Pool-BCGA: a parallelised generation-free genetic algorithm for the ab initio global optimisation of nanoalloy clusters

Search for Global Minimum Structures of P 2 n + 1 + (n = 1-15) Using xTB-Based Basin-Hopping Algorithm

A new program for searching global minimum structures of atomic clusters using basin-hopping algorithm based on the xTB method was developed here. The program can be performed with a much higher speed than its replacement directly based on DFT methods. Considering the structural varieties and complexities in finding their global minimum structures, phosphorus cluster cations were studied by the program. The global minimum structures of cationic P2n+1+ (n = 1–15) clusters are determined through the unbiased structure searching method. In the last step, further DFT optimization was performed for the selected isomers. For P2n+1+ (n = 1–4, 7), the found global minimum structures are in consistent with the ones previously reported; while for P2n+1+ (n = 5, 6, 8–12), newly found isomers are more energy-favorable than those previously reported. And those for P2n+1+ (n = 13–15) are reported here for the first time. Among them, the most stable isomers of P2n+1+ (n = 4–6, 9) are characterized by their C_3v, C_s, C_2v and C_s symmetry, in turn. But those of P2n+1+ (n = 7, 8, 10–12), no symmetry has been identified. The most stable isomers of P29+ and P31+ are characterized by single P-P bonds bridging units inside the clusters. Further analysis shows that the pnicogen bonds play an important role in the stabilization of these clusters. These results show that the new developed program is effective and robust in searching global minimum structures for atom clusters, and it also provides new insights into the role of pnicogen bonds in phosphorus clusters.

Application of Optimization Algorithms in Clusters

The structural characterization of clusters or nanoparticles is essential to rationalize their size and composition-dependent properties. As experiments alone could not provide complete picture of cluster structures, so independent theoretical investigations are needed to find out a detail description of the geometric arrangement and corresponding properties of the clusters. The potential energy surfaces (PES) are explored to find several minima with an ultimate goal of locating the global minima (GM) for the clusters. Optimization algorithms, such as genetic algorithm (GA), basin hopping method and its variants, self-consistent basin-to-deformed-basin mapping, heuristic algorithm combined with the surface and interior operators (HA-SIO), fast annealing evolutionary algorithm (FAEA), random tunneling algorithm (RTA), and dynamic lattice searching (DLS) have been developed to solve the geometrical isomers in pure elemental clusters. Various model or empirical potentials (EPs) as Lennard-Jones (LJ), Born-Mayer, Gupta, Sutton-Chen, and Murrell-Mottram potentials are used to describe the bonding in different type of clusters. Due to existence of a large number of homotops in nanoalloys, genetic algorithm, basin-hopping algorithm, modified adaptive immune optimization algorithm (AIOA), evolutionary algorithm (EA), kick method and Knowledge Led Master Code (KLMC) are also used. In this review the optimization algorithms, computational techniques and accuracy of results obtained by using these mechanisms for different types of clusters will be discussed.

A review on genetic algorithm: past, present, and future

In this paper, the analysis of recent advances in genetic algorithms is discussed. The genetic algorithms of great interest in research community are selected for analysis. This review will help the new and demanding researchers to provide the wider vision of genetic algorithms. The well-known algorithms and their implementation are presented with their pros and cons. The genetic operators and their usages are discussed with the aim of facilitating new researchers. The different research domains involved in genetic algorithms are covered. The future research directions in the area of genetic operators, fitness function and hybrid algorithms are discussed. This structured review will be helpful for research and graduate teaching.

A Genetic Algorithmic Approach to Determine the Structure of Li-Al Layered Double Hydroxides

Layered double hydroxides (LDH) demonstrate significant potential across a range of applications, including as catalysts, delivery vehicles for pharmaceuticals, environmental remediation, and supercapacitors. Explaining the mechanism of LDH action at the atomic scale in these and other applications is challenging, however, due to the difficulty in precisely defining the bulk and surface structure and chemical compositions. Here, we focus on the determination of the structure of lithium-aluminum (Li-Al) LDH, which has shown promise in the catalytic depolymerization of lignin, both directly as the catalyst and as a support for gold nanoparticles. While the relative positions of the Li and Al metals are generally well resolved by X-ray crystallography, it is the structures of the anionic layers, consisting of water and carbonate, that are less well established. Combinatorial analyses of all possible positions and rotations of the water and carbonate in the three-layered Li-AL LDH polytope reveals that the phase space is much too large to examine in any reasonable time frame in a one-by-one structure exploration. To overcome this limitation, we develop and deploy a genetic algorithm (GA) wherein fitness is determined by matching a calculated X-ray diffraction (XRD) pattern for a given structure to the known experimental XRD pattern. The GA approach results in structures of high fitness that portend the bulk Li-Al LDH structure. Importantly, the GA approach offers the potential to determine the structures of other LDH, and more generally layered materials, which are generally difficult to describe given the large chemical and structural space to be explored.

NWPEsSe: An Adaptive-Learning Global Optimization Algorithm for Nanosized Cluster Systems

Global optimization constitutes an important and fundamental problem in theoretical studies in many chemical fields, such as catalysis, materials, or separations problems. In this paper, a novel algorithm has been developed for the global optimization of large systems including neat and ligated clusters in the gas phase and supported clusters in periodic boundary conditions. The method is based on an updated artificial bee colony (ABC) algorithm method, that allows for adaptive-learning during the search process. The new algorithm is tested against four classes of systems of diverse chemical nature: gas phase Au₅₅, ligated Au₈²⁺, Au₈ supported on graphene oxide and defected rutile, and a large cluster assembly [Co₆Te₈(PEt₃)₆][C₆₀]_n, with sizes ranging between 1 and 3 nm and containing up to 1300 atoms. Reliable global minima (GMs) are obtained for all cases, either confirming published data or reporting new lower energy structures. The algorithm and interface to other codes in the form of an independent program, Northwest Potential Energy Search Engine (NWPEsSe), is freely available, and it provides a powerful and efficient approach for global optimization of nanosized cluster systems.

Simple Automatized Tool for Exchange-Correlation Functional Fitting

Density functional theory is, perhaps, the most popular and convenient tool in computational chemistry. DFT methods allow solving different chemical tasks with a good balance of accuracy and computational time. Dozens of existing functionals cover a majority of possible systems, and the development of new ones is still ongoing. However, despite the existence of different databases with accurate quantum-chemical data, the functional design remains a complicated and time-demanding task. Here, we propose a novel approach for simplifying and accelerating this process. The approach is based on a Bayesian search with stochastic sub-sampling that allows considering the 'history' of fitting steps, reduces the computational time for each step, and avoids overfitting to training data. Besides the general testing of the approach efficiency, we also showed an example of training specialized DFT functionals, outperforming the popular ones. The approach is presented as a free code with built-in analysis tools. Using the code with an appropriate reference database can help in constructing a DFT approximation for a highly specialized task.

O2 activation by AuAg clusters on a defective (100)MgO surface

In the present work, we discuss the electronic properties of supported dispersed bimetallic clusters with respect to their size, geometry, and Au_n/Ag_m (n + m = 6) composition. We have studied with supercell-density functional theory calculations the role of the charge transfer from the MgO defective support toward the cluster in the activation of O₂ by Au_nAg_m clusters. We first considered gas-phase clusters with different atomic compositions; then, we deposited all of them on a pristine (100)MgO surface and finally on a more realistic (100)MgO F-center. We performed a global and unrestricted search of the (cluster + surface) geometry. The Mexican enhanced genetic algorithm has been used to exhaustively explore the potential energy surface. Our results show that O₂ activation depends on the Au_n/Ag_m ratio. It has been found that both metals involved play different and important roles toward (a) the actual O₂ dissociation and (b) weakening of the oxygen-cluster bond, which, in turn, may promote the possibility of a catalytic process to take place, such as the oxidation process of CO and NO_x among others.

A Global Optimizer for Nanoclusters

We have developed an algorithm to automatically build the global minimum and other low-energy minima of nanoclusters. This method is implemented in PyAR (https://github.com/anooplab/pyar) program. The global optimization in PyAR involves two parts, generation of several trial geometries and gradient-based local optimization of the trial geometries. While generating the trial geometries, a Tabu list is used for storing the information of the already used trial geometries to avoid using the similar trial geometries. In this recursive algorithm, an n-sized cluster is built from the geometries of n−1 clusters. The overall procedure automatically generates many unique minimum energy geometries of clusters with size from 2 up to n using this evolutionary growth strategy. We have used our strategy on some of the well-studied clusters such as Pd, Pt, Au, and Al homometallic clusters, Ru-Pt and Au-Pt binary clusters, and Ag-Au-Pt ternary cluster. We have analyzed some of the popular parameters to characterize the clusters, such as relative energy, singlet-triplet energy difference, binding energy, second-order energy difference, and mixing energy, and compared with the reported properties.

Gold doping of tin clusters: exo- vs. endohedral complexes

We present molecular beam electric deflection experiments on neutral gold-doped tin clusters. The experimental Sn_NAu (N = 6-16) cluster beam profiles are interpreted by means of classical trajectory simulations supplied, with cluster structures generated by a genetic algorithm based on density functional theory. The combined experimental and theoretical analysis confirms that at least nine tin atoms are necessary to form a cage that is capable of encapsulating a gold atom, with high symmetry only marginally distorted by the gold atom. Two-component DFT calculations reveal that for some clusters spin-orbit effects are necessary to properly describe these species. Partial charge analysis methods predict the presence of charge transfer effects from the tin host to the dopant, resulting in a negatively charged gold atom.

Physico-Chemical Insights into Gas-Phase and Oxide-Supported Sub-Nanometre AuCu Clusters

Catalysis by AuCu nanoclusters is a promising scientific field. However, our fundamental understanding of the underlying mechanisms of mixing in AuCu clusters at the sub-nanometre scale and their physico-chemical properties in both the gas-phase and on oxide supports is limited. We have identified the global minima of gas-phase and MgO(100)-supported AuCu clusters with 3–10 atoms using the Mexican Enhanced Genetic Algorithm coupled with density functional theory. Au and Cu adatoms and supported dimers have been also simulated at the same level of theory. The most stable composition, as calculated from mixing and binding energies, is obtained when the Cu proportion is close to 50%. The structures of the most stable free AuCu clusters exhibit Cu-core/Au-shell segregation. On the MgO surface however, there is a preference for Cu atoms to lie at the cluster-substrate interface. Due to the interplay between the number of interfacial Cu atoms and surface-induced cluster rearrangement, on the MgO surface 3D structures become more stable than 2D structures. The O-site of MgO surface is found to be the most favourable adsorption site for both metals. All dimers favour vertical (V) configurations on the surface and their adsorption energies are in the order: AuCu < CuCu < AuAu < AuCu (where the underlined atom is bound to the O-site). For both adatoms and AuCu dimers, adsorption via Cu is more favourable than Au-adsorbed configurations, but, this disagrees with the ordering for the pure dimers due to a combination of electron transfer and the metal-on-top effect. Binding energy (and second difference) and HOMO-LUMO gap calculations show that even-atom (even-electron) clusters are more stable than the neighbouring odd-atom (odd- electron) clusters, which is expected for closed- and open-shell systems. Supporting AuCu clusters on the MgO(100) surface decreases the charge transfer between Au and Cu atoms calculated in free clusters. The results of this study may serve as a foundation for designing better AuCu catalysts.

GIGA: a versatile genetic algorithm for free and supported clusters and nanoparticles in the presence of ligands

We present a versatile parallelised genetic algorithm, which is able to perform global optimisation from first principles for pure and mixed free clusters in the gas phase, supported on surfaces or in the presence of one or several atomic or molecular species (ligands or adsorbates). The genetic algorithm is coupled to different quantum chemical software packages in order to permit a large variety of methods for the global optimisation. The genetic algorithm is also capable of optimising different electronic spin multiplicities explicitly, which allows global optimisation on several potential energy hypersurfaces in parallel. We employ the genetic algorithm to study ligand-passivated clusters [Cd3Se3(H2S)3]+ and to investigate adsorption of [Pt6(H2O)2]+ supported on graphene. The explicit consideration of the electronic spin multiplicity during global optimisation is investigated for nanoalloy clusters Pt4V2.

Multitribe evolutionary search for stable Cu-Pd-Ag nanoparticles using neural network models

We present an approach based on two bio-inspired algorithms to accelerate the identification of nanoparticle ground states. We show that a symbiotic co-evolution of nanoclusters across a range of sizes improves the search efficiency considerably, while a neural network constructed with a recently introduced stratified training scheme delivers an accurate description of interactions in multielement systems. The method's performance has been examined in extensive searches for stable elemental (30-80 atoms), binary (50, 55, and 80 atoms), and ternary (50, 55, and 80 atoms) Cu-Pd-Ag clusters. The best candidate structures identified with the neural network model have consistently lower energy at the density functional theory level compared with those found with traditional interatomic potentials.

The influence of support materials on the structural and electronic properties of gold nanoparticles – a DFT study

This study investigates the effect of commonly used support materials (MgO, C, CeO₂) on small gold particles using dispersion corrected density functional theory (DFT-D).

Doping effects on the geometric and electronic structure of tin clusters

In depth analysis of doping effects on the geometric and electronic structure of tin clusters via electric beam deflection, numerical trajectory simulations and density functional theory.

Are zinc clusters really amorphous? A detailed protocol for locating global minimum structures of clusters

We report the results of a conjoint experimental/theoretical effort to assess the structures of free-standing zinc clusters with up to 73 atoms. Experiment provides photoemission spectra for ZnN- cluster anions, to be used as fingerprints in structural assessment, as well as mass spectra for both anion and cation clusters. Theory provides both a detailed description of a novel protocol to locate global minimum structures of clusters in an efficient and reliable way, and its specific application to neutral and charged zinc clusters. Our methodology is based on the well-known hybrid EP-DFT (empirical potential-density functional theory) approach, in which the approximate potential energy surface generated by an empirical Gupta potential is first sampled with unbiased basin hopping simulations, and then a selection of the isomers so identified is re-optimized at a first-principles DFT level. The novelty introduced in our paper is a simple but efficient new recipe to obtain the best possible EP parameters for a given cluster system, with which the first step of the EP-DFT method is to be performed. Our method is able to reproduce experimental measurements at an excellent level for most cluster sizes, implying its ability to locate the true global minimum structures; meanwhile, if exactly the same method is applied based on the existing Gupta potential (fitted to bulk properties), it leads to wrong predicted structures with energies between 1 and 2 eV above the correct ones. Opposite to what was claimed in the past, our work unequivocally demonstrates that Zn clusters are not amorphous, and they rather adopt high symmetry structures for most sizes. We show that Zn clusters have a number of exotic, unprecedented structural and electronic properties which are not expected for clusters of a metallic element, and describe them in detail.

Phoenics: A Bayesian Optimizer for Chemistry

We report Phoenics, a probabilistic global optimization algorithm identifying the set of conditions of an experimental or computational procedure which satisfies desired targets. Phoenics combines ideas from Bayesian optimization with concepts from Bayesian kernel density estimation. As such, Phoenics allows to tackle typical optimization problems in chemistry for which objective evaluations are limited, due to either budgeted resources or time-consuming evaluations of the conditions, including experimentation or enduring computations. Phoenics proposes new conditions based on all previous observations, avoiding, thus, redundant evaluations to locate the optimal conditions. It enables an efficient parallel search based on intuitive sampling strategies implicitly biasing toward exploration or exploitation of the search space. Our benchmarks indicate that Phoenics is less sensitive to the response surface than already established optimization algorithms. We showcase the applicability of Phoenics on the Oregonator, a complex case-study describing a nonlinear chemical reaction network. Despite the large search space, Phoenics quickly identifies the conditions which yield the desired target dynamic behavior. Overall, we recommend Phoenics for rapid optimization of unknown expensive-to-evaluate objective functions, such as experimentation or long-lasting computations.

Stabilization of golden cages by encapsulation of a single transition metal atom

Golden cage-doped nanoclusters have attracted great attention in the past decade due to their remarkable electronic, optical and catalytic properties. However, the structures of large golden cage doped with Mo and Tc are still not well known because of the challenges in global structural searches. Here, we report anionic and neutral golden cage doped with a transition metal atom MAu16 (M = Mo and Tc) using Saunders 'Kick' stochastic automation search method associated with density-functional theory (DFT) calculation (SK-DFT). The geometric structures and electronic properties of the doped clusters, MAu16q (M = Mo and Tc; q = 0 and -1), are investigated by means of DFT theoretical calculations. Our calculations confirm that the 4d transition metals Mo and Tc can be stably encapsulated in the Au16- cage, forming three different configurations, i.e. endohedral cages, planar structures and exohedral derivatives. The ground-state structures of endohedral cages C2v Mo@Au16--(a) and C1 Tc@Au16--(b) exhibit a marked stability, as judged by their high binding energy per atom (greater than 2.46 eV), doping energy (0.29 eV) as well as a large HOMO-LUMO gap (greater than 0.40 eV). The predicted photoelectron spectra should aid in future experimental characterization of MAu16- (M = Mo and Tc).

A global optimization perspective on molecular clusters

Although there is a long history behind the idea of chemical structure, this is a key concept that continues to challenge chemists. Chemical structure is fundamental to understanding most of the properties of matter and its knowledge for complex systems requires the use of state-of-the-art techniques, either experimental or theoretical. From the theoretical view point, one needs to establish the interaction potential among the atoms or molecules of the system, which contains all the information regarding the energy landscape, and employ optimization algorithms to discover the relevant stationary points. In particular, global optimization methods are of major importance to search for the low-energy structures of molecular aggregates. We review the application of global optimization techniques to several molecular clusters; some new results are also reported. Emphasis is given to evolutionary algorithms and their application in the study of the microsolvation of alkali-metal and Ca²⁺ ions with various types of solvents.This article is part of the themed issue 'Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces'.

Structural evolution and metallicity of lead clusters

The evolution of the metallic state in lead clusters and its structural implications are subject to ongoing discussions. Here we present molecular beam electric deflection studies of neutral PbN (N = 19-25, 31, 36, 54) clusters. Many of them exhibit dipole moments or anomalies of the polarizability indicating a non-metallic state. In order to resolve their structures, the configurational space is searched using the Pool Birmingham Cluster Genetic algorithm based on density functional theory. Spin-orbit effects on the geometries and dipole moments are taken into account by further relaxing them with two-component density functional theory. Geometries and dielectric properties from quantum chemical calculations are then used to simulate beam deflection profiles. Structures are assigned by the comparison of measured and simulated beam profiles. Energy gaps are calculated using time-dependent density functional theory. They are compared to Kubo gaps, which are an indicator of the metallicity in finite particles. Both, experimental and theoretical data suggest that lead clusters are not metallic up to at least 36 atoms.

A Threshold-Minimization Scheme for Exploring the Energy Landscape of Biomolecules: Application to a Cyclic Peptide and a Disaccharide

We present a scheme, called the threshold-minimization method, for globally exploring the energy landscapes of small systems of biomolecular interest where typical exploration moves always require a certain degree of subsequent structural relaxation in order to be efficient, e.g., systems containing small or large circular carbon chains such as cyclic peptides or carbohydrates. We show that using this threshold-minimization method we can not only reproduce the global minimum and relevant local minima but also overcome energetic barriers associated with different types of isomerism for the example of a cyclic peptide, cyclo-(Gly)4. We then apply the new method to the disaccharide α-d-glucopyranose-1-2-β-d-fructofuranose, report energetically preferred configurations and barriers to boat-chair isomerization in the glucopyranosyl ring, and discuss the energy landscape.

Size evolution and ligand effects on the structures and stability of (AuL)n (L = Cl, SH, SCH3, PH2, P(CH3)2, n = 1–13) clusters

Size evolution on the global minimum structures of (AuCl)_n clusters at n = 1–13.

DFT global optimisation of gas-phase and MgO-supported sub-nanometre AuPd clusters

The effect of a MgO support on the structures and energies of Au–Pd clusters is calculated at the DFT level.

First principles study on the structural evolution and properties of (MCl)n (n = 1–12, M = Cu, Ag) clusters

Energetic gaps (E − E_fit) and second differences of binding energies (Δ₂E) for (CuCl)_n and (AgCl)_n clusters as a function of cluster size, n.

A comparative study of AumRhn (4 ≤ m + n ≤ 6) clusters in the gas phase versus those deposited on (100) MgO

The combined use of a genetic algorithm and Density Functional Theory (DFT) calculations allows us to explore the potential energy surface. Our results show interesting effects on the geometries of the clusters on deposition. Two-dimensional clusters in the gas phase become three-dimensional and vice versa.

Global optimization of small bimetallic Pd–Co binary nanoalloy clusters: a genetic algorithm approach at the DFT level

The global optimisation of small bimetallic PdCo binary nanoalloys are systematically investigated using the Birmingham Cluster Genetic Algorithm (BCGA).

The Birmingham parallel genetic algorithm and its application to the direct DFT global optimisation of Ir(N) (N = 10-20) clusters

A new open-source parallel genetic algorithm, the Birmingham parallel genetic algorithm, is introduced for the direct density functional theory global optimisation of metallic nanoparticles. The program utilises a pool genetic algorithm methodology for the efficient use of massively parallel computational resources. The scaling capability of the Birmingham parallel genetic algorithm is demonstrated through its application to the global optimisation of iridium clusters with 10 to 20 atoms, a catalytically important system with interesting size-specific effects. This is the first study of its type on Iridium clusters of this size and the parallel algorithm is shown to be capable of scaling beyond previous size restrictions and accurately characterising the structures of these larger system sizes. By globally optimising the system directly at the density functional level of theory, the code captures the cubic structures commonly found in sub-nanometre sized Ir clusters.

A DFT-based genetic algorithm search for AuCu nanoalloy electrocatalysts for CO₂ reduction

Using a DFT-based genetic algorithm (GA) approach, we have determined the most stable structure and stoichiometry of a 309-atom icosahedral AuCu nanoalloy, for potential use as an electrocatalyst for CO2 reduction. The identified core-shell nano-particle consists of a copper core interspersed with gold atoms having only copper neighbors and a gold surface with a few copper atoms in the terraces. We also present an adsorbate-dependent correction scheme, which enables an accurate determination of adsorption energies using a computationally fast, localized LCAO-basis set. These show that it is possible to use the LCAO mode to obtain a realistic estimate of the molecular chemisorption energy for systems where the computation in normal grid mode is not computationally feasible. These corrections are employed when calculating adsorption energies on the Cu, Au and most stable mixed particles. This shows that the mixed Cu135@Au174 core-shell nanoalloy has a similar adsorption energy, for the most favorable site, as a pure gold nano-particle. Cu, however, has the effect of stabilizing the icosahedral structure because Au particles are easily distorted when adding adsorbates.

Structural evolution of (Au2S)n (n = 1–8) clusters from first principles global optimization

First principles global optimization reveals the structural evolution and novel geometries of (Au₂S)_n nanoclusters at n = 1–8.

Perspectives on the energy landscape of Au–Cl binary systems from the structural phase diagram of AuxCly (x + y = 20)

Structural phase diagram (SPD) of Au_xCl_y (x + y = 20) clusters.