Monte Carlo-Simulated Annealing and Machine Learning-Based Funneled Approach for Finding the Global Minimum Structure of Molecular Clusters

Understanding the physical underpinnings and geometry of molecular clusters is of great importance in many fields, ranging from studying the beginning of the universe to the formation of atmospheric particles. To this end, several approaches have been suggested, yet identifying the most stable cluster geometry (i.e., global potential energy minimum) remains a challenge, especially for highly symmetric clusters. Here, we suggest a new funneled Monte Carlo-based simulated annealing (SA) approach, which includes two key steps: generation of symmetrical clusters and classification of the clusters according to their geometry using machine learning (MCSA-ML). We demonstrate the merits of the MCSA-ML method in comparison to other approaches on several Lennard-Jones (LJ) clusters and four molecular clusters-Ser8(Cl-)2, H+(H2O)6, Ag+(CO2)8, and Bet4Cl-. For the latter of these clusters, the correct structure is unknown, and hence, we compare the experimental and simulated fragmentation patterns, and the fragmentation of the proposed global minimum matches experiments closely. Additionally, based on the fragmentation of the predicted betaine cluster, we were able to identify hitherto unknown neutral fragmentation channels. In comparison to results obtained with other methods, we demonstrated a superior ability of MCSA-ML to predict clusters with high symmetry and similar abilities to predict clusters with asymmetrical structures.

Global optimization of extraframework ensembles in zeolites: structural analysis of extraframework aluminum species in MOR and MFI zeolites

Metal-modified zeolites are versatile catalytic materials with a wide range of industrial applications. Their catalytic behaviour is determined by the nature of externally introduced cationic species, i.e., its geometry, chemical composition, and location within the zeolite pores. Superior catalyst designs can be unlocked by understanding the confinement effect and spatial limitations of the zeolite framework and its influence on the geometry and location of such cationic active sites. In this study, we employ the genetic algorithm (GA) global optimization method to investigate extraframework aluminum species and their structural variations in different zeolite matrices. We focus on extraframework aluminum (EFAl) as a model system because it greatly influences the product selectivity and catalytic stability in several zeolite catalyzed processes. Specifically, the GA was used to investigate the configurational possibilities of EFAl within the mordenite (MOR) and ZSM-5 frameworks. The xTB semi-empirical method within the GA was employed for an automated sampling of the EFAl-zeolite space. Furthermore, geometry refinement at the density functional theory (DFT) level of theory allowed us to improve the most stable configurations obtained from the GA and elaborate on the limitations of the xTB method. A subsequent ab initio thermodynamics analysis (aiTA) was chosen to predict the most favourable EFAl structure(s) under the catalytically relevant operando conditions.

The Effect of Particle Size and Composition on the Optical and Electronic Properties of CdO and CdS Rocksalt Nanoparticles

Quantum confinement like behaviour in CdO and CdS nanoparticles is demonstrated through explicit evGW-BSE many-body perturbation theory calculations on 0.6-1.4 nanometre particles of these materials. However, while the lowest optical...

The effect of particle size on the optical and electronic properties of magnesium oxide nanoparticles

The quasiparticle states, fundamental gaps, optical gaps, exciton binding energies and UV-vis spectra for a series of cuboidal nanoparticles of the prototypical oxide magnesium oxide (MgO), the largest of which has 216 atoms and edges of 1 nm, were predicted using many-body perturbation theory (evGW-BSE). The evolution of the properties with the particle size was explicitly studied. It was found that, while the highest occupied and lowest unoccupied quasiparticle states and fundamental gap change with the particle size, the optical gap remains essentially fixed for all but the smallest nanoparticles, in line with what was previously observed experimentally. The explanation for these observations is demonstrated to be that, while the optical gap is associated with an exciton that is highly localised around the particle's corner atoms, the highest occupied and lowest unoccupied quasiparticle states, while primarily localised on the oxygen corner atoms (hole) and magnesium corner atoms (electron), show significant delocalisation along the edges. The strong localisation of the exciton associated with the optical gap on the corner atoms is argued to also explain why the nanoparticles have much smaller optical gaps and red-shifted spectra compared to bulk MgO. Finally, it is discussed how this non-quantum confinement behaviour, where the properties of the nanoparticles arise from surface defects rather than differences in localisation of quasiparticle or exciton states, appears typical of alkaline earth oxide nanoparticles, and that the true optical gap of bulk crystals of such materials is also probably the result of surface defects, even if unobservable experimentally.

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.

Multiple d-d bonds between early transition metals in TM2Lin (TM = Sc, Ti) superatomic molecule clusters

The synthesis and application of compounds with Cr-Cr and V-V d-d quintuple bonds (σ, 2π, 2δ) have led to new thinking about whether d-d multiple bonds also exist between early transition metals such as Sc-Sc and Ti-Ti. In this study, by extensive unbiased global search at the density functional theory level, the low-energy structures of 26e and 30e TM2Lin clusters were obtained. Based on the super valence bond (SVB) theory, the prolate double-core structure of TM2Lin clusters was regarded as a superatomic molecule, of which each half was regarded as an open-shell superatom, and the electronic shell-closure was realized by forming multiple bonds between superatoms. Then, the quintuple super bonds (2δ, 2π, σ) of the Li18Ti2, Li20Sc2, [Li17V2]+, [Li17Ti2]- clusters and the triple super bonds (2π, σ) of the Li24Sc2 and Li24Y2 clusters were confirmed via chemical-bonding analysis. This way of forming multiple bonds between early transition metals through superatomic bonding has promoted the experimental synthesis and application of early transition metal multiple bond compounds.

Structure elucidation and construction of isomerisation pathways in small to moderate-sized (6-27) MgO nanoclusters: an adaptive mutation simulated annealing based analysis with quantum chemical calculations

Determination of global minimum structures and elucidation of reaction paths or minimum energy paths between low-lying minima are of great chemical importance. To that end, we have used our own Adaptive Mutation Simulated Annealing method to determine the global minimum and the minimum energy paths for various isomerisation reactions for small to moderate-sized (MgO)n (n = 6-27) clusters, using the Born-Mayer potential with suitable parameter values. The minimum energy structures obtained by us match well with previously reported data and are used as guess structures for further optimisation at the DFT level (using the B3LYP functional and DGDZVP basis set). Our optimised structures are found to match very well with the further DFT optimised structures, where the comparison is done by determining the root mean square deviation values as well as the radial distribution function profiles. A scheme is proposed to determine the minimum energy paths for isomerisation reactions for some cluster sizes where the transition state/s obtained by us, at very low computational cost, match well with those obtained from further optimisation using DFT calculations. We have shown the efficacy of our method in determining the reaction pathways, even for cases that involve multi-step reactions.

Multiscale Computational Simulation of Amorphous Silicates’ Structural, Dielectric, and Vibrational Spectroscopic Properties

Silicates are among the most abundant and important inorganic materials, not only in the Earth’s crust, but also in the interstellar medium in the form of micro/nanoparticles or embedded in the matrices of comets, meteorites, and other asteroidal bodies. Although the crystalline phases of silicates are indeed present in nature, amorphous forms are also highly abundant. Here, we report a theoretical investigation of the structural, dielectric, and vibrational properties of the amorphous bulk for forsterite (Mg2SiO4) as a silicate test case by a combined approach of classical molecular dynamics (MD) simulations for structure evolution and periodic quantum mechanical Density Functional Theory (DFT) calculations for electronic structure analysis. Using classical MD based on an empirical partial charge rigid ionic model within a melt-quenching scheme at different temperatures performed with the GULP 4.0 code, amorphous bulk structures for Mg2SiO4 were generated using the crystalline phase as the initial guess. This has been done for bulk structures with three different unit cell sizes, adopting a super-cell approach; that is, 1 × 1 × 2, 2 × 1 × 2, and 2 × 2 × 2. The radial distribution functions indicated a good degree of amorphization of the structures. Periodic B3LYP-geometry optimizations performed with the CRYSTAL14 code on the generated amorphous systems were used to analyze their structure; to calculate their high-frequency dielectric constants (ε∞); and to simulate their IR, Raman, and reflectance spectra, which were compared with the experimental and theoretical crystalline Mg2SiO4. The most significant changes of the physicochemical properties of the amorphous systems compared to the crystalline ones are presented and discussed (e.g., larger deviations in the bond distances and angles, broadening of the IR bands, etc.), which are consistent with their disordered nature. It is also shown that by increasing the unit cell size, the bulk structures present a larger degree of amorphization.

Synthesis Target Structures for Alkaline Earth Oxide Clusters

Knowing the possible structures of individual clusters in nanostructured materials is an important first step in their design. With previous structure prediction data for BaO nanoclusters as a basis, data mining techniques were used to investigate candidate structures for magnesium oxide, calcium oxide and strontium oxide clusters. The lowest-energy structures and analysis of some of their structural properties are presented here. Clusters that are predicted to be ideal targets for synthesis, based on being both the only thermally accessible minimum for their size, and a size that is thermally accessible with respect to neighbouring sizes, include global minima for: sizes n = 9 , 15 , 16 , 18 and 24 for (MgO) n ; sizes n = 8 , 9 , 12 , 16 , 18 and 24 for (CaO) n ; the greatest number of sizes of (SrO) n clusters ( n = 8 , 9 , 10 , 12 , 13 , 15 , 16 , 18 and 24); and for (BaO) n sizes of n = 8 , 10 and 16.

Size dependent structural and polymorphic transitions in ZnO: from nanocluster to bulk

We report on an extensive survey of (ZnO)_N nanostructures ranging from bottom-up generated nanoclusters to top-down nanoparticles cuts from bulk polymorphs. The obtained results enable us to follow the energetic preferences of structure and polymorphism in (ZnO)_N systems with N varying between 10-1026. This size range encompasses small nanoclusters with 10s of atoms and nanoparticles with 100s of atoms, which we also compare with appropriate bulk limits. In all cases the nanostructures and bulk systems are optimized using accurate all-electron, relativistic density functional theory based calculations with numeric atom centered orbital basis sets. Specifically, sets of five families of (ZnO)_N species are considered: single-layered and multi-layered nanocages, and bulk cut nanoparticles from the sodalite (SOD), body centered tetragonal (BCT), and wurtzite (WZ) ZnO polymorphs. Using suitable fits to interpolate and extrapolate these data allows us to assess the size-dependent energetic stabilities of each family. With increasing size our results indicate a progressive change in energetic stability from single-layered to multi-layered cage-like nanoclusters. For nanoparticles of around 2.6 nm diameter we identify a transitional region where multi-layered cages, SOD, and BCT nanostructures are very similar in energetic stability. This transition size also marks the size regime at which bottom-up nanoclusters give way to top-down bulk-cut nanoparticles. Eventually, a final crossover is found where the most stable WZ-ZnO polymorph begins to energetically dominate at N ∼ 2200. This size corresponds to an approximate nanoparticle diameter of 4.7 nm, in line with experiments reporting the observation of wurtzite crystallinity in isolated ligand-free ZnO nanoparticles of 4-5 nm size or larger.

Electronic and vibrational second hyperpolarizabilities of (MgO)n clusters

In this work, we report results for the static second hyperpolarizability of magnesium oxide clusters including electronic and vibrational contributions. The comparison between second-order Møller-Plesset (MP2) perturbation theory and coupled cluster results to the electronic contribution points out that MP2 is a suitable method to compute this property. When computed at the MP2 level, the electronic contribution per atom converges to approximately 5000 a.u. Vibrational corrections were computed at the MP2 level through the perturbation theoretical method of Bishop and Kirtman. Results obtained showed that the term [α²]^0,0 represents around 20% of the electronic counterpart while the term [μβ]^0,0 is comparable to it. Modes that contribute significantly to [α²]^0,0 are those in which all or part of the bond lengths simultaneously increase and decrease, leading to large polarizability derivatives. In turn, modes that provide relevant contributions to [μβ]^0,0 are those in which oxygen anions move in opposite directions to the magnesium cations yielding large derivatives of the dipole moment and first hyperpolarizability.

Electro- and opto-mutable properties of MgO nanoclusters adsorbed on mono- and double-layer graphene

Inspired by recent experiments, the trapping of molecules in 2D materials has gained increasing attention due to the unique ability of the molecules to modulate the electronic and optical properties of 2D materials, which calls for fundamental understanding and predictive design strategies. Herein, we focus on mono- and double-layer graphene encapsulating various MgO clusters and explore their diverse electronic and optical properties using a number of high-level first-principles calculations. By correlating the stability of adsorption, geometry, charge transfer, band structures, optical absorption spectrum, and the van der Waals pressure, our results decode various synergies in electro- and opto-mutable properties of MgO/graphene systems. We found that 2D-MgO flakes on graphene layers exhibit surface polarization effects - in contrast to their isolated neutral flakes - and show a significant charge transfer from graphene to n-doped flakes, breaking the symmetry of graphene layers. We obtained a van der Waals pressure of ∼0.7 (0.9) GPa on bilayer graphene encapsulating MgO nanoclusters, which matches extremely well with experiment. While there is one quantum emission in the visible light region for a single MgO flake, a wide range of visible light is accessible for MgO on mono- and double-layer graphene. Overall, these findings provide new physical insights and design strategies to modulate 2D materials with several applications in optoelectronics while significantly broadening the spectrum of strategies for fabricating new hybrid 2D heterostructures by encapsulating external molecules.

An efficient genetic algorithm for structure prediction at the nanoscale

We have developed and implemented a new global optimization technique based on a Lamarckian genetic algorithm with the focus on structure diversity. The key process in the efficient search on a given complex energy landscape proves to be the removal of duplicates that is achieved using a topological analysis of candidate structures. The careful geometrical prescreening of newly formed structures and the introduction of new mutation move classes improve the rate of success further. The power of the developed technique, implemented in the Knowledge Led Master Code, or KLMC, is demonstrated by its ability to locate and explore a challenging double funnel landscape of a Lennard-Jones 38 atom system (LJ₃₈). We apply the redeveloped KLMC to investigate three chemically different systems: ionic semiconductor (ZnO)_1-32, metallic Ni₁₃ and covalently bonded C₆₀. All four systems have been systematically explored on the energy landscape defined using interatomic potentials. The new developments allowed us to successfully locate the double funnels of LJ₃₈, find new local and global minima for ZnO clusters, extensively explore the Ni₁₃ and C₆₀ (the buckminsterfullerene, or buckyball) potential energy surfaces.

ABCluster: the artificial bee colony algorithm for cluster global optimization

Global optimization of cluster geometries is of fundamental importance in chemistry and an interesting problem in applied mathematics. We apply a swarm-intelligence based heuristic algorithm, i.e. the artificial bee colony algorithm to solve this problem for various kinds of clusters.

Structure prediction of nanoclusters; a direct or a pre-screened search on the DFT energy landscape?

Atomic structure prediction, using KLMC (Lamarckian evolutionary algorithm search), and properties comparison of (KF)_n, (MgO)_n, (ZnO)_n and (CdSe)_n nanoclusters.

Stability and metastability of clusters in a reactive atmosphere: theoretical evidence for unexpected stoichiometries of MgMOx

By applying a genetic algorithm and ab initio atomistic thermodynamics, we identify the stable and metastable compositions and structures of MgMOx clusters at realistic temperatures and oxygen pressures. We find that small clusters (M≲5) are in thermodynamic equilibrium when x>M. The nonstoichiometric clusters exhibit peculiar magnetic behavior, suggesting the possibility of tuning magnetic properties by changing environmental pressure and temperature conditions. Furthermore, we show that density-functional theory with a hybrid exchange-correlation functional is needed for predicting accurate phase diagrams of metal-oxide clusters. Neither a (sophisticated) force field nor density-functional theory with (semi)local exchange-correlation functionals is sufficient for even a qualitative prediction.

Single string based global optimizer for geometry optimization in strongly coupled finite clusters: An adaptive mutation-driven strategy

We propose and implement a simple adaptive heuristic to optimize the geometries of clusters of point charges or ions with the ability to find the global minimum energy configurations. The approach uses random mutations of a single string encoding the geometry and accepts moves that decrease the energy. Mutation probability and mutation intensity are allowed to evolve adaptively on the basis of continuous evaluation of past explorations. The resulting algorithm has been called Completely Adaptive Random Mutation Hill Climbing method. We have implemented this method to search through the complex potential energy landscapes of parabolically confined 3D classical Coulomb clusters of hundreds or thousands of charges--usually found in high frequency discharge plasmas. The energy per particle (EN∕N) and its first and second differences, structural features, distribution of the oscillation frequencies of normal modes, etc., are analyzed as functions of confinement strength and the number of charges in the system. Certain magic numbers are identified. In order to test the feasibility of the algorithm in cluster geometry optimization on more complex energy landscapes, we have applied the algorithm for optimizing the geometries of MgO clusters, described by Coulomb-Born-Mayer potential and finding global minimum of some Lennard-Jones clusters. The convergence behavior of the algorithm compares favorably with those of other existing global optimizers.

Stardust silicate nucleation kick-started by SiO+TiO₂

Dust particles are quintessential for the chemical evolution of the Universe. Dust nucleates in stellar outflows of dying stars and subsequently travels through the interstellar medium, continuously evolving via energetic processing, collisions and condensation. Finally, dust particles are incorporated in the next-generation star or its surrounding planetary system. In oxygen-rich stellar outflows, silicates are observed in the condensation zone (1200-1000 K), but, in spite of several decades of experimental and theoretical study, the stardust nucleation process remains poorly understood. We have previously shown that under these conditions ternary Mg-Si-O clusters may start forming at high enough rates from SiO, Mg and H₂O through heteromolecular association processes. In this reaction scheme, none of the possible initial association reactions was thermodynamically favourable owing to the large entropy loss at these temperatures. Here, we follow a previous idea that the incorporation of TiO₂ could help to initiate stardust nucleation. In contrast to these studies, we find that there is no need for TiO₂ cluster seeds-instead, one molecule of TiO₂ is sufficient to kick-start the subsequent nucleation of a silicate dust particle.

Structure determination of neutral MgO clusters--hexagonal nanotubes and cages

Structural information for neutral magnesium oxide clusters has been obtained by a comparison of their experimental vibrational spectra with predictions from theory. (MgO)(n) clusters with n = 3-16 have been studied in the gas phase with a tunable IR-UV two-color ionization scheme and size-selective infrared spectra have been measured. These IR spectra are compared to the calculated spectra of the global minimum structures predicted by a hybrid ab initio genetic algorithm. The comparison shows clear evidence that clusters of the composition (MgO)(3k) (k = 1-5) form hexagonal tubes, which confirm previous theoretical predictions. For the intermediate sizes (n≠ 3k) cage-like structures containing hexagonal (MgO)(3) rings are identified. Except for the cubic (MgO)(4) no evidence for bulk like structures is found.

Atomistic and electronic structure of (X 2 O 3 ) n nanoclusters; n =1–5, X=B, Al, Ga, In and Tl

The stable and metastable, as measured using an all-electron density functional theory approach, stoichiometric clusters of boron, aluminium, gallium, indium and thallium oxide are reported. Initial candidate structures were found using an evolutionary algorithm to search the energy landscape, defined using classical interatomic potentials, for alumina and india followed by data mining or rescaling. Characterization of the refined structures was performed by electronic structure techniques at the hybrid density functional and many-body GW levels of theory. We make accurate predictions of the spectroscopic properties represented by mean ionization potentials of 11.4, 9.9, 9.8, 8.8 and 8.4 eV and electron affinities of 0.05, 1.1, 1.6, 1.9 and 2.5 eV for boria, alumina, gallia, india and thallia, respectively. The changes in the global minima, atomistic and electronic properties with respect to the cluster and cation size are discussed.