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 build by families of NTs which connect the same stationary points. For different exit saddles of different height we discuss the corresponding pulling corridors; mainly by simple two-dimensional surfaces models. If there are different exit saddles then there can exist saddles of index two, at least, in between. Then the case that ∗To whom correspondence should be addressed †Leipzig University ‡Universitat de Barcelona ¶Universitat de Barcelona


INTRODUCTION
The application of mechanical stress to molecular systems has recently attracted significant interest as a means of controlling chemical reactions. Under stress the minimums and saddle points (SP) of the potential energy surface (PES) change their relations. This concerns the energy height as well as the molecular geometry. Curves for such changes in the configuration space of a molecule are the aim of this paper. In theoretical chemistry a basic concept already exists underlying many widely used models, namely, the reaction path (RP). The RP is a one-dimensional description of a chemical reaction through a sequence of molecular geometries in an N-dimensional configuration space. A reaction mechanism is described through the PES model by means of RPs. We use N = 3n − 6 for the number of non-redundant internal coordinates, r, and n is the number of the atoms of the molecular system. Such a simplified one-dimensional description generally makes it possible to move on the PES. A well known type of RP models is the Newton trajectory (NT). [1][2][3][4][5][6][7][8] For this type of RP holds that at every point of the curve the gradient of the PES points into the same direction, a direction of a prescribed search vector. On the other hand, this property is the central idea of models of mechanochemical stress applied to a molecular system where the search direction is now the direction of the stress vector. This is the reason why NTs should be taken into account as the basic models of a great variety of mechanochemical problems. 9 The present study is a follow-up of a first investigation of the realm of mechanochemistry with the NTs tool. 10,11 In the last years, the phenomena of the action of a mechanical stress over a molecular system have motivated experimental and theoretical researches (see Refs. 12-23 and references therein) We will here quantify molecular stress geometrically. Basically, the model employed to rationalize mechanochemical experiments was created early by Eyring et al., 24 by Bell 25 in 1978, and it was used, for example by Bustamante et al. 26 in 2004. More recently, it was recreated and treated by Ong et al., 27 by Ribas-Ariño et al. 28 and by Wolinski and Baker 29 in the same year 2009. The generally accepted model 15,19,30 consists in a first order perturbation on the associated PES of the unperturbed molecular system, V (r), due to a stress or pulling force vector, f, by where the symbol T means transposition of a vector or a matrix. r is the coordinate vector, r o is any fixpoint, and f is the force vector. In this equation, being the starting point for the theory of mechanochemistry, the scalar product ∑ N i f i (r i − r oi ) with the force vector, f, ensures that only the part of the coordinate vector, (r − r o ), in the direction of f acts. The subtrahend in the equation is a hyperplane in the space IR N+1 whith slope in the direction of the force, f ∈ IR N . V f is named the force-transformed PES, 15,28,31,32 or the effective PES. The displacement of the stationary points of the new effective PES is described by NTs. [9][10][11] The force, f, can be a pull or a push. It is worth mentioning that a similar model was discussed by Thornton 33 for the prediction of the effect of substituent changes on the transition state (TS) geometry.
The formalism employed to study mechanochemistry can also be applied to explore how external electric fields can catalyze and control reactions [34][35][36] when these fields are constant in time and position. 34 One could even think of using this formalism as a first-order approximation to study catalytic and enzymatic processes. Notice that in the present model, the direction and the magnitude of f are constant. (Note that also non-constant forces are studied in the past. [37][38][39] ) The used constant force here is certainly a strong approximation, but it might be useful when it comes to understand and design catalysts and enzymes. In this case, the direction and the magnitude of vector f are fixed for every special catalyst. Indeed, if we can quantify all the electrostatic and nonelectrostatic parts of the action of a catalyst (an enzyme in biochemistry), into a force represented mathematically by a vector, f, then we can handle the change of the PES by the ansatz of Eq. (1).
The action of the catalyst will then be explicitly calculable and interpretable by the displacement of a TS and a reactant minimum of the original PES, V (r), to the effective PES, V f (r).
In the present context it is important to consider that historically the problem of enzyme catalysis was discussed early by Haldane 40 and Pauling. 41 There should be an 'active region of the surface of the enzyme which is closely complementary in structure not to the substrate molecule itself, ..., but rather to the substrate molecule in a strained configuration...', meaning the transition state (TS). '... and caused by the forces of attraction to assume the strained state which favors the chemical reaction...' 41 Thus Pauling proposed that the enzyme binds the TS. It was named TS stabilization (TSS), in contrast to reactant destabilization. [42][43][44] The activation energy is lowered for the catalyzed versus the uncatalyzed process, giving rise to large rate accelerations. It is equivalent to the action of Eq. (1) for pulling which leads us to the remark that this equation not only describes the pulling process, but can generally describe catalysis as well. 10,11,26 A second look back has to appreciate the early guess of Warshel 42,[45][46][47] that the catalytic effect of enzymes has an electrostatic origin. An electrostatic force of the enzyme stabilizes the TS of the accelerated reaction: this expresses in words what Eq. (1) expresses as a linear ansatz by a formula.
Maybe this ansatz is too simple for a given problem, but it explains many observed and studied results and now remains the task to determinate the force, f, for every enzyme. The methodology reported in the next section allows us to calculate explicitly the lowering of the activation energy by the enzyme, if f is known (and if we can calculate, or approximate the PES, or the free energy surface of the reaction under consideration).
At this point, it should be mentioned that mechanochemistry and catalysis should not be considered as two different facets of chemistry under the point of view of Eq. (1). Indeed, a direct combination of electrostatical and mechanical forces is found in molecular motors. These two facets also meet in the research area of mechanical activation of catalysts 48,49 and on experiments in which an enzyme acts on a substrate subject to tensile stress. 50,51 In this article, we present the chemical consequences of a topological study of generic PESs based on NTs that are related with the basic ansatz Eq. (1). For this reason, the conclusions drawn are completely general, independent of the type of quantum mechanical calculation of the PES.
Nevertheless, chemical examples from the literature are discussed under this new view corroborating the present theory. Our analysis demonstrates the existence of streams of NTs (which will be referred to as chemical corridors) that are crucial to understand and model the acting forces in mechanochemistry and catalytic phenomena. We will unveil the unforeseen challenges and complexity embodied in Eq. (1). The article is organized as follows. Next we explain the theory of NTs for the understanding of a chemical pulling/pushing process, or an action of a catalyst (enzyme).
This theoretical part is a summary of that published in references 9,10, and is briefly exposed and explained, in this way the article is self-contained. Then we use two-dimensional test surfaces to explain the impact of stress on the topography and the stationary points of the effective PES. We mainly treat the 'strong multidimensional case' 22 with competing reaction pathways. We show that already for two dimensions, if at least two competing RPs exist, we can find rather complicated areas for the movement of the stationary points on the PES under stress. Nevertheless, though pure theoretical, these generic surfaces are taken from well tested models of specific chemical systems.
The paper presents at the end a Discussion, and some Remarks with a Conclusion.

METHODOLOGY
For a given force, f, we have an effective PES, V f (r), which has a new inherent chemistry with respect to the unperturbed PES, V (r), such as other reaction rates and other chemical properties.
Of course, the linear perturbation in Eq. (1) is the simplest model. 52 Meanwhile, time was used to derive new basic ideas of mechanochemistry. 15,[52][53][54][55][56][57] The force f in Eq. (1) may be determined by the change of a distance between two pulling points of the molecule, 58 or by any other experimental setup. Formerly it was associated with one of the N = 3n − 6 internal coordinates 58 or a linear combination of them. 27 Throughout the present work we assume for simplicity that the plane of two intrinsic coordinates (x, y) is the stage where the pulling works. These coordinates may describe the weakest point of a molecule where it breaks preferentially. 59 We note that the theory and the use of NTs can be applied to N-dimensional systems with any large n for the number of included atoms.
Due to the external force, the stationary points are located at different positions on the effective potential, V f (r), 56 with respect to the unperturbed potential, V (r), where it holds ∇ r V (r) = g(r) = 0. g(r) is the gradient of the unperturbed PES. The stationary points on the effective potential have to satisfy the analogous condition, ∇ r V f (r) = 0. Since V f (r) is the one given in Eq. (1) there follows that the new minimums or SPs should satisfy One searches a point where the gradient of the unperturbed PES, g, has to be equal to the mechanochemical force, f, being the force that induces the chemical process. If the mechanical stress in a defined direction is f = Fl with a fixed unit vector, l, then it is l = g/|g| and F = |g| is the magnitude at the stationary points since from Eq. (2) we have, 0 = l (|g| − F), being satisfied if the former equality is also satisfied. Eq. (2) means that the tangential hyperplane to the original PES, characterized by the gradient, g, is equal to the hyperplane of the pulling force, f T r, in Eq. (1). The case F = 0 is named the pure thermal limit. 15 This is the case without a mechanical load. Now we treat a fixed direction of l, but different magnitudes of the forces, F. For changing magnitudes, F, we get a series of effective PESs (1), and on every effective PES we get its stationary points. So we get a path following for the curve of the force displaced stationary points (FDSPs). 30 The minimums and the SPs of any index of the effective PES (1) change and they are on the FDSPs curve, because at these points holds ±|g| = F. A FDSPs is the path connecting the stationary states of the perturbed PESs. Each effective PES of this ensemble corresponds to a given magnitude F of the force but to the same direction l of application.
The FDSPs are on the solution of the differential equation of Branin 60 which we can use in N = 3n − 6 nonredundant coordinates 9,10 t is a curve parameter and the matrix A is constructed as a product of the determinant of the Hessian, H, of the unperturbed PES with the inverse Hessian, At the given g, the Branin equation yields a "Newton Trajectory" (NT) corresponding to the FDSPs for this g. The Branin equation is nothing but a simple strategy to generate the points on the FDSPs without solving g/|g| = l and ±|g| = F for several values of the force magnitude.
The signs '±' in Eq. (3) Note that H is always the Hessian of the original zero-force PES, V (r), because the model Eq. The definition of an optimal pulling direction 10 is also derived: the mechanical force to be applied to the molecule of interest is f = F l where F = g T g. To each NT belongs the corresponding lvector, and by varying it we have different NT curves. All the regular NTs, that leave the minimum and arrive at the same SP, cross at least once a Det(H) = 0-line. The Det(H) = 0-line that each NT crosses gives the BBP of this NT, the maximal rupture force, F max . This is due to the fact that at the BBP point of a regular NT, the gradient norm |g|, takes its maximum value along this NT curve.
If we compare all NTs of such a set, then the NT which gives the lowest value of F max is named the optimal NT. It coincides with a gradient extremal (GE) exactly at the intersection point with the Det(H) = 0-line. 10 In this special BBP, the Det(H) = 0-line, the GE and the optimal regular NT meet. The optimal BBP is a stationary point on the function |g| crossed by the optimal NT. On x y Figure 1: Direct chemical corridor of type 1 of the PES of Eq.(S1) in SI. All the regular NTs between the two singular NTs (in blue and orange color) form this corridor. Some regular NTs (in gray) are shown. The BBP condition, Det(H) = 0, is the green line. A singular NT is the Newton trajectory that passes through a VRI point, whereas a regular NT does not pass through this type of points. The spreading of pulling directions can be large. 65 All NTs describe pulling directions to different families of effective PESs. The surface without load is shown in the background for comparison.
Historically, the BBP was named the breaking point distance (for single bonds), 15

MODELS FOR CHEMICAL CORRIDORS
The theory herein presented aims to rationalize both mechanochemistry and catalysis. It is based on the topology or, more specifically, the topography of the common PES associated to different types of chemical mechanisms. In the following, we shall analyze the topography of the basic PES models, namely, (i) a PES describing an elementary pathway, (ii) a PES describing a process with two different reaction channels, and (iii) a PES describing a process with two different reaction channels that include intermediates. This will permit us to draw generic topological conclusions which are important for the general applicability of the present theory.

Potential energy surface with an elementary pathway
We present as an example a simple PES with a one-step mechanism to analyze the nature of the corridors associated to this type of PES. As a generic representative of this class of PES, we use the skew double-minimum PES, 6,10 whose mathematical description is given in the Supplementary Information (by Eq.(S1), the PES is shown in Figure 1). This PES has two minima, connected by one SP of index one (i.e, a transition state).
In Figure 1  subset are reaction pathways satisfying the condition of the minimum energy path. 73 The gradient norm at the BBP line is the maximal force at which a reaction is enforced. 26,65 Going along an RP the gradient norm achieves the maximum value at the intersection point with the BBP line. 10 Of course, the maximal force significantly depends on the pulling direction.
The movement of the effective SP e f f is manifested in many enzymatic reactions. 26 Figure 2: Schematic transformation of (Z)-Merocyanine to Spiropyrane. Depending on the conformational isomer of (Z)-Merocyanine, an enantiomer of Spiropyrane is obtained. The conformational interconversion of (Z)-Merocyanine takes place through a planar transition state indicated at the bottom.
Starting a pulling at the minimum at the point (-1,-1) in Figure 1, we may observe that the spreading of the tangent directions of the possible regular NTs is large. In contrast, starting a pulling at the minimum at (1,1) in contrary directions, we can observe that the spreading of the tangent directions of the possible NTs is quite smaller. We recall that any NT of the corridor follows one and the same gradient direction through its evolution. Thus both starting points at the minimums encircle an equal range of gradient directions. This also happens at the common SP of index 1 which all NTs cross. From a chemical point of view, this topological conclusion means that if a given vectorial force is able to drive the system from reactants to products in a corridor of b) The charge transfer, Φ CT , and product, Φ P , valence-bond configurations projected from the molecular orbitals wave function for the −O − attack of a > C = N + < bond. c) The bond breaking point BBP 1 labeled according Figure 2. BBP 1 has the major contribution of Φ CT , more than BBP 2 . Shown is the external electric field, −F O−C , along the "reaction coordinate" (O · · · C). type 1, the reverse force will necessarily drive the system from products to reactants.
An example of this type of corridor within the context of mechanochemistry concerns the intramolecular addition of an electrophile to a nucleophile, such as the rearrangement of (Z)-Merocyanine to Spiropyrane, see Figure 2. Each of the elementary pathways of Figure 2 (either the left-handside one or the right-handside one) corresponds to a type 1 corridor.In this particular example, we will consider that the PES is perturbed by an electric field rather than a mechanical force.
A valence bond diagram of this rearrangement is displayed in It is important to take into account that the Φ CT state has a major contribution in the description of the electronic structure of BBP 1 , shown in panel (c) of Figure 3, in comparison to BBP 2 .
Let us deeper consider Figure 3 (a). When an external electric field is applied along the "reaction coordinate" C · · · O, a negative F O−C field stabilizes Φ CT , which shifts the TS towards the An example of another type 1 corridor applied to catalysis is provided by the 1,2-sigmatropic Hshift rearrangement of cyclopentadiene. This reaction was computationally studied by Schleyer 77 and it was found that Li + was able to catalyze it. We take this reaction as an example of the present model. This chemical example is also proposed as a description of a primitive version of an algorithm based on Eq. (1) to find the optimal catalyst for a specific reaction. The first step consists to locate BBPs on the IRC path joining the reactants and products through the transition state. Since we are going from the reactant to the product we are only interested in the BBP located in the reactant valley. Note that according to the section Methodology this BBP usually is not exactly the optimal, however, it may be close to the optimal one. In the second step we compute and analyze the BBP gradient vector. In the next step we look for a chemical species that makes a force to the substrate in the direction to the BBP gradient vector. Note that in this situation, dV e f f (r) /dt ≈ 0, because, |g| − F ≈ 0, as explained in section 2. In our particular example the BBP structure and the corresponding gradient vector is shown in Figure 4, part (b).
The selected chemical species which partially satisfies the above requirements is the Li + which was located below the molecular plane, that is to say in the opposite side where the H-shift takes place. According to the present theory the direction of the gradient vector at BBP in the presence of Li + should overlap as maximal as possible with the gradient vector of the BBP without the cation, otherwise, it is not the best catalyst. In the present case these two vectors form an angle of approximately 30 • . We conclude that Li + is an efficient catalyst but is not the optimal. The structure of the BBP and the corresponding gradient vector is reported in Figure 4, If we regard Figure 5 below, the half-family of NTs forms a contrast. Its branches start at the lower minimum at the point (-1,-1). They cross neither the next BBP line nor the SP of index 1.
They characterize a forbidden chemical corridor for a pulling purpose. Note that in this case the pulling is not impossible but useless and leads in every case to an enforced reaction from the upper minimum to the lower one. We call this half-corridor a useless chemical corridor. for the two minimums of the reactant, R, and of the product, P and still more for the two SP of index 1, the low SP l and the upper SP u . The PES has a heigh center, an SP of index two, labeled by Max, see Figure 6. As we can see with the level lines, on this PES, the upper reaction pathway taking place through the SP u is energetically very higher with respect to the lower one that occurs through the SP l .
There are two VRI points between the reactant, R, and the maximum, as well as between the product, P, and the maximum, see Figure 6. Two singular NTs connect all these points shown in Starting at the reactant, R, we find three families of NTs for a pulling to the two SP of index 1. One direct family is shown in Figure 8, and in Figure 9 we develop to one NT of the family the corresponding effective surfaces. Shown are three cases: panel (a) shows a low force, panel (b) a medium, and panel (c) shows a higher force. Figure 10 also explains a pathway to the lower SP l .
Another way over the upper SP u is treated by Figure 11. The first family in Figure 8 realizes a direct connection between reactant, R, and product, P.
This kind of direct chemical corridor is of type 1. In our two-dimensional problem is the corridor characterized by the set of all regular NTs located between the two singular NTs (which are depicted in blue and orange colors). The singular NTs are the borders of the corridor, compare the former Section. The chemical corridor contains the two optimal BBPs of the pulling, as well as of the reverse pulling, enforcing the optimal reactions from R to P, or from P to R. We recall that the optimal BBPs are at the intersection of a GE curve with the Det(H) = 0 or BBP manifold (green lines). These points are in the reactant or in the product valley. 10   The set of regular NTs that belongs to this type 1 chemical corridor satisfy the MEP condition 73 since these NTs are located in the valley containing the minimums R and P and the SP of index 1. Pulling along any regular NT of this corridor leads to a BBP that is near to the optimal BBP, that is BBP 1 if we are pulling from R, BBP 2 if we are pulling from P. If we start a pulling at the R side, then the SP l and the minimum R coalesce under this pulling, and the reaction is enhanced.
We show in Figure 9 the action of a pulling along the thick gray, regular NT of Figure 8. It is named NT low . It is observed that the pulling force results in a displacement of the reactant and SP l along this NT. As the magnitude of the force, F, increases, these two stationary points get closer and, eventually, they coalesce above a given external force (see Figure 9). The force direction is  Note that the force employed to induce the coalescence between reactant and SP l in Figure 9 is larger than the optimal force, that is to say, the minimum force that is needed to favor the process.
Interestingly, in this particular case, the optimal NT coincides with the singular NT depicted in orange in Figure 8, which defines the border of this corridor. The optimal F value there is still smaller: F max = 2.956. The direction of the optimal NT through BBP 1 is (0.878,-0.478) T . It is clear that the pulling is reversible by an inverse pulling direction. Under the inverse pulling, the lower SP l and the product minimum move together and eventually coalesce. 10,11 The VRI points,  98 All the NTs connect the minimums R and P in an indirect manner, i.e., they go through the SP l , Max, and SP u . Two singular NTs are the borders of this corridor; they are colored by blue and orange. The blue singular NT crosses the VRI 2 two times in the product valley, the orange singular NT touches two VRIs outside of the Figure, near the x = 0-line. The black points are VRIs. The red points are the optimal BBPs. The surface without load is shown in the background for comparison. enhancing the reaction rate. But one should expect that pulling into the other, inverse direction makes that the product and this same SP move together and again form a TSS, now for the enhanced back-reaction. The reason for this expectation is simply the SP structure: in both directions of the SP valley the potential energy decreases along the steepest descent. However, we find below that the direct pulling corridor (of type 1) is usually very small, and that there other types exist.
Under the first corridor in Figure 8, the second family of NTs starts from minimum, R, to the lower SP l , see Figure 10 where again the border NTs are colored by blue and orange. However, its NTs turn after the lower SP uphill to the maximum in between the two SPs, then go to the upper SP, and further they turn back to the product, P. The possibility of a connection by regular NTs of an SP of index one and an SP of index two is given by the index theorem for NTs. 99 Observe that the NTs of this corridor, especially the parts on the ridge between the SP l and the SP u , are by no  Figure 11: An asymmetric type 3 chemical corridor on the modified Eckhardt surface. 98 All the NTs connect the minimums R and P in an indirect manner, i.e., they go through SP u , Max, and SP l . Two singular NTs are the borders of this corridor; they are colored by blue and orange. The blue NT touches the VRI 1 two times, the orange NT touches two VRIs outside of the figure, on the x = 0-line. The red points are the optimal BBPs. the previous studied case, namely, the reactant, R, and the SP l move together to form a TSS and coalesce at least. Now also the SP u and the maximum move together. 10,11 With this behavior, the family of NTs defines a new type of chemical corridor, labeled as chemical corridor of type 3.
Note that this corridor is indirect. In other words, this corridor does not go directly to the product.
In this type, an inversion of the pulling direction starting from the second minimum, P, does not lead to the SP l , it leads to the SP u . Thus the pulling along the type 3 corridor shows an intrinsic hysteresis. Of course, the hysteresis concerns here the mechanical corridor, its curve of FDSPs, but not possible chemical reactions over SP l . The reverse direction of pulling can turn the reaction of the molecule over the upper saddle. There the original SP height and the F max at the respective BBP 4 are much higher.
A representation of the effective PESs for different forces, F, for a pulling direction from a corridor of type 3 is given elsewhere. 10,11 An important aspect emerging from the analysis presented above is that small changes in the direction of the external force in any chemical system statisfying an Eckhardt-like PES can result in drastically different scenarios due to the switch of the type of corridors. Starting from the minimum, R, only small variations of the pulling direction, l, can decide between a pulling effect of the type 1 or type 3. The pulling itself works like in the first case: the SP l and the reactant, R, move together and coalesce, at least. However, at the same time, the SP u and the maximum move together. 10 Supplementary Information Eq.(S3).) The upper panels shows the NTs through the points VRI 1 , VRI 3 , and VRI 5 , but the lower panels shows the NTs through the points VRI 2 , VRI 4 , and VRI 6 . There are further VRI points outside of the region displayed, see Supplementary Information for further details about these VRIs.
Contrary to the lower SP l , to attain the higher SP u from R by a pulling, we have to chose directions from a next family of NTs, shown in Figure 11. It is the mirror picture to the corridor of Figure 10. The mirror line is the y-axis for x = 0. The optimal BBP 3 of the family is at point (-1.22,1.45). Its F max = 13.182 is more than four times larger than the F max at the lower BBP 1 . The direction of the optimal NT is (0.502, 0.865) T . After crossing the SP u , the optimal NT turns uphill and connects the two SPs, the SP u and the SP l and it still crosses the maximum, the SP of index two in between, before entering the product. If one pulls the reactant in any of the directions of  Figure 10 and Figure 11, as well as in the right minimum, P. Thus, here "forbidden" 11 or useless directions do not exist. The case is newly discussed in the next subsection.
The topography of the Eckhardt surface might be relevant, for instance, to rationalize the chemistry associated with the cis↔trans isomerization of azobenzene derivatives, which constitute one of the most important classes of photoswitchable molecules. It has been observed that this isomerization can take place by means of two different reaction pathways: a rotation mechanism and an inversion mechanism. 100 Remarkably, pronounced hysteresis was detected in simulations in which the isomerization was induced mechanically (either with a pull or push force). 101 The ring opening in benzocyclobutene (BCB) is a next chemical example. 31 BCB acts as a twofold proof of our concept.
(i) An appropriate pulling force can enforce a symmetry forbidden reaction path, a conrotatory ring opening against the Woodward-Hoffmann rules. Thus, the respective pulling drives the molecule over a higher SP of the PES.
(ii) By the use of two experimentally different pulling directions one gets two different amounts of force or a literally opening of the molecule, F 1 =1400 pN against F 2 =900 pN. We assume that both directions are in the same corridor. However, the height of their BBP energies can be very different. 11 Direction l 1 is the pulling mainly along a single bond, l 2 is the pulling mainly along a double bond. The forces are applied by a direct anchor or a so-called lever-arm polymer anchor of the molecule. 31

Alternate Pathways with Intermediates: The Rhee-Pande PES 102
The examples studied in the previous subsection concern alternate pathways without intermediates.
In this subsection we study the case of alternate pathways with an intermediate and its possible effect on the pulling. We consider the case of the Rhee-Pande PES. 102 As before the surface has two alternate pathways 86,93 where we can assume that different cases of a directed force, f, again enable two different pulling pathway families with respective FDSPs. One sort of corridors leads over the low energy SP, but the other SP is energetically disfavored in the forceless case. 59 Of course, the two-pathway-model is still in the framework of minimal models, 55  x y Figure 13: The direct small corridor between the two deep minimums, R and P. The border parts of the two singular NTs are drawn in blue (the NT throughVRI 1 ) and orange color (the NT throughVRI 2 ). A usual regular NT is drawn by a gray line in between. It has a second branch through Max and SP u .
There is a direct corridor named pulling of type 1, shown in Figure 13, like in the former examples. This particular corridor is circumscribed by two singular NTs that mark the limits of a very small region of the configuration space. It is quenched between the two singular NTs through VRI 1 and VRI 2 , compare the left two panels of Figure 12. An NT inside the corridor is also shown in gray color; it has a second, closed branch through the maximum, and near the two VRIs, and through the SP u . Of course, this second branch concerns the movement of the SP u and of the maximum under a pulling along the direction of the included NTs of the type 1 corridor. The SP u also moves downhill in direction of the minimums, R or P, along the NT. The movement however is smaller than that of the minimums, R or P, and the SP l together. The small changes in SP u thus do not have relevant consequences from a chemical point of view when a force is applied to the corridor going through SP l . The corridor itself concerns the pulling over the low SP l . The type of corridors displayed in Figure 15 and Figure 16 show a complicated behavior but follow the features of the precedent corridors. These corridors include useless half-corridors like that found in the description of the corridor of type 2. However, they are more complicated due to the higher complexity of the underlying PES. Nevertheless, starting at reactant, R, and using such a corridor enforces the reaction over SP l in case of Figure 15, or over SP u in case of Figure 16.
In these corridors the former relation between reactant, R, and the product, P, is fully disconnected. This kind of corridors is named pulling of type 4. They start in the reactant region, at R , but after crossing the region of the maximum, they do not turn down to the product, however, they escape uphill into the mountains of the surface. The first case is the corridor between VRI 3 (a) The closed circular branches of NTs which form the loop 'behind' VRI 7 form a region of the PES which is not connected to the minimum, R, by a continuous NT. Such compact branches of NTs are described elsewhere. 103 The unreachability of special regions, seen from the minimums, also concerns the regions 'behind' the VRI 3 and the VRI 4 .
The other corridor in the right panel of Figure 15 is the small corridor between VRI 4 and VRI 6 .
It goes further uphill after VRI 6 .
If one looks to the product side, there is a disconnected second half-corridor in the product valley, in both cases. (Disconnected with the exception of the one singular NT forming the border: at the corresponding singular VRI point, the two corridors touch.) The regular NTs of the product side do not cross the (green) BBP line. It means that an inverse pulling is quite useless: it moves the product minimum in its valley, forward or backward, but not to a coalescence with one of the SPs. A pulling in such a direction cannot enforce the disappearance of the product side, independent from the amount of the force, F.
The last corridor is a circular region shown in Figure 16. For a pulling in this roundabout corridor we find a simple switch between two competing SPs.
They are enforced by the same force, only in forward or backward direction. Of course, the BBPs of special NTs can be of quite different height.
If one looks to the product side, there is a truly disconnected half-corridor in the product valley.
The NTs of the product side do not cross the (green) BBP line. It means that an inverse pulling is useless from point of view of the product side. It moves the product minimum in its valley, for or back, but not to a coalescence with one of the SPs. A pulling cannot enforce a disappearance of the product side, independent from the amount of the force, F. The type 2 pulling corridor is in every case of a corresponding force a 'one-way' enforced reaction R→P. In the forward direction, say with F > 0, it may lead over SP l . In the backward direction, say with F < 0, it may lead over SP u . No F can enforce the back-reaction, P→R. or backward directions (F < 0) describe a useful pulling, or a useless one. Thus, "forbidden", or useless directions do not exist for the reactant, 11 but thoroughly for the product side, compare the Figure 15 and Figure 16.
We emphasize that for directions of all types of corridors we can here enforce the reaction R→P. All pullings make that the reactant, R, and a respective SP move together, along the respective NT, and coalesce at least, if F max at the BBP is fulfilled. The different types of corridors concern, for this PES model, a possible question for a back-reaction only.
A key chemical process whose mechanism can be explained through a Rhee-Pande-like surface is the Diels-Alder reaction under the influence of an external electric field. 34,76 The electric field applies directly through Eq. (1) on the PES of the system. 29 When a negative electric field is oriented along the reaction axis, the reaction takes place via a two-step mechanism with a zwitterionic intermediate if and only if the electric field is larger (in absolute value) than a given threshold. 76 According to the present theory, this threshold is associated with the force of the optimal BBP point. On the contrary, for smaller fields or fields pointing in the opposite direction, the reaction takes place via a concerted mechanism. 76  An example that shows the application of the present theory to a catalytic process based on the Rhee-Pande model is the Cope rearrangement, 104 see Figure 17. The Cope reaction stands as an example of [3,3]-sigmatropic rearrangement. It involves a stereospecific and concerted migration of an allyl group along a second allyl fragment within a 1,5-diene with concomitant σ -and π-bond reorganization. 105 This reaction is mainly reversible and very often needs strong reaction conditions. For the reason it is important to develop catalytic variants of the Cope rearrangement.
Such developments are the metal-catalysts by Pd(II) 106 and gold by bonding to the diene π-system and the substituents attached to the 1,5-diene. In the latest case the catalytic rate process can be enhanced if the substituents are electron-withdrawing with the addition of Brønsted acids. 107 Examples of these substituents are carbonyls and the iminium ions recently reported by Gleason and Kaldre. 108, 109 We proof by doing the present theory that electron-withdrawing groups transformed in ions enhance its catalytic rate. For the purpose we compute at HF/6-31G level of theory the Cope rearrangement of 2-methylenehex-5-en-1-imine, see Figure 17, both, the concerted and the non-concerted pathways. The non-concerted path is the lowest energy path, passing through a biradical intermediate, see Figure 18  they are not necessarily the optimal ones. They might not belong to a gradient extremal curve.
The structures of the six BBPs are given in Figure 19 with the corresponding components of their gradient vector. According to the present theory, the analysis of these gradient vectors gives us the optimal external force to be applied to enhance the rate of the process. According to that explained in references 108,109 the protonated form enhances the rate of reaction. We proof that this experimental observation is in accord to the present theory. First we add a proton to each BBP structure and compute the gradient. The protonated BBP structures are the corresponding BBPs  With this result, finally we compute the reaction pathways of 2-methylenehex-5-en-1-iminium ion  Figure 19: Structures of the BBPs for the concerted and non-concerted pathways corresponding to the 2-methylenehex-5-en-1-imine Cope rearrangement. The labels are that given in Figure 18 (a). The arrows are the gradient components.
at the same level of theory. The energy profiles for the concerted and non-concerted pathways are reported in Figure 18 panel (b). The form of this profile is in accordance to the behavior described in Figure 14 panel (a) of the Rhee-Pande surface. From these results we conclude that the protonation is an efficient catalyst however it is not the optimal catalyst according to the present theory.

Alternate pathways with different products: a modified BQC PES 110
In this subsection we deal with the case of alternative pathways, each of them leading to a different product. Additionally, the products can be connected through another pathway. Such PESs are related with mechanisms of reactions where the reactant converts to different products.
The particular PES employed in this subsection, which is a modified version of a surface proposed by us, 110 is shown in Figure 20  R to P 1 right below, and a second reaction path opens from R to P 2 right top, and also a valley exists between P 1 and P 2 . In sum we have a curvilinear triangle of reaction pathways. The SPs are separated by VRI points. The center is again an SP of index two. There is a slight asymmetry for the two SPs: the SP 1 is slightly lower in energy than the SP 2 .
In contrast to the first modified Eckhardt case, here we find three direct corridors of type 1, along the three assumed reaction pathways. To get the corridors, we have to set first the borders, the singular NTs of this PES. They are shown in Figure 21. There are valley-ridge inflection points between the reactant and the maximum, as well as between the two product minimums and the maximum, see Figure 21. Panel (a) shows the singular NT through VRI 1   shown in Figure 22. Panel (a) shows the valley over SP 1 , but panel (b) shows the valley over SP 2 The corridor between the two product minimums is discussed later. Note that every of the NTs which define these corridors are divided into two branches. One branch describes the FDSPs in the reactant valley of interest. The other branch is a loop in the contrary product valley. It describes the behaviour of the contrary SP and the contrary product minimum under a respective pulling.
The contrary SP moves to the maximum and its energy increases, what is discussed elsewhere. 10 Again, here emerges another type of corridor, the type 3, for NT directions outside the valley corridors. The first one is the corridor between the VRI 1 right below, and a VRI 4 at the top corner of the shown region, see Figure 23 (a). Contrary to the former PES example (Rhee-Pande PES), the corridor is only at the beginning a type 3 pathway (for example, from R to SP 1 , then over the maximum to SP 2 , but then down to P 2 . Thus, it goes not directly from SP 2 to the aim, the P 1 minimum. However, the minimum P 2 is at the same time enforced by a pulling in a respective direction to P 1 because the pathway P 2 to P 1 is an additional continuation here, a valley corridor of type 1. So to say a type mixture happens between the type 3 corridor for the pulling of the reactant, and a type 1 corridor for an additional pulling of a possible state at P 2 to the aim P 1 . The enforced reaction would go from R over an effective SP 1 , but also a molecule at state P 2 would be enforced to react to P 1 . Note the spread of NTs over a wide region after SP 2  for the reversion. An inverse direction of the force would turn the P 1 to P 2 by a decrease of the difference between P 1 and SP p , but at the same time also the SP 2 back to the product, P 2 . The back reaction then would be enforced over a new effective SP 2 directly to the reactant, R. The last step would be accompanied by a move together of SP 1 and maximum thus closing this col. 10 The last type of corridor is here a roundabout one, see Figure 24. It is of type 2. Its respective NTs are composed from two parts. The loops are now the interesting branches in the reactant valley. The other branches connect the two product minimums. If a pulling in such a direction is done, say to the SP 1 , then this SP 1 and R coalesce, at least, what enforces the reaction to product P 1 . The SP 2 and the maximum move together, at the same time. The inverse direction turns the situation and interchanges the respective stationary points. Note that many NTs in this corridor have a TP. It means that they belong usually not to the optimal region of the pathways to the respective SP. 9,11 From point of view of a backreaction, P 1 → R, or P 2 → R, the directions of the corridor are forbitten.

DISCUSSION OF FURTHER EXAMPLES
If only one SP exists between two minimums, (a very simple case which may be restricted to small molecular systems) e.g., a large set of pericyclic reactions, then there is a corridor of possible pulling directions for an enforced reaction, which is described elsewhere. 11 The corridor may usually cover directions over a broad range.
For two SPs thus two competing RPs there may also be corridors of type 1. A chemical example of a type 1 corridor may be that reported by Shaik et al. 35,36 in the study of the selectivity control of enzymatic-like bond activation by an external electric field. Two different oxidation pathways were considered: an oxidation over a > C = C< double bond, epoxidation, and an oxidation over a C-H bond, hydroxylation. When these reactions take place on the PES of a doublet state, they occur without intermediates. On the contrary, when they take place on the PES of a quartet state, an intermediate appears.
According to the study, 35  corresponding spin density depends on the direction of this field. In one special direction, the field leads indeed to two inverse cases of a TSS with a ∆E ‡ difference of -1.3 kcal mol −1 , and -2.0 kcal mol −1 in another direction with an applied force in both special directions of F = ±0.01 au. 35 This field effect could correspond to a corridor of type 1.
When the electronic state of the system is a doublet, (2S+1=2), with the corresponding spin density, the application of the external electric field breaks the behavior of a type 1 chemical corridor, as we will explained below.
A second reaction of propene to hydroxylation, again in both of the spin states 2 or 4, does also not show the corridor type 1 behavior because the energy can increase after the TS. 35 Possibly a simple PES model can explain the results of Shaik et al, 35 see Figure 25. The figure shows the PES without load, and some NTs for the possible pathways of the FDSPs. The formula is given in the Supplementary Information, Eq.(S4). The way to the product P 1 may be the epoxidation, but the way to the product P 2 may be the hydroxylation, of propene. The spin state is that of a doublet, (2S+1=2). In this case no intermediates emerge. The force to pull (in whatever x-direction 35 ) along a certain NT on SP 1 leads to a ∆E ‡ = 0 for field strength +0.01 au, but a TSS for the case −0.01au by ∆E ‡ =-3.5 kcal mol −1 . We guess that the NT-pathway for the effective SP moves here only for the negative field downward. For the positive field the effective SP moves along a level line. That is possible in a type 2 corridor, at least. If one looks for the SP 1  restricted form f k =(0,...,0, f k,i ,0,...0, f k, j ,0...0) T with two times three components f k,i = − f k, j = 0 for the Cartesian coordinates, k = 1, 2, 3, of the anchor atoms, i, j. A similar ansatz is used elsewhere. 117 Usually, a pulling could use a more filled vector, f, for example for a full force vector of an electrical field. 29 If the structure of the catalytic side is known, one can determine the direction of the electrostatic force, see for example the Figure 2 in Ref. 118.
The use of Cartesian coordinates in the given example, of Ref. 116, should not mean that we undergo in our theory a coordinate transformation. In contrast, the Eq.(1) can be simply applied in the form of Cartesians; the N former r i coordinates become simply N + 6 coordinates x j with j = 1, ..., N + 6. It is another, again a linear ansatz. However, note: the theory of NTs can be used independent on the coordinate system. 119 Note that there is an application of a non-linear ansatz of a force vector, instead of Eq.(1), the Artificial Force Induced Reaction (AFIR) method. [37][38][39] However, the version is used only for numerical reasons to calculate transition states. It is described as a method without a physical meaning. Some phenomenons there observed are similar to our results.
This article has not given a full proof for the applicability of the used models to mechanochemical or biochemical processes. It proposes to deepen an already discussed model, Eq. (1), by some better mathematical curves, the NTs, which are to use for the curve of the force displaced stationary points, FDSPs. Armed with the knowledge of the mathematical structure of NTs, and if we use the simplest possible ansatz, Eq. (1), the linear one, then we can predict the transition state stabilization (TSS) by NTs, or respectively also a destabilization. We use here only two-dimensional, minimal parsimonious models with two or three minimums, and with one, two, or three SPs. However, we can propose a new, very interesting behavior of the curve of FDSPs on different kinds of energy surfaces.
In order to find the optimal force to be applied to a molecular system for a particular rearrangement it is crucial to know the direction of this force. Previous works by us demonstrated that this optimal direction can be rigorously and elegantly found using the Newton Trajectory model and Catastrophe theory. Drawing on this theoretical framework we present here a complete and rigorous classification of different effects resulting from different force directions (the so-called 'pulling corridors'). This classification spans many possible phenomena or cases so far reported in the field of mechanochemistry and electric field induced reactions (catch-bonds, chemical hysteresis phenomena, enzymatic reaction enhanced by external electric field, mechanophores, molecular motors, etc...).
We summarize this study with some remarks: 1. The results are of topological nature as mentioned in the Introduction section and other parts of the article. Thus the reasoning is general and it is not based on the type of calculation used to obtain the PES or the free energy surface.
2. According to the results previously exposed, the mechanochemistry and the enzymatic catalysis (or inhibition) can be seen as catastrophes (transitions) of the original reaction to the modified reaction. This is based on the topological theory of Catastrophes. [69][70][71] Note that the manifold Det (H) = 0 is crucial in the present context and determines the type of catastrophe. 10 3. The previous theory is variational in nature or optimal control theory. One can see that the vector, l, controls the optimal push or pull of the process. This control vector is associated to the optimal BBP, 10 which is a point of an optimal NT. The nature of this curve is variational, see Ref. 8. The NT curve is the background of the present study.
4. The optimal BBP and the associated l-vector give all the information needed for pullingpushing mechanochemistry or for a construction of an optimal catalyst (enzyme) for a reaction. Taking this information and a set of libraries about materials or aminoacids one can build by dynamic programing the optimal catalyst (inhibitor) or enzyme for the reaction under consideration. This is the next future.
5. For enzymes holds: if Eq. (1) is a valid model for the intrinsic structure of the enzyme activity, then we can slightly change the early view of Halder 40 and Pauling 41 that the bulk of the usual very big enzymes exists to maintaining the active site in a geometry faithful to the transition state. 120 We now introduce the hypothesis that the bulk of the enzyme causes the active site developing an optimal force vector for the substrate. The enzyme is optimal for the best possible movement of the effective minimum and of the effective SP together on the free energy surface. We do believe that the classification reported and deeply analyzed in this paper provides a new language and new rules that will be employed in future to rationalize mechanochemical and enzyme phenomena.
We end this article paraphrasing Carpenter et al., 121 "It is also irresistible to comment that experimental test of the predictions in the present work would be most welcome!"

SUPPORTING INFORMATION AVAILABLE
The mathematical formulas of the diverse PES of the paper are available in the Supporting Information. This material is to get free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGMENT
Financial support from the Spanish Ministerio de Economía y Competitividad, Project CTQ2016-76423-P, is acknowledged. We give thanks to Mr. Sergio Pablo García Carrillo for assistance in preparing some graphical material.
Graphical TOC Entry P R NT BBP BBP An asymmetric chemical corridor by Newton trajectories (NT) from left to right minimum (R->P). The red points are the optimal barrier breakdown points (BBP). .