Comment on: Exploring the potential energy landscape of the Thomson problem via Newton homotopies

We show that the Newton homotopy used in the paper [D. Mehta et al., J. Chem. Phys. 142, 194113 (2015)] is related to the Newton trajectory method. With the theory of the Newton trajectories at hand, we can sharpen some findings of the paper

Embedding of the saddle point of index two on the PES of the ring opening of cyclobutene

The ring opening of cyclobutene is characterized by a competition of the two different pathways: a usual pathway over a saddle of index one (SP1) along the conrotatory behavior of the end groups, as well as a 'forbidden' pathway over a saddle point of index two (SP2) along the disrotatory behavior of the end CH2 groups. We use the system of ordinary differential equations for the method of the gentlest ascent dynamics (GAD) to determine saddle points of the potential energy surface (PES) of the ring opening of cyclobutene to cis-butadiene. We apply generalized GAD formulas for the search of a saddle point of index two. To understand the relation of the different regions of the PES (around minimums, around SPs of index one or two) we also calculate valley-ridge inflection (VRI) points on the PES using Newton trajectories (NT). VRIs and the corresponding singular NTs subdivide the regions of 'attraction' of the different SPs. We calculate the connections of the SP2 (in its different symmetry versions) with different SPs of index one of the PES by different 'reaction pathways.' We compare the possibilities of the tool of the GAD curves for the exploration of PESs with these of NT. The barrier of the disrotatory SP2 is somewhat higher than the barrier of the conrotatory SP1, however, pathways across the slope to the SP2 open additional reaction valleys

An Analysis of Some Properties and the Use of the Twist Map for the Finite Frenkel-Kontorova Model

We discuss the twist map, with a special interest in its use for the finite Frenkel-Kontorova model. We explain the meaning of the tensile force in some proposed models. We demonstrate that the application of the twist map for the finite FK model is not correct, because the procedure ignores the necessary boundary conditions

Some mathematical reasoning on the artificial force induced reaction method

There are works of the Maeda-Morokuma group, which propose the artificial force induced reaction (AFIR) method (Maeda et al., J. Comput. Chem. 2014, 35, 166 and 2018, 39, 233). We study this important method from a theoretical point of view. The understanding of the proposers does not use the barrier breakdown point of the AFIR parameter, which usually is half of the reaction path between the minimum and the transition state which is searched for. Based on a comparison with the theory of Newton trajectories, we could better understand the method. It allows us to follow along some reaction pathways from minimum to saddle point, or vice versa. We discuss some well-known two-dimensional test surfaces where we calculate full AFIR pathways. If one has special AFIR curves at hand, one can also study the behavior of the ansatz

Locating saddle points of any index on potential energy surfaces by the generalized gentlest ascent dynamics

The system of ordinary differential equations for the method of the gentlest ascent dynamics (GAD) is tested to determine the saddle points of the potential energy surface of some molecules. The method has been proposed earlier [E and Zhou in Nonlinearity 24:1831 (2011)]. We additionally use the metric of curvilinear internal coordinates. By a number of examples, we explain the possibilities of a GAD curve; it can find the transition state of interest by a gentlest ascent, directly or indirectly, or not. A GAD curve can be a model of a reaction path, if it does not contain a turning point for the energy. We further discuss generalized GAD formulas for the search of saddle points of a higher index. We calculate diverse examples

Newton Trajectories for the tilted Frenkel‐Kontorova model

Newton trajectories are used for the Frenkel-Kontorova model of a finite chain with free-end bound- ary conditions. We optimise stationary structures, as well as barrier breakdown points for a critical tilting force were depinning of the chain happens. These special points can be obtained straight for- wardly by the tool of Newton trajectories. We explain the theory and add examples for a finite-length chain of a fixed number of 2, 3, 4, 5 and 23 particles

Comment on 'Exploring potential energy surface with external forces'

Recently, a work (Wolinski, K., J. Chem. Theory Comput. 2018, 14, 6306, 10.1021/acs.jctc.8b00885) was published in which the SEGO method (standard and enforced geometry optimization) was proposed to find new minimums on potential energy surfaces. We study this important method from a theoretical point of view. Up to now, the understanding of the proposer does not take into account the barrier breakdown point on a SEGO path being usually half of the path, which is searched for. However, a better understanding of the method allows us to follow along the reaction pathway from a minimum to a saddle point or vice versa. We discuss the well-known two-dimensional MB test surface where we calculate full SEGO pathways. If one has special SEGO curves at hand, one can also detect some weaknesses of the ansatz

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

It is shown that the path described by the gentlest ascent dynamics to nd transition states [W. E and X. Zhou, Nonlinearity 24, 1831 (2011)] is an example of a quickest nautical path for a given stationary wind or current, the so-called Zermelo navigation variational problem. In the present case the current is the gradient of the potential energy surface. The result opens the possibility to propose new curves based on Zermelo's theory for two tasks: locate transition states and de ne reaction paths. The relation between a minimal variational character, that some former reaction pathways possess, and the minimum energy path is discussed

Level sets as progressing waves: an example for wake-free waves in every dimension

The potential energy surface of a molecule can be decomposed into equipotential hypersurfaces of the level sets. It is a foliation. The main result is that the contours are the wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, as well as with a corresponding eikonal equation. The energy seen as an additional coordinate plays the central role in this treatment. A solution of the wave equation can be a sharp front in the form of a delta distribution. We discuss a general Huygens' principle: there is no wake of the wave solution in every dimension

Sliding paths for series of Frenkel-Kontorova models - A contribution to the concept of 1D-superlubricity

Newton trajectories are used to calculate low energy pathways for a series of Frenkel-Kontorova models with 6 and up to 69 particles thus up to a medium chain, and an expedition to 101 particles. The model is a nite chain with free-end boundary conditions. It has two competing potentials and an additional, external force. We optimize stationary structures and calculate the low energy paths between global minimums for a movement of the chain over its on-site potential, if an external tilting by a push- and/or pull direction is applied. We propose to understand a low energy path for a possibility of a superlubricity of the chain. We compare di erent mis t parameters. The result is that the minimums di er only little, however, the critical length of the chain, Ncr, depends on the mis t parameter. Ncr describes the end of a `good' calculability of the Newton trajectory which follows the low energy pathway of the chain through the potential energy surface, for a movement of the chain along the axis. We discuss reasons for the boundary of an Ncr. However, we assume that the low energy paths exist beyond their calculability by NTs