but in many case quite scattered distance to the center, suggesting structures of very low symmetry. For 8 e N e 24, all radial distances take quite large values suggesting that the structure of these clusters resembles a hollow cage. The existence of such structures was actually predicted recently.47 Also, the results of density-functional calculations by Gu et al.48 on AuN clusters with 32 e N e 35 point to the existence of cage-like structures, which, however, here only marginally is supported in this size range. We shall now offer an explanation for the occurrence of planar structures for the smallest values of N and of cage-like structures for somewhat larger N.

Since our total-energy expression is not of the form of eq 1, we cannot explain our results as being due to the range of the interatomic interactions. On the other hand, our computational approach includes relativistic effects (except for spin-orbit couplings). Moreover, as we shall argue, our results support the consensus that gold atoms can form stable structures containing low-coordinated atoms. To see this, we define a cutoff distance (we use the average of the nearest- and nextnearest-neighbor distances in the fcc crystal) that determines whether two atoms are considered being bonded. Subsequently, we calculate the coordinations for all atoms for all optimized structures. The main results are summarized in Figure 6. From N around 15-20, the most frequent coordinations are 5, 6, and 7, and the average coordination then hardly increases as a function of N. Moreover, for none of the optimized structures do we find 12-fold coordinated atoms (which is the value for the fcc crystal). Finally, Li et al.17 also report structures for small clusters where the atoms have fairly low coordination. Thus, our results, as well as the existence of planar and cage-like structures, all suggest that gold structures with low coordination often are more stable than what usually is the case for metals. The fact that gold nanostructures have low-coordinated atoms has been held responsible for the enhanced catalytic activity of such systems (see, e.g., refs 49-52). Thus, our findings suggest that the existence of low-coordinated atoms is a general

Bi - B R0 | ri ) |R

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J. Phys. Chem. C, Vol. 111, No. 34, 2007 12533

Figure 6. Various properties related to the number of nearest neighbors for the atoms of AuN clusters as functions of N. The lowest, middle, and upper curve show the smallest, average, and highest number of nearest neighbors, respectively.

phenomenon for gold nanostructures and, accordingly, that gold nanostructures have promising properties within the field of catalysis. The stability of structures containing low-coordinated gold atoms may be related to the stability of chains of gold atoms. During the past decade, chains of atoms have been studied in break-junction experiments (see, e.g., refs 53 and 54). It has been found that it is easier to create chains of certain elements than of others. The difference was related to the ability of gold to form strong bonds in low-coordinated systems55 as well as to surface reconstruction and to a competition between s and d bonding.56 Thereby, the particular stability of chains of Au atoms was rationalized. Our finding that the total energy per atom for the AuN clusters rapidly reaches a saturation and our finding of the occurrence of cage-like and planar structures are in agreement with the explanations for the occurrence of gold chains, and we therefore suggest that they are different manifestations of the same properties of gold, that is, its ability to form strong binding in low-coordinated systems. Moreover, it has been argued that this ability of gold in turn is responsible for the occurrence of amorphous structures for the clusters.23,32 For several clusters, one or more of the largest radial distances are significantly larger than the radius of the spherical jellium for the same cluster size. The reason is that these clusters have a particularly low symmetry and are very far from being spherical. Such clusters often possess atoms on the surface that are bonded to only one or two nearest neighbors. That existing structural element has also been seen in the density-functional study of Remacle and Kryachko on smaller clusters.13 Figure 5 confirms that the cluster for N ) 33 is particularly symmetric. Fa et al.15 suggested the occurrence of structures related to either fragments of the crystal or to tube-like structures. Our results, including those of Figure 5, do not support the latter suggestion. In order to address the first suggestion in more detail, we use two different approaches. In the first, we use so-called similarity functions defined as follows. For a given N, we sort all of the radial distances. Simultaneously, we construct a spherical fragment of the crystal (with a fixed choice of the center) and sort its radial distances, {rfcc i }. From

q)

[

1

]

N

2 (ri - rfcc ∑ i ) N i)1

1/2

(8)

the similarity function is defined as

S)

1 1 + q/ul

(9)

(ul ) 1 au), which approaches 1 (0) if the two structures are very similar (different). This function is shown in Figure 7,

Figure 7. Each panel shows the similarity function for the AuN clusters when comparing with (a) an icosahedral cluster, and (b-d) a spherical fragment of the fcc crystal when the center of the fragment is placed at (b) the position of an atom, (c) the middle of a nearest-neighbor bond, and (d) the center of the cube, respectively.

where we also compare it with the structure of an icosahedron. Here, we have compared the radial distances of the AuN cluster with those of an icosahedral Au147 cluster in the same way as above. Our experience for other systems57,58 suggests that only when the similarity function is well above 0.8 the two structures that are compared are structurally related to each other. Thus, the results of the present study do not at all confirm the suggestions of Fa et al.15 Alternatively, we have also considered the common-neighbor analysis.59 However, also, this analysis does not at all suggest that fragments of the crystal are found. An interesting cluster is the Au55 cluster which has been the subject of numerous experimental and theoretical studies18. As seen in Table 1, we find that it has a relatively low symmetry (and not an icosahedral, cuboctahedral, or dodecahedral symmetry, as often assumed) with one atom at the center of a slightly distorted high-symmetric structure. Accordingly, this, as well as other structures that according to Table 1 have a low symmetry, appears to be as close to high-symmetric upon visual inspection as shown for the Au18, Au33, and Au55 clusters in Figure 4. The radial distances of Figure 5 suggest that certain structural motifs develop as a function of cluster size, that is, that the cluster with N atoms is similar to that with N - 1 atoms plus an extra atom. Some deviations may be found for the smallest values of N as well for N ) 33. In order to quantify this suggestion, we consider similar functions that are obtained by

12534 J. Phys. Chem. C, Vol. 111, No. 34, 2007 comparing the cluster, having N - 1 atoms with each of the N fragments, with N - 1 atoms of the cluster with N atoms. Both for the AuN-1 cluster and for each of the fragments, we calculate and sort either the radial distances or the interatomic distances and calculate subsequently a q value analogous to that above. The smallest value of the N q values is used in defining the similarity function S. The results suggest, however, that hardly any of the clusters can be related to the one of 1 atom less. Thus, here indeed, each atom counts! In particular, the Au33 cluster is exceptional and different from the other ones. Finally, we have seen that the total energy per atom is almost constant for N above approximately 20. This has as a further consequence, when considering the dissociation processes, AuN f AuN-K + AuK; and seeking that value of K * 0 that requires the smallest energy, we find quite scattered values. Thus, whereas K ) 1 or K ) 2 for N e 20, K takes much larger values for larger N, implying that many of the larger clusters may preferably split into two not too different parts. This is a marked difference from what we have found for other metal clusters using the embedded-atom method.57,58 IV. Conclusions Using an unbiased structure-optimization method (based on genetic algorithms) in combination with a parametrized densityfunctional method we have studied electronic and structural properties for the whole series of AuN clusters with N e 58. This study represents the first such one where also electronic degrees of freedom explicitly are included for all N, which indeed turns out to be important. Most other related studies have been carried out using simpler descriptions for the interatomic interactions without explicitly including electronic-orbital interactions. Since the latter are those being responsible for directional interactions, it may not surprise that most, previous studies have found structures characterized by close packing. This includes the finding of particularly stable, highly symmetric clusters (i.e., magic numbers). When including orbital interactions, not only packing but also directional interactions determine the optimal structure, and therefore in most cases, our optimized structures do not have a very high symmetry; that is, atoms bonded to just one or two neighbors are often found. In this respect, gold seems to be special. For other metals, packing effects are often dominating, whereas for covalently bonded elements, the effects due to directional bonds are dominating. We suggest that for gold there is a competition between the two leading to low-symmetry, although often quite compact structures of the AuN clusters. In particular, this competition stabilizes structures with gold atoms that are fivefold to sevenfold coordinated. Admittedly, by using a parametrized (and not first-principles) method, our results may be connected with some uncertainty. Thus, the fact that we find the transition from planar to threedimensional structures for a much too small N may be explained from this. On the other hand, low-symmetry structures have been found in other, more accurate studies on selected clusters, as discussed above. The fact that electronic effects are important was indicated by the results of the calculations for the spherical-jellium model. In particular, the stability function from these calculations had a somewhat larger amplitude than was the case for the DFTB results. Once again, the lowering of the symmetry is one reason for this difference. Furthermore, in some cases we could correlate the occurrence of particularly stable clusters with large gaps of the electronic orbitals around the Fermi level.

Dong and Springborg In agreement with recent results, we found that the most stable structures for 10 e N e 20 correspond to shell-like structures. We suggest that both the occurrence of planar structures for the smallest clusters and that of cage-like structures for slightly larger clusters have the same reasons as the occurrence of chains of gold atoms in a break-junction experiment; that is, it can be related to a particular stability of structures containing lowcoordinated gold atoms, to surface reconstructions, and to a competition of s and d bonding. Moreover, we predict that Au33 should be particularly stable. Surprisingly, for N > 20, the total energy per atom changes only little, which as a consequence means that the energetically favored dissociation channels for these clusters often are those where the cluster splits into two larger fragments. Furthermore, the structures showed hardly any resemblance with fragments of either crystalline gold or an icosahedron. Nevertheless, the structures did show some regular patterns like the building up of atomic shells for clusters larger than around 20 atoms. In total, we hope to have demonstrated that the properties of gold clusters are surprising and that they can only be understood in details if incorporating explicitly electronic effects. Whether an accurate description of relativistic effects is necessary, we will leave as an open question. Acknowledgment. This work was supported by the SFB 277 at the University of Saarland. The authors are very grateful to the Fonds dear Chemischen Industrie for generous support. References and Notes (1) Baletto, F.; Ferrando, R. ReV. Mod. Phys. 2005, 77, 371. (2) Yoon, B.; Ha¨kkiinen, H.; Landman, U.; Wo¨rz, A. S.; Antonietti, J.-M.; Abbet, S.; Judai, K.; Heiz, U. Science 2005, 307, 403. (3) Guzman, J.; Gates, B. C. J. Am. Chem. Soc. 2004, 126, 2672. (4) Sachenz, A.; Abbet, S.; Heiz, U.; Schneider, W. D.; Ha¨kkinen, H.; Barneet, R. N.; Landman, U. J. Phys. Chem. A 1999, 103, 9573. (5) Ha¨kkinen, H.; Yoon, B.; Landman, U.; Li, X.; Zhai, H.-J.; Wang, L.-S. J. Phys. Chem. A 2003, 107, 6168. (6) Gilb, S.; Weis, P.; Furche, F.; Ahlrichs, R.; Kappes, M. M. J. Chem. Phys. 2002, 116, 4094. (7) Neumaier, M.; Weigend, F.; Hampe, O.; Kappes, M. M. J. Chem. Phys. 2005, 122, 104702. (8) Bravo-Pe´rez, G.; Garzo´n, I. L.; Novaro, O. J. Mol. Struc. 1999, 493, 225. (9) Gro¨nbeck, H.; Andreoni, W. Chem. Phys. 2000, 262, 1. (10) Ha¨kkinen, H.; Landman, U. Phys. ReV. B 2000, 62, 2287. (11) Wang, J.; Wang, G.; Zhao, J. Phys. ReV. B 2002, 66, 035418. (12) Walker, A. V. J. Chem. Phys. 2005, 122, 094310. (13) Remacle, F.; Kryachko, E. S. J. Chem. Phys. 2005, 122, 044304. (14) Ferna´ndez, E. M.; Soler, J. M.; Balba´s, L. C. Phys. ReV. B 2006, 73, 235433. (15) Fa, W.; Luo, C.; Dong, J. Phys. ReV. B 2005, 72, 205428. (16) Bonacˇic´-Koutecky´, V.; Burda, J.; Mitric´, R.; Ge, M. J. Chem. Phys. 2002, 117, 3120. (17) Li, X.-B.; Wang, H.-Y.; Yang, X.-D.; Zhu, Z.-H.; Tang, Y.-J. J. Chem. Phys. 2007, 126, 084505. (18) Ha¨kkinen, H.; Moseler, M.; Kostko, O.; Morgner, N. Hoffmann, M. A.; van Issendorff, B. Phys. ReV. Lett. 2004, 93, 093401. (19) Xiao, L.; Tollberg, B.; Hu, X.; Wang, L. J. Chem. Phys. 2006, 124, 114309. (20) Gao, Y.; Zeng, X. C. J. Am. Chem. Soc. 2005, 127, 3698. (21) Apra, E.; Ferrando, R.; Fortunelli, A. Phys. ReV. B 2006, 73, 205414. (22) Garzo´n, I. L.; Michaelian, M.; Beltra´n, M. R.; Posada-Amarillas, A.; Ordejo´n, P.; Artacho, E.; Sa´nchez-Portal, D.; Soler, J. M. Eur. Phys. J. D 1999, 9, 211. (23) Ferna´ndez, E. M.; Soler, J. M.; Garzo´n, I. L.; Balba´s, L. C. Phys. ReV. B 2004, 70, 165403. (24) Doye, J. P. K.; Wales, D. J. New J. Chem. 1998, 22, 733. (25) Wilson, N. T.; Johnston, R. L. Eur. Phys. J. D 2000, 12, 161. (26) Alamanova, D.; Dong, Y.; Rehman, H. u.; Springborg, M.; Grigoryan, V. G. Comp. Lett. 2005, 1, 319. (27) Grigoryan, V. G.; Alamanova, D.; Springborg, M. Eur. Phys. J. D 2005, 34, 187.

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