The variational nature of the gentlest ascent dynamics and the relation of a variational minimum of a curve and the minimum energy path

Locating saddle points using gradient extremals on manifolds adaptively revealed as point clouds

Steady states are invaluable in the study of dynamical systems. High-dimensional dynamical systems, due to separation of time scales, often evolve toward a lower dimensional manifold M. We introduce an approach to locate saddle points (and other fixed points) that utilizes gradient extremals on such a priori unknown (Riemannian) manifolds, defined by adaptively sampled point clouds, with local coordinates discovered on-the-fly through manifold learning. The technique, which efficiently biases the dynamical system along a curve (as opposed to exhaustively exploring the state space), requires knowledge of a single minimum and the ability to sample around an arbitrary point. We demonstrate the effectiveness of the technique on the Müller-Brown potential mapped onto an unknown surface (namely, a sphere). Previous work employed a similar algorithmic framework to find saddle points using Newton trajectories and gentlest ascent dynamics; we, therefore, also offer a brief comparison with these methods.

Restricted-Variance Constrained, Reaction Path, and Transition State Molecular Optimizations Using Gradient-Enhanced Kriging

Gaussian process regression has recently been explored as an alternative to standard surrogate models in molecular equilibrium geometry optimization. In particular, the gradient-enhanced Kriging approach in association with internal coordinates, restricted-variance optimization, and an efficient and fast estimate of hyperparameters has demonstrated performance on par or better than standard methods. In this report, we extend the approach to constrained optimizations and transition states and benchmark it for a set of reactions. We compare the performance of the newly developed method with the standard techniques in the location of transition states and in constrained optimizations, both isolated and in the context of reaction path computation. The results show that the method outperforms the current standard in efficiency as well as in robustness.

Stochastic dynamics of an active particle escaping from a potential well

Active matter systems are driven out of equilibrium by the energy directly supplied at the level of constituent active particles that are self-propelled. We consider a model for an active particle in a potential well, characterized by an active velocity with a constant magnitude but a random orientation subject to white noises. We are interested in the escape of the active particle from the potential well in multiple-dimensional space. We investigate two distinct optimal paths, namely, the shortest arrival-time path and the most probable path, by using the analytical and numerical techniques from optimal control and rare event modeling. In particular, we elucidate the relationship between these optimal paths and the reachable set using the Hamiltonian dynamics for the shortest arrival-time path and the geometric minimum action method for the most probable path, respectively. Numerical results are presented by applying these techniques to a two-dimensional double-well potential.

Interplay between the Gentlest Ascent Dynamics Method and Conjugate Directions to Locate Transition States

An algorithm to locate transition states on a potential energy surface (PES) is proposed and described. The technique is based on the GAD method where the gradient of the PES is projected into a given direction and also perpendicular to it. In the proposed method, named GAD-CD, the projection is not only applied to the gradient but also to the Hessian matrix. Then, the resulting Hessian matrix is block diagonal. The direction is updated according to the GAD method. Furthermore, to ensure stability and to avoid a high computational cost, a trust region technique is incorporated and the Hessian matrix is updated at each iteration. The performance of the algorithm in comparison with the standard ascent dynamics is discussed for a simple two dimensional model PES. Its efficiency for describing the reaction mechanisms involving small and medium size molecular systems is demonstrated for five molecular systems of interest.

Conformational analysis of enantiomerization coupled to internal rotation in triptycyl-n-helicenes

We present a computational study of a reduced potential energy surface (PES) to describe enantiomerization and internal rotation in three triptycyl-n-helicene molecules, centering the discussion on the issue of a proper reaction coordinate choice. To reflect the full symmetry of both strongly coupled enantiomerization and rotation processes, two non-fixed combinations of dihedral angles must be used, implying serious computational problems that required the development of a complex general algorithm. The characteristic points on each PES are analyzed, the intrinsic reaction coordinates are calculated, and finally they are projected on the reduced PES. Unlike what was previously found for triptycyl-3-helicene, the surfaces for triptycyl-4-helicene and triptycyl-5-helicene contain valley-ridge-inflection (VRI) points. The reaction paths on the reduced surfaces are analyzed to understand the dynamical behaviour of these molecules and to evaluate the possibility of a molecule of this family exhibiting a Brownian ratchet behaviour.

Analysis of the Acting Forces in a Theory of Catalysis and Mechanochemistry

The theoretical description of a chemical process resulting from the application of mechanical or catalytical stress to a molecule is performed by the generation of an effective potential energy surface (PES). Changes for minima and saddle points by the stress are described by Newton trajectories (NTs) on the original PES. From the analysis of the acting forces we postulate the existence of pulling corridors built by families of NTs that connect the same stationary points. For different exit saddles of different height we discuss the corresponding pulling corridors; mainly by simple two-dimensional surface models. If there are different exit saddles then there can exist saddles of index two, at least, between. Then the case that a full pulling corridor crosses a saddle of index two is the normal case. It leads to an intrinsic hysteresis of such pullings for the forward or the backward reaction. Assuming such relations we can explain some results in the literature. A new finding is the existence of roundabout corridors that can switch between different saddle points by a reversion of the direction. The findings concern the mechanochemistry of molecular systems under a mechanical load as well as the electrostatic force and can be extended to catalytic and enzymatic accelerated reactions. The basic and ground ansatz includes both kinds of forces in a natural way without an extra modification.