Theory and Examples of Catch Bonds.

We discuss slip bonds, catch bonds, and the tug-of-war mechanism using mathematical arguments. The aim is to explain the theoretical tool of molecular potential energy surfaces (PESs). For this, we propose simple 2-dimensional surface models to demonstrate how a molecule under an external force behaves. Examples are selectins. Catch bonds, in particular, are explained in more detail, and they are contrasted to slip bonds. We can support special two-dimensional molecular PESs for E- and L-selectin which allow the catch bond property. We demonstrate that Newton trajectories (NT) are powerful tools to describe these phenomena. NTs form the theoretical background of mechanochemistry.

Optimal Oriented External Electric Fields to Trigger a Barrierless Oxaphosphetane Ring Opening Step of the Wittig Reaction.

The Wittig reaction is one of the most important processes in organic chemistry for the asymmetric synthesis of olefinic compounds. In view of the increasingly acknowledged potentiality of the electric fields in promoting reactions, here we will consider the effect of the oriented external electric field (OEEF) on the second step of Wittig reaction (i. e. the ring opening oxaphosphetane) in a model system for non-stabilized ylides. In particular, we have determined the optimal direction and strength of the electric field that should be applied to annihilate the reaction barrier of the ring opening through the polarizable molecular electric dipole (PMED) model that we have recently developed. We conclude that the application of the optimal external electric field for the oxaphosphetane ring opening favours a Bestmann-like mechanism.

An algorithm to find the optimal oriented external electrostatic field for annihilating a reaction barrier in a polarizable molecular system.

The use of oriented external electric fields (OEEFs) to promote and control chemical reactivity has motivated many theoretical and computational studies in the last decade to model the action of OEEFs on a molecular system and its effects on chemical processes. Given a reaction, a central goal in this research area is to predict the optimal OEEF (oOEEF) required to annihilate the reaction energy barrier with the smallest possible field strength. Here, we present a model rooted in catastrophe and optimum control theories that allows us to find the oOEEF for a given reaction valley in the potential energy surface (PES). In this model, the effective (or perturbed) PES of a polarizable molecular system is constructed by adding to the original, non-perturbed, PES a term accounting for the interaction of the OEEF with the intrinsic electric dipole and polarizability of the molecular system, so called the polarizable molecular electric dipole (PMED) model. We demonstrate that the oOEEF can be established by locating a point in the original PES with unique topological properties: the optimal barrier breakdown or bond-breaking point (oBBP). The essential feature of the oBBP structure is the fact that this point maintains its topological properties for all the applied OEEFs, also for the unperturbed PES, thus becoming much more relevant than the commonly used minima and transition state structures. The PMED model proposed here has been implemented in an open access package and is shown to successfully predict the oOEEF for two processes: an isomerization reaction of a cumulene derivative and the Huisgen cycloaddition reaction.

Controlling Chemical Reactivity with Optimally Oriented Electric Fields: A Generalization of the Newton Trajectory Method.

The use of oriented external electric fields (OEEF) as a tool to accelerate chemical reactions has recently attracted much interest. A new model to calculate the optimal OEEF of the least intensity to induce a barrierless chemical reaction path is presented. A suitable ansatz is provided by defining an effective potential energy surface (PES), which considers the unperturbed or original PES of the molecular reactive system and the action of a constant OEEF on the overall dipole moment of system. Based on a generalization of the Newton Trajectories (NT) method, it is demonstrated that the optimal OEEF can be determined upon locating a special point of the potential energy surface (PES), the so-called "optimal bond-breaking point" (optimal BBP), for which two different algorithms are proposed. At this point, the gradient of the original or unperturbed PES is an eigenvector of zero eigenvalue of the Hessian matrix of the effective PES. A thorough discussion of the geometrical aspects of the optimal BBP and the optimal OEEF is provided using a two-dimensional model, and numerical calculations of the optimal OEEF for a SN2 reaction and the 1,3-dipolar retrocycloaddition of isoxazole to fulminic acid plus acetylene reaction serve as a proof of concept. The knowledge of the orientation of optimal OEEF provides a practical way to reduce the effective barrier of a given chemical process.

Barnes Update Applied in the Gauss-Newton Method: An Improved Algorithm to Locate Bond Breaking Points.

A mechanochemical reaction is a reaction induced by mechanical energy. A general accepted model for this type of reaction consists of a first-order perturbation on the associated potential energy surface (PES) of the unperturbed molecular system due to mechanical stress or pulling force. Within this theoretical framework, the so-called optimal barrier breakdown points or optimal bond breaking points (BBPs) are critical points of the unperturbed PES where the Hessian matrix has a zero eigenvector that coincides with the gradient vector. Optimal BBPs are "catastrophe points" that are particularly important because their associated gradient indicates how to optimally harness tensile forces to induce reactions by transforming a chemical reaction into a barrierless process. Building on a previous method based on a nonlinear least-squares minimization to locate BBPs (Bofill et al., J. Chem. Phys. 2017, 147, 152710-10), we propose a new algorithm to locate BBPs of any molecular system based on the Gauss-Newton method combined with the Barnes update for a nonsymmetric Jacobian matrix, which is shown to be more appropriate than the Broyden update. The efficiency of the new method is demonstrated for a multidimensional model PES and two medium size molecular systems of interest in enzymatic catalysis and mechanochemistry.

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. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.

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.

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.

An algorithm to locate optimal bond breaking points on a potential energy surface for applications in mechanochemistry and catalysis.

The reaction path of a mechanically induced chemical transformation changes under stress. It is well established that the force-induced structural changes of minima and saddle points, i.e., the movement of the stationary points on the original or stress-free potential energy surface, can be described by a Newton Trajectory (NT). Given a reactive molecular system, a well-fitted pulling direction, and a sufficiently large value of the force, the minimum configuration of the reactant and the saddle point configuration of a transition state collapse at a point on the corresponding NT trajectory. This point is called barrier breakdown point or bond breaking point (BBP). The Hessian matrix at the BBP has a zero eigenvector which coincides with the gradient. It indicates which force (both in magnitude and direction) should be applied to the system to induce the reaction in a barrierless process. Within the manifold of BBPs, there exist optimal BBPs which indicate what is the optimal pulling direction and what is the minimal magnitude of the force to be applied for a given mechanochemical transformation. Since these special points are very important in the context of mechanochemistry and catalysis, it is crucial to develop efficient algorithms for their location. Here, we propose a Gauss-Newton algorithm that is based on the minimization of a positively defined function (the so-called σ-function). The behavior and efficiency of the new algorithm are shown for 2D test functions and for a real chemical example.

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.

Reaction rates in a theory of mechanochemical pathways.

If one applies mechanical stress to a molecule in a defined direction then one generates a new, effective potential energy surface (PES). Changes for minima and saddle points (SP) by the stress are described by Newton trajectories on the original PES (Quapp and Bofill, Theor. Chem. Acc. 2016, 135, 113). The barrier of a reaction fully breaks down for the maximal value of the norm of the gradient of the PES along a pulling Newton trajectory. This point is named barrier breakdown point (BBP). Depending on the pulling direction, different reaction pathways can be enforced. If the exit SP of the chosen pulling direction is not the lowest SP of the reactant valley, on the original PES, then the SPs must change their role anywhere: in this case the curve of the log(rate) over the pulling force of a forward reaction can show a deviation from the normal concave curvature. We discuss simple, two-dimensional examples for this model to understand more deeply the mechanochemistry of molecular systems under a mechanical stress. © 2016 Wiley Periodicals, Inc.

Comment on "Analyses of bifurcation of reaction pathways on a global reaction route map: A case study of gold cluster Au5" [J. Chem. Phys. 143, 014301 (2015)].

We demonstrate by an example that the search for valley-ridge transition (VRT) points used in the commented paper does not always indicate a bifurcation of a reaction path. This was claimed.

A Relation Between the Eikonal Equation Associated to a Potential Energy Surface and a Hyperbolic Wave Equation.

The potential energy surface (PES) of a molecule can be decomposed into equipotential hypersurfaces. We show in this article that the hypersurfaces are the wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, or the steepest descent, or the steepest ascent lines of the PES. The energy seen as a reaction coordinate plays the central role in this treatment.

The Variational Structure of Gradient Extremals.

The gradient extremals can be taken as a representation of reaction paths. We prove that this type of curve possesses a variational nature. We report the conditions such that these curves have the character of a minimal curve. Finally we discuss the relations between the points of these curves being turning points with respect to other special points of the potential energy surface, like the valley-ridge inflection points.

Variational nature, integration, and properties of Newton reaction path.

The distinguished coordinate path and the reduced gradient following path or its equivalent formulation, the Newton trajectory, are analyzed and unified using the theory of calculus of variations. It is shown that their minimum character is related to the fact that the curve is located in a valley region. In this case, we say that the Newton trajectory is a reaction path with the category of minimum energy path. In addition to these findings a Runge-Kutta-Fehlberg algorithm to integrate these curves is also proposed.

A comment to the nudged elastic band method.

The minimum energy path (MEP) is an important reaction path concept of theoretical chemistry, and the nudged elastic band (NEB) method with its many facets is a central method to determine the MEP. We demonstrate in this comment that the NEB does not have to lead to a steepest descent pathway (as always assumed). In contrast, as long as it is used without spring forces, it can lead to a gradient extremal.

Transition state theory with Tsallis statistics.

We discuss the rate of an elementary chemical reaction. We use the reaction path and especially its saddle point on the potential energy surface. The reaction path connects reactant and product of a reaction over the transition state (TS). Usually, the TS is assumed near or at the single saddle point of the reaction path. By means of comparison of the statistics of states at the reactant and at the TS, one can estimate the reaction rate by the Eyring theory. We propose to use the Tsallis statistics at the TS, a statistics of seldom accidents. Thus, we propose to generalize the well-known Boltzmann-Gibbs statistics, which is the limiting case of the Tsallis statistics. We use features of this nonextensive thermostatistics. The basic properties of the statistics are used to derive (approximated) partition functions, and they are applied on reaction rates. The approximation includes a factorization of the partition functions. The theory is applied to HCN isomerization to HNC, and to the reaction H(2) + CN --> H + HCN. It allows an accordance with experimental estimations of the reaction rates.

Finding the transition state of quasi-barrierless reactions by a growing string method for newton trajectories: application to the dissociation of methylenecyclopropene and cyclopropane.

A method for finding a transition state (TS) between a reactant minimum and a quasi-flat, high dissociation plateau on the potential energy surface is described. The method is based on the search of a growing string (GS) along reaction pathways defined by different Newton trajectories (NT). Searches with the GS-NT method always make it possible to identify the TS region because monotonically increasing NTs cross at the TS or, if not monotonically increasing, possess turning points that are located in the TS region. The GS-NT method is applied to quasi-barrierless and truly barrierless chemical reactions. Examples are the dissociation of methylenecyclopropene to acetylene and vinylidene, for which a small barrier far out in the exit channel is found, and the cycloaddition of singlet methylene and ethene, which is barrierless for a broad reaction channel with Cs-symmetry reminiscent of a mountain cirque formed by a glacier.