Differences in the Abilities to Mechanically Eliminate Activation Energies for Unimolecular and Bimolecular Reactions

Mechanochemistry, i.e. the application of forces, F, at the molecular level, has attracted significant interest as a means of controlling chemical reactions. The present study uses quantum chemical calculations to explore the abilities to mechanically eliminate activation energies, ΔE‡, for unimolecular and bimolecular reactions. The results demonstrate that ΔE‡ can be eliminated for unimolecular reactions by applying sufficiently large F along directions that move the reactant and/or transition state (TS) structures parallel to the zero-F reaction coordinate, S0. In contrast, eliminating ΔE‡ for bimolecular reactions requires the reactant to undergo a force-induced shift parallel to S0 irrespective of changes in the TS. Meeting this requirement depends upon the coupling between F and S0 in the reactant. The insights regarding the differences in eliminating ΔE‡ for unimolecular and bimolecular reactions, and the requirements for eliminating ΔE‡, may be useful in practical efforts to control reactions mechanochemically.

However, the ability to mechanically render reactions barrierless may apply to unimolecular reactions in a more general sense. Essentially, the reactant and TS structures for unimolecular reactions are separated by a finite distance along S in the absence of an applied force, and hence it should be possible in principle to eliminate this separation via mechanical manipulation as long as the system can sustain sufficiently large forces applied along a direction that causes the reactant and TS structures to converge. The ability to eliminate ΔE ‡ in bimolecular reactions occurring under mechanochemical conditions is not as clear. This is particularly true in the case of bimolecular reactions that involve infinitely separated reacting species. In such cases, the infinite distance between the reactants and TS along S precludes the convergence of these structures via mechanical manipulation (or any other means). However, many bimolecular reactions include elementary reaction steps that involve progression from a complex to the TS, and may be amenable to the mechanochemical convergence of the reactant and TS structures. In light of these apparent distinctions in the abilities to activate unimolecular and bimolecular reactions, we ask: do differences exist in the abilities to mechanically eliminate ΔE ‡ for unimolecular and bimolecular reactions, and if so, what limitations exist in the ability to render each type of reaction barrierless? The present study answers these questions by using quantum chemical calculations to examine how the thermal energy barriers to a set of representative unimolecular and bimolecular reactions are affected by F. The force-induced changes in the thermal energy barriers are interpreted in terms of changes in the separation of the reactant and TS structures along S, and shed light on the conditions that must be met to mechanically eliminate ΔE ‡ .

Results
The force-dependent reactant and TS structures were evaluated for the series of reactions outlined in Fig. 1. Reactions U1 through U4 correspond to unimolecular reactions involving bond scission (U1), changes in hybridization as the system passes through the TS (U2), or changes in conformation without bond dissociation (U3 and U4). Reactions B1 through B5 are bimolecular reactions. Reactions B1 and B2 correspond to S N 2 reactions in which the reacting species form a complex. Reactions B3 through B5 are cycloaddition reactions, with reactions B3 and B4 proceeding via the formation of a reactant complex, and the reacting components being infinitely separated for reaction B5. Collectively, these specific reactions are representative of general classes of reactions used extensively in chemical research. For instance, pericyclic reactions (U1), substitution reactions (B1 and B2), and cycloadditions (B3 through B5) are used extensively in organic synthesis [40][41][42] . Furthermore, conformational changes (U2 through U4) play a role in many pharmaceutical and materials applications, with particular connections to chirality 43,44 . In addition, pericyclic reactions like U1 and S N 2 reactions like B1 and B2 have been activated mechanically under experimental conditions 28,37,45 .
The effects of applied forces on the reactant and TS structures for the reactions in Fig. 1 were evaluated along with the force-dependent changes in ΔE ‡ . In all cases, F was applied between a pair of atoms, called pulling points (PPs) hereafter, that were selected based on preliminary analyses that indicated applying F at these locations had the potential to eliminate ΔE ‡ . The magnitude of F was increased until TS structures possessing a single normal mode with an imaginary frequency could no longer be located. It is noted that these PPs and magnitudes of F may not be accessible in experimental studies of these reactions, but their use was necessary to explore the limits in the ability to mechanically eliminate ΔE ‡ . For the same reasons, it was assumed that the reaction mechanism does not change when F is applied. Reactions B1 through B4 correspond to bimolecular reactions in which the two reacting components form a complex before reaching the TS. The interactions between these components are indicated with dashed lines. Reaction B5 corresponds to an addition reaction in which the reacting components are infinitely separated in the reactants. External forces are applied in a tensile manner in all cases, with arrows indicating the atoms used as PPs and the directions of those arrows approximating the direction in which F is applied to the PPs.  (Fig. 2b) show that increasing F increases the similarity of the reactant and TS structures, and that Δx RMS should reach zero at finite F in all cases if sufficiently large F can be supported by the systems. The values of Δx RMS obtained at the highest F considered are near zero for reactions U1 and U2 (0.020 and 0.029 Å, respectively), which is consistent with the observation that these reactions become effectively barrierless at high F. The high-force values of Δx RMS for reactions U3 and U4 seem large considering the data in Fig. 2a indicate these reactions become nearly barrierless. This is a result of evaluating the structural similarities using the Cartesian coordinates of the atoms when the structural changes are better described in terms of internal coordinates. Moving from the reactants to the TS for these reactions involves rotation about the central C-C bond of the biaryl. The change in the associated torsion upon moving from the reactants to the TS at the highest F considered is relatively small (10.3° and 13.0° for reactions U3 and U4, respectively), yet leads to relatively large changes in the Cartesian positions of the atoms residing far from the rotational axis.
The data in Fig. 2 are consistent with the Hammond postulate, with reductions in ΔE ‡ being associated with decreasing values of Δx RMS . The force-induced convergence of the reactant and TS can be achieved through changes in the structure of either one of these species with respect to their locations on the zero-force PES. The displacements of these species parallel (Hammond effects) and perpendicular (anti-Hammond effects) to the zero-force minimum energy path, S 0 , were evaluated as follows. First, S 0 was evaluated via an intrinsic reaction coordinate calculation. A stationary point, i.e. a reactant or TS, optimized on the force-modified PES for that reaction was selected and the Δx RMS values based on the difference between the structure of that stationary point and each structure along S 0 were evaluated. The structure, j, along S 0 that yielded the lowest value of Δx RMS was deemed most similar to the stationary point. This value of Δx RMS , ∆ ⊥ x RMS , was also used to quantify the amount the stationary point had moved perpendicular to S 0 . The value of Δx RMS associated with the difference between structure j and the zero-force reactant, ∆x RMS , was used to quantify the force-induced deformation of the stationary point parallel to S 0 . This procedure was repeated for the stationary points optimized at several F for all reactions.

RMS
R MS positions of the stationary points for the unimolecular reactions are given in Fig. 3 and show that the force-induced convergence of the reactant and TS structures is due to a combination of Hammond and anti-Hammond effects. For reactions U1 and U2, F shifts the positions of both the reactant and TS structures with respect to S 0 . The data for reactions U3 and U4 show that the reactants undergo significant force-induced shifts parallel and perpendicular to S 0 , while the TS structure does not move parallel to S 0 at all. This behavior can be attributed to S 0 being dominated by rotation about the central C-C bond of the biaryl for these reactions. In the TS, the torsion defined by this bond and the two atoms acting as PPs is 180°, and thus further rotation to move the TS parallel to S 0 would move the PPs closer to one another. Such changes in structure would require work to be performed against F, and thus does not occur.

Bimolecular reactions.
The results of quantum chemical calculations of the thermal energy barriers for the bimolecular reactions are reported and examined in what follows. The force-dependent changes in ΔE ‡ and the ability of F to increase the similarities of the reactant and TS structures are considered first. The ability of F to eliminate ΔE ‡ are then considered in terms of the changes in the deformation and interaction energies of the reacting components as they proceed along the reaction coordinate. The changes illustrate that the reactants must undergo a force-induced shift toward the TS in order for ΔE ‡ to be eliminated mechanically in bimolecular reactions. The ability of the reactants to undertake such a force-induced shift is then considered in terms of the coupling between the direction along with F is applied and the zero-force reaction coordinate.
The ΔE ‡ s for the bimolecular reactions are plotted versus 1/F in Fig. 4a. For reactions B1 through B3, ΔE ‡ is below 0.25 kcal/mol at the highest F that could be supported by the TS. The ΔE ‡ s for reactions B4 and B5 do not reach zero at the highest values of F considered, with the lowest ΔE ‡ s achieved for these reactions being 2.3 and 6.9 kcal/mol, respectively. However, the trends in the data suggest that ΔE ‡ should reach zero at finite F for both of these reactions. This is reasonable for reaction B4, which involves a reactant complex, yet is inconsistent with the infinite separation of the reacting species for reaction B5.
The force-induced changes in the reactant and TS structures for the bimolecular reactions relative to S 0 are plotted in Fig. 4b. The data for reactions B1 through B3 show that applying F causes the reactants and TS to move toward each other along curved paths in the ∆ ∆ ⊥ ( ) The changes in the reactant and TS structures are relatively symmetric for reactions B1 and B3, whereas the force-induced changes of the TS for reaction B2 is greater than that for the reactant. The data for reaction B4 show that F induces structural changes in the reactant are nearly perpendicular to the zero-force reactant structure. As a result, any force-induced reduction in the distance between the reactants and TS for this reaction is due primarily to changes in the TS structure. Finally, the TS for reaction B5 is shifted towards the reactants with increasing F; however, the reactants remain infinitely far away.
The force-dependent ΔE ‡ s can be represented as the sum of a term corresponding to the change in energy, ΔE def , associated with structural and electronic changes that occur within each reacting component upon moving from the reactant to TS geometries and another term associated with the change in the interaction energy, ΔE int, between the two reacting species as the system moves from the reactants to the TS. The deformation energy is given by:  where A and B designate each of the reacting species, x TS and x r are the nuclear coordinates of the TS and reactants, respectively, and R TS and R r are the distances between the PPs in the TS and reactant, respectively. ΔE def is necessarily greater than zero in all cases where A and B are infinitely separated in the reactants, which is true for reaction B5. Conversely, the reacting species are present in complexes for reactions B1 through B4, and ΔE def is calculated relative to the complex structures for those reactions. In such cases, ΔE def is not strictly bound to be greater than zero, but this limit likely still applies because the distance along the reaction coordinate between the complex and completely separated reactants is lower than that between the TS and the completely separated reactants. Once ΔE def is calculated, ΔE int can be evaluated as . This quantity can take on positive or negative values.
The values of ΔE def and ΔE int are plotted versus F for reactions B1 through B5 in Fig. 5a. These data show that ΔE def is positive and ΔE int is negative at low values of F for all reactions. For reactions B1 through B3, increasing F lowers ΔE def and increases ΔE int monotonically such that these quantities each approach zero from opposite sides as F approaches the value at which the reactions become barrierless. Similar force-induced changes in ΔE int and ΔE def occur for reactions B4 and B5; however, in these cases, increasing F to the highest values considered causes ΔE int to become positive without ΔE def reaching zero. The steady reduction in ΔE def for these reactions suggests that increasing F causes the reacting components to adopt increasingly similar structures in the reactants and TS. In contrast, the adoption of positive values of ΔE int indicates that the interactions between these components stabilize the reactants more than the TS at high F.
, these results suggest that reactions B4 and B5 should not become barrierless with increasing F.
The results in Fig. 5a highlight that maintaining a negative or zero ΔE int is necessary to eliminate ΔE ‡ for bimolecular reactions. The data in Figs 4b and 5a suggest that ΔE int remains negative or zero if the reactant experiences a large shift parallel to S 0 , e.g. reactions B1 through B3, yet becomes positive if the reactant does not move a significant distance parallel to S 0 irrespective of structural changes in the TS, e.g. reactions B4 and B5. This observation can be understood by examining how the interaction energy, E int , between the two reacting components changes as the system progresses along S 0 (Fig. 5b). The plots in Fig. 5b show that for all reactions, E int increases slowly as the system progresses from the reactants toward the TS before reaching a maximum and then decreasing again as the system moves closer to the TS. In all cases, E int is lower at the TS than it is at the reactant, which leads to the negative zero-force values of ΔE int for all reactions (Fig. 5a). However, ΔE int is a relative quantity that will be affected by the distortions of the reactant and TS structures parallel to S 0 . As such, ΔE int can become positive if E int is greater in the TS than it is in the reactant. This situation can occur if the TS is shifted toward the reactant without a sufficiently large shift of the reactant toward the TS. Interestingly, the shapes of the E int curves indicate that positive ΔE int should not occur if the TS does not move parallel to S 0 . As a whole, these observations lead to the conclusion that the reactant must undergo force-induced change in structure that are directed toward the TS in order to eliminate ΔE ‡ for bimolecular reactions, whereas force-induced changes in the TS structure are not required to eliminate ΔE ‡ .
The requirement for the reactant to move toward the TS to eliminate ΔE ‡ is based on the assumption that E int -S 0 curves possessing the features of those shown in Fig. 5b are common to bimolecular reactions occurring at arbitrary F. The origin and generality of these features can be understood by examining the structural changes that occur as the system moves from the reactants to the TS along S 0 . This progression can be described in terms of the change in the separation of the reacting species and the deformation of the internal structures of the reacting species. The contributions of each of these types of structural distortions along S 0 were quantified by evaluating the changes in the atomic coordinates of each structure, j, along S 0 relative to the reactant complex, r, and using these changes to define the total structural change, where A indicates the reacting components, n A is the number of atoms in component A, n t is the total number of atoms in the system, x i j indicates the Cartesian coordinates of the atoms in the structure containing all atoms present in the complex, and x i A j , indicates the coordinates of the atoms in the individual reacting components. The rates at which Δx sep and Δx def change along S 0 are compared in Fig. 5c. In all cases, these data show that changes in Δx sep dominate near the reactants while changes in Δx def dominate closer to the TS. The approach of the two reacting species in the region where Δx sep dominates leads to steric and electrostatic repulsion between valence electrons in these components, which causes E int to increase. As the system continues toward the TS and enters the region where Δx def dominates, the initiation of bonding between the reacting species occurs and reduces E int . This process occurs for the vast majority of bimolecular reactions, and thus the features of the E int -S 0 curves in Fig. 5b are largely general in nature. The arguments above regarding the changes in ΔE int are based on force-induced changes in structure parallel to S 0 ; however, the data in Fig. 4b indicate that the reactants and TS experience structural changes perpendicular to S 0 . Despite these anti-Hammond effects, the progression from a region where Δx sep dominates near the reactants to another where Δx def dominates near the TS occurs for bimolecular reactions at higher F if the reactant does not shift significantly toward the TS. The retention of these features at high F causes the E int -S curves at higher F to resemble those shown in Fig. 5c (see Supplementary  Information for examples). In fact, anti-Hammond effects contribute to the steady decrease in ΔE def toward zero while ΔE int increases that is evident at high F for reactions B4 and B5 in Fig. 5a. Such behavior suggests that ΔE ‡ may even begin to increase at higher F for these reactions.
The importance of the force-induced distortions of the reactant structure parallel to S 0 in the context of eliminating thermal energy barriers to bimolecular reactions is most apparent in the cases of reactions B3 and B4, which are both pericyclic additions in which the reagents form a complex. Despite this similarity, the differences in the force-induced changes to the reactant structures for these reactions suggest that the nuclear DOFs in these complexes couple to F in different ways. This coupling was assessed by calculating the overlap between F and S 0 and examining the directions in which the atomic positions in the complexes change due to F. The overlap was obtained by evaluating the tangent to S 0 at each point, i, along S 0 as: and normalizing as τ = τ τ / i i 46 . The normalized vector, = R R R / i i , along which F is applied was then projected onto τ at each point along S 0 .
The resulting overlaps are plotted against S 0 in Fig. 5d for reactions B3 and B4. These data show that the overlap between R and τ is nearly zero at the reactant structure and increases as the system moves toward the TS. This is consistent with the ability of F to move the TS parallel to S 0 for both reactions, but suggests that direct overlap between F and S 0 is not responsible for the distortion of the reactant complex parallel to S 0 observed for reaction B3. Instead, the structural changes of the reactant complexes are determined by the manner in which F couples to other nuclear degrees of freedom (DOFs) in the complexes. This is illustrated by the structures in Fig. 5d, which correspond to the zero-force reactant complexes for these reactions with arrows indicating the magnitudes and directions of the force-induced displacements of each atom. The arrows in each of the reacting components in the complex for reaction B3 point toward each other, indicating that F will shift these components closer to one another, which is consistent with structural changes parallel to S 0 . The force-induced changes in the structure show that the coupling between R and the other nuclear DOFs causes the C-C distances associated with the formation of products to change at rates of − 2.3 × 10 −2 Å/nN. Conversely, the arrows on the complex for reaction B4 do not point toward one another, which would be required for changes in structure parallel to S 0 , with the structural changes indicating that the C-O and C-C distances associated with product formation changing at rates of − 1.9 × 10 −3 and − 2.2 × 10 −3 Å/nN, respectively. Similar analyses show that the reactants and TSs for reactions B1 and B2 exhibit analogous features to those illustrated for reaction B3. Of course, F has no effect on Δx sep for reaction B5 due to the infinite separation of the reactants.
In a general sense, the ability of F to displace the reactant along S 0 in the direction of the TS can assessed by examining the molecular compliance matrix, i.e. the inverse of the molecular Hessian evaluated in terms of internal coordinates [47][48][49] , and τ. Specifically, one can use τ to determine which DOFs change significantly as the system follows the reaction coordinate starting at the reactant. Once the relevant DOFs are identified, the elements of the compliance matrix that couple R to other nuclear DOFs can be used to quantify the displacement of the system along S 0 when F is applied. This approach can also be used to identify PPs that lead to force-induced distortions of the reactant toward the TS.

Discussion
Collectively, the results of the calculations show that F can be used to eliminate ΔE ‡ for elementary reactions by causing the reactant and TS structures to converge through a combination of Hammond and anti-Hammond effects. However, differences in the abilities of F to induce the necessary structural convergence exist for unimolecular and bimolecular reactions.
The analysis of the force-induced changes of the reactants and TS structures with respect to S 0 indicates that coupling between F and DOFs that cause the reactant structure to shift along S 0 toward the TS is necessary for the elimination of ΔE ‡ for bimolecular reactions irrespective of the changes in the TS structure. This coupling is absent in the case of bimolecular reactions in which the reactants consist of two infinitely separated molecules. As a result, there will not exist a set of PPs that drives the reactant toward the TS through force-induced reductions in Δx sep (assuming the PPs both reside in one of the reacting molecules), and thus it will not be possible to mechanically eliminate ΔE ‡ for such reactions without changing the mechanism.
In bimolecular reactions in which the reagents form a complex prior to reacting, a finite distance separates the reactants and TS along S 0 and some degree of interaction exists between the two reacting species. As such, these reactions are not necessarily prevented from being rendered barrierless under mechanochemical conditions. In cases where applying F to one reagent can drive the entire reactant complex toward the TS due to coupling between F and other DOFs that are aligned with S 0 , e.g. reactions B1 through B3, the system can retain negative values of ΔE int and ΔE ‡ can be eliminated. When the interactions between the components in the complex do not lead to sufficiently coupling between F and any DOFs that move the reactant parallel to the reaction coordinate, e.g. reaction B4, the system must rely on the changes in the TS structure for the convergence of the reactants and TS to be achieved. As this occurs, the TS samples structures in which ΔE int increases without a compensating reduction in ΔE def , which may even cause ΔE ‡ to increase for certain ranges of increasing F and allow ΔE ‡ to reach zero only if the system can sustain sufficiently large F to drive the TS all the way to the reactant. However, the lack of coupling between S and F in the vicinity of the reactant will likely prevent the convergence of the reactant and TS structures as the TS approaches the reactant.
The presence of the reacting species as a single unit in unimolecular reactions renders arguments related to changes in E int inapplicable and obviates the need to identify PPs that drive the reactants toward the TS in order to eliminate ΔE ‡ . Instead, unimolecular elementary reactions can be rendered barrierless by force-induced changes in either the reactant or TS structures. The presence of strong coupling between the atoms in the molecule undergoing a unimolecular reaction, i.e. coupling sufficiently strong to deem the reactant a single molecule, ensures that it should always be possible in principle to eliminate ΔE ‡ for unimolecular reactions; although, the PPs and F required to achieve this may not be experimentally accessible.
Overall, the results of the calculations provide new insights into the ability to mechanically activate reactions by delineating the scenarios in which ΔE ‡ can be mechanically eliminated for common classes of reactions, and relating those abilities to the changes in the reactant and TS structures with respect to S 0 . The results are consistent with studies of the unimolecular reactions, which show that the elimination of ΔE ‡ occurs in conjunction with a convergence of the reactant and TS structures 33,34 . For bimolecular elementary reactions, the results indicate that the Hammond effects associated with the distortion of the reactant toward the TS are practically necessary to eliminate ΔE ‡ mechanochemically, whereas the changes in the TS structure are less critical in this context. These results can be considered in light of experimental studies showing that reaction barriers can be reduced through force-induced Hammond and anti-Hammond effects 49,50 . Such interpretations are based on force-induced changes the separation of the reactant and TS structures along the reaction coordinate, which is sufficient to reduce ΔE ‡ . However, the results of the present study indicate that eliminating ΔE ‡ requires the displacement of the reactant along S 0 instead of a consideration of relative structures of the reactants and TS. In addition, models for predicting force-induced changes in ΔE ‡ focus on relative values of properties of the reactant and TS structures 48,49,51 . Indeed, such models suggest that having a rigid reactant is beneficial for lowering ΔE ‡ , whereas the results of the present study suggest that such a scenario would preclude the elimination of ΔE ‡ for some bimolecular reactions. Highlighting the importance of coupling between F and DOFs that undergo force-induced motions parallel to S 0 in the reactant may be of practical use in the development of mechanophores and the selection of locations to act as PPs in such molecules. Moreover, illustrating the limitations of the ability to mechanically eliminate ΔE ‡ for bimolecular reactions is of fundamental value.
Of course, the calculations reported above were performed under idealized conditions in which all conceivable sets of PPs could be accessed, constant F with magnitudes exceeding those accessible experimentally were applied, and the reaction mechanisms were unaffected by F. Despite this idealized scenario, the results of this study are general in nature and apply broadly to the mechanical activation of reactions. Rather deviations from these idealized conditions simply place additional limitations on the ability to mechanically activate reactions. Additionally, the calculations also excluded enthalpic and entropic contributions to reaction kinetics. These factors were not considered in detail through the calculations reported here because of the limited abilities to access enthalpies and free energies via static calculations; however, the mechanical elimination of enthalpic and free energy barriers are subject to the same requirements regarding the convergence of the reactant and TS structures discussed above. In addition, the reactions considered were performed on small model systems that are amenable to computation. Nonetheless, these types of reactions are used extensively in basic and industrial chemical applications [52][53][54] .

Methods
The calculations employed the External Force Explicitly Included formalism 51,52 in which F is applied along the vector connecting two PPs. Within this formalism, the potential energy of a system on the force-modified PES is: where x represents the force-dependent nuclear positions of the atoms in the molecule, E BO is the energy of the system on the Born-Oppenheimer PES and R is the distance between the PPs. All calculations were performed using Kohn-Sham density functional theory 55,56 with a version of the NWChem 6.1 software package 57 that we modified to permit single-point energy calculations, geometry optimizations, and frequency calculations on the force-modified PES defined by Eq. (3). The B3LYP exchange-correlation functional 58,59 was used in conjunction with a 6-31G(d,p) basis set for unimolecular reactions and a 6-31+ + G(d,p) basis set for bimolecular reactions.
Calculations with other basis sets and exchange-correlation functionals for selected reactions yielded ΔE ‡ versus F relationships that were qualitatively consistent with those reported in the main text. All stationary points were characterized as minima or first-order saddle points on the force-modified PES via harmonic frequency analysis. Values of ΔE ‡ for reaction B5 were corrected for basis set superposition errors using the counterpoise procedure 60 .