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Novel genetic algorithm search procedure for LEED surface structure determination

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 225005 (8pp)

doi:10.1088/0953-8984/26/22/225005

Novel genetic algorithm search procedure for LEED surface structure determination M L Viana1, D D dos Reis2, E A Soares2, M A Van Hove3, W Moritz4 and V E de Carvalho2 1

  Departamento de Ciências Exatas, Instituto Federal Minas Gerais—Campus Bambuí, Bambuí, Minas Gerais, Brazil 2   Departamento de Física, ICEx, UFMG, Belo Horizonte, Minas Gerais, Brazil 3   Institute of Computational and Theoretical Studies, and Department of Physics, Hong Kong Baptist University, 224 Waterloo Road, Kowloon, Hong Kong 4   Department of Earth and Environmental Sciences, Ludwig-Maximilians-Universität München, ­Theresienstr. 41, 80333 Münich, Germany E-mail: [email protected] Received 27 January 2014, revised 20 March 2014 Accepted for publication 2 April 2014 Published 13 May 2014 Abstract

Low Energy Electron Diffraction (LEED) is one of the most powerful experimental techniques for surface structure analysis but until now only a trial-and-error approach has been successful. So far, fitting procedures developed to optimize structural and nonstructural parameters—by minimization of the R-factor—have had a fairly small convergence radius, suitable only for local optimization. However, the identification of the global minimum among the several local minima is essential for complex surface structures. Global optimization methods have been applied to LEED structure determination, but they still require starting from structures that are relatively close to the correct one, in order to find the final structure. For complex systems, the number of trial structures and the resulting computation time increase so rapidly that the task of finding the correct model becomes impractical using the present methodologies. In this work we propose a new search method, based on Genetic Algorithms, which is able to determine the correct structural model starting from completely random structures. This method—called here NGA-LEED for Novel Genetic Algorithm for LEED—utilizes bond lengths and symmetry criteria to select reasonable trial structures before performing LEED calculations. This allows a reduction of the parameter space and, consequently of the calculation time, by several orders of magnitude. A refinement of the parameters by least squares fit of simulated annealing is performed only at some intermediate stages and in the final step. The method was successfully tested for two systems, Ag(1 1 1)(4 × 4)-O and Au(1 1 0)-(1 × 2), both in theory versus theory and in theory versus experiment comparisons. Details of the implementation as well as the results for these two systems are presented. Keywords: surface structure by LEED, optimization method for LEED, genetic algorithm applied to LEED analysis, NGA-LEED approach (Some figures may appear in colour only in the online journal)

1. Introduction

related to surface structural properties. Thus, the detailed knowledge of the atomic positions of the outermost layers of solid crystals is crucial to better understand these properties. Low energy electron diffraction (LEED) [1–3] is one of the most powerful experimental techniques in surface structure determination of solid crystals. The diffraction pattern of back

Several physical and chemical processes which are very important for the development of new technologies such as electron emission, adsorption, corrosion, oxidation, friction, heterogeneous catalysis, and epitaxial growth are strongly 0953-8984/14/225005+8$33.00

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J. Phys.: Condens. Matter 26 (2014) 225005

scattered electrons can reveal the symmetry, the orientation of the unit cell and the presence of super lattices and/or disorder at the sample surface. In addition, the intensity of the diffracted electron beams as function of the kinetic energy of the incident beam, the so-called I(V) curves, contains information about the atomic coordinates, which determine the surface structure. However, a generally applicable direct method to extract this information has not yet been developed. Several approaches have been proposed to construct a hologram from the diffracted intensities where the interference between a reference wave and an object wave is used [4, 5]. Although in a few cases a direct image of adsorbed atoms or molecules could be obtained, the method is still not generally applicable, in particular for more complicated adsorption structures. Further developments of holographic methods have been proposed and applied for the related method of Photoelectron Diffraction [6] and may become applicable for LEED in the future. The prevalent way in which structure analysis by LEED is performed is simulation of the I(V) curves followed by a comparison with the experimental data and then repetition with other atomic coordinates. Thus, the full structural determination involves a search for the surface atomic positions which generate theoretical I(V) curves that present the best agreement with the experimental ones. The quantification of the agreement between the theoretical and experimental I(V) curves is usually performed by means of a reliability factor (R-factor) [1, 7]. Several different ways to calculate R-factors have been proposed; they are designed to give low values if the curves are very similar (zero for completely coincident curves) and high values if there is no coincidence. Therefore, the surface structure determination by LEED becomes a minimization of the R-factor with the aim of finding the global minimum. However, the R-factor topography as a function of the surface structural parameters (atomic coordinates), and non-structural parameters (Debye temperature, inner potential, etc.) presents a large number of local minima which increases rapidly with the complexity of the system. The problem in identifying the global minimum among this set of local minima becomes a very hard task when analyzing complex structures, such as oxide compounds, metallic alloys or organic systems. This usually has restricted the technique to simple systems with small unit cells. In order to solve this problem, many local and global optimization methods have been proposed with limited success. Among the local methods fare steepest descent [8], a combination of tensor LEED [9] with the gradient method [10] and least squares [11–13]. The local methods work very well for refinement, requiring a previously known structural model; they perform the search by varying the surface parameters in small ranges around pre-defined atomic positions. The convergence radius can be enlarged with a simultaneous optimization method which uses subsets of the experimental data set [1, 14] or by a fast approximation for the calculation of the multiple scattering amplitudes in the vicinity of the reference structure [9–11, 15] As was mentioned before, the R-factor topography presents a set of local minima, so that local methods will be able to reach the global minimum only if the search starts near it.

Therefore, as an attempt to overcome this difficulty, global optimization methods have been developed and tested, such as simulated annealing [16–18] and genetic algorithms [­ 19–21]. However, the global methodologies implemented so far [18, 19, 21] still require a pre-defined structural model which is used as starting point for the search. That remains a serious problem since for complex systems the number of possible models could be huge. One idea to solve this problem is "to evolve" initial clusters of atoms in order to obtain an atomic structure. In fact this has already been explored. For example, Deaven and Ho [23] and Glass et al [22] apply the genetic algorithm ideas for bulk crystal structure prediction using DFT free energy evaluations to obtain the structures of fullerene (C60) and for various other systems respectively. Based on these ideas we propose in this work a method, named the NGA-LEED algorithm (NGA standing for Novel Genetic Algorithm), which is able to create completely new trial structures and to do the comparison between them in order to find the best final structure. The procedure is based only on symmetry and bond-length conservation as discussed below and is independent of model assumptions. The method presented here differs from the previous application to LEED of genetic algorithms by applying constraints to the models and the crossover procedure (defined in the next section) and also by refinement at intermediate steps, such that the parameter space and the number of full LEED calculations are drastically reduced. In this paper we present details of the algorithmic implementation of NGA-LEED as well as its application to the Ag(1 1 1)-(4  ×  4)-O and Au(1 1 0)-(1  ×  2) systems. The Ag(1 1 1)-(4  ×  4)-O superstructure was first observed in a LEED study in 1974 [25], but its structure was not solved by that time. The correct model was found in two independent studies, one using STM combined with LEIS and DFT calculations [26], and the other combining STM with x-ray diffraction [27]. This model could be confirmed by a LEED-I(V) analysis using conventional parameter refinement [28]. We show here that a direct solution of this relatively complicated model is possible using a genetic algorithm. The calculation starts from random atom positions without model assumptions. The second example of the reconstructed Au(1 1 0) surface is a much simpler model, but contains atomic displacements in three layers which have a large influence on the I(V) curves [24, 29] . We show here that the correct model can be clearly determined by the genetic algorithm. In section 2 we show how the NGA-LEED code can create slabs of surface atoms which define structural models by applying symmetry and bond length criteria, how the search engine works and additional details of the implementation. In section 3 we present results for the Ag(1 1 1)(4 × 4)-O and Au(1 1 0)-(1 × 2) systems in theory versus theory comparison that is crucial to evaluate the code performance. Section 4 shows the results for the afore mentioned systems using real experimental data. The structural results are in agreement with the previous ones for both systems [28, 29]. Finally, in the last section we present the summary and conclusions of this work. 2

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2.  Details of the method implementation

For pre-defined sites the lateral positions are fixed, while the z-coordinates are variable on a grid. In the case of Ag(1 1 1)-(4 × 4)-O it would have been reasonable—were the structure not known—to assume that Ag atoms in the reconstructed layer(s) sit in top, hollow or bridge sites of the nonreconstructed substrate; the same assumption is reasonable for Au(1 1 0)-(1 × 2). Section 3 presents the evolution of the search for both methodologies in these two systems. The code searches for the correct structures starting from randomly chosen atomic configurations. However, it is also possible to include previously defined structures in the first generation. This capability of the program allows one to restart the search at the same point after any interruption whether accidental or intentional, and also to include structures suggested by other experimental or theoretical techniques, by similar solved systems or by previous runs. In this way, the program is able to use prior knowledge about the system to speed up the calculations. After this initial generation of structures is created, the search operators produce the next generations of trial structures; this will later be repeated to produce further generations of trial structures, until a stop criterion terminates the search. The main search operator is the crossover, which creates new members for the next generation by combining pairs of existing structures. Other operators are also included, such as elitism, mutation, and two alternative non-GA methods, least squares fitting, and simulated annealing. These search operators are explained in more detail next. Crossover: The ‘crossover’ selects a pair of individuals (structural models) obeying probabilities of selection that are calculated using their R-factors values so as to favor lowR-factor structures. A random number of atoms is inherited from the first individual of the couple and the other atoms are substituted by atoms of the second individual of the pair, as shown in figure 1. If the atomic positions of some new atom violates the bond length criterion, it is moved to another random position and the bond length criterion is checked again. This process continues until the criterion is fulfilled. Elitism: The ‘elitism’ operation copies the M best individuals—the M structures having the lowest R-factors—to the next generation, where M is previously defined. Usually M = 1 works well. Elitism guarantees that the next generation has at least one individual of fitness equal to the best one of the previous generation. Mutation: This operation randomly selects one individual and randomly modifies its atomic coordinates, respecting the given symmetry and bond length criteria. This procedure is applied with a pre-defined probability and maintains the diversity of the population by reducing the chances that the search is trapped in a local minimum. Refinement: The code also allows local refinement of the best obtained structures. Two methods are available: least squares fitting (already included in the LEEDFIT package) and simulated annealing [18] (implemented in the main code). In order to save computational time, the refinements are applied only for a pre-defined number of best individuals, only after a pre-defined number of generations, and/or for structures with an R-factor value lower than a pre-defined value.

The approach proposed in this work consists basically in creating structural models randomly by applying search operators, calculating the theoretical I(V) curves and evaluating the R-factor for each of them. This process is repeated until identifying the structure that gives the best agreement between the experimental and theoretical data. In the present implementation we have built a code based on the Genetic Algorithm where the theoretical LEED I(V) curve calculations were performed by using the LEEDFIT package [12, 13, 30] working as a subroutine called by the main program. The NGA-LEED algorithm starts the search from a population of randomly defined structures. If the atomic surface coverage (density) was previously determined by some spectroscopy technique like XPS or AES [31], the code can use this information in order to keep the correct number of each kind of atom inside the unit cell. Nonetheless, if there is some doubt about the coverage, it is possible to let the number of atoms vary, that is, the coverage can also be fitted throughout the search. Since the number of possibilities for the atomic positions inside the unit cell is infinite in a continuous space of parameters it is strictly necessary to reduce the search space in order to allow the identification of the global minimum in a reasonable period of time. Thus, some physical constraints are imposed on the randomly generated structures before submitting them to R-factor evaluations. The first constraint concerns symmetry: the overall symmetry of the correct structure is assumed (based on the symmetries of the substrate, the adsorbate and the diffraction pattern), and the atoms are randomly distributed only inside the asymmetric part of the unit cell.The symmetry is automatically preserved in this way. If a new atom position is chosen close to a rotational axis or mirror plane, then the position is moved either onto the axis or onto the mirror plane, or further away, such that too short distances between symmetrically equivalent atoms are avoided. If such a shift is not possible then the model is discarded. The second constraint concerns bond lengths: for each atom in the asymmetric unit the interatomic distances to all its neighbors are checked. If at least one of these distances is smaller than a pre-defined minmum bond length for that atom pair then new coordinates for the last atom are randomly chosen until all distances satisfy the bond length criterion. The atomic distance evaluations are much faster than the full LEED calculations, which are thus considerably reduced by avoiding R-factor calculations for unacceptable structures. In order to reduce the infinite number of structural possibilities due to continuous variables, we have used two different methodologies: restricting to a grid of coordinates with pre-defined step size and pre-defining the allowed sites for the atoms. The first methodology can be made as accurate as desired by reducing the step size. However, GAs are not efficient for refined searches, which are better handled with local optimization methods such as steepest descent [21]. The second methodology reduces the search space substantially, however, it is limited to cases where pre-defined sites are reasonable, such as for simple relaxations, substitutional replacement, alloy re-ordering and obvious adsorbate sites. 3

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Parents

B

A 2 1 5

8

7

6

9

4

10

3

12

11

Offspring 8 1 5 3

10

12

Figure 1. Schematic representation of a crossover operation, with atoms numbered in an asymmetrical 2D unit (dotted lines) of the unit cell (continuous lines). The parents (A and B) are recombined to generate the offspring (bottom). In this example the offspring contains the atoms 1, 3 and 5 from the parent A and the atoms 8, 10 and 12 from the parent B.

Therefore the NGA-LEED approach works according to the following steps: (i) A population of N individuals—structural models—is created using random atomic coordinates that satisfy the symmetry and bond length criteria. This initial population can also include pre-defined structures. (ii) The R-factors are evaluated for all the individuals by running the LEEDFIT code. (iii) The individuals are ranked by the R-factor values. Elitism is applied and the M best individuals are selected. For those individuals with a sufficiently small value of the R-factor, refinement is applied using a local method. (vi) Stopping criterion is applied to the lowest R-factor obtained in the refinement procedure (if any) of step (iii). Two criteria are used: a) R-factor lower than a predefined value and/or b) R-factor does not decrease after a pre-defined number of generations. If the criterion is not satisfied the search continues in step (v). (v) A new generation is created including the M individuals selected (refined or not) in step (iii) and the N-M individuals created by crossover and mutation according to a pre-defined probability. (vi) Back to step (ii) The algorithm is explained in the schematic flowchart shown in figure 2.

Figure 2. Schematic flowchart of the NGA-LEED algorithm.

Figure 3. Ag(1 1 1)(4 × 4)-O model, with one unit cell outlined. Small red circles: oxygen; large blue circles: top Ag atoms [28].

and Au(1 1 0)-(1 × 2) systems. In a previous LEED study by Reichelt et al [28] the reconstruction model had been confirmed for the Ag(1 1 1)(4 × 4)-O system containing two units of six triangularly arranged Ag atoms and a stacking fault in one half of the unit cell. The structural and thermal parameters were refined in a least squares optimization. The model agrees with earlier DFT, STM and x-ray studies [26, 27]. The six O atoms per unit cell occupy sites in the trenches between six-Ag triangles as shown in figure 3. We used this solution model as input for LEEDFIT to create theoretical I(V) curves, which were used as ‘pseudo-experimental’ curves in the theory versus

3.  Theory versus theory comparison In order to test the applicability and the performance of the NGA-LEED code we have applied it in a theory versus theory comparison for the previously analyzed Ag(1 1 1)(4  ×  4)-O

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Figure 4. Left:bulk-like Au(1 1 0)-(1 × 1); right:Au(1 1 0)-(1 × 2) reconstruction (reproduced with permission from Elsevier, copyright 2010) [29]. Outermost Au atoms are shown in yellow.

theory analysis. The same procedure was carried out for the Au(1 1 0)-(1 × 2) system shown in figure 4. The surface structure was recently revisited by dos Reis et al [29] using a conventional LEED analysis. The obtained missing row structure of Au(1 1 0)-(1 × 2) was used as an input to generate, through LEEDFIT, the ‘pseudo-experimental’ I(V) curves. In this kind of search it is possible to find an R-factor value equal to zero, consisting in a trustworthy test for the method. In the Ag(1 1 1) (4 × 4)-O analysis the square R-factor (R2) [1] was used and for the Au(1 1 0)-(1 × 2) the Pendry R-factor (Rp) [7] was used. In both tests we have used a population with 10 individuals, where the first best individual was always allowed to be cloned to the next generation. Every time 50 generations were created we applied ‘least squares fitting’ and ‘simulated annealing’ refinement to the best individual once it presents an R-factor value less than 0.12. For each generation we let mutation take place with a probability of 70%. We assumed three-fold symmetry for the Ag(1  1  1) (4  ×  4)-O system and the coverage in the surface slab was previously known to consist of four Ag and two O atoms in the asymmetric unit. We have distributed the atoms in the asymmetric unit in two ways:

• Using a grid of coordinates where we let atomic coordinates to have values randomly defined in steps of 0.1 Å for the planar coordinates and 0.05 Å for the perpendicular coordinate in the range of (8.1 × 8.1 × 0.72) Å3. Thus the number of possible sites for an atom's location is 52488. Once again only configurations that obey the bond length criterion are allowed. In the case of the Au(1 1 0) surface it has been shown that the clean surface exhibits a missing row reconstruction (for details see references [24] and [29]). In order to test the capability of the algorithm we used a (2 × 2) cell instead of a (1 × 2) unit cell to place up to 11 Au atoms in random positions in the first layer, keeping the others fixed. For each atom an occupation factor of 0 or 1 was randomly chosen, where 0 means no atom and 1 means one Au atom. In this way, it was possible to test another capability of the NGA-LEED algorithm, that is, to fit also the coverage. Once again we have tested both approaches to place the atoms as described next: • For this system 25 pre-defined sites of high symmetry were allowed, which includes top, hollow and bridge sites. Thus, the number of possible configurations is around 3.4 × 107. Once again this number is substantially reduced by applying the bond length criterion. • The second approach lets the planar atomic coordinates vary in steps of 0.0385  Å and the perpendicular coordinates in steps of 0.0250 Å in the range of (8.16  ×  8.16  ×  1.026)  Å3. Here the number of possible

• Using pre-defined sites in the first layer, which the atoms are allowed to occupy. This leads to 49 sites of high symmetry inside the asymmetric unit, including top, hollow and bridge sites. Thus, the number of possible configurations is around 496, or 1.3 × 1010, but it is substantially reduced by applying the bond length criterion.

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Figure 5. Best R-factor in each generation for the two systems using pre-defined sites (left panel) and a grid of coordinates (right panel).

Figure 6. An example of the evolution over successive generations of the best structure found along the grid-of-coordinates search for the Au(1 1 0)-(1 × 2) system. On the left hand side is the representation of the structure obtained after the first generation. On the right hand side is the final obtained missing-row structure. In the middle is the ‘zig-zag’ missing-row structure achieved after 12 generations which is held up to 100 generations (large plateau in figure 5).

system a Pendry R-factor of 2.7  ×  10−18 was found after 7 generations or 70 trial structures using pre-defined sites and 5.3  ×  10−3 after 138 generations or 1380 trial structures using grid of coordinates. In this case refinement was not required. Figure 5 shows the best R-factor in each generation for the two tested systems using both approaches and in figure 6 we can see an example of how the structure changes during a grid search. Both approaches were able to find the correct structures for the two tested systems, the same structures which were used to create the ‘pseudo-experimental’ data. However the implementation with pre-defined sites showed itself much faster than the implementation with grid of coordinates as expected. Normally we can not expect to have previous knowledge about the allowed sites for the more complex systems; nonetheless, the use of pre-defined sites could always help to obtain some information that could be used as a starting point for a more

sites for an atom allocation is 1 842 704, which generates a huge number of possible structures. However, due the bond length criterion, the number of physically acceptable structures is again substantially reduced. The code was able to find the correct structures for both tested systems using pre-defined sites and a grid of coordinates. For the Ag(1 1 1)(4 × 4)-O system the analysis was performed using the R2-factor and a value of 3.0  ×  10−6 was obtained after 34 generations or 340 trial structures by using pre-defined sites and an R2-factor of 0.07 for the grid of coordinates, which already indicated the correct structural model, after 700 generations or 7000 trial structures. This structure was refined by least squares fitting, included in the NGA-LEED implementation, and an R2-factor of 4.5  ×  10−4 was obtained after a few minutes using a single 3.0 GHz processor. For the Au(1 1 0)-(1 × 2) 6

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Figure 7. Structural model found by the NGA-LEED for real experimental data using (a) pre-defined sites after 2000 tries, and (b) using grid of coordinates after 10 000 tries (small red circles: oxygen; large blue circles: top Ag atoms).

Rp-factor of 0.26 and the correct structure were obtained using pre-defined sites and grid of coordinates, but the first approach took 6 generations or 60 trials while the second took 143 generations or 1430 trials. The grid search takes less than 2  weeks using a single 3.0  GHz processor. This would take months if done in the conventional way, in which many different structures must be imagined and input by hand.

accurate and lengthy kind of search, such as, for example, a grid search. 4.  Application using experimental data The same approaches have been tested using real experimental data for the same Ag(1 1 1)(4 × 4)-O and Au(1 1 0)-(1 × 2) systems. Despite the fact that these systems have already been solved by conventional LEED analysis [28, 29], the application of the NGA-LEED code to them is quite important to test the capability of the method. The analysis of real experimental data is harder than that of ‘pseudo-experimental’ or theoretical data and we can not expect to obtain R-factors very close to zero. For the Ag(1 1 1)(4 × 4)-O system we have used the same data set that was used by Reichelt et al [28]. The LEED intensity measurements were performed at 220 K, where the diffraction spots could be measured up to E = 150 eV. The data set consists of 14 symmetrically inequivalent beams in the energy range from 12 to 150 eV. Both approaches, using predefined sites and a grid of coordinates, were used in the same way as described in the previous section. Using pre-defined sites the code was able to find the correct structure (figure 3) fast after testing around 2000 structures the code found the correct positions for the 6 inequivalent atoms, obtaining a R2factor of 0.28 what is in agreement with the results obtained by Reichelt et al [28] (figure 7(a)). Otherwise, by using a grid of coordinates a R2-factor of 0.32 was found after trying more than 10 000 structures. In this case the code was able to find the correct positions for the four Ag atoms, but the two O atoms were still in wrong positions as is shown in figure 7(b). The fact that the correct positions were not obtained for the O atoms should not be considered as a failure of the method. It is important to realize that the LEED intensities are much more sensitive to the Ag positions than to those of the weakly scattering oxygen atoms. The oxygen positions could be easily found in a final refinement. A second test was performed using the same experimental data set reported by dos Reis et al [29] for the Au(1 1 0)-(1 × 2) system. LEED measurements of the diffracted beam intensity were recorded for the 50–400 eV energy range, with the sample at 178 K. The I(V) curves were recorded for 11 symmetrically distinct beams covering a total energy range of 3118 eV. In this analysis we have used the same approaches as described in last section for the theory versus theory comparison. An

5. Conclusions The NGA-LEED algorithm has proven itself to be a powerful tool for LEED structure determination: it was able to find the correct structural model for both our tested systems starting from completely random structures. Two methods, using predefined sites and a grid of coordinates, were able to find the correct structure in the theory versus theory comparison. The first one is much faster than the second, however it is limited to cases where the occupation of high symmetry sites seems plausible. The refinement with least squares methods is very important to allow the program to produce the final structures, mainly for the grid implementation. For the theory versus experiment comparison the code was able to find at least the correct positions for the Ag atoms in the Ag(1 1 1)(4 × 4)-O system by using grid of coordinates. Using pre-defined sites the correct positions for all Ag and O atoms were found. For the Au(1 1 0)-(1  ×  2) system, the code was successful also when tested for real experimental data. The results of this work suggest that the method may be a promising approach for the determination of the surface atomic structure of complex systems, but more investigations are still needed in order to have a better understanding of its performance. It is clear that direct methods and automatic search methods are necessary for more complex structures which cannot be solved by the conventional ‘trial and error’ methods used in LEED until now. These are mainly structures with large reconstructions or adsorbed molecules. Pre-defined sites are not really useful in these cases, and a grid with step sizes around 0.05 Åwould lead to a number of models exceeding the present calculation capabilities. However, the description of molecular structures can be simplified using constraints like rigid atom groups, bond angles and torsion angles [32]. The NGA-LEED algorithm can easily be extended to handle crossover of different groups of parameters which should speed up the calculation substantially. 7

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[14] Blanco-Rey M, Heinz K and de Andres P L 2008 J. Phys.: Condens. Matter 20 304201 [15] Yu Z X and Tong S Y 2005 Phys. Rev. B 71 161404 [16] Rous P J 1993 Surf. Sci. 296 358 [17] Nascimento V B, de Carvalho V E, de Castilho C M C, Costa B V and Soares E A 2001 Surf. Sci. 478 15 [18] Correia E R, Nascimento V B, de Castilho C M C, Esperidião A S C, Soares E A and de Carvalho V E 2005 J. Phys.: Condens. Matter 17 1–16 [19] Van Hove M A and Döll R 1996 Surf. Sci. 355 L393 [20] Viana M L, Diez Muiño R, Soares E A, Van Hove M A and de Carvalho V E 2007 J. Phys.: Condens. Matter 19 446002 [21] Viana M L, Simões e Silva W, Soares E A, de Carvalho V E, de Castilho C M C and Van Hove M A 2008 Surf. Sci. 602 3395 [22] Glass C W, Organov A R and Hansen N 2006 Comput. Phys. Commun. 175 713 [23] Deaven D M and Ho K M 1995 Phys. Rev. Lett. 75 288 [24] Moritz W and Wolf D 1985 Surf. Sci. 163 L655 [25] Rovida G, Pratesi F, Maglietta M and Ferroni E 1974 Surf. Sci. 43 230 [26] Schmid M et al 2006 Phys. Rev. Lett. 96 146102 [27] Schnadt J, Michaelides A, Knudsen J, Vang R T, Reuter K, Laegsgaard E E, Scheffler M and Besenbacher F 2006 Phys. Rev. Lett. 96 146101 [28] Reichelt R, Günther S, Wintterlin J, Moritz W, Aballe L and Mentes T O 2007 J. Chem. Phys. 127 134706 [29] dos Reis D D, Negreiros F R, de Carvalho V E and Soares E A 2010 Surf. Sci. 604 568 [30] Over H, Ketterl U, Moritz W and Ertl G 1992 Phys. Rev. B 46 15438 [31] Briggs D and Seah M P 1983 Practical Surface Analysis by Auger and x-ray Photoelectron Spectroscopy (Chichester: Wiley) p 553 [32] Harris K D M, Johnston R L and Kariuki B M 1998 Acta Cryst. A54 632

Acknowledgments The authors would to thank FAPEMIG, CNPQ and CAPES (Brazilian Research Agencies), and DAAD (German Research Agency) for financial support. MAVH was supported by the HKBU Strategic Development Fund. References [1] Van Hove M A, Weinberg W H and Chan C M 1986 Low Energy Electron Diffraction: Experiment, Theory and Structural Determination vol 6 Springer Series in Surface Sciences (Berlin: Springer) [2] Soares E A, de Castilho C M C and de Carvalho V E 2011 J. Phys.: Condens. Matter 23 303001 [3] Pussi K and Diehl R D 2012 Low-Energy Electron Diffraction Characterization of Materials 2nd edn, vol 3, ed E N Kaufmann (New York: Wiley) pp 1841–53 [4] Wu H, Xu S, Ma S, Lau W P, Xie M H and Tong S Y 2002 Phys. Rev. Lett. 89 216101 [5] Seubert A, Heinz K and Saldin D K 2003 Phys. Rev. B 67 125417 [6] Suzuki A, Hashimoto A, Nojima M, Owari M and Niheib Y 2008 Surf. Interface Anal. 40 1627 [7] Pendry J B 1980 J. Phys. C 13 937 [8] Powell P G and de Carvalho V E 1987 Surf. Sci. 187 175 [9] Rous P J, Pendry J B, Saldin D K, Heinz K, Müller K and Bickel N 1986 Phys. Rev. Lett. 57 2951 [10] Rous P J, Van Hove M A and Somorjai G A 1990 Surf. Sci. 226 15 [11] Pendry J B and Heinz K 1990 Surf. Sci. 230 137 [12] Kleinle G, Moritz W, Adams D L and Ertl G 1989 Surf. Sci. 219 L637 [13] Kleinle G, Moritz W and Ertl G 1990 Surf. Sci. 238 119

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Novel genetic algorithm search procedure for LEED surface structure determination.

Low Energy Electron Diffraction (LEED) is one of the most powerful experimental techniques for surface structure analysis but until now only a trial-a...
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