Dynamics on statistical samples of potential energy surfaces

An improved model of homogeneous nucleation for high supersaturation conditions: aluminum vapor

A novel model of stationary nucleation, treating the thermodynamic functions of small clusters, has been built. The model is validated against the experimental data on the nucleation rate of water vapor obtained in a broad range of supersaturation values (S = 10-120), and, at high supersaturation values, it reproduces the experimental data much better than the traditional classical nucleation model. A comprehensive analysis of the nucleation of aluminum vapor with the usage of developed stationary and non-stationary nucleation models has been performed. It has been shown that, at some value of supersaturation, there exists a double potential nucleation barrier. It has been revealed that the existence of this barrier notably delayed the establishment of a stationary distribution of subcritical clusters. It has also been demonstrated that the non-stationary model of the present work and the model of liquid-droplet approximation predict different values of nucleation delay time, τ_s. In doing so, the liquid-droplet model can underestimate notably (by more than an order of magnitude) the value of τ_s.

Where macro meets micro

Reconciling or somehow linking the macroscopic and microscopic approaches to chemical and physical processes has been a challenge unaddressed for many years. One approach, presented here, treats the issue by examining individual phenomena well described by a macro approach that fails when applied to small systems. The key to the approach is determining the approximate system size below which the breakdown of the macro description is observable. The most developed example is the failure of the Gibbs phase rule for sufficiently small atomic clusters. Other examples, such as the onset, at sufficient size, of the insulator-to-metal transition, are discussed, as are some still more challenging phenomena.

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 15≤N≤65. 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.

Principal component analysis of potential energy surfaces of large clusters: allowing the practical calculation of the master equation

The number of variables in many-particle systems is typically unmanageably large; some way to reduce that number and still retain access to the important information about the system of interest is one of the great challenges in the broad topic of complexity. Principal components and principal coordinates provide a powerful means to extract--from unwieldy, large data sets--a reduced collection of variables that provide the information one needs, in a relatively efficient way and useful form. We investigate the application of principal components to the analysis of kinetics of the atomic motions in atomic clusters, particularly of clusters that are large enough so that a full description in terms of the entire high-dimensional potential surface is entirely impractical. A specific application is the use of principal components linking minima with their adjacent saddles, permitting the evaluation of rate coefficients (in the context of transition state theory) as ratios of partition functions of only one or two key variables.

Determination, prediction, and understanding of structures, using the energy landscapes of chemical systems – Part II

In the past decade, new theoretical approaches have been developed to determine, predict and understand the struc-ture of chemical compounds. The central element of these methods has been the investigation of the energy landscape of chemical systems. Applications range from extended crystalline and amorphous compounds over clusters and molecular crystals to proteins. In this review, we are going to give an introduction to energy landscapes and methods for their investigation, together with a number of examples. These include structure prediction of extended and mo-lecular crystals, structure prediction and folding of proteins, structure analysis of zeolites, and structure determination of crystals from powder diffraction data.

Pathways for conformational change in nitrogen regulatory protein C from discrete path sampling

Pathways corresponding to the conformational change in nitrogen regulatory protein C are calculated using the CHARMM19 force field with an implicit solvation model. Our analysis employs the discrete path sampling approach to grow a database of local minima and transition states from the potential energy surface that contains kinetically relevant pathways. The pathways with the largest contribution to the phenomenological two-state rate constants are found to exhibit a number of structural features that agree with experimental observations. Further details of the calculated pathways for conformational change may therefore provide useful predictions of how this large-scale motion is achieved.

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.

Characterizing Potential Surface Topographies through the Distribution of Saddles and Minima

Three related clusters of thirteen particles bound by pairwise Morse potentials with different ranges are the vehicles for relating the dynamics and kinetics of these clusters to the topographies of their energy landscapes. The analyses are based on the distributions of minima and saddles, on the asymmetries of the barriers and the kinetics of passage among the energy bands that the distributions of minima display. While all three of the examples are essentially structure-seekers, the extent of this character is clearly related to the range of the potential.

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.

Kinetics of model energy landscapes: an approach to complex systems

An idealized potential energy surface (PES), simply a PES-like network of stationary points, is demonstrated to be a useful tool to study kinetic relaxation of complex energy landscapes. Combined with a master equation, we show that if constructed with proper regularity, the kinetics of the PES is easy to predict and understand by carefully examining the eigenmodes of the master equation. By modifying the idealized PES model to make it more and more complicated, we demonstrate a systematic method to study the complex kinetics on large PES. The idealized PES model is used to explore the feasibility and the robustness of statistical sampling of large PES. We develop several sampling strategies, such as the "rough topography sampling" and the "low barrier saddle sampling" in the idealized PES model and find they are also applicable to a realistic PES of the 13-atom Morse cluster with range parameter rho= 6. To measure the robustness of the sampling methods, we compare the eigenvalue spectra, the eigenvector similarity and the relaxation times of the total energy of the full and sample PESs.

Energy barriers and activated dynamics in a supercooled Lennard-Jones liquid

We study the relation of the potential energy landscape (PEL) topography to relaxation dynamics of a small model glass former of Lennard-Jones type. The mechanism under investigation is the hopping between superstructures of PEL minima, called metabasins (MBs). Guided by the idea that the mean durations <tau> of visits to MBs should reflect the local PEL structure, we first derive the effective depths of MBs from dynamics, by the relation E(app)=d ln<tau>/dbeta, where beta=1/k(B)T. Second, we establish a connection of E(app) to the barriers that surround MBs. As the consequence of a rugged PEL, it turns out that escapes from MBs do not happen by single hops between PEL minima, but correspond to complicated multiminima sequences. We introduce the concept of return probabilities to the bottom of the MBs in order to judge when the attraction range of a MB has been left. The energy barriers overcome can then be identified. These turn out to be in good agreement with the effective depths E(app), calculated from dynamics. We are thus able to relate MB lifetimes to their local structure. Moreover, we can trace back the overall diffusive dynamics to the population of MBs and to their local topology, i.e., to purely thermodynamic and structural quantities. Single energy barriers are identified with the help of a new method, which accurately performs a descent along the ridge between two minima. We analyze the population of transition regions between minima, called basin borders. No indication for the mechanism of diffusion to change around the mode-coupling temperature can be found. We discuss the question whether the one-dimensional reaction paths connecting two minima are relevant for the calculation of reaction rates at the temperatures under study.

Taboo search by successive confinement: surveying a potential energy surface

A taboo search for minima on a potential energy surface (PES) is performed by means of confinement molecular dynamics: the molecular dynamics trajectory of the system is successively confined to various basins on the PES that have not been sampled yet. The approach is illustrated for a 13-atom Lennard-Jones cluster. It is shown that the taboo search radically accelerates the process of surveying the PES, with the probability of finding a new minimum defined by a propagating Fermi-like distribution.