An investigation of two approaches to basin hopping minimization for atomic and molecular clusters

Energy landscapes for clusters of hexapeptides

We present the results for energy landscapes of hexapeptides obtained using interfaces to the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) program. We have used basin-hopping global optimization and discrete path sampling to explore the landscapes of hexapeptide monomers, dimers, and oligomers containing 10, 100, and 200 monomers modeled using a residue-level coarse-grained potential, Mpipi, implemented in LAMMPS. We find that the dimers of peptides containing amino acid residues that are better at promoting phase separation, such as tyrosine and arginine, have melting peaks at higher temperature in their heat capacity compared to phenylalanine and lysine, respectively. This observation correlates with previous work on the same uncapped hexapeptide monomers modeled using atomistic potential. For oligomers, we compare the variation in monomer conformations with radial distance and observe trends for selected angles calculated for each monomer. The LAMMPS interfaces to the GMIN and OPTIM programs for landscape exploration offer new opportunities to investigate larger systems and provide access to the coarse-grained potentials implemented within LAMMPS.

CO organization at ambient pressure on stepped Pt surfaces: first principles modeling accelerated by neural networks

Step and kink sites at Pt surfaces have crucial importance in catalysis. We employ a high dimensional neural network potential (HDNNP) trained using first-principles calculations to determine the adsorption structure of CO under ambient conditions (T = 300 K and P = 1 atm) on these surfaces. To thoroughly explore the potential energy surface (PES), we use a modified basin hopping method. We utilize the explored PES to identify the adsorbate structures and show that under the considered conditions several low free energy structures exist. Under the considered temperature and pressure conditions, the step edge (or kink) is totally occupied by on-top CO molecules. We show that the step structure and the structure of CO molecules on the step dictate the arrangement of CO molecules on the lower terrace. On surfaces with (111) steps, like Pt(553), CO forms quasi-hexagonal structures on the terrace with the top site preferred, with on average two top site CO for one multiply bonded CO, while in contrast surfaces with (100) steps, like Pt(557), present a majority of multiply bonded CO on their terrace. Short terraced surfaces, like Pt(643), with square (100) steps that are broken by kink sites constrain the CO arrangement parallel to the step edge. Overall, this effort provides detailed analysis on the influence of the step edge structure, kink sites, and terrace width on the organization of CO molecules on non-reconstructed stepped surfaces, yielding initial structures for understanding restructuring events driven by CO at high coverages and ambient pressure.

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.

Structure prediction of crystals, surfaces and nanoparticles

We review the current techniques used in the prediction of crystal structures and their surfaces and of the structures of nanoparticles. The main classes of search algorithm and energy function are summarized, and we discuss the growing role of methods based on machine learning. We illustrate the current status of the field with examples taken from metallic, inorganic and organic systems. This article is part of a discussion meeting issue 'Dynamic in situ microscopy relating structure and function'.

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.

Chemically directed structure evolution for crystal structure prediction

The chemically directed structure evolution method uses chemical models to quantify the environment of atoms and vacancy sites in a crystal structure with that information used to inform how to modify the structure for crystal structure prediction.

Crystal Structure Prediction via Basin-Hopping Global Optimization Employing Tiny Periodic Simulation Cells, with Application to Water-Ice

A crystal structure prediction algorithm for use in periodic boundary conditions with empirical rigid models is presented, which employs (i) unrestricted cutoff radii for the real-space interactions, thus allowing the treatment of even very small unit cells, and (ii) a global-optimization algorithm based on the basin-hopping method of Wales et al. (D. J. Wales and J. P. K. Doye, J. Phys. Chem. A 1997, 101, 5111). The algorithm is then applied to the TIP4P model of water (W. L. Jorgensen et al., J. Chem. Phys. 1983, 79, 926.) in order to find the lowest enthalpy water-ice crystalline structures in the pressure region 0-8000 bar, in unit cells holding in the range of 1-16 molecules, and a database of the 10 lowest enthalpy structures found at pressures 0, 4000, and 8000 bar is presented. The algorithm finds many of the ice polymorphs and, in particular, finds that the lowest energy structure at zero pressure is almost exactly tied between an ice Ic (cubic ice) and ice Ih (hexagonal ice) structure, having near-identical energies.

Exploring energy landscapes: from molecular to mesoscopic systems

We review a comprehensive computational framework to survey the potential energy landscape for systems composed of rigid or partially rigid molecules. Illustrative case studies relevant to a wide range of molecular clusters and soft and condensed matter systems are discussed.

A survey of the potential energy surface for the (benzene)13 cluster

We report on a survey of the potential energy surface for the 13-molecule benzene cluster, (C(6)H(6))(13), bound by an atom-atom intermolecular potential developed from first principles. The potential, which has an anisotropic repulsion term, is found to support distinct pairs of structures of C(3), C(i), and S(6) symmetry as low-lying minima, including a C(3) global minimum. The organisation of the low-lying region of the potential energy surface suggests that one of the S(6) structures is likely to act as a kinetic trap, hindering efficient relaxation to the global minimum, in agreement with the hypothesis that two isomers coexist in this system, suggested by spectroscopic experiments.

Electronic structure and bonding of ozone

The ground and low-lying states of ozone (O(3)) have been studied by multireference variational methods and large basis sets. We have constructed potential energy curves along the bending coordinate for (1,2) (1)A('), (1,2) (1)A("), (1,2) (3)A('), and (1,2) (3)A(") symmetries, optimizing at the same time the symmetric stretching coordinate. Thirteen minima have been located whose geometrical and energetic characteristics are in very good agreement with existing experimental data. Special emphasis has been given to the interpretation of the chemical bond through valence-bond-Lewis diagrams; their appropriate use captures admirably the bonding nature of the O(3) molecule. The biradical character of its ground state, adopted long ago by the scientific community, does not follow from a careful analysis of its wave function.

Potential energy and free energy landscapes

Familiar concepts for small molecules may require reinterpretation for larger systems. For example, rearrangements between geometrical isomers are usually considered in terms of transitions between the corresponding local minima on the underlying potential energy surface, V. However, transitions between bulk phases such as solid and liquid, or between the denatured and native states of a protein, are normally addressed in terms of free energy minima. To reestablish a connection with the potential energy surface we must think in terms of representative samples of local minima of V, from which a free energy surface is projected by averaging over most of the coordinates. The present contribution outlines how this connection can be developed into a tool for quantitative calculations. In particular, stepping between the local minima of V provides powerful methods for locating the global potential energy minimum, and for calculating global thermodynamic properties. When the transition states that link local minima are also sampled we can exploit statistical rate theory to obtain insight into global dynamics and rare events. Visualizing the potential energy landscape helps to explain how the network of local minima and transition states determines properties such as heat capacity features, which signify transitions between free energy minima. The organization of the landscape also reveals how certain systems can reliably locate particular structures on the experimental time scale from among an exponentially large number of local minima. Such directed searches not only enable proteins to overcome Levinthal's paradox but may also underlie the formation of "magic numbers" in molecular beams, the self-assembly of macromolecular structures, and crystallization.

Structural isomers and reactivity for Rh6 and Rh6+

The structure, energetics, and interconversion of isomers of Rh(6) and Rh(6)(+) are studied by using density functional theory with Gaussian basis sets, using guess structures derived from basin-hopping simulations, and obtained by using the Sutton-Chen potential. A large range of spin multiplicities is considered for each isomer. Our calculations suggest two low-lying structures as possible structural isomers: a square bipyramid and a trigonal prism. The reactivity of these two candidate structural isomers with respect to adsorption of nitric oxide is studied via location of reaction transition states and calculation of reaction barriers. Similarities and differences with surface reaction studies are highlighted. These data provide powerful evidence that structural isomerism, and not different spin states, is responsible for the observed biexponential reaction kinetics.

Global optimization of ionic Mg(n)F(2n) (n=1-30) clusters

The global optimization basin-hopping (BH) method has been used to locate the global minima (GM) of Mg(n)F(2n) (n=1-30) clusters using a Born-Mayer-type potential. Some of the GM were particularly difficult to find, requiring more than 1.5 x 10(4) BH steps. We have found that both the binding energy per MgF2 unit and the effective volume of the GM isomers increase almost linearly with n, and that cluster symmetry decreases with cluster size. The data derived from the BH runs reveal a growing density of local minima just above the GM as n increases. Despite this, the attraction basin around each GM is relatively large, since after all their atomic coordinates are randomly displaced by values as high as 2.0 bohrs, the perturbed structures, upon reoptimization, relax back to the GM in more than 50% of the cases (except for n=10 and 11). The relative stabilities derived from energy second differences suggest that n=8,10,13,15, and 20 are probably the magic numbers for these systems. Mass spectrum experiments would be very useful to clarify this issue.

Relaxation dynamics and structural isomerism in Nb10 and Nb10+

The structure, energetics, and interconversion of isomers of Nb10 and Nb10+ are studied using density functional theory with Gaussian basis sets, using guess structures derived from basin-hopping simulations with the Finnis-Sinclair [Philos. Mag. A 50, 45 (1984)] potential. These results are used as input to a master equation approach to model the relaxation of these clusters. Ionization potentials are calculated for all relevant minima, as are the infrared spectra. On the basis of these data, and known experimental results, plausible explanations are given for the biexponential reaction kinetics observed for Nb10 and Nb10+ with respect to small molecule adsorbates. In principle, this approach could be extended to investigate any midsized transition metal cluster that exhibits structural isomerism.

Equilibrium thermodynamics from basin-sampling

We present a "basin-sampling" approach for calculation of the potential energy density of states for classical statistical models. It combines a Wang-Landau-type uniform sampling of local minima and a novel approach for approximating the relative contributions from local minima in terms of the volumes of basins of attraction. We have employed basin-sampling to study phase changes in atomic clusters modeled by the Lennard-Jones potential and for ionic clusters. The approach proves to be efficient for systems involving broken ergodicity and has allowed us to calculate converged heat capacity curves for systems that could previously only be treated using the harmonic superposition approximation. Benchmarks are also provided by comparison with parallel tempering and Wang-Landau simulations, where these proved feasible.

Evolution of the Properties of AlnNn Clusters with Size

A global optimization of stoichiometric (AlN)(n) clusters (n = 1-25, 30, 35, ..., 95, 100) has been performed using the basin-hopping (BH) method and describing the interactions with simple and yet realistic interatomic potentials. The results for the smaller isomers agree with those of previous electronic structure calculations, thus validating the present scheme. The lowest-energy isomers found can be classified in three different categories according to their structural motifs: (i) small clusters (n = 2-5), with planar ring structures and 2-fold coordination, (ii) medium clusters (n = 6-40), where a competition between stacked rings and globular-like empty cages exists, and (iii) large clusters (n > 40), large enough to mix different elements of the previous stage. All the atoms in small and medium-sized clusters are in the surface, while large clusters start to display interior atoms. Large clusters display a competition between tetrahedral and octahedral-like features: the former lead to a lower energy interior in the cluster, while the latter allow for surface terminations with a lower energy. All of the properties studied present different regimes according to the above classification. It is of particular interest that the local properties of the interior atoms do converge to the bulk limit. The isomers with n = 6 and 12 are specially stable with respect to the gain or loss of AlN molecules.

Infrared spectrum of cyclic ozone: A theoretical investigation

The infrared absorption spectrum of cyclic ozone is calculated by means of a new ab initio potential energy surface, the dipole moment function, and exact quantum mechanical dynamics calculations. Five different isotopomers are considered. The absorption line for excitation of the bending fundamental near 800 cm(-1) is by far the strongest band; all other bands are more than one order of magnitude less intense. This spectral pattern as well as the isotope shifts for the various isotopomers are important for identifying cyclic ozone. Several possibilities for accessing the ring minimum of cyclic ozone are also discussed on the basis of recent electronic structure calculations.

Phase changes in Lennard-Jones mixed clusters with composition ArnXe6−n (n=0,1,2)

We have carried out parallel tempering Monte Carlo calculations on the binary six-atom mixed Lennard-Jones clusters, Ar(n)Xe(6-n) (n=0,1,2). We have looked at the classical configurational heat capacity C(V)(T) as a probe of phase behavior. All three clusters show a feature in the heat capacity in the region of 15-20 K. The Ar(2)Xe(4) cluster exhibits a further peak in the heat capacity near 7 K. We have also investigated dynamical properties of the Ar(2)Xe(4) cluster as a function of temperature using molecular dynamics. We report the interbasin isomerization rate and the bond fluctuation parameter obtained from these calculations. At 7 K, the isomerization rate is on the order of 0.01 ns(-1); at 20 K, the isomerization rate is greater than 10 ns(-1). Furthermore, at 7 K, the bond fluctuation parameter is less than 3%; at 20 K, it is in the range of 10-15% (depending on the sampling time used). Using this information, together with Monte Carlo quenching data, we assign the 15-20 K feature in the heat capacity to a solid-liquid phase change and the 7-K peak to a solid-solid phase change. We believe this is the smallest Lennard-Jones cluster system yet shown to exhibit solid-solid phase change behavior.

Interaction of lead atom with atmospheric dioxygen and ozone: Quantic study of the structure and the stability of resulting Pb(On) (n=1,2,3) compounds

Lead reaction with oxygen and ozone molecules is of a great importance for the study of the impact of this metal in the atmosphere medium. Stable species, intermediates, and transition states of possible resulting complexes have been studied with the three parameter hybrid B3LYP exchange-correlation DFT method, and coupled cluster with single, double, and triple excitation methods. Geometry, and spectroscopic and thermodynamic properties obtained for the different species are presented, discussed, and compared to available experimental data. On the basis of the knowledge of the thermal dissociation enthalpies and the absorbed wave length edges calculated for each species, we have drawn the most probable oxygenated complexes present in the atmosphere.

A dynamic lattice searching method for fast optimization of Lennard-Jones clusters

A highly efficient unbiased global optimization method called dynamic lattice searching (DLS) was proposed. The method starts with a randomly generated local minimum, and finds better solution by a circulation of construction and searching of the dynamic lattice (DL) until the better solution approaches the best solution. The DL is constructed adaptively based on the starting local minimum by searching the possible location sites for an added atom, and the DL searching is implemented by iteratively moving the atom located at the occupied lattice site with the highest energy to the vacant lattice site with the lowest energy. Because the DL can greatly reduce the searching space and the number of the time-consuming local minimization procedures, the proposed DLS method runs at a very high efficiency, especially for the clusters of larger size. The performance of the DLS is investigated in the optimization of Lennard-Jones (LJ) clusters up to 309 atoms, and the structure of the LJ(500) is also predicted. Furthermore, the idea of dynamic lattice can be easily adopted in the optimization of other molecular or atomic clusters. It may be a promising approach to be universally used for structural optimizations in the chemistry field.

Unbiased global optimization of Lennard-Jones clusters for N < or =201 using the conformational space annealing method

We apply the conformational space annealing method to the Lennard-Jones clusters and find all known lowest energy configurations up to 201 atoms, without using extra information of the problem such as the structures of the known global energy minima. In addition, the robustness of the algorithm with respect to the randomness of initial conditions of the problem is demonstrated by ten successful independent runs up to 183 atoms. Our results indicate that this method is a general and yet efficient global optimization algorithm applicable to many systems.

Effect of compression on the global optimization of atomic clusters

Recently, Locatelli and Schoen proposed a transformation of the potential energy that aids the global optimization of Lennard-Jones clusters with nonicosahedral global minima. These cases are particularly difficult to optimize because the potential energy surface has a double funnel topography with the global minimum at the bottom of the narrower funnel. Here we analyze the effect of this type of transformation on the topography of the potential energy surface. The transformation, which physically corresponds to a compression of the cluster, first reduces the number of stationary points on the potential energy surface. Secondly, we show that for a 38-atom cluster with a face-centered-cubic global minimum the transformation causes the potential energy surface to become increasingly dominated by the funnel associated with the global minimum. The transformation has been incorporated in the basin-hopping algorithm using a two-phase approach.

Global optimization of clusters, crystals, and biomolecules

Finding the optimal solution to a complex optimization problem is of great importance in many fields, ranging from protein structure prediction to the design of microprocessor circuitry. Some recent progress in finding the global minima of potential energy functions is described, focusing on applications of the simple "basin-hopping" approach to atomic and molecular clusters and more complicated hypersurface deformation techniques for crystals and biomolecules. These methods have produced promising results and should enable larger and more complex systems to be treated in the future.