A global optimization approach for Lennard‐Jones microclusters

Trilateration-Based Multilevel Method for Minimizing the Lennard-Jones Potential

Simulating atomic evolution for the mechanics and structure of materials presents an ever-growing challenge due to the huge number of degrees of freedom borne from the high-dimensional spaces in which increasingly high-fidelity material models are defined. To efficiently exploit the domain-, data-, and approximation-based hierarchies hidden in many such problems, we propose a trilateration-based multilevel method to initialize the underlying optimization and benchmark its application on the simple yet practical Lennard-Jones potential. We show that by taking advantage of a known hierarchy present in this problem, not only a faster convergence, but also a better local minimum can be achieved comparing to random initial guess.

Arbitrarily tight α BB underestimators of general non-linear functions over sub-optimal domains

In this paper we explore the construction of arbitrarily tight α BB relaxations of C 2 general non-linear non-convex functions. We illustrate the theoretical challenges of building such relaxations by deriving conditions under which it is possible for an α BB underestimator to provide exact bounds. We subsequently propose a methodology to build α BB underestimators which may be arbitrarily tight (i.e., the maximum separation distance between the original function and its underestimator is arbitrarily close to 0) in some domains that do not include the global solution (defined in the text as "sub-optimal"), assuming exact eigenvalue calculations are possible. This is achieved using a transformation of the original function into a μ -subenergy function and the derivation of α BB underestimators for the new function. We prove that this transformation results in a number of desirable bounding properties in certain domains. These theoretical results are validated in computational test cases where approximations of the tightest possible μ -subenergy underestimators, derived using sampling, are compared to similarly derived approximations of the tightest possible classical α BB underestimators. Our tests show that μ -subenergy underestimators produce much tighter bounds, and succeed in fathoming nodes which are impossible to fathom using classical α BB.

Continuous symmetry measures for complex symmetry group

Symmetry is a fundamental property of nature, used extensively in physics, chemistry, and biology. The Continuous symmetry measures (CSM) is a method for estimating the deviation of a given system from having a certain perfect symmetry, which enables us to formulate quantitative relation between symmetry and other physical properties. Analytical procedures for calculating the CSM of all simple cyclic point groups are available for several years. Here, we present a methodology for calculating the CSM of any complex point group, including the dihedral, tetrahedral, octahedral, and icosahedral symmetry groups. We present the method and analyze its performances and errors. We also introduce an analytical method for calculating the CSM of the linear symmetry groups. As an example, we apply these methods for examining the symmetry of water, the symmetry maps of AB4 complexes, and the symmetry of several Lennard-Jones clusters.

Shape-tailored polymer colloids on the road to become structural motifs for hierarchically organized materials

Anisometric polymer colloids are likely to behave differently when compared with centrosymmetric particles. Their study may not only shine new light on the organization of matter; they may also serve as building units with specific symmetries and complexity to build new materials from them. Polymer colloids of well-defined complex geometries can be obtained by packing a limited number of spherical polymer particles into clusters with defined configurations. Such supracolloidal architectures can be fabricated at larger scales using narrowly dispersed emulsion droplets as templates. Assemblies built from at least two different types of particles as elementary building units open perspectives in selective targeting of colloids with specific properties, aiming for mesoscale building blocks with tailor-made morphologies and multifunctionality. Polymer colloids with defined geometries are also ideal to study shape-dependent properties such as the diffusion of complex particles.

Complex colloidal microclusters from aerosol droplets

In this Article, we report on a packing scheme of monodisperse colloidal nanospheres by aerosol-assisted clustering. When an aqueous suspension of colloidal nanospheres was sprayed into aerosol droplets by an ultrasonic nebulizer, the nanospheres were encapsulated in the aerosol droplets and the evaporation of water from the droplets at high temperatures led to the packings of nanospheres. The configurations of the colloidal nanospheres minimized the interparticle potential energy or the second moment of mass distribution depending on the number of the constituting nanospheres. Other types of nonspherical microparticles or hollow microclusters were also produced by self-organizing organic-inorganic binary colloids of different sizes in an aerosol spray pyrolysis reactor. The aerosol-assisted fabrication of colloidal clusters was very effective as compared to the method based on oil-in-water or water-in-oil emulsions, in which the removal of residual oil was not easy and time-consuming.

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.

Global minimum structure searches via particle swarm optimization

Novel implementation of the evolutionary approach known as particle swarm optimization (PSO) capable of finding the global minimum of the potential energy surface of atomic assemblies is reported. This is the first time the PSO technique has been used to perform global optimization of minimum structure search for chemical systems. Significant improvements have been introduced to the original PSO algorithm to increase its efficiency and reliability and adapt it to chemical systems. The developed software has successfully found the lowest-energy structures of the LJ(26) Lennard-Jones cluster, anionic silicon hydride Si(2)H(5) (-), and triply hydrated hydroxide ion OH(-) (H(2)O)(3). It requires relatively small population sizes and demonstrates fast convergence. Efficiency of PSO has been compared with simulated annealing, and the gradient embedded genetic algorithm.

Self-Organization of Bidisperse Colloids in Water Droplets

Most of the colloidal clusters have been produced from oil-in-water emulsions with identical microspheres dispersed in oil droplets. Here, we present new types of binary colloidal clusters from phase-inverted water-in-oil emulsions using various combinations of two different colloids with several size ratios: monodisperse silica or polystyrene microspheres for larger particles and silica or titania nanoparticles for smaller particles. Obviously, a better understanding of how finite groups of different colloids self-organize in a confined geometry may help us control the structure of matter at multiple length scales. In addition, since aqueous dispersions have much better phase stability, we could produce much more diverse colloidal materials from water-in-oil emulsions rather than from oil-in-water emulsions. Interestingly, the configurations of the large microspheres were not changed by the presence of the small particles. However, the arrangement of the smaller particles was strongly dependent on the nature of the interparticle interactions. The experimentally observed structural evolutions were consistent with the numerical simulations calculated using Surface Evolver. These clusters with nonisotropic structures can be used as building blocks for novel colloidal structures with unusual properties or by themselves as light scatterers, diffusers, and complex adaptive matter exhibiting emergent behavior.

ASTRO-FOLD: a combinatorial and global optimization framework for Ab initio prediction of three-dimensional structures of proteins from the amino acid sequence

The field of computational biology has been revolutionized by recent advances in genomics. The completion of a number of genome projects, including that of the human genome, has paved the way toward a variety of challenges and opportunities in bioinformatics and biological systems engineering. One of the first challenges has been the determination of the structures of proteins encoded by the individual genes. This problem, which represents the progression from sequence to structure (genomics to structural genomics), has been widely known as the structure-prediction-in-protein-folding problem. We present the development and application of ASTRO-FOLD, a novel and complete approach for the ab initio prediction of protein structures given only the amino acid sequences of the proteins. The approach exhibits many novel components and the merits of its application are examined for a suite of protein systems, including a number of targets from several critical-assessment-of-structure-prediction experiments.

Stochastic formulation of sampling dynamics in generalized ensemble methods

The fundamental relations of the sampling process in the multicanonical ensemble and simulated tempering have been studied in the expanded ensembles formalism. The simulated tempering is identified as the multicanonical sampling with the generalized weight determined by a Laplace transform of the temperature weight. The characteristic dynamics of both sampling methods has been verified in the stochastic formulation of the sampling process. Our study gives a necessary and sufficient condition for the weights to realize a uniform sampling in the energy and temperature spaces. Based on the stochastic model, robust force biased iteration schemes have been proposed to allow automatic determinations of uniform sampling weights.

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.

Determination of multicanonical weight based on a stochastic model of sampling dynamics

Based on the stochastic interpretation of the sampling process modeled by a Langevin equation, we present an effective iteration scheme to determine the weight in multicanonical molecular dynamics. Our method enables an automatic determination of the weight producing a uniform energy sampling via an iterative cancellation of the deterministic force in a Langevin equation. The deterministic force has been calculated from the energy trajectory by identifying the moments of the transition probability of a Fokker-Planck equation associated with a Langevin equation. The intimate relationship between the sampling process and the stochastic dynamics has been verified by applying the iteration scheme to a helix-coil transition of the 8-polyalanine system in a gas phase.

Dynamical origin of uniform sampling in multicanonical ensemble

The stochastic model describing the sampling process in multicanonical ensemble has been derived by considering the sampling process as an overdamped Brownian motion on the free energy surface. The essential dynamics of the multicanonical sampling has been characterized by a Langevin equation in a piecewise multivalleyed free energy landscape, modulated by a temperature-dependent curvature. Based on the stochastic model we showed that the multicanonical weight can be determined by interpolating maximum probability energy points of the canonical samplings at different temperatures.

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.