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The evolution of many systems is dominated by rare activated events that occur on timescale ranging from nanoseconds to the hour or more. For such systems, simulations must leave aside the full thermal description to focus specifically on mechanisms that generate a configurational change. We present here the activation relaxation technique (ART), an open-ended saddle point search algorithm, and a series of recent improvements to ART nouveau and kinetic ART, an ART-based on-the-fly off-lattice self-learning kinetic Monte Carlo method.

There has been considerable interest, in the last two decades, in the development of accelerated numerical methods for sampling the energy landscape of complex materials. The goal is to understand the long-time kinetics of chemical reactions, self-assembly, defect diffusion, and so forth associated with high-dimensional systems counting many tens to many thousands of atoms.

Some of these methods are extension of molecular dynamics (MD), such as Voter’s hyperdynamics [

Once a method is available for finding saddle points and adjacent minima, we must decide what to do with this information. A simple approach is to sample these minima and saddle points and classify the type of events that can take place, providing basic information on the possible evolution of these systems. This approach has been applied, using ART nouveau or other open-ended methods, on a number of materials ranging from Lennard-Jones clusters to amorphous silicon and proteins [

In the long run however, most researchers are interested in understanding the underlying kinetics controlling these complex systems. To this end, it is no longer sufficient to collect these saddle points: they must be ordered and connected in some fashion to reconstruct at least a reduced representation of the energy landscape. Wales and collaborators used a different approach: starting from a large catalog of events, they construct a connectivity matrix that links together minima through saddle points, and they apply a master equation to solve the kinetics [

A more straightforward approach to generate kinetics with these methods is to apply an on-the-fly kinetic Monte Carlo procedure: in a given local minimum, a finite number of open-ended event searches are launched and the resulting barriers are used to estimate a probable timescale over which an event will take place. This approach was applied to a number of systems using various open-ended methods such as the dimer [

While we are fully aware of other similar algorithms, we focus in this paper on the methods developed by our group: ART nouveau and kinetic ART. First, we present the most recent implementation of ART nouveau, which has been extensively optimized and is now faster than most comparable methods [

The ART nouveau method is an open-ended algorithm for first-order saddle point search. This algorithm has been developed over the last 15 years. Based on the activation-relaxation technique proposed in 1996 [

ART nouveau proceeds in three steps.

These three steps form an

After providing this short overview, it is useful to describe with some details the three basic steps defining ART nouveau.

The selection of a direction for leaving the harmonic well is crucial for finding rapidly a characteristic distribution of saddle points surrounding the initial minimum. This harmonic well is described as the region of the energy landscape surrounding a local minimum with only positive curvature.

It is, therefore, tempting to use the eigenvectors of the Hessian matrix, corresponding to the second derivative of the energy:

Sketch of minimum-energy (steepest descent) pathways from various first-order saddle points to the local energy minimum (

In the absence of a discriminating basis in the selection of a given direction for launching the saddle-point search, it is best to simply use randomly chosen directions of deformation. While global random deformations are appropriate, numerous tests have shown that these lead to an oversampling of the most favorable events, generally associated with low-energy barriers, making the sampling of the energy landscape relatively costly [

More precisely, the procedure implemented in ART nouveau for leaving the harmonic well is the following:

an atom is first selected at random from a predefined set of regions, that can include the whole system but can be limited to, for example, the surface of a slab or a region containing defects;

this atom and its neighbors, identified using a cut-off distance that can be limited to the first-shell neighbors or run through many layers, are then moved iteratively along a random direction;

at every step, a slight minimisation in the perpendicular hyperplane is applied to avoid collisions and allow the full system to react to this deformation; this minimisation is not complete, as this would bring back the system onto the direction of lowest curvature of the harmonic well, and it is selected to be just sufficient to avoid unphysical conformations;

at every step, we use a Lanczos procedure [

Once a direction of negative curvature has been identified, it is relatively straightforward to bring the system at a first-order saddle point: the system has to be pushed along this direction while the energy is minimised in the perpendicular hyperplane. This ensures that the system will reach a first-order transition state as the force vanishes.

While straightforward, the activation can be very costly as it requires a partial knowledge of the Hessian matrix. If, for a system counting a few atoms, it is appropriate to simply compute and diagonalize the Hessian, this approach is not feasible with larger problems. To avoid this expensive calculation, a number of algorithms have been proposed for this search, including approximate projections [

Irrespective of system size, we find that a

Near the saddle point, the dual approach of activation with Lanczos and minimization in the perpendicular hyperplane with a different algorithm is not optimal. At this point, we find it often preferable to use an integrated algorithm that can converge efficiently on inflection points. As described in [

In summary, the activation and convergence to a first-order saddle point phase can be described with the following steps:

using the Lanczos algorithm, the direction of negative curvature is obtained;

the system is pushed slightly along this direction, with a displacement decreasing as a square root of the number of iterations, to facilitate convergence onto the saddle point;

the energy is relaxed in the perpendicular hyperplane; in the first iterations after leaving the harmonic well, only a few minimization steps are taken, to avoid going back into the harmonic well and losing the direction of negative curvature;

if DIIS is not used, the first three steps are repeated until the total force falls below a predefined threshold, indicating that a first-order saddle point has been reached or until the lowest eigenvalue becomes positive, indicating that the system has found its way back into the harmonic well;

if DIIS is applied near the transition state, we apply the Lanczos algorithm until the negative eigenvalue has reached a minimum and has gone up for 4 sequential iterations (this criterion and the DIIS implementation are discussed in [

Once the configuration has reached a saddle point, it is necessary to nudge slightly over to allow it to converge into a new minimum. There is clearly some flexibility here. In ART nouveau, the configuration is generally pushed over a distance of 0.15 times the distance between the saddle and the initial minimum along the eigenvector and away from the initial minimum. The system is then brought into an adjacent local energy minimum using FIRE [

ART nouveau has been applied with success to a range of problems from Lennard-Jones clusters to proteins and amorphous silicon [

Olsen et al. also noted that for the same method, the number of required force evaluations to reach a saddle point grows with the effective dimensionality of the event taking place. Looking at Pt diffusion on a Pt(111) surface, they found that, as more atoms were allowed to move, the number of force evaluations for their ART nouveau implementation increased from 145, when a single atom was free, to 2163 with 175 free atoms [

In a recent paper, Machado-Charry et al. [

Efficiency of various open-ended and double-ended saddle-point searching algorithms. The first four columns show results on four different systems using the latest version of ART nouveau as described by Machado et al. [^{1}Effective number of degrees of freedom. ^{2}Starting near the saddle point and not from a minimum. ^{3}Approximate number extracted from Figure

Algo. | ART nouveau | ART | Dimer method | Improved dimer | GSM | ||||||

Ref. | Machado et al. [ | Olsen et al. [ | [ | [ |
[ | ||||||

System |
V_{Si} |
C_{20} | SiC | Pt(111) | Pt(111) | Pt(111) | Pt(111) | PHBH/H_{2}O | _{2} | ||

BC | Bulk | Bulk | Isol. | Surf. | Surf. | Surf. | Surf. | Surf. | Isol. | Sol. | Isol. |

Pot. | SW | DFT | DFT | DFT | Morse | Morse | Morse | Morse | DFT | QM/MM | DFT |

Method | PBE | LDA | PBE | B3LYP | AM1 | B3LYP | |||||

DOF | 3000 |
12^{1} | 60 |
222^{1} | 3 | 525 | 3 | 525 | 48 | 144 | 12^{1} |

235 | 210 | 322 | 262 | 145 | 372 | 204 | 335 | 384^{2} | 425^{3} | 330 | |

670 | 302 | 718 | 728 | 145 | 2163 | 204 | 2148 | — | — | — |

Table

ART nouveau can be applied to sample the energy landscape of complex systems and search for minimum-energy states as can be seen in the two applications presented here, to interstitial self-diffusion in iron and protein folding, both described with semiempirical potentials.

The ART nouveau method using an empirical potential is applied to the systematic search in the energy landscape of small self-interstitial clusters in iron from monointerstitial,

ART nouveau revealed a large number of distinct configurations which increase rapidly with cluster size, exceeding 400, 1100, and 1500 for

(a) Histogram of the relative formation energies and associated barrier energies (red squares and left vertical axis) of the ART nouveau-generated configurations for tetrainterstitials in iron modeled by the Ackland-Mendelev potential. The zero energy is the energy of the most stable configuration. The vertical axis on the right-hand side (associated with the bars) corresponds to the number of distinct minima found by ART nouveau. (b) Transition pathway for the unfaulting mechanism of the ring configuration of the tetrainterstitial,

ART nouveau was also used for finding transition pathways from high-energy configurations to lower-energy configurations. We have ensured a broader exploration of different ways to escape from the starting minima by performing ART nouveau simulations with different values of the fictitious Metropolis temperature, namely, 100 K, 400 K, and 800 K. Hence, we are able to provide the complete path for the unfaulting mechanisms of the self-trapped ring configurations _{3-4}. These configurations were recently proposed for the interstitial clusters [

ART nouveau can also be applied to molecules such as proteins, to characterize folding [

Among others, ART nouveau was applied to study folding of protein A, a fast-folding 60-residue sequence that adopts a three-helix bundle conformation in its native state. Because of its relatively low complexity, protein A has been extensively studied to understand protein folding (e.g. [

Evolution of trajectory #18 (a), #32 (b), #35 (c), and #25 (d). Top graphs: formation of the helical regions H1 (green), H2 (blue), and H3 (magenta) into helices and evolution of the energy level (red). The blue box of (d) indicates that a lower temperature criterion was used for that part of the simulation. All lines are smoothed by a Gaussian of width

Of the 52 trajectories, 36 adopt a structured conformation with an energy below

The structure of lowest energy found through independent simulations at 900 K followed by a energy refining simulations at 600 K. (a) The left-handed native-like bundle found in 4 simulations; (b) the right-handed mirror image found 7 times; (c)

As a general rule, although ART nouveau does not provide kinetic information regarding trajectories, it can provide information regarding the crucial steps in relaxation, folding, and aggregation patterns for relatively large systems that are not dominated by entropic considerations. In protein A, ART nouveau managed to identify metastable ordered structures of higher energy that can shed light on close homological sequences. Finally, ART nouveau is also a very interesting tool for exploring conformation of large biological systems, if it is coupled with internal coordinates and multiscale representations [

While ART nouveau is very efficient for finding saddle points, it cannot be applied directly to evaluate the dynamics of a system as the inherent bias for selecting a specific barrier over the others is not known. In the absence of such characterization, ART nouveau is ideal to sample possible events that can then serve either in a master equation or in a kinetic Monte Carlo scheme.

As mentioned in the introduction, the first approach has been attempted by Wales [

Recently, a number of kinetic Monte Carlo algorithms have been proposed to address some of these concerns by constructing a catalog: the kinetic ART [

Our approach attempts to minimise the computational efforts while preserving the most correct long-time kinetics, including long-range elastic effects. To do so, it is necessary to generate an event catalog that can be expanded and reused as simulations progress or new simulations are launched and, contrary to the vast majority of KMC schemes available, can handle off-lattice atomic positions. This catalog must be as compact as possible, that is, offer an efficient classification scheme, yet be sufficiently flexible to handle alloys, surfaces, and disordered environments. In an off-lattice situation, no catalog can provide precise energy barriers, as short- and long-range elastic deformations will affect the local environment. Rates, if derived from the catalog, must therefore be reevaluated at each step to include elastic effects.

This is what kinetic ART achieves by using ART nouveau for generating new events and update energy barriers, coupled with a topological classification for local environments, which allows the construction of a discrete event catalog based on a continuous geometry even in the most complex environment.

Before going into details, it is useful to present the main steps of the algorithm.

Starting from a local energy-minimum, the local topology associated with each atom is analyzed. For each topology, ART nouveau searches are launched and new events are added to the catalog of events and attached to this topology. In order to ensure that the catalog is complete, event searches are not limited to new topologies: the number of searches is proportional to the logarithm of the frequency with which a topology appears in a given simulation. All events associated with the configuration at the local energy-minimum are added to the tree.

All low-energy events are reconstructed and the transition state is refined to take the impact of long-range elastic deformations on the rate into account.

With all barriers known, the average Poisson rate is determined according to the standard KMC algorithm; the clock is pushed forward and an event is selected at random, with the proper weight [

We go back to

Each of these steps involves, of course, a number of elements, which we describe in the following sections.

In order to construct a catalog of events, it is necessary that these be discretized to ensure that events are recognized uniquely. For lattice-constrained motion, this requirement is straightforward to implement by focusing simply on the occupancy of the various crystalline sites. Geometry, however, is no longer a satisfactory criterion for reconstructed sites, defective and disordered systems, where atoms cannot easily be mapped back to regular lattice sites. Our solution is to use a topological description of local environments for mapping events uniquely, allowing us to describe environments of any chemical and geometrical complexity. This is done by first selecting a central atom and all its neighbors within a given cut-off radius. Atoms are then connected, following a predefined procedure that can be based loosely on first-neighbor distance or a Voronoi construction, forming a graph that is then analysed using the software NAUTY, a powerful package developed by McKay [

For this discretization to work, it is necessary that a one-to-one correspondence exist between the topological description and the real-space geometry of this local region. This correspondence is enforced by the fact that the graph has to be embedded into a well-defined surrounding geometry and must correspond to either a local minimum or a first-order saddle with the given forcefield. These two elements ensure that the one-to-one correspondence is valid in the vast majority of conformations.

This correspondence may fail in two cases: for very flat minima or saddle points, there might be more than one topology associated with the geometry. In this case, the corresponding events might be overcounted. For this reason, we compare the barrier height as well as the absolute total displacement and the direction of motion of the main atom (the one that moves most during an event) both between initial and final state and between initial state and the saddle point. We only add events to the catalog that are sufficiently different from previously found ones.

It is also possible that a single topology corresponds to more than one geometry. Most frequently, this problem appears in highly symmetric systems: two topologically identical events might differ only in the direction of motion. To resolve this issue, we reconstruct geometry and ensure that motion takes place in different directions. In this case, events are flagged as different and additional ART nouveau searches are performed to identify all topologically equivalent but geometrically different moves.

In some cases, a topology is associated with fundamentally different geometries. Here, this failure is identified automatically, since it leads to incorrect saddle-point reconstruction that cannot be converged. When this occurs, the algorithm automatically modifies the first-neighbor cut-off used for constructing the topology until the multiple geometries are separated into unique topologies. In most systems, this is extremely rare, and it is generally a symptom of an inappropriately short topology cutoff radius.

Once a new topology is identified, ART nouveau saddle point searches are launched a number of times from the central atom associated with this topology (to increase the efficiency of finding topologies with multiple geometries, events are launched from a randomly selected set of atoms characterized by the same topology). Depending on the system, we use between 25 and 50 searches for every

These events, which are stored in the catalog, are called

While representative, the transition states and energy barriers in the catalog cannot be applied exactly as they are to new configurations, as short- and long-range elastic deformations affect every barrier differently, creating favored directions even for formally isotropic defects, for example.

To include these effects, every generic event should be reconstructed for each atom and fully relaxed. This is what kinetic ART does, with one approximation: to limit the effort of refining generic into specific events, only the kinetically relevant events are reconstructed and relaxed.

After each KMC step, the event list is ordered according to the barrier energy and only the lowest-energy barrier events, representing up to given threshold (we use typically 99.9 or 99.99%) of the total rate are fully reconstructed and refined. Depending on the system and the temperature, this means that only one to ten percent of all barriers in the catalog are refined specific events. The remaining events, which contribute very little to the rate, are cloned from the generic events without adjusting the barrier.

This local reconstruction takes place in two steps. First, using the reference atomic positions of the generic event, the geometric transformation necessary to map the initial state onto the current configuration is determined. This operation is then applied to the atomic displacements between initial state and saddle point. From this first reconstruction of the saddle point geometry, the saddle point (and with it the energy barrier) is refined. In the second step, the system is relaxed into the final state by pushing it over the saddle point. If it is impossible to map the initial state onto the current configuration even though the central atoms have the same topology, we know that the one-to-one correspondence between topology and geometry has failed and apply the corrections mentioned in the previous section.

The reconstruction of specific events ensures that all elastic and geometrical deformations are taken into account when constructing the list of events that is used in the kinetic Monte Carlo algorithm. This list now contains both refined low-energy specific events and clones of generic events with higher barrier energies.

Rates are determined according to transition state theory:

These rates are then combined, following Bortz et al. [

The local properties of kinetic ART allow a very efficient implementation of the algorithm. First, it is straightforward to dispatch the event searching and refining to separate processors. For a typical system, between 20 and 40 processors can be used for these tasks (up to 128 when building the initial catalog), with an efficiency of 90% or more. Moreover, as events are inherently local, there is no need to compute global forces during an ART nouveau event: only nonzero forces need to be computed most of the time, with a final global minimization needed only at the end of each KMC step. This renders the algorithm for computing an event almost independent of system size for a fixed number of defects and order

In KMC, the residence time

We first demonstrate the capability of kinetic ART by applying it to the relaxation of a large Si box with ion implantation. In a previous work by Pothier et al. [

Kinetic ART is applied at 300 K on this complex system. Figure

(a) The total potential energy of the bombarded c-Si crystal during annealing at 300 K. We show three simulations that have the same initial configuration. (b) The energy barriers crossed during a c-Si anneal at 300 K are represented by the stars. The number of topologies where we performed ART searches is shown as a continuous line.

Meanwhile, Figure

While standard KMC simulations easily run for millions of steps, regularly reaching timescale of minutes or more, fundamental restrictions have limited them to simple lattice-based problems such as metal on metal growth. The possibility, with kinetic ART, to now apply these accelerated approaches to complex materials such as ion-bombarded Si, oversimulated timescale many orders of magnitude longer than what can be reached by MD, opens the door to the study in numerous systems of the long-time atomistic dynamics that was long considered out of reach. This is the case for fully disordered systems.

Here, we show that even such complex materials such as amorphous silicon (a-Si), a system that has been used as a test case for ART over the years [

Disordered systems are characterized by an extremely large number of possible configurations that exclude studies with traditional lattice-based Monte Carlo methods. Event catalogues will therefore be large, as almost each atom in a box of a few thousand atoms has its own topology. However, since the topological classification is based on the local environment, it still provides for a real gain given enough sampling.

Figure

Simulation of an a-Si model containing 1000 atoms at 300 K. The simulation was started using a previously built event catalogue of over 87 000 events. Top: The number of new topologies searched and the total runtime as a function of simulated time. Bottom: The change in energy and the squared total displacement as a function of simulated time.

In a system such as a-Si, the continuous distribution of activated barriers means that there is not a clear separation between frequent and rare events. The energy cutoff for bac-MRM is then chosen so that the inverse of the associated rate is small compared to the desired simulated time. Since in an amorphous system flicker-like events can occur at any energy scale, the value must be adjusted as a function of the degree of relaxation and temperature. In the present case, the cutoff value was 0.35 eV, meaning that while the thermodynamics is accurate at all scales, internal dynamics on timescales lower than 120 ns is ignored. This simulation required 617 CPU-hours over six processors for a total run-time of 103 hours. As shown in Figure

Starting from a well-relaxed configuration, the simulation quickly reaches the

Even though disordered systems require more computational efforts than crystalline-based defective systems, the application of kinetic ART to such systems allows the study of kinetics on timescales well beyond what could have been reached until now using more traditional molecular dynamics, as can be seen here, in a short test simulation.

We have presented the activation-relaxation technique, ART nouveau, a very efficient open-ended search method for transition state, and have shown how it can be applied to extensively search for diffusion mechanisms and pathways as well as low-energy configurations with systems as diverse as interstitials in iron and a 60-residue protein. With recent improvements, finding a transition can take as little as 210 force evaluations and, when lost events are taken into account, between 300 and 700 force evaluations, allowing for extensive searches, with either empirical or

As the event-selection bias is unknown in ART nouveau, it cannot be used directly to study the kinetics of atomistic systems. This limitation is lifted by coupling it to a kinetic Monte Carlo scheme. Kinetic ART goes beyond this simple coupling, however, by introducing a topological classification for catalog building while fully preserving the impact of elastic deformations on the kinetics while being fully off-lattice. The efficiency of this approach was demonstrated by characterizing the relaxation of a 30 000-atom ion-bombarded box of silicon and a 1000-atom box of amorphous silicon.

Although not presented here, the algorithm also readily handles other systems such as metals and alloys. At the moment, this is done using a fixed prefactor for the attempted frequency in transition state theory. As discussed previously, such an approximation is fairly good for a number of systems. However, we are currently implementing a version of the algorithm that also evaluates prefactors. While this is more costly, it could be essential to describe correctly the kinetics of complex materials.

Both ART nouveau and kinetic ART open the door to the study of questions that were out of reach only a few years ago, either to identify diffusion mechanisms and catalytic reactions or to recreate full diffusion and relaxation pathways in complex materials such as alloys and glasses on timescale that had been out of reach until now. As discussed previously, a number of other accelerated techniques have been proposed recently and it is too early to determine whether one method will really stand out. Irrespective of this, ART nouveau and kinetic ART have already shown what they can be used for.

E. Machado-Charry, P. Pochet, and N. Mousseau acknowledge support by Nanosciences Fondation in the frame of MUSCADE Project. Partial funding was also provided by the Natural Science and Engineering Research Council of Canada, the Fonds Québécois de Recherche Nature et Technologie, and the Canada Research Chair Foundation. This work was performed using HPC resources from GENCI-CINES (Grant 2010-c2010096323) and from Calcul Québec (CQ). The ART nouveau code is freely distributed in a format appropriate for both empirical and quantum forces. Kinetic ART is still under development but a version should be available for distribution in the near future. To receive the code or for more information, please contact the authors.

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