New AuN (N = 27â30) Lowest Energy Clusters ... - ACS Publications

Subscriber access provided by University of Newcastle, Australia

Article

New AuN (N = 27 - 30) Lowest Energy Clusters Obtained by Means of an Improved DFT–Genetic Algorithm Methodology Jorge Alberto Vargas, Fernando Buendia, and Marcela R. Beltran J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b12848 • Publication Date (Web): 20 Mar 2017 Downloaded from http://pubs.acs.org on March 22, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

New AuN (N=27–30) Lowest Energy Clusters Obtained by Means of an Improved DFT–Genetic Algorithm Methodology Jorge A. Vargas, Fernando Buendía, and Marcela R. Beltrán∗ Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Circ. ext. s/n Apdo. Postal 70-360, C.P. 04510, Cd. de México, México E-mail: [email protected] Phone: +52 (55)56224624. Fax: +52 (55)56161251

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract The aim of the current study is to explore the structural evolution of neutral and negatively charged, gas phase, gold clusters in the size range of 27–30 atoms. We conducted an improved Genetic Algorithm global search combined with the Density Functional Theory study, which we present and explain. New and lower in energy structures are reported, these new structures are found to exhibit core-shell structures with the number of core atoms increasing with cluster size. Important discrepancies were observed in the energy ranking and energy differences of the obtained geometries when two different functionals (TPSS and PBE) are used. In general terms Au− 27 and Au− 28 clusters form nanotube-like structures, whereas that geometry changes into cage− like structures for Au− 29 and Au30 . On the other hand, structures related to a fcc

fragment were found to own lower energies for the neutral systems for Au27 , Au28 and Au29 , while for Au30 a non-symmetric cage with one core atom was the lowest-energy isomer. The obtained structures reveal generic structural motifs and structural trends for gold clusters in this size range, which will be valuable for future studies.

Introduction Matter at the nanoscopic scale often present interesting physical and chemical properties which can be different than their bulk counterparts, these properties are the consequence of particular geometrical structures, lower atomic coordination, large surface to volume ratios and quantum confinement effects. These special properties can be exploited in a great number of applications. 1,2 Gold clusters are probably the most extensively studied systems, since the strong relativistic effects and significant s-d hybridization lead to atypical structures and properties. 3–5 A great number of studies have been devoted to a deeper understanding of the structures on size-selected gold clusters and their size evolution. This knowledge is critical for the precise determination of the their electronic and optical properties as well as the structure2


Page 2 of 29

Page 3 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


catalytic activity relationship. In brief, over the last decade many structural issues of small gold clusters have been resolved. The study of the transition from two-dimensional (2D) to three-dimensional (3D) structures for gold clusters has a long history 6–11 and has been carried out by different methodologies. Up to date, it is acknowledged that the 2D→3D transition for AuN happens at N = 12, 11 and 8 for the anionic, 12–14 neutral 15,16 and cationic 17 clusters respectively. The smallest clusters with the shell-like or hollow cage structures also depends on the charge. They have been reported for anions (N = 16, 17, 18), 12,18–21 neutrals (N = 17, 18) 22 and cations (N = 18, 19). 17 The structural determination of the Au20 cluster 23 is a milestone that triggered many other studies of medium-size gold clusters. It is a highly symmetric tetrahedral pyramid with a large gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Moreover, it has a single-peak infrared spectrum. 24 It is very stable for the anionic and neutral systems, whereas a mixture of two isomers is reported for the cation. 25 The next structural transition for anionic and neutral gold clusters happens at N = 24, a size for which it has been proposed hollow cage structures. 26,27 Some of them can be considered as nanotubes. 12,28 Other size that has attracted some attention is the Au28 , since the proposal from Garzón et al. to be a chiral and disordered cluster. 29,30 More recent theoretical studies state that Au27 and Au28 have tubular motifs for the neutral systems, 28,31 while for the anions, core-shell structures were found to fit better the exper32 imental data as well as for Au− As far as we know, there are not theoretical studies of 30 .

Au29 clusters. As mentioned above, gold clusters exhibit very unusual geometries making necessary to perform a good Global Minimum (GM) search of the potential energy surface to find the correct structures. About two decades ago, they were carried out using semi-empirical potentials. 11,33–37 Unfortunately, despite the fact that important progresses were done by those methods, the semi-empirical potentials are unable to reproduce the complexity of the electronic structure in bare gold clusters and can mislead the search of the most stable isomers

3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

even when the lowest-energy isomers were refined using ab initio methods. 29,38 However, we have to note that recently important efforts have been made to improve the parametrization within the density functional tight binding, for example. 39 For that reason, the GM search started to be carried out through codes directly coupled to DFT calculations, using methods like basin-hopping 18,22,40 or genetic algorithms. 41–45 The main ideas, performances and case studies of these have been reviewed by Heiles and Johnston. 46 In this work we analyze the lowest-energy isomers of anionic and neutral clusters consisting of 27–30 gold atoms. They were obtained through a GM search using a DFT Genetic Algorithm (GA). The starting point of our implementation was the Birmingham Pool Genetic Algorithm (BPGA) developed by Johnston et al. 47 sharing several characteristics with it. However, the code was completely redesigned to make it more efficient and flexible as well as to ease the output analysis. Therefore, we start with a detailed description of the employed genetic algorithm in the next section. Then, we give the calculation details, followed by the AuN (N = 27 − 30) results and their discussion. Finally, we summarize the findings in the conclusions.

Description of the Genetic Algorithm In the literature, there are several genetic algorithm implementations applied to the problem of geometry optimization of a wide range of chemical species. 42–45,48 The method itself has evolved for the sake of efficiency and accuracy. 49 In this section we give a detailed description of the GA employed in this work. Of course, some main ideas were taken from different sources which will be specified. However, other features are the result of our effort to improve it. Therefore, we decided to call it Mexican Enhanced Genetic Algorithm (MEGA). In the development of MEGA, many tests were carried out, in which the most stable structures reported in the literature for AuN clusters (N ≤ 20) were obtained, proving the adequacy of the methodology described in the next section.

4


Page 4 of 29

Page 5 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The MEGA code shares the main functionalities with that of the BPGA 47 in the sense that both (written in Python) are capable to carry out independent and parallel relaxations synchronized with a global database or pool, which contains the energy and the atomic positions of the current most stable clusters for a given composition and is the central object of these implementations. The pool size is set at the beginning and it is kept fixed throughout the whole process. The flow chart in Figure 1 shows the overall algorithm and can be used as a guide for the following explanation. In order to ease the understanding, we use color cycles. In the blue cycle, new structures are generated, relaxed and checked that all the atoms are bonded together. If so, an appropriate candidate goes to the red cycle where the selection rules are accomplished to determine if it enters the pool or not. There is not a fixed convergence criterion, instead the number of cycles (or generated structures) is given in the input file.

Figure 1: Flow chart of MEGA.

5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Starting Modes MEGA can start from scratch, i.e., without any previous structures. In that case, the pool is filled out with relaxed random structures, which are generated within a sphere of radius R = 2N 1/3 ra , with N being the number of atoms and ra , the atomic radius given in the input. This means a filling fraction of 1/8. One of the added features, is the possibility to feed in previous structures as an input to fill either partially or completely the pool after their relaxation. This includes the case when the previous structures have less atoms. In that case, the cluster is completed adding atoms randomly to the cluster surface. This option really speeds up the initial part of the search and the method is robust enough to get out from an eventual starting bias. MEGA, as the BPGA, can be restarted at any time, even if there is/are one or more processes still running. In fact, this is the way how the independent relaxations are carried out in parallel.

Fixing the Overlap When a new structure is created its important to check that there is not any pair of atoms too close to each other because otherwise the DFT calculation will suffer and waste time. Checking is easy but repairing it can be tricky because it may lead to infinite loops. MEGA uses a smart way to fix this “overlap”. First, the cluster is centered in its center of mass. Then, the atoms are sorted in increasing distances to the center. The first atom (closest to the center) is left unchanged and for the i-th atom, the distances with the (i − 1) previous atoms are checked. Whenever it is too close to other atom (dij < 2ra ), the i-th atom is displaced radially to fix the overlap. In this manner, the subroutine does exactly N (N − 1)/2 distance checks and the overlaps are repaired in just one cycle.

6


Page 6 of 29

Page 7 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Fitness and Selection In the evolution theory jargon, the fitness is a measure of the individual suitability. Here, we use a dynamic fitness scaling, 49 which means that after every pool update the fitnesses of all structures are scaled relative to the best and worst members of the current population. This is achieved by using a normalised value of the energy,

ρi =

Vi − Vmin , Vmax − Vmin

(1)

where Vmin and Vmax are the lowest and the highest cluster energies in the current pool, respectively. Thus, the fitness of the i-th cluster is defined through the following expression,

fi = 1/2[1 − tanh(2ρi − 1)].

(2)

All the fitnesses are then normalised to obtain the probabilities to choose the corresponding structures for mate or mutation. In this manner, a roulette wheel selection is carried out.

Mate or Crossover This is the milestone of the GA method. It is the way in which two parents pass on their genetic information to the offspring. We use a variant of the “cut and splice” crossover operator of Deaven and Ho. 34 In our implementation, the parent clusters are first randomly rotated and then they are cut and the complementary fragments are spliced together. Each parent cluster contribute to the offspring with a number of atoms determined by its relative fitness. Thus, the offspring structure resembles more that of the lowest-energy parent.

Mutations The mutation operators are important to increase population diversity by introducing new genetic material, which in our case means new substructures. Normally, a mutation involves 7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

only a restricted number of atoms or part of a cluster from the pool in order to keep some favourable geometries. Once the pool is complete, the blue cycle starts with a choice of operation. The probability to perform a mutation, Pmut , is set in the input file (commonly 20%). All other new structures are generated by crossover. The mutation operator can be always the same or randomly chosen among the following options: Move: 25% of the cluster atoms are displaced randomly by varying each position coordinate a value between −ra and +ra , with ra being the atomic radius. In contrast with the usual move operators, MEGA do not choose randomly all the atoms to be displaced, but only the first one and all the others are its nearest neighbours. In this way, the mutation is localized while most part of the cluster remains unchanged. Rotate: 25% of the cluster atoms are rigidly rotated about a randomly directed axis, which pass by the center of mass. Since in this case, the atoms are selected randomly, this is the most drastic mutation to search other potential-energy basins. Twist: Similar to the previous mutation, but in this case one half of the cluster rotates a random angle respect to the other half. This means that each half keeps its geometry with the probable exception of the interface between them. Atom Inversion: One of the farthest atoms from the center of mass is passed to the other side by changing the signs of its 3 coordinates. This is a novel proposal very useful for medium-sized clusters, as those studied here.

Selection Rules Once a cluster has emerged from the blue cycle there are several checks to decide if it goes into the pool or not. For that, the new cluster should have lower energy than at least one structure of the pool. Then, in order to identify if the new cluster is not already in the pool, a sorted list created with all the interatomic distances is compared with a corresponding list of each pool structure. If all elements of both lists differ in less than 5%, the clusters are considered similar and that with the highest energy is discarded. Otherwise, the new cluster 8


Page 8 of 29

Page 9 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


replace the member with the highest energy of the entire pool. It is of crucial importance to keep diversity in the pool for several reasons. First, it improves the exploration of the potential energy surface of the geometries hyperspace. Second, it reduces the possibility to get trapped in a local minimum basin. And last but not least, with a single run one obtains different structures to further be tested with other functionals or levels of theory, which not necessarily coincide in the energy ranking and/or the simulated spectra. The method described above works very well and it is fairly fast. For medium size clusters (more than 20 atoms) it may identify different geometries as similar. However, the procedure to keep that with lower energy alleviates the problem. Other GA’s in the literature use energy comparisons to keep the diversity of the population, 34,36 this can however be dangerous because two configurations can be very different and still be close in energy. For that reason, some structural comparisons have been proposed 43 like radial distribution of atoms, average and variance of atomic distances, the eigenvalues of inertia tensors or even the direct comparison of the structures by millions of rotations. The latter may be the most accurate but it also is the most time consuming. Vilhelmsen and Hammer 45 used a similar criterion as us with the sorted list of interatomic distances and they tested it with and without an additional energy check finding a better performance when only the geometric check is done. We also carried out this test and agreed with them that the energy comparison can be a hindrance to the diversity check because it sometimes led to false positives. Thus, we remained only with the geometric check.

The Au20 Test As it is mentioned in the introduction, there is a plenty number of studies of small gold clusters in the literature. Some of them encompass up to Au20 which has been a benchmark since it was discovered. 23 Thus, it represents a good test for the performance of MEGA directly with the plane wave DFT methodology employed in this work. Starting from scratch, 9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment

Page 10 of 29

Page 11 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Calculation Details More than 99.9% of computer time used by MEGA is spent in the relaxations. We decided to do this crucial step with a DFT code in order to have the correct trends from the beginning. The DFT calculations were performed using The Vienna Ab-initio Simulation Package VASP 50–53 by means of the Perdew-Burke-Erzenhof generalized gradient approximation revised for soilds (PBEsol), 54 as Johansson et al. suggest for gold clusters. 13 We used Projected Augmented Wave (PAW) pseudopotentials, 55 with 230 eV as energy cut-off. These pseudopotentials include scalar relativistic effects, since they were obtained through a fully relativistic calculation for the core electrons and treat valence electrons in a scalar relativistic approximation. 56 Gamma point calculations were performed within a cubic cell leaving at least 10 Å between periodically repeated clusters to avoid cluster-cluster interaction. We started the GA process with 12 random structures of Au28 clusters and generating iteratively more than 500 new geometries. The enhanced design of MEGA helped us to find general trends within relatively few steps. Nevertheless, at certain points, we modified manually some structures to ensure that the best geometries get tested. Taking some of these results as input, partially filled pools were prepared for the 4 sizes (27–30 gold atoms) and independent GA cycles were started for the anionic and neutral systems. We normally ran 2 or 3 parallel processes, each using 16 cores. All the minima here reported were obtained within less than 300 generated structures. The mutation rate (Pmut ) was 20% using all mutation types randomly chosen each time. The other 80% of the new candidates were produced by crossover. When we considered it necessary, some modified geometries were introduced by hand. Additionally, some reported structures were reproduced and compared. With this procedure, several symmetric structures were found to have lower energies than the non-symmetric ones. However, in certain cases the opposite was true. One parameter used to determine whether we have run enough cycles or not is the binding

11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 29

energy calculated from the following expression,

Eb =

EAuN − N EAu , N

(3)

where EAuN is the total energy of the AuN cluster and EAu is the energy of single, spinpolarized Au atom, and N is the total number of atoms. We have to mention that VASP gives already the numerator and we just had to divide it by N . It is known that the normal behaviour of Eb is to approach exponentially to the bulk value as the number of atoms in the clusters is increased. Thus, we ensured that Eb27 < Eb28 < Eb29 < Eb30 at least for the minima, with EbN being the binding energy of the AuN cluster. We stopped the GA processes when at least the five structures with the lowest energies did not change in more than 50 steps. Then, the charged structures were compared with the neutral ones and additional calculations with the neutral (or charged) clusters were carried out if needed. Subsequently, the lowest-energy structures were reoptimized with GAUSSIAN09, 57 using both the PBE 58 and TPSS 59 functionals and the DEF2TZVP basis set, which takes into account the scalar relativistic effects through the effective core potentials. 60 In this step, all the geometries were first relaxed in the anionic state and the result was reoptimized for the neutral state, in order to calculate the Vertical Detachment Energy (VDE), which is obtained as the energy difference between the relaxed anion cluster and the neutral one with the same geometry. In this case, the binding energies were obtained according to the equation (3), where EAu was calculated for the two different functionals but the same basis set (the DEF2TZVP). At this stage, the zero point energy correction was taken into account. It ranges from 0.40 eV for Au27 to 0.45 eV for Au30 , added to all structures with deviations about 0.01 eV, i.e., it did not change the reported energy ordering. On the other hand, it has been reported that the spin-orbit coupling changes the energy differences between isomers and it may even change the energy ordering. 13,61,62 However, it has little effect on the relaxed geometries and is much more computationally expensive.

12


Page 13 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Considering the good geometrical agreement obtained in the test calculations, we are not including the spin-orbit effects in this study. Nevertheless, results accounting for them are generally in better agreement with some experimental data like that obtained by PhotoElectron Spectroscopy (PES). 20,32

Results The obtained AuN (N = 27−30) clusters are shown in the Figures 2–5. They are at least the two lowest-energy isomers according to the PBE and TPSS functionals for both, the anionic and neutral systems. The first thing to observe is that there are several similar structures which we can group into three “families” to ease the discussion: Nanotubes, Cages and those derived from the trigonal bipyramid (TBP) structure of Fig. 6G, which has the geometry of the famous Au20 cluster, but mirror-replicated upon a tetrahedral face. However, at certain point, the distinction between families becomes tricky. Additionally, sometimes the clusters possess a single- (SAC) or double-atom core (DAC) highly coordinated, while in other cases they are hollow. Most of the structures shown in the figures have some symmetry, but also there are some non-symmetric among the lowest-energy structures. It is worth to mention that in all the cases there is a structural relaxation when the total charge of the cluster is changed, or even by using other functional, but normally it is too slight to see it by naked eyes. Anyway, the pictures shown in the figures are the anionic systems relaxed with the TPSS functional. The atomic coordinates, three views of the structures and simulated spectra are given in the Supporting Information. The results for the anionic and neutral systems are summarized in Tables 1 and 2 respectively, where the minima (with the different functionals) are highlighted. It is interesting to note that, in about half of the systems studied here, the minima obtained with PBE do not coincide with those obtained with TPSS. Moreover, for the four sizes the minima of the anions have different geometry than the minima of the neutral clusters. In all of them, it

13




Page 14 of 29

Page 15 of 29





Page 16 of 29

Page 17 of 29




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

different, as it can be observed in Fig. 6. The cluster in Fig. 6A (the TPSS minimum) is a non-symmetric DAC cage, whereas the structure in Fig. 6E (the PBE minimum) is a symmetric hollow cage. In turn, for the neutral system, the lowest-energy isomer according to the TPSS functional is the cluster in Fig. 6B, which is a non-symmetrical SAC cage. It also has low energy with PBE but it is surpassed by the cluster in Fig. 6D, a symmetric SAC cage. Here, it is observed a similar behaviour as for the cluster in Fig. 3A, since the structures in Figs. 6D and 6F exhibit a concavity when they are relaxed with the TPSS functional, while the PBE functional favour the convex structures (not shown in the Figure). Interestingly, in the DAC cage cluster shown in Fig. 6C, the core atoms are displaced from a more symmetric position through the relaxation. On the other hand, the two clusters at the bottom of Fig. 6 have opposite behaviour with the two functionals. The TBP cluster (Fig. 6G) has lower energies with TPSS for the charged and the neutral systems, while the non-symmetric hollow cage (Fig. 6H) is much more stable according to PBE.

Conclusions We have developed a Genetic Algorithm code departing from the BPGA, which has more functionalities than its predecessor. For example, a more efficient sampling of the potential energy surface requiring less calls of the DFT optimization to obtain a similar result. Partially, this efficiency comes from better population rules maintaining the necessary diversity in the pool to generate new different structures yet keeping characteristics which are energetically favourable. In this work, we report several lowest-energy isomers of gold, concentrating ourselves in a not enough explored region (27–30 atoms), where only the experimental-theoretical data of Shao et al. 32 is available. The new geometries reported here are based on the fact that the GA search is done on the potential energy surface at the DFT level. From our results, the first thing to remark is that for these intermediate-size clusters,

18


Page 18 of 29

Page 19 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1: Symmetry, total energy difference respect to the minimum, binding and vertical detachment energies, obtained for the anionic system using the TPSS (PBE) functional. N 27

28

29

30

Clus A B C D E F G A B C D E F G A B C D E F G A B C D E F G H

Sym C2v C2v Cs C1 C1 C2v C2v C2v Cs C3v Cs C1 C1 Cs C1 Cs C2v C1 Cs C1 C3v C1 C1 Cs Cs Cs Cs C3h C1

∆E (eV) 0.04 (0.01) 0.51 (0.00) 0.52 (0.02) 0.69 (0.18) 0.00 (0.35) 0.66 (0.36) 0.23 (0.47) 0.00 (0.00) 0.01 (0.02) 0.81 (0.31) 0.32 (0.35) 0.18 (0.41) 0.39 (0.38) 0.47 (0.12) 0.00 (0.00) 0.15 (0.16) 0.82 (0.21) 0.16 (0.47) 0.24 (0.53) 0.39 (0.22) 0.46 (0.27) 0.00 (0.65) 0.12 (0.54) 0.22 (0.77) 0.23 (0.53) 0.40 (0.00) 0.32 (0.69) 0.27 (0.94) 0.49 (0.09)

19

Eb (eV/atom) 2.485 (2.408) 2.487 (2.409) 2.468 (2.408) 2.461 (2.402) 2.487 (2.396) 2.462 (2.396) 2.478 (2.392) 2.488 (2.411) 2.487 (2.410) 2.458 (2.400) 2.476 (2.398) 2.481 (2.396) 2.474 (2.397) 2.471 (2.406) 2.504 (2.421) 2.498 (2.415) 2.476 (2.414) 2.498 (2.405) 2.496 (2.403) 2.490 (2.413) 2.488 (2.411) 2.502 (2.407) 2.498 (2.411) 2.495 (2.403) 2.492 (2.411) 2.489 (2.429) 2.491 (2.406) 2.493 (2.398) 2.486 (2.426)


VDE (eV) 3.89 (4.19) 4.05 (4.17) 3.95 (4.08) 3.82 (3.95) 3.89 (4.00) 3.29 (4.33) 3.89 (4.00) 3.63 (3.78) 3.67 (3.81) 3.47 (3.62) 3.47 (3.62) 3.24 (3.49) 3.11 (3.38) 3.49 (3.72) 3.82 (3.95) 3.56 (3.75) 3.86 (4.02) 3.87 (4.00) 3.77 (3.91) 3.93 (4.05) 3.79 (3.92) 3.74 (3.88) 3.28 (3.43) 3.35 (3.57) 3.30 (3.44) 4.13 (4.23) 3.53 (3.47) 3.19 (3.37) 3.96 (4.06)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 2: Symmetry, total energy difference respect to the minimum and binding energies, obtained for the neutral system using the TPSS (PBE) functional. N 27

28

29

30

Clus A B C D E F G A B C D E F G A B C D E F G A B C D E F G H

Sym C2v C2v Cs C1 Cs C2v C2v C2v Cs C3v Cs Cs C1 Cs C1 Cs C2v C1 Cs C1 C3v C1 C1 C1 Cs Cs Cs C3h C1

∆E (eV) 0.65 (0.11) 1.17 (0.06) 1.08 (0.00) 1.11 (0.02) 0.50 (0.23) 0.40 (0.44) 0.00 (0.36) 0.14 (0.14) 0.18 (0.20) 0.86 (0.13) 0.43 (0.33) 0.00 (0.21) 0.10 (0.00) 0.50 (0.15) 0.14 (0.08) 0.00 (0.00) 0.96 (0.30) 0.37 (0.59) 0.20 (0.56) 0.66 (0.39) 0.39 (0.31) 0.35 (0.46) 0.00 (0.01) 0.14 (0.31) 0.10 (0.00) 1.11 (0.25) 0.25 (0.19) 0.06 (0.32) 1.05 (0.19)

20

Eb (eV/atom) 2.339 (2.254) 2.320 (2.256) 2.323 (2.258) 2.322 (2.258) 2.345 (2.250) 2.349 (2.242) 2.363 (2.245) 2.365 (2.278) 2.364 (2.276) 2.339 (2.273) 2.355 (2.271) 2.370 (2.275) 2.367 (2.283) 2.353 (2.278) 2.374 (2.286) 2.379 (2.289) 2.346 (2.278) 2.366 (2.268) 2.372 (2.269) 2.357 (2.275) 2.366 (2.278) 2.379 (2.283) 2.391 (2.298) 2.387 (2.288) 2.386 (2.299) 2.354 (2.290) 2.383 (2.293) 2.389 (2.288) 2.356 (2.292)


Page 20 of 29

Page 21 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the choice of functional (PBE/TPSS) leads to quite different results, with some identificable tendencies. In about half of the cases studied here, the same structures were found to be the lowest-energy isomers with both functionals. However, important discrepancies were observed in the energy ranking and energy differences of the obtained geometries. Never− theless, a size evolution is observed. For the anions, we can say that the Au− 27 and Au28

clusters have the tendency to form nanotubes, whereas that geometry changes into cage-like − structures for Au− 29 and Au30 with one or two highly coordinated core atoms. On the other

hand, for neutrals, the observed trend is towards a faceted fcc fragment for Au27 and Au28 and a closely related structure for Au29 , while the transition to a SAC cage was obtained for Au30 .

Acknowledgments We are very thankful with Prof. Roy L. Johnston for the helpful discussions and also for sharing the BPGA code. The GA calculations were performed at the Center for Computational Chemistry and Materials Science Saar (C3 MSaar) in Germany. We acknowledge Prof. Michael Springborg for that facility. The Gaussian calculations and other refinements were carried out in Miztli at the UNAM’s supercomputer center and in some machines of the IIM (Instituto de Investigaciones en Materiales). We thank Alberto López Vivas for his technical support. We acknowledge support from PAPIIT IN100515, UNAM project. F.B.Z. acknowledges support from CONACYT (financial support No. 379750). J.V.T. acknowledges a postdoctoral grant from DGAPA UNAM.

Associated Content The atomic coordinates, three views of the structures, electronic binding energies and simulated vibrational spectra are given in the Supporting Information. The electronic binding energies are thought to be compared with experimental PES spectra. Therefore, we plot 21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

the negative of the highest occupied molecular orbitals broadened with lorentzians using a full width at half maximum (FWHM) of 0.02 eV. They were shifted such that the calculated VDE coincides with the first binding energy. Similarly, for the simulated vibrational spectra, the frequencies were broadened with lorentzians using a FWHM of 1 cm−1 .

References (1) Haruta, M. Size and Support Dependency in the Catalysis of Gold. Catal. Today 1997, 36, 153 – 166. (2) Kovalenko, M. V.; Manna, L.; Cabot, A.; Hens, Z.; Talapin, D. V.; Kagan, C. R.; Klimov, V. I.; Rogach, A. L.; Reiss, P.; Milliron, D. J. et al. Prospects of Nanoscience with Nanocrystals. ACS Nano 2015, 9, 1012–1057. (3) Pyykkö, P. Theoretical Chemistry of Gold. Angew. Chem. Int. Ed. 2004, 43, 4412–4456. (4) Pyykkö, P. Theoretical Chemistry of Gold II. Inorg. Chim. Acta 2005, 358, 4113 – 4130. (5) Häkkinen, H. Atomic and Electronic Structure of Gold Clusters: Understanding Flakes, Cages and Superatoms From Simple Concepts. Chem. Soc. Rev. 2008, 37, 1847–1859. (6) Häkkinen, H.; Landman, U. Gold Clusters AuN, (N = 2-10) and Their Anions. Phys. Rev. B 2000, 62, R2287–R2290. (7) Gilb, S.; Weis, P.; Furche, F.; Ahlrichs, R.; Kappes, M. M. Structures of Small Gold Cluster Cations (AuN+, N

New AuN (N = 27â30) Lowest Energy Clusters ... - ACS Publications

Recommend Documents

New AuN (N = 27â30) Lowest Energy Clusters ... - ACS Publications