Global cluster geometry optimization by a phenotype algorithm with Niches: Location of elusive minima,and low-order scaling with cluster size

The problem of global geometry optimization of clusters is addressed with a phenotype variant of the method of genetic algorithms, with several novel performance enhancements. The resulting algorithm is applied to Lennard–Jones clusters as benchmark system, with up to 150 atoms. The well-known, difficult cases involving nonicosahedral global minima can be treated reliably using the concept of niches. The scaling of computer time with cluster size is approximately cubic, which is crucial for future applications to much larger clusters. © 1999 John Wiley & Sons, Inc. J Comput Chem 20: 1752–1759, 1999     

Free energy of solvation of a small Lennard–Jones particle

In molecular dynamics (MD) and Monte Carlo (MC) free energy calculations, the choices of the thermodynamic paths from state a to state b affect the accuracy of the result and the efficiency of the programs. Most of the problems occur at the initial stages of growing in a new particle into a solvent. Based on statistical mechanical perturbation theory, an accurate and efficient direct calculation of inserting a small Lennard–Jones particle into solvent is derived. This eliminates the need for calculation of the initial stages of growing in a new particle by MD or MC simulation. Examples are given to show the utility of direct calculation. The recommended procedure is to use direct calculation for a small Lennard–Jones particle and then use MD or MC simulations to calculate the ΔG of changing the small Lennard–Jones particle into the target molecule. © 1994 by John Wiley & Sons, Inc.     

A strategy to find minimal energy nanocluster structures

An unbiased strategy to search for the global and local minimal energy structures of free standing nanoclusters is presented. Our objectives are twofold: to find a diverse set of low lying local minima, as well as the global minimum. To do so, we use massively the fast inertial relaxation engine algorithm as an efficient local minimizer. This procedure turns out to be quite efficient to reach the global minimum, and also most of the local minima. We test the method with the Lennard–Jones (LJ) potential, for which an abundant literature does exist, and obtain novel results, which include a new local minimum for LJ₁₃, 10 new local minima for LJ₁₄, and thousands of new local minima for . Insights on how to choose the initial configurations, analyzing the effectiveness of the method in reaching low‐energy structures, including the global minimum, are developed as a function of the number of atoms of the cluster. Also, a novel characterization of the potential energy surface, analyzing properties of the local minima basins, is provided. The procedure constitutes a promising tool to generate a diverse set of cluster conformations, both two‐ and three‐dimensional, that can be used as an input for refinement by means of ab initio methods. © 2013 Wiley Periodicals, Inc.     

Viscosity Prediction for Oxygen,Nitrogen and Their Mixtures at Zero and Moderately Dense Regimes via Semi‐Empirically Based Assessment

The viscosity coefficients for the gaseous states of N₂ and O₂ and their mixtures are determined at zero and moderately density regimes. The Lennard‐Jones 12–6 (LJ 12–6) potential energy function is used as the initial model potential required y the technique. The interaction potential energies from the inversion procedure reproduce the viscosity commensurate to the best measurements. The initial density dependence of gaseous viscosity coefficient according to the Rainwater‐Friend theory, which was given by Najafi et al., has been considered for pure N₂ and pure O₂.     

Global optimization using bad derivatives: Derivative-free method for molecular energy minimization

A general method designed to isolate the global minimum of a multidimensional objective function with multiple minima is presented. The algorithm exploits an integral “coarse-graining” transformation of the objective function, U, into a smoothed function with few minima. When the coarse-graining is defined over a cubic neighborhood of length scale ϵ, the exact gradient of the smoothed function, 𝒰_ϵ, is a simple three-point finite difference of U. When ϵ is very large, the gradient of 𝒰_ϵ appears to be a “bad derivative” of U. Because the gradient of 𝒰_ϵ is a simple function of U, minimization on the smoothed surface requires no explicit calculation or differentiation of 𝒰_ϵ. The minimization method is “derivative-free” and may be applied to optimization problems involving functions that are not smooth or differentiable. Generalization to functions in high-dimensional space is straightforward. In the context of molecular conformational optimization, the method may be used to minimize the potential energy or, preferably, to maximize the Boltzmann probability function. The algorithm is applied to conformational optimization of a model potential, Lennard–Jones atomic clusters, and a tetrapeptide. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1445–1455, 1998     

A detailed investigation on the global minimum structures of mixed rare‐gas clusters: Geometry,energetics, and site occupancy

We performed a global minimum search of mixed rare‐gas clusters by applying an evolutionary algorithm (EA), which was recently proposed for binary atomic systems (Marques and Pereira, Chem. Phys. Lett. 2010, 485, 211). Before being applied to the potentials used in this work, the EA was further tested against results previously reported for the Ar_NXe_38?N clusters and several new putative global minima were discovered. We employed either simple Lennard‐Jones (LJ) potentials or more realistic functions to describe pair interactions in Ar_NKr_38?N, Ar_NXe_38?N, and Kr_NXe_38?N clusters. The long‐range tail of the pair‐potentials shows some influence on the energetic features and shape of the structure of clusters. In turn, core–shell type structures are mostly observed for global minima of the binary rare‐gas clusters, for both accurate and LJ potentials. However, the long‐range tail of the potential may have influence on the type of atoms that segregate on the surface or form the core of the cluster. While relevant differences for the preferential site occupancy occur between the two potentials for Ar_NKr_38?N (for N > 21), the type of atoms that segregate on the surface for Ar_NXe_38?N and Kr_NXe_38?N clusters is unaffected by the accuracy of the long‐range part of the interaction in almost all cases. Moreover, the global minimum search for model‐potentials in binary systems reveals that the surface‐site occupancy is mainly determined by the combination of two parameters: the size ratio of the two types of particles forming the cluster and the minimum‐energy ratio corresponding to the pair‐interactions between unlike atoms. © 2012 Wiley Periodicals, Inc.     

PDECO: Parallel differential evolution for clusters optimization

The optimization of the atomic and molecular clusters with a large number of atoms is a very challenging topic. This article proposes a parallel differential evolution (DE) optimization scheme for large‐scale clusters. It combines a modified DE algorithm with improved genetic operators and a parallel strategy with a migration operator to address the problems of numerous local optima and large computational demanding. Results of Lennard–Jones (LJ) clusters and Gupta‐potential Co clusters show the performance of the algorithm surpasses those in previous researches in terms of successful rate, convergent speed, and global searching ability. The overall performance for large or challenging LJ clusters is enhanced significantly. The average number of local minimizations per hit of the global minima for Co clusters is only about 3–4% of that in previous methods. Some global optima for Co are also updated. We then apply the algorithm to optimize the Pt clusters with Gupta potential from the size 3 to 130 and analyze their electronic properties by density functional theory calculation. The clusters with 13, 38, 54, 75, 108, and 125 atoms are extremely stable and can be taken as the magic numbers for Pt systems. It is interesting that the more stable structures, especially magic‐number ones, tend to have a larger energy gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital. It is also found that the clusters are gradually close to the metal bulk from the size N > 80 and Pt₃₈ is expected to be more active than Pt₇₅ in catalytic reaction. © 2013 Wiley Periodicals, Inc.     

Size‐guided multi‐seed heuristic method for geometry optimization of clusters: Application to benzene clusters

Since searching for the global minimum on the potential energy surface of a cluster is very difficult, many geometry optimization methods have been proposed, in which initial geometries are randomly generated and subsequently improved with different algorithms. In this study, a size‐guided multi‐seed heuristic method is developed and applied to benzene clusters. It produces initial configurations of the cluster with n molecules from the lowest‐energy configurations of the cluster with n − 1 molecules (seeds). The initial geometries are further optimized with the geometrical perturbations previously used for molecular clusters. These steps are repeated until the size n satisfies a predefined one. The method locates putative global minima of benzene clusters with up to 65 molecules. The performance of the method is discussed using the computational cost, rates to locate the global minima, and energies of initial geometries. © 2018 Wiley Periodicals, Inc.     

Exploring the first-shell and second-shell structures arising in the microsolvation of Li+ by rare gases

An evolutionary algorithm was used to search for the low-energy structures of Li⁺Ar_n and Li⁺Kr_n (n = 1 − 14). Two functions were used to describe the interaction potential at the CCSD(T)/aug-cc-pVQZ level of theory: one is based on a sum of all pair potentials, whereas the other includes three-body interactions. In general, the global minimum structures are similar for both Li⁺Ar_n and Li⁺Kr_n. Modifications in the octahedral structure of the first solvation shell lead to a high-energy penalty. Conversely, the second solvation shell shows a panoply of minima with similar energies that are likely to be interconverted. Post-optimization at the MP2 level confirmed that, for n = 2 and 3, one has to include three-body terms in the potential to reproduce the low-energy structures. Additionally, MP2 calculations indicate that energy reorder of the global minimum structure observed for Li⁺Kr₈ is related to the Kr₃ Axilrod-Teller-Muto term included in the potential.     

MC(JBW): Simple but smart Monte Carlo algorithm for free energy simulations of multiconformational molecules

Many of the most common molecular simulation methods, including Monte Carlo (MC) and molecular or stochastic dynamics (MD or SD), have significant difficulties in sampling the space of molecular potential energy surfaces characterized by multiple conformational minima and significant energy barriers. In such cases improved sampling can be obtained by special techniques that lower such barriers or somehow direct search steps toward different low energy regions of space. We recently described a hybrid MC/SD algorithm [MC(JBW)/SD] incorporating such a technique that directed MC moves of selected torsion and bond angles toward known low energy regions of conformational space. Exploration of other degrees of freedom was left to the SD part of the hybrid algorithm. In the work described here, we develop a related but simpler simulation algorithm that uses only MC to sample all degrees of freedom (e.g., stretch, bend, and torsion). We term this algorithm MC(JBW). Using simulations on various model potential energy surfaces and on simple molecular systems (n-pentane, n-butane, and cyclohexane), MC(JBW) is shown to generate ensembles of states that are indistinguishable from the canonical ensembles generated by classical Metropolis MC in the limit of very long simulations. We further demonstrate the utility of MC(JBW) by evaluating the room temperature free energy differences between conformers of various substituted cyclohexanes and the larger ring hydrocarbons cycloheptane, cyclooctane, cyclononane, and cyclodecane. The results compare favorably with available experimental data and results from previously reported MC(JBW)/SD conformational free energy calculations. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1736–1745, 1998     

Do intelligent configuration search techniques outperform random search for large molecules?

We compare three global configuration search methods on a scalable model problem to measure relative performance over a range of molecule sizes. Our model problem is a 2-D polymer composed of atoms connected by rigid rods in which all pairs of atoms interact via Lennard–Jones potentials. The global minimum energy can be calculated analytically. The search methods are all hybrids combining a global sampling algorithm with a local refinement technique. The sampling methods are simulated annealing (SA ), genetic algorithms (GA ), and random search. Each of these uses a conjugate gradient (CG ) routine to perform the local refinement. Both GA and SA perform progressively better relative to random search as the molecule size increases. We also test two other local refinement techniques in addition to CG , coupled to random search as the global method. These are simplex followed by CG and simplex followed by block-truncated Newton. For small problems, the addition of the intermediate simplex step improved the performance of the overall hybrid method. © 1992 John Wiley & Sons, Inc.     

Planar Hypercoordinate Carbons in Alkali Metal Decorated CE32− and CE22− Dianions

After exploring the potential energy surfaces of M_mCE₂^p (E=S−Te, M=Li−Cs, m=2, 3 and p=m-2) and M_nCE₃^q (E=S−Te, M=Li−Cs, n=1, 2, q=n-2) combinations, we introduce 38 new global minima containing a planar hypercoordinate carbon atom (24 with a planar tetracoordinate carbon and 14 with a planar pentacoordinate carbon). These exotic clusters result from the decoration of V-shaped CE₂²⁻ and Y-shaped CE₃²⁻ dianions, respectively, with alkali counterions. All these 38 systems fulfill the geometrical and electronic criteria to be considered as true planar hypercoordinate carbon systems. Chemical bonding analyses indicate that carbon is covalently bonded to chalcogens and ionically connected to alkali metals.     

Conformational search using a molecular dynamics–minimization procedure: Applications to clusters of coulombic charges,Lennard–Jones particles,and waters

A new conformational search method, molecular dynamics–minimization (MDM), is proposed, which combines a molecular dynamics sampling strategy with energy minimizations in the search for low-energy molecular structures. This new method is applied to the search for low energy configurations of clusters of coulombic charges on a unit sphere, Lennard–Jones clusters, and water clusters. The MDM method is shown to be efficient in finding the lowest energy configurations of these clusters. A closer comparison of MDM with alternative conformational search methods on Lennard–Jones clusters shows that, although MDM is not as efficient as the Monte Carlo–minimization method in locating the global energy minima, it is more efficient than the diffusion equation method and the method of minimization from randomly generated structures. Given the versatility of the molecular dynamics sampling strategy in comparison to Monte Carlo in treating molecular complexes or molecules in explicit solution, one anticipates that the MDM method could be profitably applied to conformational search problems where the number of degrees of freedom is much greater. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 60–70, 1998     

Structural evolution and electronic properties of cerium doped germanium anionic nanocluster CeGen− (n = 5–17): Theoretical investigation

The rare earth element doped germanium cluster represents a fundamental nanomaterial and exhibits potential in next-generation industrial electronic nanodevices and applied semiconductors. Herein, the cerium-doped germanium anionic nanocluster CeGe_n⁻ (n = 5–17) has been comprehensively investigated by the double hybrid density functional theory of mPW2PLYP associated with the unbiased global searching technique of artificial bee colony algorithm. The cluster's growth pattern undergoes three stages: n = 5–9 with the replaced structure, n = 10–15 with the linked structure, and n ≥ 16 forming a Ce-encapsulated in Ge inner cage motif. The clusters' PES, IR, and Raman spectra were simulated, and their HOMO-LUMO gap, magnetism, charge transfer, and relative stability were predicted. These theoretical values can serve as a reference for future experiments to some extent. Moreover, the special D_2d symmetry cage geometry of CeGe₁₆⁻ leads to a higher stability and preferred energy gap, making it an ideal candidate for further studies on its aromaticity, UV–vis spectra, and chemical bonding characteristics. In summary, CeGe₁₆⁻ has excellent optical activity that can be potentially employed as a building block in the development of optoelectronic functional materials.     

Derivation of memory function from its equation of motion

A new form of Memory Function (MF) appearing in the Mori's formalism has been derived using plausible approximations. In addition to the fact that present form of MF satisfies sum-rules upto sixth order, it has special characteristic of presence of one more adjustable parameter. It is also found that the present form of MF behaves as sech ^v (bt) under suitable conditions. The utility of the present MF is exemplified by studying time evolution of velocity auto correlation function and transport coefficients of Lennard–Jones fluids.     

Optimization of bimetallic Cu–Au and Ag–Au clusters by using a modified adaptive immune optimization algorithm

A modified adaptive immune optimization algorithm (AIOA) is designed for optimization of Cu–Au and Ag–Au bimetallic clusters with Gupta potential. Compared with homoatom clusters, there are homotopic isomers in bimetallic cluster, so atom exchange operation is presented in the modified AIOA. The efficiency of the algorithm is tested by optimization of Cu_nAu_38‐n (0 ≤ n ≤ 38). Results show that all the structures with the putative global minimal energies are successfully located. In the optimization of Ag_nAu_55‐n (0 ≤ n ≤ 55) bimetallic clusters, all the structures with the reported minimal energies are obtained, and 36 structures with even lower potential energies are found. On the other hand, with the optimized structures of Cu_nAu_55‐n, it is shown that all 55‐atom Cu–Au bimetallic clusters are Mackay icosahedra except for Au₅₅, which is a face‐centered cubic (fcc)‐like structure; Cu₅₅, Cu₁₂Au₄₃, and Cu₁Au₅₄ have two‐shell Mackay icosahedral geometries with I_h point group symmetry. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

Exponential repulsion improves structural predictability of molecular docking

Molecular docking is a powerful tool for theoretical prediction of the preferred conformation and orientation of small molecules within protein active sites. The obtained poses can be used for estimation of binding energies, which indicate the inhibition effect of designed inhibitors, and therefore might be used for in silico drug design. However, the evaluation of ligand binding affinity critically depends on successful prediction of the native binding mode. Contemporary docking methods are often based on scoring functions derived from molecular mechanical potentials. In such potentials, nonbonded interactions are typically represented by electrostatic interactions between atom‐centered partial charges and standard 6–12 Lennard–Jones potential. Here, we present implementation and testing of a scoring function based on more physically justified exponential repulsion instead of the standard Lennard–Jones potential. We found that this scoring function significantly improved prediction of the native binding modes in proteins bearing narrow active sites such as serine proteases and kinases. © 2016 Wiley Periodicals, Inc.     

New optimization method for conformational energy calculations on polypeptides: Conformational space annealing

A new optimization method is presented to search for the global minimum-energy conformations of polypeptides. The method combines essential aspects of the build-up procedure and the genetic algorithm, and it introduces the important concept of “conformational space annealing.” Instead of considering a single conformation, attention is focused on a population of conformations while new conformations are obtained by modifying a “seed conformation.” The annealing is carried out by introducing a distance cutoff, D_cut, which is defined in the conformational space; D_cut effectively divides the whole conformational space of local minima into subdivisions. The value of D_cut is set to a large number at the beginning of the algorithm to cover the whole conformational space, and annealing is achieved by slowly reducing it. Many distinct local minima designed to be distributed as far apart as possible in conformational space are investigated simultaneously. Therefore, the new method finds not only the global minimum-energy conformation but also many other distinct local minima as by-products. The method is tested on Met-enkephalin, a 24-dihedral angle problem. For all 100 independent runs, the accepted global minimum-energy conformation was obtained after about 2600 minimizations on average. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1222–1232     

Study of Cl−(H2O)n (n = 1–4) using basin‐hopping method coupled with density functional theory

Cl^?(H₂O)_n (n = 1–4) clusters were investigated using a basin‐hopping (BH) algorithm coupled with density functional theory (DFT). Structures, energetics, thermodynamics, vertical detachment energies, and vibrational frequencies were obtained from high‐level ab initio calculations. Through comparisons with previous theoretical and experimental data, it was demonstrated that the combination of the BH method and DFT could accurately predict the global and local minima of Cl^?(H₂O)_n (n = 1–4). Additionally, to optimize larger Cl^?(H₂O)_n (n > 4) clusters, several popular density functionals as well as DF‐LMP2 (Schütz et al., J. Chem. Phys. 2004, 121, 737) (second‐order Møller‐Plesset perturbation theory using local and density fitting approximations) were tested with appropriate basis sets through comparisons with MP2 optimized results. DF‐LMP2 will be used in future studies because its overall performance in describing the relative binding energies and the geometrical parameters of Cl^?(H₂O)_n (n = 1–4) was outstanding in this study. © 2013 Wiley Periodicals, Inc.     

Extraction of Intercalated O2 from Aligned Carbon Nanotubes: The Breaking of Intertube Paths and Exponential Changes in Resistance

In carbon nanotube films, the alignment of carbon nanotubes creates Lennard–Jones potentials at intertube junctions and trapped O₂ appears to oscillate at elevated temperatures. Electrical measurements reveal a low hopping barrier along the transverse direction and an underlying mechanism that involves intercalated molecules acting as charge carriers between tubes. Ab initio calculations support dynamic intercalation and charge transfer through O₂ bouncing between tubes.