Global minimum structure searches via particle swarm optimization

On assessing the carbon capture performance of graphynes with particle swarm optimization

Tackling climate change is one of the greatest challenges of current times and therefore the development of efficient technologies to limit anthropogenic emissions is of utmost urgency. Recent research towards this goal has alluded to the use of carbon-based solid sorbents for carbon capture. Graphynes (GYs), an interesting class of porous carbon membranes, have recently proven their potential as excellent membranes for gas adsorption and separation. Herein, we explored the CO2 and N2 adsorption characteristics and CO2/N2 selectivities of a class of GYs, namely γ-GY-1, γ-GY-2 and γ-GY-4. We investigated the putative global minimum geometries of adsorbed unary (n = 2-10) and binary (n : m; n, m ∈ [1, 8]) clusters of CO2 and N2 by employing a stochastic global optimization method called particle swarm optimization in conjunction with empirical intermolecular force field formulations. The intervening interactions are modeled using various pairwise potentials, including Lennard-Jones potential, improved Lennard-Jones potential, Buckingham potential and Coulombic potential. The binding energies for both unary and binary clusters are highest for adsorption on γ-GY-1, followed by γ-GY-2. The putative global minimum geometries suggested that N2 molecules preferred binding over the pore centres while CO2 molecules showed higher clustering propensity than any binding site preference. The predicted interaction energies suggested higher selectivity for CO2 over N2 for all the three γ-GYs.

Active-learning for global optimization of Ni-Ceria nanoparticles: The case of Ce4-xNixO8- x (x = 1, 2, 3)

Ni-CeO2 nanoparticles (NPs) are promising nanocatalysts for water splitting and water gas shift reactions due to the ability of ceria to temporarily donate oxygen to the catalytic reaction and accept oxygen after the reaction is completed. Therefore, elucidating how different properties of the Ni-Ceria NPs relate to the activity and selectivity of the catalytic reaction, is of crucial importance for the development of novel catalysts. In this work the active learning (AL) method based on machine learning regression and its uncertainty is used for the global optimization of Ce(4-x)NixO(8-x) (x = 1, 2, 3) nanoparticles, employing density functional theory calculations. Additionally, further investigation of the NPs by mass-scaled parallel-tempering Born-Oppenheimer molecular dynamics resulted in the same putative global minimum structures found by AL, demonstrating the robustness of our AL search to learn from small datasets and assist in the global optimization of complex electronic structure systems.

Performance of Particle Swarm Optimization in Predicting the Orientation of π-Conjugated Molecules Inside Carbon Nanotubes Compared with Density Functional Theory Calculations

Particle swarm optimization (PSO) was employed to obtain the global minimum of host-guest structures consisting of a triiodobenzene molecule (BzI3) inside an armchair (m,m) nanotube (BzI3@(m,m)), whose host-guest interactions are approximated by Lennard-Jones (LJ) potentials. The host-guest structures obtained using the PSO-LJ method were then compared with those obtained through dispersion-corrected density functional theory (DFT) calculations to evaluate the performance of the PSO-LJ approach in predicting the guest orientation inside a tube. When the inner space of the host tube is limited for guest encapsulation, the PSO-LJ method can reproduce the DFT results of BzI3@(m,m) in terms of the guest orientation. Conversely, in nanotubes with a sufficiently large space to allow a guest to freely move, corresponding to weak tube confinement, the PSO-LJ method yields guest orientations that are different from those obtained through DFT calculations; however, both methods obtain energetically close guest orientations. Accordingly, PSO-LJ method-assisted DFT calculations can quickly provide energetically stable guest orientations in BzI3@(m,m) in weak tube confinement, which can be ignored in DFT calculations, where a single initial geometry is typically used.

Bonobo Optimizer: A New Tool Toward the Global Optimization of Small Atomic Clusters

In the realm of structural and bonding investigations within chemical systems, elucidating global minimum energy configurations stands as a paramount goal. As the systems increase in size and complexity, this pursuit becomes progressively challenging. Herein, we introduce Bonobo optimizer (BO), a metaheuristic algorithm inspired by the social and reproductive behaviors of bonobos, to the domain of chemical problem solving. Focusing on small carbon clusters, this study systematically evaluates BO's performance, showcasing its robustness and efficiency. Parametric studies highlight the algorithm's adaptability, consistently converging to global minimum structures. Rigorous statistical validation supports the results, and a comparative analysis against established global optimization algorithms underlines BO's superior efficiency. This exploration extends the applicability of BO to the optimization of atomic clusters, providing a promising avenue for future advancements in computational chemistry.

Novel Computational Chemistry Infrastructure for Simulating Astatide in Water: From Basis Sets to Force Fields Using Particle Swarm Optimization

Using the example of astatine, the heaviest naturally occurring halogen whose isotope At-211 has promising medical applications, we propose a new infrastructure for large-scale computational models of heavy elements with strong relativistic effects. In particular, we focus on developing an accurate force field for At- in water based on reliable relativistic density functional theory (DFT) calculations. To ensure the reliability of such calculations, we design novel basis sets for relativistic DFT, via the particle swarm optimization algorithm to optimize the coefficients of the new basis sets and the polarization-consistent basis set idea's extension to heavy elements to eliminate the basis set error from DFT calculations. The resulting basis sets enable the well-grounded evaluation of relativistic DFT against "gold-standard" CCSD(T) results. Accounting for strong relativistic effects, including spin-orbit interaction, via our redesigned infrastructure, we elucidate a noticeable dissimilarity between At- and I- in halide-water force field parameters, radial distribution functions, diffusion coefficients, and hydration energies. This work establishes the framework for the systematic development of polarization-consistent basis sets for relativistic DFT and accurate force fields for molecular dynamics simulations to be used in large-scale models of complex molecular systems with elements from the bottom of the periodic table, including actinides and even superheavy elements.

Stability and Equilibrium Structures of Unknown Ternary Metal Oxides Explored by Machine-Learned Potentials

Ternary metal oxides are crucial components in a wide range of applications and have been extensively cataloged in experimental materials databases. However, there still exist cation combinations with unknown stability and structures of their compounds in oxide forms. In this study, we employ extensive crystal structure prediction methods, accelerated by machine-learned potentials, to investigate these untapped chemical spaces. We examine 181 ternary metal oxide systems, encompassing most cations except for partially filled 3d or f shells, and determine their lowest-energy crystal structures with representative stoichiometry derived from prevalent oxidation states or recommender systems. Consequently, we discover 45 ternary oxide systems containing stable compounds against decomposition into binary or elemental phases, the majority of which incorporate noble metals. Comparisons with other theoretical databases highlight the strengths and limitations of informatics-based material searches. With a relatively modest computational resource requirement, we contend that heuristic-based structure searches, as demonstrated in this study, offer a promising approach for future materials discovery endeavors.

Structure search method for atomic clusters based on the dividing rectangles algorithm

The Dividing Rectangles (DIRECT) algorithm is a deterministic optimization method to explore optimal solutions by repeatedly dividing a given hyperrectangle search space into subhyperrectangles. Herein, we propose a structure search method for atomic clusters based on the DIRECT algorithm in combination with a gradient-based local optimizer to enable an efficient structure search in high-dimensional search spaces. We use the Z-matrix representation for defining the hyperrectangle search space, in which the bond lengths, bond angles, and dihedral angles specify a cluster structure. To evaluate its performance, we applied the proposed method to the Lennard-Jones clusters and two kinds of real atomic clusters with many metastable structures, i.e., phosphorus and sulfur clusters, and compared the results with those of conventional methods. The proposed method exhibits a higher efficiency than random search and a comparable efficiency to basin hopping.

Exploring Metal Nanocluster Catalysts for Ammonia Synthesis Using Informatics Methods: A Concerted Effort of Bayesian Optimization, Swarm Intelligence, and First-Principles Computation

This paper details the use of computational and informatics methods to design metal nanocluster catalysts for efficient ammonia synthesis. Three main problems are tackled: defining a measure of catalytic activity, choosing the best candidate from a large number of possibilities, and identifying the thermodynamically stable cluster catalyst structure. First-principles calculations, Bayesian optimization, and particle swarm optimization are used to obtain a Ti8 nanocluster as a catalyst candidate. The N2 adsorption structure on Ti8 indicates substantial activation of the N2 molecule, while the NH3 adsorption structure suggests that NH3 is likely to undergo easy desorption. The study also reveals several cluster catalyst candidates that break the general trade-off that surfaces that strongly adsorb reactants also strongly adsorb products.

The global minimum of Ag30: a prolate spheroidal structure predicted using a genetic algorithm with incomplete local optimizations at the DFT level

Genetic algorithms have been widely used to explore global minimum points of atomic clusters, and their incorporation with ab initio calculations (including density functional theory methods) as local optimization approaches increases their ability to accurately locate the global minimum points on complicated potential energy surfaces. However, the local optimizations using ab initio calculations significantly increase the computational cost relative to those based on empirical or semi-empirical calculations. Herein, we develop a genetic algorithm program with an incomplete local optimization strategy at the DFT level. Using several representative clusters as test examples, this program showed high efficiency in locating their global minimum points. The low-lying isomers of Ag₃₀ were explored using this program, and the determined global minimum is a prolate spheroidal structure. The elongated spheroidal shape causes degeneracy lifting of the free electron shells, and endows Ag₃₀ with a large HOMO-LUMO gap. The sharp increase of silver clusters' reactivity around the sizes with 30 valence electrons observed in our previous experiments could be correlated with this theoretical figure.

GAMaterial-A genetic-algorithm software for material design and discovery

Genetic algorithms (GAs) are stochastic global search methods inspired by biological evolution. They have been used extensively in chemistry and materials science coupled with theoretical methods, ranging from force-fields to high-throughput first-principles methods. The methodology allows an accurate and automated structural determination for molecules, atomic clusters, nanoparticles, and solid surfaces, fundamental to understanding chemical processes in catalysis and environmental sciences, for instance. In this work, we propose a new genetic algorithm software, GAMaterial, implemented in Python3.x, that performs global searches to elucidate the structures of atomic clusters, doped clusters or materials and atomic clusters on surfaces. For all these applications, it is possible to accelerate the GA search by using machine learning (ML), the ML@GA method, to build subsequent populations. Results for ML@GA applied for the dopant distributions in atomic clusters are presented. The GAMaterial software was applied for the automatic structural search for the Ti₆ O₁₂ cluster, doping Al in Si₁₁ (4Al@Si₁₁ ) and Na₁₀ supported on graphene (Na₁₀ @graphene), where DFTB calculations were used to sample the complex search surfaces with reasonably low computational cost. Finally, the global search by GA of the Mo₈ C₄ cluster was considered, where DFT calculations were made with the deMon2k code, which is interfaced with GAMaterial.

Enhancing the Performance of Global Optimization of Platinum Cluster Structures by Transfer Learning in a Deep Neural Network

The global optimization of metal cluster structures is an important research field. The traditional deep neural network (T-DNN) global optimization method is a good way to find out the global minimum (GM) of metal cluster structures, but a large number of samples are required. We developed a new global optimization method which is the combination of the DNN and transfer learning (DNN-TL). The DNN-TL method transfers the DNN parameters of the small-sized cluster to the DNN of the large-sized cluster to greatly reduce the number of samples. For the global optimization of Pt₉ and Pt₁₃ clusters in this research, the T-DNN method requires about 3-10 times more samples than the DNN-TL method, and the DNN-TL method saves about 70-80% of time. We also found that the average amplitude of parameter changes in the T-DNN training is about 2 times larger than that in the DNN-TL training, which rationalizes the effectiveness of transfer learning. The average fitting errors of the DNN trained by the DNN-TL method can be even smaller than those by the T-DNN method because of the reliability of transfer learning. Finally, we successfully obtained the GM structures of Pt_n (n = 8-14) clusters by the DNN-TL method.

Algebraic Graph-Based Machine Learning Model for Li-Cluster Prediction

In cluster research, determining the ground-state structure of medium-sized clusters is hindered by a large number of local minimum on potential energy surfaces. The global optimization heuristic algorithm is time-consuming due to the use of DFT to determine the relative size of the cluster energy. Although machine learning (ML) is proved to be a promising way to reduce the DFT computational costs, a suitable method to represent clusters as input vectors is one of the bottlenecks in the application of ML to cluster research. In this work, we proposed a multiscale weighted spectral subgraph (MWSS) as an effective low-dimension representation of clusters and build an MWSS-based ML model to discover the structure-energy relationships in lithium clusters. We combine this model with the particle swarm optimization algorithm and DFT calculations to search for globally stable structures of clusters. We have successfully predicted the ground-state structure of Li20.

Growth Pattern of Large Morse Clusters with Medium-Range Potentials

Due to the extremely complex potential energy surfaces of large Morse clusters with medium-range potentials (i.e., ρ = 6 and 10), global optimization studies in the literature are limited to a cluster size (N) of ≤240. Starting from completely random structures, we successfully systematically studied Morse clusters with up to 700 atoms using our unbiased fuzzy global optimization (FGO) method. While all of the putative global minima reported previously have been efficiently obtained, new global minima with lower energies are identified for N values of 176, 258, 485, 561, 817, and 923 with ρ = 6 and for N values of 151, 202, 226, and 229 with ρ = 10. A detailed growth pattern and magic clusters are obtained. For the first time, we find that a central vacancy is present in Morse clusters containing 542, 543, 548, and 922 atoms with ρ = 6. FGO has achieved high performance in large clusters with different interatomic interaction ranges, thus showing great application potential in the global structure optimization of general clusters.

Elucidation of noble gas cluster configurations bound on graphdiyne: A metaheuristic approach

Graphynes are a class of all-carbon two-dimensional membranes that have been intensely researched for various membrane-based technologies on account of their unique pore architectures. Herein, we report an investigation of the mechanism and energetics of adsorption of noble gases (He, Ne and Ar) on graphdiyne (GDY), the most popular form of graphynes. Two global optimization techniques, namely particle swarm optimization (PSO) and differential evolution are employed to predict the putative global minima configurations of rare gas clusters in the size range 1-30 when adsorbed on GDY. We use the 12-6 Lennard-Jones potential to represent the pairwise non-covalent interactions between various interacting atoms. Initially, the gas atoms adsorb as monolayers on GDY at the centers of the triangular pores until all the triangular pores are filled. This is followed by a second layer formation on top of the hexagonal pore centers or on top of the C-C bonds. The findings from the empirical approach are further validated by performing density functional theory calculations on the predicted adsorbed cluster configurations. We have also looked into the adsorption of noble gas clusters on bilayer GDY systems and have found that the intercalation of gas atoms within the bilayers is feasible. Our study suggests that the stochastic nature of the swarm intelligence technique, PSO can assist in an effective search of the potential energy surfaces for the global minima, eventually enabling large-scale simulations.

Ensemble representation of catalytic interfaces: soloists, orchestras, and everything in-between

Catalytic systems are complex and dynamic, exploring vast chemical spaces on multiple timescales. In this perspective, we discuss the dynamic behavior of fluxional, heterogeneous thermal and electrocatalysts and the ensembles of many isomers which govern their behavior. We develop a new paradigm in catalysis theory in which highly fluxional systems, namely sub-nano clusters, isomerize on a much shorter timescale than that of the catalyzed reaction, so macroscopic properties arise from the thermal ensemble of isomers, not just the ground state. Accurate chemical predictions can only be reached through a many-structure picture of the catalyst, and we explain the breakdown of conventional methods such as linear scaling relations and size-selected prevention of sintering. We capitalize on the forward-looking discussion of the means of controlling the size of these dynamic ensembles. This control, such that the most effective or selective isomers can dominate the system, is essential for the fluxional catalyst to be practicable, and their targeted synthesis to be possible. It will also provide a fundamental lever of catalyst design. Finally, we discuss computational tools and experimental methods for probing ensembles and the role of specific isomers. We hope that catalyst optimization using chemically informed descriptors of ensemble nature and size will become a new norm in the field of catalysis and have broad impacts in sustainable energy, efficient chemical production, and more.

Machine learning accelerated random structure searching: Application to yttrium superhydrides

The search for new superhydrides, promising materials for both hydrogen storage and high temperature superconductivity, made great progress, thanks to atomistic simulations and Crystal Structure Prediction (CSP) algorithms. When they are combined with Density Functional Theory (DFT), these methods are highly reliable and often match a great part of the experimental results. However, systems of increasing complexity (number of atoms and chemical species) become rapidly challenging as the number of minima to explore grows exponentially with the number of degrees of freedom in the simulation cell. An efficient sampling strategy preserving a sustainable computational cost then remains to be found. We propose such a strategy based on an active-learning process where machine learning potentials and DFT simulations are jointly used, opening the way to the discovery of complex structures. As a proof of concept, this method is applied to the exploration of tin crystal structures under various pressures. We showed that the α phase, not included in the learning process, is correctly retrieved, despite its singular nature of bonding. Moreover, all the expected phases are correctly predicted under pressure (20 and 100 GPa), suggesting the high transferability of our approach. The method has then been applied to the search of yttrium superhydrides (YH x) crystal structures under pressure. The YH6 structure of space group Im-3m is successfully retrieved. However, the exploration of more complex systems leads to the appearance of a large number of structures. The selection of the relevant ones to be included in the active learning process is performed through the analysis of atomic environments and the clustering algorithm. Finally, a metric involving a distance based on x-ray spectra is introduced, which guides the structural search toward experimentally relevant structures. The global process (active-learning and new selection methods) is finally considered to explore more complex and unknown YH x phases, unreachable by former CSP algorithms. New complex phases are found, demonstrating the ability of our approach to push back the exponential wall of complexity related to CSP.

An anisotropic dressed pairwise potential model for the adsorption of noble gases on boron nitride sheets

Development of empirical potentials with accurate parameterization is indispensable while modeling large-scale systems. Herein, we report accurate parameterization of an anisotropic dressed pairwise potential model (PPM) for probing the adsorption of noble gases, He, Ne, Ar and Kr on boron nitride sheets. For the noble gas binding on B₄₈N₄₈H₂₄, we carried out a least-squares fit analysis of the dispersion and dispersionless contributions of the interaction potential separately. The transferability of the parameters for a range of molecular model systems of boron nitride is further established. The dressed PPM is then used in conjunction with a global optimization technique, namely particle swarm optimization (PSO) to assess the possibility of performing large-scale simulations with the PPM-PSO methodology. The results obtained for the adsorption of 2-5 noble gases on BN sheets establish the proof-of-concept, encouraging the pursuit of large-scale simulations using the PPM-PSO approach.

A symmetry-orientated divide-and-conquer method for crystal structure prediction

Crystal structure prediction has been a subject of topical interest but remains a substantial challenge especially for complex structures as it deals with the global minimization of the extremely rugged high-dimensional potential energy surface. In this paper, a symmetry-orientated divide-and-conquer scheme was proposed to construct a symmetry tree graph, where the entire search space is decomposed into a finite number of symmetry dependent subspaces. An artificial intelligence-based symmetry selection strategy was subsequently devised to select the low-lying subspaces with high symmetries for global exploration and in-depth exploitation. Our approach can significantly simplify the problem of crystal structure prediction by avoiding exploration of the most complex P1 subspace on the entire search space and has the advantage of preserving the crystal symmetry during structure evolution, making it well suitable for predicting the complex crystal structures. The effectiveness of the method has been validated by successful prediction of the candidate structures of binary Lennard-Jones mixtures and the high-pressure phase of ice, containing more than 100 atoms in the simulation cell. The work therefore opens up an opportunity toward achieving the long-sought goal of crystal structure prediction of complex systems.

Automatic structural elucidation of vacancies in materials by active learning

The artificial intelligence method based on active learning for the automatic structural elucidation of vacancies in materials. This is implemented in the quantum machine learning software/agent for material design and discovery (QMLMaterial).

Molecular dynamics simulation study of gold nanosheet as drug delivery vehicles for anti-HIV-1 aptamers

The adsorption process of three aptamers with gold nanosheet (GNS) as a drug carrier has been investigated with the help of molecular dynamics simulations. The sequencing of the considered aptamers are as (CUUCAUUGUAACUUCUCAUAAUUUCCCGAGGCUUUUACUUUCGGGGUCCU) and (CCGGGUCGUCCCCUACGGGGACUAAAGACUGUGUCCAACCGCCCUCGCCU) for AP1 and AP2, respectively. AP3 is a muted version of AP1 in which nucleotide positions 4, 6, 18, 28 and 39 have C4A, U6G, A18G, G28A, and U39C mutations. At positions 24, and 40, a deletion mutation is seen to eliminate U24 and U40 bases. These aptamers are inhibitors for HIV-1 protease and can be candidates as potential pharmaceutics for treatment of AIDS in the future. The interactions between considered aptamers and GNS have been analyzed in detail with help of structural and energetic properties. These analyses showed that all three aptamers could well adsorb on GNS. Overall, the final results show that the adsorption of AP2 on the GNS is more favorable than other considered ones and consequently GNS can be considered as a device in order to immobilize these aptamers.

DFT Study of Microsolvated [NO3·(H2O)n]- (n = 1-12) Clusters and Molecular Dynamics Simulation of Nitrate Solution

Investigation of the process of the NO₃^- anion solvation is central to understanding the chemical and physical properties of its aqueous solutions. The importance of this topic can be seen in atmospheric chemistry, as well as in nuclear waste processing research. In this work, we used a particle swarm optimization technique driven by density functional theory to sample the potential energy surface of various microsolvated [NO₃·(H₂O)_n]^- (n = 1-12) clusters. We found that the charge transfer plays a crucial role in the stabilization of the investigated species. Moreover, by conducting ab initio molecular dynamics simulations, we showed that at low concentrations (∼0.2 M) the NO₃^- species tend to be located on the surface of water solution. We also observed that the contact ion pair K⁺-NO₃^- undergoes a fast dissociation and each of the ions is solvated separately. As a result, from our calculations, we expect that at low concentration there could be oppositely signed concentration gradients for NO₃^- and K⁺ ions in a thin water film.

Kick-Fukui: A Fukui Function-Guided Method for Molecular Structure Prediction

Here, we introduce a hybrid method, named Kick-Fukui, to explore the potential energy surface (PES) of clusters and molecules using the Coulombic integral between the Fukui functions in the first screening of the best individuals. In the process, small stable molecules or clusters whose combination has the stoichiometry of the explored species are used as assembly units. First, a small set of candidates has been selected from a large and stochastically generated (Kick) population according to the maximum value of the Coulombic integral between the Fukui functions of both fragments. Subsequently, these few candidates are optimized using a gradient method and density functional theory (DFT) calculations. The performance of the program has been evaluated to explore the PES of various systems, including atomic and molecular clusters. In most cases studied, the global minimum (GM) has been identified with a low computational cost. The strategy does not allow to identify the GM of some silicon clusters; however, it predicts local minima very close in energy to the GM that could be used as the initial population of evolutionary algorithms.

Bonding of C1 fragments on metal nanoclusters: a search for methane conversion catalysts with swarm intelligence

There is a need for a catalyst that can directly convert methane into useful substances. The use of Ni as a catalyst for the steam reforming of methane has led us to look at Ni nanoclusters as potential candidates for the direct conversion of methane. Fe, Co, Cu, and Zn nanoclusters are also focused on. How the type of C1 fragments (CH4, CH3, CH2, CH, and C) stabilized by the metal nanoclusters as adsorbed species varies with metal species is theoretically investigated. The particle swarm optimization algorithm, which is based on swarm intelligence, as well as density functional theory, is used for this calculation. The Ni nanoclusters are found to preferentially adsorb C as a stable species, the Fe and the Co nanoclusters both CH and CH3, and the Cu nanoclusters CH3; the Zn nanoclusters are found not to chemisorb any C1 fragment. The methane activation capacity can be ranked in the order of Ni > Fe > Co > Cu > Zn. The highest methane activation capacity of Ni is due to the strongest covalent nature of the interaction between Ni and the adsorbed species. The ionicity of the bond between Fe and the adsorbed species is higher than that between Co and the adsorbed species, while the covalent nature of the bonds is comparable for both. The weak methane activation ability of Cu compared to Fe, Co, and Ni is found to be due to the fact that both the covalent and ionic bond strengths between Cu and the adsorbed species are weak. Zn and the adsorbed species form neither ionic nor covalent bonds. These results indicate that the Fe and the Co nanoclusters as well as the Ni may lead to the over-oxidation of methane, whereas the Zn nanoclusters cannot activate methane in the first place; therefore, their application to direct methane conversion catalysts is unlikely. Since the Cu nanoclusters do not adsorb C and CH as stable species, but CH3 stably, the Cu nanoclusters are expected to work as a catalyst for the direct conversion of methane.

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.

From Infection Clusters to Metal Clusters: Significance of the Lowest Occupied Molecular Orbital (LOMO)

In this paper, the nature of the lowest-energy electrons is detailed. The orbital occupied by such electrons can be termed the lowest occupied molecular orbital (LOMO). There is a good correspondence between the Hückel method in chemistry and graph theory in mathematics; the molecular orbital, which chemists view as the distribution of an electron with a specific energy, is to mathematicians an algebraic entity, an eigenvector. The mathematical counterpart of LOMO is known as eigenvector centrality, a centrality measure characterizing nodes in networks. It may be instrumental in solving some problems in chemistry, and also it has implications for the challenge facing humanity today. This paper starts with a demonstration of the transmission of infectious disease in social networks, although it is unusual for a chemistry paper but may be a suitable example for understanding what the centrality (LOMO) is all about. The converged distribution of infected patients on the network coincides with the distribution of the LOMO of a molecule that shares the same network structure or topology. This is because the mathematical structures behind graph theory and quantum mechanics are common. Furthermore, the LOMO coefficient can be regarded as a manifestation of the centrality of atoms in an atomic assembly, indicating which atom plays the most important role in the assembly or which one has the greatest influence on the network of these atoms. Therefore, it is proposed that one can predict the binding energy of a metal atom to its cluster based on its LOMO coefficient. A possible improvement of the descriptor using a more sophisticated centrality measure is also discussed.

Systematic cluster growth: a structure search method for transition metal clusters

The systematic cluster growth (SCG) method is a biased structure search strategy based on a seeding process for investigating the structural evolution and growth pattern of transition metal clusters.

Reliability of semiempirical and DFTB methods for the global optimization of the structures of nanoclusters

In this work, we explore the possibility of using computationally inexpensive electronic structure methods, such as semiempirical and DFTB calculations, for the search of the global minimum (GM) structure of chemical systems. The basic prerequisite that these inexpensive methods will need to fulfill is that their lowest energy structures can be used as starting point for a subsequent local optimization at a benchmark level that will yield its GM. If this is possible, one could bypass the global optimization at the expensive method, which is currently impossible except for very small molecules. Specifically, we test our methods with clusters of second row elements including systems of several bonding types, such as alkali, metal, and covalent clusters. The results reveal that the DFTB3 method yields reasonable results and is a potential candidate for this type of applications. Even though the DFTB2 approach using standard parameters is proven to yield poor results, we show that a re-parametrization of only its repulsive part is enough to achieve excellent results, even when applied to larger systems outside the training set.

A swarm intelligence modeling approach reveals noble gas cluster configurations confined within carbon nanotubes

Confinement of atoms and molecules brings forth fascinating properties to chemical systems that are otherwise not known in the bulk. Carbon nanotubes (CNTs) and fullerenes are excellent hosts for probing the confinement effects. Herein, we explore the potential energy surfaces of large noble gas clusters, Ngn (Ng = He, Ne and Ar; n = 10, 20, 30, 40, and 50), in the confines of CNTs of various lengths. Our implementation involves integrating the continuum approximation for CNTs with the well-known swarm intelligence technique, particle swarm optimization (PSO), followed by a deterministic local optimization. Global search techniques such as PSO have been increasingly utilized in recent times to track down minimum energy configurations on highly rugged potential energy surfaces. Aside from the position vectors of the noble gas atoms, we have considered the radius of the CNTs as a design variable. Such an approach enabled us to predict the optimal CNT radii for the encapsulation of each of the clusters. Confined cluster geometries ranging from linear, zig-zag, and double-helical to spiral configurations are obtained on encapsulation, in sharp contrast to their bare cluster geometries. On increasing the CNT length, our approach yielded quasi-linear geometries, suggesting that the length of the CNTs plays a crucial role in determining the stable cluster configurations on confinement.

Topology-Based Machine Learning Strategy for Cluster Structure Prediction

In cluster physics, the determination of the ground-state structure of medium-sized and large-sized clusters is a challenge due to the number of local minimal values on the potential energy surface growing exponentially with cluster size. Although machine learning approaches have had much success in materials sciences, their applications in clusters are often hindered by the geometric complexity clusters. Persistent homology provides a new topological strategy to simplify geometric complexity while retaining important chemical and physical information without having to "downgrade" the original data. We further propose persistent pairwise independence (PPI) to enhance the predictive power of persistent homology. We construct topology-based machine learning models to reveal hidden structure-energy relationships in lithium (Li) clusters. We integrate the topology-based machine learning models, a particle swarm optimization algorithm, and density functional theory calculations to accelerate the search of the globally stable structure of clusters.

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.

ReB6-: A Metallaboron Analog of Metallabenzenes

Metallabenzenes are a class of molecules in which a CH unit in benzene is replaced by a functionalized transition-metal atom. While all-boron analogues of aromatic and antiaromatic hydrocarbons are well-known, there have not been any metallaboron analogs. We have produced and investigated two metal-doped boron clusters, ReB₆^- and AlB₆^-, using high-resolution photoelectron imaging and quantum chemical calculations. Vibrationally resolved photoelectron spectra have been obtained and compared with the theoretical results. The ReB₆^- cluster is found to be perfectly planar with a B-centered hexagonal structure (C_2v, ¹A₁), while AlB₆^- is known to have a similar structure, but with a slightly out-of-plane distortion (C_s, ¹A'). Chemical bonding analyses show that the closed-shell ReB₆^- is doubly σ- and π-aromatic, while AlB₆^- is known to be σ-aromatic and π-antiaromatic. The out-of-plane distortion in AlB₆^- is due to antiaromaticity, akin to the out-of-plane distortion of the prototypical antiaromatic cyclooctatetraene. The π-bonding in ReB₆^- is compared with that in both benzene and rhenabenzene [(CO)₄ReC₅H₅], and remarkable similarities are found. Hence, ReB₆^- can be viewed as the first metallaboron analog of metallabenzenes and it may be viable for syntheses with suitable ligands.

Augmenting Basin-Hopping With Techniques From Unsupervised Machine Learning: Applications in Spectroscopy and Ion Mobility

Evolutionary algorithms such as the basin-hopping (BH) algorithm have proven to be useful for difficult non-linear optimization problems with multiple modalities and variables. Applications of these algorithms range from characterization of molecular states in statistical physics and molecular biology to geometric packing problems. A key feature of BH is the fact that one can generate a coarse-grained mapping of a potential energy surface (PES) in terms of local minima. These results can then be utilized to gain insights into molecular dynamics and thermodynamic properties. Here we describe how one can employ concepts from unsupervised machine learning to augment BH PES searches to more efficiently identify local minima and the transition states connecting them. Specifically, we introduce the concepts of similarity indices, hierarchical clustering, and multidimensional scaling to the BH methodology. These same machine learning techniques can be used as tools for interpreting and rationalizing experimental results from spectroscopic and ion mobility investigations (e.g., spectral assignment, dynamic collision cross sections). We exemplify this in two case studies: (1) assigning the infrared multiple photon dissociation spectrum of the protonated serine dimer and (2) determining the temperature-dependent collision cross-section of protonated alanine tripeptide.

Modified Particle Swarm Optimization Algorithms for the Generation of Stable Structures of Carbon Clusters, Cn (n = 3-6, 10)

Particle Swarm Optimization (PSO), a population based technique for stochastic search in a multidimensional space, has so far been employed successfully for solving a variety of optimization problems including many multifaceted problems, where other popular methods like steepest descent, gradient descent, conjugate gradient, Newton method, etc. do not give satisfactory results. Herein, we propose a modified PSO algorithm for unbiased global minima search by integrating with density functional theory which turns out to be superior to the other evolutionary methods such as simulated annealing, basin hopping and genetic algorithm. The present PSO code combines evolutionary algorithm with a variational optimization technique through interfacing of PSO with the Gaussian software, where the latter is used for single point energy calculation in each iteration step of PSO. Pure carbon and carbon containing systems have been of great interest for several decades due to their important role in the evolution of life as well as wide applications in various research fields. Our study shows how arbitrary and randomly generated small Cn clusters (n = 3-6, 10) can be transformed into the corresponding global minimum structure. The detailed results signify that the proposed technique is quite promising in finding the best global solution for small population size clusters.

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.

TGMin: An efficient global minimum searching program for free and surface-supported clusters

In this article, we introduce an efficient global-minimum structural search program named Tsinghua Global Minimum 2 (TGMin-2), which is the successor of the original TGMin algorithm that was developed in our group in 2011. We have introduced a number of new features and improvements into TGMin-2, including a symmetric structure generation algorithm that can produce good initial seeds for small- and medium-size clusters, the duplicated structure identification algorithm, and the improved structure adaption algorithm that was implemented in the original TGMin code. To predict the simulated photoelectron spectrum (PE spectrum) automatically, we also implemented a standalone program named AutoPES (Auto Photoelectron Spectroscopy), which can be used to simulate PE spectra and compare them with experimental results automatically. We have demonstrated that TGMin-2 and AutoPES are powerful tools for studying free and surface-supported molecules, clusters, and nanoclusters. © 2018 Wiley Periodicals, Inc.

On-the-Fly Machine Learning of Atomic Potential in Density Functional Theory Structure Optimization

Machine learning (ML) is used to derive local stability information for density functional theory calculations of systems in relation to the recently discovered SnO_{2}(110)-(4×1) reconstruction. The ML model is trained on (structure, total energy) relations collected during global minimum energy search runs with an evolutionary algorithm (EA). While being built, the ML model is used to guide the EA, thereby speeding up the overall rate by which the EA succeeds. Inspection of the local atomic potentials emerging from the model further shows chemically intuitive patterns.