Multicanonical basin hopping: a new global optimization method for complex systems

We introduce a new optimization algorithm that combines the basin-hopping method, which can be used to efficiently map out an energy landscape associated with minima, with the multicanonical Monte Carlo method, which encourages the system to move out of energy traps during the computation. As an example of implementing the algorithm for the global minimization of a multivariable system, we consider the Lennard-Jones systems containing 150-185 particles, and find that the new algorithm is more efficient than the original basin-hopping method.     

Optimal covering of C(60) fullerene by rare gases

Putative global energy minima of clusters formed by the adsorption of rare gases on a C(60) fullerene molecule, C(60)X(N) (X=Ne, Ar, Kr, Xe; N ≤ 70), are found using basin-hopping global optimization in an empirical potential energy surface. The association energies per rare gas atom as a function of N present two noticeable minima for Ne and Ar and just one for Kr and Xe. The minimum with the smallest N is the deepest one and corresponds to an optimal packing monolayer structure; the other one gives a monolayer with maximum packing. For Kr and Xe, optimal and maximum packing structures coincide. By using an isotropic average form of the X-C(60) interaction, we have established the relevance of the C(60) surface corrugation on the cluster structures. Quantum effects are relevant for Ne clusters. The adsorption of these rare gases on C(60) follows patterns that differ significantly from the ones found recently for He by means of experimental and theoretical methods.     

Global potential energy minima of C60(H2O)n clusters

Likely candidates for the global potential energy minima of C60(H2O)n clusters with n < or = 21 are found using basin-hopping global optimization. The potential energy surfaces are constructed using the TIP4P intermolecular potential for the water molecules, a Lennard-Jones water-fullerene potential, and a water-fullerene polarization potential, which depends on the first few nonvanishing C60 multipole polarizabilities. This combination produces a rather hydrophobic water-fullerene interaction. As a consequence, the water component of the lowest C60(H2O)n minima is quite closely related to low-lying minima of the corresponding TIP4P (H2O)n clusters. In most cases, the geometrical substructure of the water molecules in the C60(H2O)n global minimum coincides with that of the corresponding free water cluster. Exceptions occur when the interaction with C60 induces a change in geometry. This qualitative picture does not change significantly if we use the TIP3P model for the water-water interaction. Structures such as C60@(H2O)60, in which the water molecules surround the C60 fullerene, correspond to local minima with much higher potential energies. For such a structure to become the global minimum, the magnitude of the water-fullerene interaction must be increased to an unphysical value.     

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.     

计算机模拟原子簇的稳定构型和能量性质

用密度泛函(DFT)方法研究了铜原子簇Cu~n(n=2,3,4,6)的稳定几何构型和电子结构。通过拟合从头算势能面构造铜原子簇势能函数的双体、三体及四体项,并利用该函数和全局优化“Basin-Hopping”算法得到较大铜原子簇(n=13～56)能量极小的结构,计算结果与实验及其它计算结果相一致。     

Monte Carlo temperature basin paving with effective fragment potential: an efficient and fast method for finding low-energy structures of water clusters (H2O)20 and (H2O)25

Determining low-energy structures of large water clusters is a challenge for any optimization algorithm. In this work, we have developed a new Monte Carlo (MC)-based method, temperature basin paving (TBP), which is related to the well-known basin hopping method. In the TBP method, the Boltzmann weight factor used in MC methods is dynamically modified based on the history of the simulation. The states that are visited more are given a lower probability by increasing their temperatures and vice versa. This allows faster escapes from the states frequently visited in the simulation. We have used the TBP method to find a large number of low-energy minima of water clusters of size 20 and 25. We have found structures energetically same to the global minimum structures known for these two clusters. We have compared the efficiency of this method to the basin-hopping method and found that it can locate the minima faster. Statistical efficiency of the new method has been investigated by running a large number of trajectories. The new method can locate low-energy structures of both the clusters faster than some of the reported algorithms for water clusters and can switch between high energy and low-energy structures multiple times in a simulation illustrating its efficiency. The large number of minima obtained from the simulations is used to get both general and specific features of the minima. The distribution of minima for these two clusters based on the similarity of their oxygen frames shows that the (H(2)O)(20) can have different variety of structures, but for (H(2)O)(25), low-energy structures are mostly cagelike. Several (H(2)O)(25) structures are found with similar energy but with different cage architectures. Noncage structures of (H(2)O)(25) are also found but they are 6-7 kcal/mol higher in energy from the global minimum. The TBP method is likely to play an important role for exploring the complex energy landscape of large molecules.     

Comparison of stochastic optimization methods for all-atom folding of the Trp-Cage protein.

The performances of three different stochastic optimization methods for all-atom protein structure prediction are investigated and compared. We use the recently developed all-atom free-energy force field (PFF01), which was demonstrated to correctly predict the native conformation of several proteins as the global optimum of the free energy surface. The trp-cage protein (PDB-code 1L2Y) is folded with the stochastic tunneling method, a modified parallel tempering method, and the basin-hopping technique. All the methods correctly identify the native conformation, and their relative efficiency is discussed.     

Geometry optimization of bimetallic clusters using an efficient heuristic method

In this paper, an efficient heuristic algorithm for geometry optimization of bimetallic clusters is proposed. The algorithm is mainly composed of three ingredients: the monotonic basin-hopping method with guided perturbation (MBH-GP), surface optimization method, and iterated local search (ILS) method, where MBH-GP and surface optimization method are used to optimize the geometric structure of a cluster, and the ILS method is used to search the optimal homotop for a fixed geometric structure. The proposed method is applied to Cu(38-n)Au(n) (0 ≤ n ≤ 38), Ag(55-n)Au(n) (0 ≤ n ≤ 55), and Cu(55-n)Au(n) (0 ≤ n ≤ 55) clusters modeled by the many-body Gupta potential. Comparison with the results reported in the literature indicates that the present method is highly efficient and a number of new putative global minima missed in the previous papers are found. The present method should be a promising tool for the theoretical determination of ground-state structure of bimetallic clusters. Additionally, some key elements and properties of the present method are also analyzed.     

Global optimization of binary Lennard-Jones clusters using three perturbation operators

Global optimization of binary Lennard-Jones clusters is a challenging problem in computational chemistry. The difficulty lies in not only that there are enormous local minima on the potential energy surface but also that we must determine both the coordinate position and the atom type for each atom and thus have to deal with both continuous and combinatorial optimization. This paper presents a heuristic algorithm (denoted by 3OP) which makes extensive use of three perturbation operators. With these operators, the proposed 3OP algorithm can efficiently move from a poor local minimum to another better local minimum and detect the global minimum through a sequence of local minima with decreasing energy. The proposed 3OP algorithm has been evaluated on a set of 96 × 6 instances with up to 100 atoms. We have found most putative global minima listed in the Cambridge Cluster Database as well as discovering 12 new global minima missed in previous research.     

New algorithm in the basin hopping Monte Carlo to find the global minimum structure of unary and binary metallic nanoclusters

The basin-hopping Monte Carlo algorithm was modified to more effectively determine a global minimum structure in pure and binary metallic nanoclusters. For a pure metallic Ag55 nanocluster, the newly developed quadratic basin-hopping Monte Carlo algorithm is 3.8 times more efficient than the standard basin-hopping Monte Carlo algorithm. For a bimetallic Ag42Pd13 nanocluster, the new algorithm succeeds in finding the global minimum structure by 18.3% even though the standard basin-hopping Monte Carlo algorithm fails to achieve it.     

Growth and structural properties of Mg(N) (N = 10-56) clusters: density functional theory study

Using the minima hopping global geometry optimization method on density functional potential energy surface, we have studied the structural and electronic properties of magnesium clusters for a size range of Mg(N) where N = 10-56. Our exhaustive search reveals that most of our global minima are nonsymmetric in the size range above N = 20. We elucidate the evolutionary trend of the entire series and present more details about the peculiar growth of the clusters. For N > 20, it is possible to divide the cluster into two regions: the core region and the surface region. It turns out that the growth follows a peculiar cyclic pattern where the core and surface grow alternatively. The surface energy, as a function of number of atoms shows a clear signature as the number of atoms in the core increases by one. We have also carried out stability analysis and the stable sizes(magic numbers) agree very well with the experimental magic numbers reported by Diederich [J. Chem. Phys. 2011, 134, 124302]. We point out the similarities and differences between our results and sodium clusters.     

Physical properties of small water clusters in low and moderate electric fields

Likely candidates for the lowest minima of water clusters (H(2)O)(N) for N ≤ 20 interacting with a uniform electric field strength in the range E ≤ 0.6 V/? have been identified using basin-hopping global optimization. Two water-water model potentials were considered, namely TIP4P and the polarizable Dang-Chang potential. The two models produce some consistent results but also exhibit significant differences. The cluster internal energy and dipole moment indicate two varieties of topological transition in the structure of the global minimum as the field strength is increased. The first takes place at low field strengths (0.1 V/? 10) usually forming helical structures.     

Folding funnels: the key to robust protein structure prediction

Natural proteins fold because their free energy landscapes are funneled to their native states. The degree to which a model energy function for protein structure prediction can avoid the multiple minima problem and reliably yield at least low-resolution predictions is also dependent on the topography of the energy landscape. We show that the degree of funneling can be quantitatively expressed in terms of a few averaged properties of the landscape. This allows us to optimize simplified energy functions for protein structure prediction even in the absence of homology information. Here we outline the optimization procedure in the context of associative memory energy functions originally introduced for tertiary structure recognition and demonstrate that even partially funneled landscapes lead to qualitatively correct, low-resolution predictions.     

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.     

Exploring the similarities between potential smoothing and simulated annealing

Simulated annealing and potential function smoothing are two widely used approaches for global energy optimization of molecular systems. Potential smoothing as implemented in the diffusion equation method has been applied to study partitioning of the potential energy surface (PES) for N‐Acetyl‐Ala‐Ala‐N‐Methylamide (CDAP) and the clustering of conformations on deformed surfaces. A deformable version of the united‐atom OPLS force field is described, and used to locate all local minima and conformational transition states on the CDAP surface. It is shown that the smoothing process clusters conformations in a manner consistent with the inherent structure of the undeformed PES. Smoothing deforms the original surface in three ways: structural shifting of individual minima, merging of adjacent minima, and energy crossings between unrelated minima. A master equation approach and explicit molecular dynamics trajectories are used to uncover similar features in the equilibrium probability distribution of CDAP minima as a function of temperature. Qualitative and quantitative correlations between the simulated annealing and potential smoothing approaches to enhanced conformational sampling are established. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 531–552, 2000     

Effect of salt bridges on the energy landscape of a model protein

The effect of introducing salt bridges (gatekeepers) into an off-lattice three-color, 46-bead model protein is investigated in terms of the effect on global optimization statistics. The global minima for all the gatekeepers that exhibited faster folding in previous molecular dynamics studies are located more rapidly than for the original potential, although the global minimum itself may change. Visualization of the underlying potential energy surface using disconnectivity graphs reveals that the gatekeepers exhibit structure intermediate between the original potential and a Go model. Competition between low-lying minima and the global minimum is reduced in the gatekeepers compared to the original potential, and interconversion barriers are generally smaller.     

Global perspectives on the energy landscapes of liquids, supercooled liquids, and glassy systems: the potential energy landscape ensemble

In principle, all of the dynamical complexities of many-body systems are encapsulated in the potential energy landscapes on which the atoms move--an observation that suggests that the essentials of the dynamics ought to be determined by the geometry of those landscapes. But what are the principal geometric features that control the long-time dynamics? We suggest that the key lies not in the local minima and saddles of the landscape, but in a more global property of the surface: its accessible pathways. In order to make this notion more precise we introduce two ideas: (1) a switch to a new ensemble that deemphasizes the concept of potential barriers, and (2) a way of finding optimum pathways within this new ensemble. The potential energy landscape ensemble, which we describe in the current paper, regards the maximum accessible potential energy, rather than the temperature, as a control variable. We show here that while this approach is thermodynamically equivalent to the canonical ensemble, it not only sidesteps the idea of barriers it allows us to be quantitative about the connectivity of a landscape. We illustrate these ideas with calculations on a simple atomic liquid and on the Kob-Andersen [Phys. Rev. E 51, 4626 (1995)] of a glass-forming liquid, showing, in the process, that the landscape of the Kob-Anderson model appears to have a connectivity transition at the landscape energy associated with its empirical mode-coupling transition. We turn to the problem of finding the most efficient pathways through potential energy landscapes in our companion paper.     

Efficient moves for global geometry optimization methods and their application to binary systems

We show that molecular dynamics based moves in the minima hopping method are more efficient than saddle point crossing moves. For binary systems we incorporate identity exchange moves in a way that allows one to avoid the generation of high energy configurations. Using this modified minima hopping method, we re-examine the binary Lennard-Jones benchmark system with up to 100 atoms and we find a large number of new putative global minima.     

Computer optimization of small atomic clusters

We have successfully identified stable configurations of both rare-gas and NaCl clusters with a new optimization procedure. An initial cluster configuration is prepared in a so-called shoot-and-stay process. Its total energy is then minimized with respect to the atomic coordinates. To prevent the system from being locked in local minima, the step size of each move is chosen as the width of the energy well at a higher level. As the system evolves, the global minimum is contained in the volume bounded by the decreasing value of step sizes. We have also carried out the optimization of NaCl clusters by the simulated annealing technique, for comparison. The results show that for such heterogeneous systems, the latter method cannot always find the global minimum, because of large energy gaps between different catchment regions in phase space.     

Search for global minimum geometries for medium sized germanium clusters: Ge12-Ge20

We have performed an unbiased search for the global minimum geometries of small-to-medium sized germanium clusters Gen(12< or =n< or =18) as well as a biased search (using seeding method) for Gen(17< or =n< or =20). We employed the basin-hopping algorithm coupled with the plane-wave pseudopotential density functional calculations. For each size, we started the unbiased search with using several structurally very different initial clusters, or we started the biased search with three different seeds. Irrespective of the initial structures of clusters we found that the obtained lowest-energy clusters of the size n=12-16 and 18 are the same. Among them, the predicted global minima of Gen(12< or =n< or =16) are identical to those reported previously [Shvartsburg et al., Phys. Rev. Lett. 83, 167 (1999)]. For n=17-20, we have identified two or three nearly isoenergetic low-lying isomers (for each size) that compete for the global minimum. Nearly all the low-lying clusters in the size range of 12< or =n< or =20 contain the tri-caped trigonal prism motif and are all prolate in geometry, in agreement with the experiment.