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ERROR: type should be string, got "https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 ACCESS\nMetrics & More Without zero-point vibration these reactions would be very slow, which is a manifestation of a\nunique quantum effect. Reactions where the reactivity is determined by dynamical factors such as the vibrationally enhanced\nattraction are found to be beyond the range of applicability of Polanyi’s rules. ■INTRODUCTION The Role of Zero-Point Vibration and Reactant Attraction in\nExothermic Bimolecular Reactions with Submerged Potential\nBarriers: Theoretical Studies of the R + HBr →RH + Br (R = CH3, HO)\nSystems The Role of Zero-Point Vibration and Reactant Attraction in\nExothermic Bimolecular Reactions with Submerged Potential\nBarriers: Theoretical Studies of the R + HBr →RH + Br (R = CH3, HO)\nSystems Published as part of The Journal of Physical Chemistry virtual special issue “125 Years of The Journal of\nPh\nl Ch\n” Published as part of The Journal of Physical Chemistry virtual special issue “125 Years of The Journal of\nPhysical Chemistry”. Benjámin Csorba, Péter Szabó, Szabolcs Góger, and György Lendvay*\nCite This: J. Phys. Chem. A 2021, 125, 8386−8396\nRead Online\nACCESS\nMetrics & More\nArticle Recommendations\n*\nsı\nSupporting Information\nABSTRACT: The dynamics of the reactions CH3 + HBr →CH4 + Br and\nHO + HBr →H2O + Br have been studied using the quasiclassical trajectory\nmethod to explore the interplay of the vibrational excitation of the breaking\nbond and the potential energy surface characterized by a prereaction van der\nWaals well and a submerged barrier to reaction. The attraction between the\nreactants is favorable for the reaction, because it brings together the reactants\nwithout any energy investment. The reaction can be thought to be controlled\nby capture. The trajectory calculations indeed provide excitation functions\ntypical to capture: the reaction cross sections diverge when the collision energy\nis reduced toward zero. Excitation of reactant vibration accelerates both\nreactions. The barrier on the potential surface is so early that the coupling\nbetween the degrees of freedom at the saddle point geometry is negligible. However, the trajectory calculations show that when the breaking bond is\nstretched at the time of the encounter, an attractive force arises, as if the radical approached a HBr molecule whose bond is partially\nbroken. As a result, the dynamics of the reaction are controlled more by the temporary “dynamical”, vibrationally induced than by\nthe “static” van der Waals attraction even when the reactants are in vibrational ground state. The cross sections are shown to drop to\nvery small values when the amplitude of the breaking bond’s vibration is artificially reduced, which provides an estimate of the\nreactivity due to the “static” attraction. Without zero-point vibration these reactions would be very slow, which is a manifestation of a\nunique quantum effect. Reactions where the reactivity is determined by dynamical factors such as the vibrationally enhanced\nattraction are found to be beyond the range of applicability of Polanyi’s rules. Benjámin Csorba, Péter Szabó, Szabolcs Góger, and György Lendvay* Article ACCESS\nMetrics & More ABSTRACT: The dynamics of the reactions CH3 + HBr →CH4 + Br and\nHO + HBr →H2O + Br have been studied using the quasiclassical trajectory\nmethod to explore the interplay of the vibrational excitation of the breaking\nbond and the potential energy surface characterized by a prereaction van der\nWaals well and a submerged barrier to reaction. The attraction between the\nreactants is favorable for the reaction, because it brings together the reactants\nwithout any energy investment. The reaction can be thought to be controlled\nby capture. The trajectory calculations indeed provide excitation functions\ntypical to capture: the reaction cross sections diverge when the collision energy\nis reduced toward zero. Excitation of reactant vibration accelerates both\nreactions. The barrier on the potential surface is so early that the coupling\nbetween the degrees of freedom at the saddle point geometry is negligible. However, the trajectory calculations show that when the breaking bond is\nstretched at the time of the encounter, an attractive force arises, as if the radical approached a HBr molecule whose bond is partially\nbroken. As a result, the dynamics of the reaction are controlled more by the temporary “dynamical”, vibrationally induced than by\nthe “static” van der Waals attraction even when the reactants are in vibrational ground state. The cross sections are shown to drop to\nvery small values when the amplitude of the breaking bond’s vibration is artificially reduced, which provides an estimate of the\nreactivity due to the “static” attraction. Without zero-point vibration these reactions would be very slow, which is a manifestation of a\nunique quantum effect. Reactions where the reactivity is determined by dynamical factors such as the vibrationally enhanced\nattraction are found to be beyond the range of applicability of Polanyi’s rules. However, the trajectory calculations show that when the breaking bond is\nstretched at the time of the encounter, an attractive force arises, as if the radical approached a HBr molecule whose bond is partially\nbroken. As a result, the dynamics of the reaction are controlled more by the temporary “dynamical”, vibrationally induced than by\nthe “static” van der Waals attraction even when the reactants are in vibrational ground state. The cross sections are shown to drop to\nvery small values when the amplitude of the breaking bond’s vibration is artificially reduced, which provides an estimate of the\nreactivity due to the “static” attraction. The Journal of Physical Chemistry A According to high-level ab initio calculations, the\nattraction between the reactants modifies this picture: the\ngeometry of the reactants at the saddle point corresponding to\nH-abstraction on the PES becomes so reactant-like that the\nbreaking H−Br bond is hardly longer than the equilibrium\ndistance (1.487 vs 1.413 Å for both reactions), and the forming\nbond is as long as 1.7 and 1.551 Å for reactions R19,20 and R2,\nrespectively.23 Moreover, the potential barrier submerges\nbelow the reactant level, as shown in Figure 1, which means\nthat the primary factor determining the dynamics, especially at\nlow collision energies, can be assumed to be the attractive\npotential leading to the prereaction van der Waals well. In\ngeneral, the dynamical consequence of the long-range\nattraction is that the reaction cross sections are very large at\nlow collision energy and decrease quickly when the latter\nincreases. The phenomenon when the reactants attract each\nother, and there is no barrier to the reaction so that the\nmajority of collisions lead to reaction, is called capture.24\nCapture-type behavior is characterized by excitation functions\ndiverging with decreasing collision energy. The rate coefficients\ncorresponding to this kind of excitation function are\ncharacterized by negative activation energy, as are those\nobserved for both reactions R1 and R2. For reaction R1,\ndiverging cross sections at decreasing collision energy have\nbeen observed in recent quasiclassical trajectory and reduced-\ndimensional quantum dynamical calculations.25 For reaction\nR2, de Oliveira et al.23 reported excitation functions that seem\nto show extremely fast divergence of the reaction cross sections\nwhen the collision energy decreases, which was supported by\nthe quantum dynamical calculations of D. Wang et al.26 The\ncross sections observed in molecular beam experiments by\nKasai and co-workers27,28 also increase with decreasing\ncollision energy. For reactions with an early barrier, such as\nreactions R1 and R2, Polanyi rules of reaction dynamics29\npredict that vibrational excitation of the reactants promotes the Useful information also comes from studies of reactions\nrelated to R2. Guo and co-workers32,33 studied the effect of\nreactant vibrational excitation of two reactions of the HO +\nHX family, with X = F and with X = Cl. The reaction of HF is\nendothermic with a high, late barrier, while that of HCl is\nexothermic with a low, early barrier. The Journal of Physical Chemistry A According to high-level ab initio calculations, the\nattraction between the reactants modifies this picture: the\ngeometry of the reactants at the saddle point corresponding to\nH-abstraction on the PES becomes so reactant-like that the\nbreaking H−Br bond is hardly longer than the equilibrium\ndistance (1.487 vs 1.413 Å for both reactions), and the forming\nbond is as long as 1.7 and 1.551 Å for reactions R19,20 and R2,\nrespectively.23 Moreover, the potential barrier submerges\nbelow the reactant level, as shown in Figure 1, which means\nthat the primary factor determining the dynamics, especially at\nlow collision energies, can be assumed to be the attractive\npotential leading to the prereaction van der Waals well. In\ngeneral, the dynamical consequence of the long-range\nattraction is that the reaction cross sections are very large at\nlow collision energy and decrease quickly when the latter\nincreases. The phenomenon when the reactants attract each\nother, and there is no barrier to the reaction so that the\nmajority of collisions lead to reaction, is called capture.24\nCapture-type behavior is characterized by excitation functions\ndiverging with decreasing collision energy. The rate coefficients\ncorresponding to this kind of excitation function are\ncharacterized by negative activation energy, as are those\nobserved for both reactions R1 and R2. For reaction R1,\ndiverging cross sections at decreasing collision energy have\nbeen observed in recent quasiclassical trajectory and reduced-\ndimensional quantum dynamical calculations.25 For reaction\nR2, de Oliveira et al.23 reported excitation functions that seem\nto show extremely fast divergence of the reaction cross sections\nwhen the collision energy decreases, which was supported by\nthe quantum dynamical calculations of D. Wang et al.26 The\ncross sections observed in molecular beam experiments by\nKasai and co-workers27,28 also increase with decreasing\ncollision energy. For reactions with an early barrier, such as\nreactions R1 and R2, Polanyi rules of reaction dynamics29 the experiments reported negative activation energy.15,18,19\nThe curiosity of these reactions is that, because a weak H−Br\nbond is broken and a much stronger C−H or O−H bond is\nformed, they are significantly exothermic. The potential barrier\nto reaction is early and low, in agreement with the expectation\nfor exoergic reactions, based on the Bell−Evans−Polányi21,22 to reaction is early and low, in agreement with the expectation\nfor exoergic reactions, based on the Bell−Evans−Polányi21,22\nprinciple. ■INTRODUCTION the reactions of F atoms with H2O10 and with methane,11 but\nlittle is known about the general rules. In the class of reactions\nof hydrogen halides with alkyl and other radicals, those\ninvolving Br atoms attracted attention because of their\nimportance in atmospheric chemistry12,13 and flame retarda-\ntion.14 The experimental studies of H atom-abstraction\nreactions from HBr were instrumental in the determination\nof heats of formation of the radicals.15−18 For the reactions of\nHBr with methyl radicals Hydrogen abstraction reactions by halogen atoms and by OH\nradicals from small molecules are prototypes of bimolecular\natom-transfer reactions. As they lend themselves to detailed\nexperimental and theoretical reaction dynamical studies, they\nalso serve as testing grounds of theories on the general rules of\nreaction dynamics. A characteristic feature of these reactions is\nthat on their potential energy surfaces (PESs) shallow potential\nwells appear due to the attractive long-range forces between\nthe two reactant as well as the two product particles.1 When\nboth reactants are polar, the depth of the well corresponding to\nthe prereaction complex generally exceeds 10 kJ/mol,2−5\nespecially when well-defined hydrogen bonds can be formed.6\nWhen only one of the reactants is polar, then the well is\nsignificantly less deep,7−9 and the attractive forces acting at a\nlong-range are rather weak. The presence of the potential well\ncan influence the dynamics of the reaction, as demonstrated for Received:\nJune 30, 2021\nPublished: September 20, 2021 © 2021 The Authors. Published by\nAmerican Chemical Society 8386 Figure 1. Potential energy profiles of the (a) CH3 + HBr (R1) and (b) HO + HBr (R2) reactions. The classical energy levels of the stationary\npoints characterizing the Czakó-Góger-Szabó-Lendvay20 potential energy surface are given in kJ/mol. S.P. denotes the saddle point. The Journal of Physical Chemistry A\npubs.acs.org/JPCA\nArticle https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 The Journal of Physical Chemistry A The Journal of Physical Chemistry A Article pubs.acs.org/JPCA Figure 1. Potential energy profiles of the (a) CH3 + HBr (R1) and (b) HO + HBr (R2) reactions. The classical energy levels of the stationary\npoints characterizing the Czakó-Góger-Szabó-Lendvay20 potential energy surface are given in kJ/mol. S.P. denotes the saddle point. CH\nHBr\nCH\nBr\n3\n4\n+\n→\n+\n(R1)\nand with OH radicals\nHO\nHBr\nH O\nBr\n2\n+\n→\n+\n(R2) CH\nHBr\nCH\nBr\n3\n4\n+\n→\n+\n(R1)\nand with OH radicals\nHO\nHBr\nH O\nBr\n2\n+\n→\n+\n(R2) reaction less efficiently than an equal amount of translational\nenergy. The rules do not exclude that reactant vibration (in\nfact, that of the breaking bond) enhances the reactivity; the\nfocus is on the relative efficiency with respect to translation. These rules apply to reactions with positive potential barriers,\nand not much is known about how the presence of the\nprereaction potential well interferes with them. For reaction\nR1, the currently available information on how the vibrational\nexcitation of the HBr molecule, i.e., the breaking bond,\ninfluences the reaction cross sections comes from the\ntheoretical study by Y. Wang et al.,30 who performed\nreduced-dimensional quantum scattering calculations on\nreaction R1 on an analytical potential energy surface.31 On\nthe PES they used, the barrier to reaction is positive, and\naccordingly, the excitation functions they obtained for the\nreaction of HBr in the vibrational ground state correspond to\nwhat is referred to as activated behavior: the cross sections are\nzero below a threshold and rise slowly above it. However, for\nHBr excited by one vibrational quantum, the shape of the\nexcitation function is qualitatively different: the cross sections\nare very large at low collision energy and decrease quickly with\nrising collision energy, i.e., they display capture-type behavior. The observation of this kind of excitation function seems to\ncontradict the presence of the potential barrier on the PES. (R1) (R2) the experiments reported negative activation energy.15,18,19\nThe curiosity of these reactions is that, because a weak H−Br\nbond is broken and a much stronger C−H or O−H bond is\nformed, they are significantly exothermic. The potential barrier\nto reaction is early and low, in agreement with the expectation\nfor exoergic reactions, based on the Bell−Evans−Polányi21,22\nprinciple. ■METHODS The dynamics of reactions R1 and R2 have been studied by\nstandard quasiclassical trajectory (QCT) calculations.41,42\nInitial conditions were generated by the Raff−Porter−Miller\nscheme43 for the diatomic reactants and by normal mode\nsampling for the CH3 radical.44−46 For testing the possible\ntemporal evolution of the internal state of CH3 during the\ninitial free flight, we utilized the observation47,48 that if sets of\ntrajectories are started at various initial reactant separations,\nthus allowing different initial flight time, the reaction cross\nsections oscillate or systematically increase or decrease if\nplotted against the initial distance. The details of the test have\nbeen described in ref 25. Briefly, sets of 16 000 trajectories\nwere integrated starting from different center-of-mass reactant\ndistances in the range of covering a 12 to 32 Å at two collision\nenergies. This range of initial distances corresponds to a flight\ntime range at least 50 periods of C−H vibration. The reaction\ncross sections were found to fluctuate within an 8% range of\nthe average, without any tendency. The phenomenon was also observed for the very endothermic\nH + HF37,39 and for the almost thermoneutral H + HCl40\nhydrogen-abstraction reactions. According to quasiclassical\ntrajectory calculations, the switch of the excitation function\nfrom activated to capture-type was found to take place as soon\nas the energy content of the HX vibration exceeded the\npotential barrier, which occurred at excitation by about 2.5\nquanta for HF and below 2 quanta for HCl. The reason for the\ndrastic change of the excitation functions was traced back to\nthe shape of the potential energy surface. When the breaking\nbond is highly excited, the amplitude of the X−H bond length\noscillation is very large. When the vibration is at the outer\nturning point (OTP), the H atom is so far from the X atom\nthat the bond will behave as partially broken, which makes it\neasy for the approaching reactant to abstract the H atom. In\nterms of the PES, this is manifested when one calculates the\npotential energy as a function of the length of the forming\nbond at fixed values of the breaking bond length: As shown in\nFigure S1 for the reaction of HF with H, at equilibrium H−X\ndistance the potential energy increases monotonously as the\nreactants approach each other. ■METHODS At low collision energies, maximum impact parameters\nas large as 14 Å were needed to get converged reaction\nprobabilities and cross sections, while at large relative\nvelocities, 4.5 Å is satisfactory. These values are similar to\nthose observed for the H + H2O(vstretch = 4) reaction.36−38 p\ng\nReactant vibration with large amplitude can induce\nunexpected features to the dynamics of reactions in which\nthe potential barrier to reaction is submerged as a reef and is\nlocated very close in the configuration space to the minimum\nof a pre-reaction well. In such cases, the reacting system has\nenough energy to glide over the potential barrier. If the\nreactants are captured in the well for a long enough time, and\nenergy is completely redistributed, then the outcome of the\nencounter depends only on statistical factors. In such cases,\nonly the magnitude of the total energy made available to the\nreactants counts; its source does not (subject to angular\nmomentum conservation). However, considering that the\ncoupling between the degrees of freedom in a van der Waals\ncomplex is weak, one cannot expect complete energy\nredistribution. Thus, there is a good chance that the effect of\nvibrational and translational energy on the reactivity is\ndifferent. We intend to explore the influence of vibrational excitation\nof the reactants on the dynamics of reactions R1 and R2 and\nfind an explanation to the unexpectedly fast divergence of the\ncross sections at low collision energy. In the rest of the\nmanuscript, first we briefly summarize the methodology, and\nthen present the excitation functions and opacity functions at\ndifferent conditions. This will be followed by an analysis of the\npotential energy surfaces and the role of the amplitude of the\ninitial reactant vibration, with special attention to its\nmagnitude in the vibrational ground state. We will vary the\namplitude with various methods, changing the vibrational\nquantum number beyond that dictated by semiclassical The Journal of Physical Chemistry A In both reactions, when\nthe HX reactant is in vibrational ground state, the cross\nsections are zero below a threshold associated with the\npotential barrier, in agreement with the expected activated\nbehavior. However, when the HX reactant is vibrationally\nexcited, the change of the excitation functions differs for the\ntwo reactions. For the reaction of HF, with respect to the\nground-state reaction, the threshold energy is reduced by\napproximately the energy of the vibrational quantum and the\ncross sections grow an order of magnitude faster. On the other\nhand, for the reaction of HCl, reactant vibrational excitation\nchanges the character of the excitation function from activated\nto capture-type. That vibrational excitation can switch the character of the\nexcitation function is not unknown in the literature. Smith and\nco-workers34 observed extreme, 16 orders of magnitude speed-\nup for the H + H2O →H2 + OH reaction when the local O−H 8387 https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 The Journal of Physical Chemistry A Article pubs.acs.org/JPCA quantization. Finally, we briefly discuss the connection\nbetween the observed vibrational effect and Polanyi’s rules. stretch mode of water was excited by three or four vibrational\nquanta. The reaction is endothermic and has a late barrier, so\nin light of Polanyi’s rule, the fact that the rate increases by\nvibrational excitation of the reactant is not surprising, but the\nmagnitude is. The explanation, the appearance of capture-type\nexcitation functions at vibrational excitation by more than 2\nquanta, was provided by the related theoretical studies.35−38 ■METHODS When the length of the\nbreaking H−X bond is increased, the rate of the potential\nenergy increase slows down, and above a certain H−X\ndistance, the curves become attractive (see Figure S1b). When at the time of the encounter of the approaching\nreactants the breaking bond is near the OTP of its vibration,\nthis attraction pulls the reactants together. g ,\ny\ny\nThe maximum impact parameter was varied according to the\ncollision energy. The production calculations were performed\nwith impact parameters that were large enough to ensure that\nno reactive collisions occur in the 0.5 Å wide outer ring of the\ntarget. At low collision energies, maximum impact parameters\nas large as 14 Å were needed to get converged reaction\nprobabilities and cross sections, while at large relative\nvelocities, 4.5 Å is satisfactory. These values are similar to\nthose observed for the H + H2O(vstretch = 4) reaction.36−38\nTrajectories were integrated with the velocity−Verlet and the\nRunge−Kutta−Gill method for reactions R1 and R2,\nrespectively, with time steps 0.1 or 0.07 fs, ensuring energy\nconservation to better than 0.05 kJ/mol. All reactive collisions\nwere included in the cross-section calculations, without any\nweighting. This method resulted in good agreement with\nexcitation functions provided by reduced-dimensional quan-\ntum scattering calculations25 and with the experimental\nthermal rate coefficients23 for reaction R1. For the calculation\nof excitation functions, 124 000 trajectories were integrated at\neach collision energy and vibrational state. Opacity functions\nwere determined by running 50 000 trajectories at each impact\nparameter The statistical errors of the quantities obtained with\nthe Monte Carlo QCT method were calculated using the\nstandard prescription,41,42 and in most cases are as small as the\nsize of symbols shown in the plots. The calculations were\nperformed using an extensively modified version40,49−51 of the\nVENUS code.52 The potential energy surface developed\nearlier,9,23 referred to as CGSL PES was used for the\nsimulations of reaction R1, while the trajectory calculations\nfor reaction R2 were performed using the PES of de Oliveira et\nal.23 The maximum impact parameter was varied according to the\ncollision energy. The production calculations were performed\nwith impact parameters that were large enough to ensure that\nno reactive collisions occur in the 0.5 Å wide outer ring of the\ntarget. ■RESULTS Cross Sections. The excitation functions for reactions R1\nand R2 are plotted in Figure 2 for the ground and first and\nsecond excited states of the HBr reactant. The reactive cross\nsections diverge quickly as the collision energy decreases,\nwhich is the origin of the negative activation energy observed\nboth experimentally15−18 and in simulations.25 If the\nsubmerged barrier were not present, the divergence could be 8388 https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 Figure 2. Excitation functions of the (a) CH3 + HBr(v) and (b) HO\n+ HBr(v) reaction for v = 0, 1, and 2. The methyl and HO radicals are\nin their vib-rotational ground state. The error bars calculated with the\nstandard Monte Carlo formula are smaller than the symbol size. Figure 3. Opacity functions of the (a) CH3 + HBr and (b) HO + HBr\nreaction at various collision energies. All reactants are in vib-rotational\nground state. The Journal of Physical Chemistry A\npubs.acs.org/JPCA\nArticle Figure 3. Opacity functions of the (a) CH3 + HBr and (b) HO + HBr\nreaction at various collision energies. All reactants are in vib-rotational\nground state. pubs.acs.org/JPCA\nArticle Figure 3. Opacity functions of the (a) CH3 + HBr and (b) HO + HBr\nreaction at various collision energies. All reactants are in vib-rotational\nground state. pubs.acs.org/JPCA\nArticle The Journal of Physical Chemistry A The Journal of Physical Chemistry A Article pubs.acs.org/JPCA p\ng Figure 2. Excitation functions of the (a) CH3 + HBr(v) and (b) HO\n+ HBr(v) reaction for v = 0, 1, and 2. The methyl and HO radicals are\nin their vib-rotational ground state. The error bars calculated with the\nstandard Monte Carlo formula are smaller than the symbol size. Figure 3. Opacity functions of the (a) CH3 + HBr and (b) HO + HBr\nreaction at various collision energies. All reactants are in vib-rotational\nground state. Individual Trajectories. As described in the Introduction,\nfor several typical reactions taking place on potential surfaces\nwith a positive barrier, capture-type excitation functions were\nobserved at low collision energies when the breaking bond was\nvibrationally sufficiently highly excited. The phenomenon was\ntraced back to the behavior of the reacting system near the\ncorner region of the PES. Figure 4 is intended to show that\nreaction R2 displays the same qualitative behavior (the\ndynamics of reaction R1 are very similar). ■RESULTS Representative\nreactive trajectories for reaction R2 are projected on the r(O−\nH)−r(H−Br) plane, together with the contour plots of the cut\nof the multidimensional PES along the same plane, with all\nother coordinates being fixed at the saddle point values. At\nrelatively large reactant separation, where the interaction is\nweak, the trajectories oscillate between the equipotential lines\ncorresponding to the zero-point energy of the HBr vibration. At large O−H distances, the equipotentials are parallel to the\nhorizontal axis, which reflects that the interaction is small, thus\nthe internal potential of HBr does not change. considered to be the natural consequence of the long-range\nattraction between the reactants. The appearance of divergent\ncross sections indicates that the decisive factorat least at low\ncollision energiesis the attractive potential, and the potential\nbarrier plays a secondary role. The magnitude of the reactivity\nenhancement with decreasing collision energy, however, seems\nto be rather large if one considers that the PES on the reactant\nside is flat, and the small long-range attraction leads to a\nshallow potential well instead of the many kJ/mol deep minima\nin ion−molecule and radical−radical reactions. Vibrational excitation of HBr by one quantum is favorable\nfor both reactions. The shapes of the excitation functions\nremain similar to those for the ground state, but they run\nabove the latter. The enhancement of reactivity is not\nsurprising, since the bond to be broken is excited. The\nenhancement factor, the ratio of the cross sections for v(HBr)\n= 1 to those at v(HBr) = 0 at identical collision energies\n(shown in Figure S2), is the largest at around Ecoll = 10 kJ/mol,\nwhere the cross sections for v(HBr) = 1 are larger than for\nv(HBr) = 0 by about a factor of 1.6 and 4 for reactions R1 and\nR2, respectively, and decreases when the collision energy\nincreases or decreases. When HBr is excited by a second\nvibrational quantum, the additional enhancement of the cross\nsections is significantly smaller than that induced by the first. The v(HBr) = 2 to v(HBr) = 1 enhancement factor is not\nlarger than 1.16 and 1.11 for reactions R1 and R2, respectively,\nat any collision energy. At low collision energy, the three\nexcitation functions converge. This indicates that the efficiency\nof reactant vibrational excitation does not increase without\nlimits with the vibrational energy; instead, a saturation effect\ncan be observed. https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 ■RESULTS Accordingly, the amplitude and the classical action of the\nH−Br vibration remain constant during the approach of the\nreactants. At relatively small reactant separation (at around\nr(O−H) = 2.5 Å), the equipotentials at energies corresponding\nto the zero-point vibrational energy of HBr and above start to\nturn away from the horizontal axis when r(O−H) decreases. The trajectories near the outer turning points of the H−Br\noscillation keep touching the same lines, which means that the\nvibration remains adiabatic. Since the equipotentials are bent,\nwhen the trajectory arrives close to them, a force component\narises parallel to the forming bond, which slightly accelerates\nthe approach of the reactants. This is reflected in the growing\nhorizontal distance between the successive OTPs. This is a\nmanifestation of the same attraction that was observed for\nother reactions when the breaking bond was vibrationally\nhighly excited (see Figure S1). When the trajectory reaches the\nequipotential corresponding to the available energy in the (so\nfar adiabatic) vibration at a point where its curvature is large,\nthen, due to the force component parallel to the r(OH) axis,\nthe trajectory is reflected into the product valley (see Figure 4a\nand b). The motion hardly depends on the presence of the\nbarrier, because it is low. As a result, in many cases the\npotential barrier is crossed “in the wrong direction”, for\nexample, perpendicular to the minimum energy path (Figure\n4c), as if the trajectory “were dropped from above” on the\nsaddle point region. What is remarkable is that in reactions R1\nand R2, the phenomenon appears at the lowest physically Opacity Functions. In Figure 3, the opacity functions for\nreactions R1 and R2 are plotted at selected collision energies. The change of the shape is typical for reactions with attractive\npotential. At very low collision energy, the reaction\nprobabilities are large, around 0.7 in the entire impact\nparameter range extending, for example, up to 12 Å at Ecoll =\n0.01 kJ/mol for reaction R1, where they suddenly drop. This\nlimit marks the location of the (orientation averaged)\ncentrifugal barrier. Capture, in fact, does not guarantee that\nreaction occurs, even though the energetic requirements are\nfulfilled. With increasing collision energy, the opacity functions\nshrink, and for reaction R1 a maximum appears near the high-\nimpact-parameter end of the curves, which disappear when the\ncollision energy is above about 5 kJ/mol. 8389 https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. ■RESULTS A 2021, 125, 8386−8396 Figure 4. (a−c) Representative trajectories for the HO + HBr (ν = 0) reaction, together with the contour representation of the potential energy\nsurface plotted as a function of the lengths of the forming and breaking bonds. The rest of the geometrical parameters are set to their value at the\nsaddle point of the PES. The collision energy is 0.05 kJ/mol. The contour lines are drawn in 1 kJ/mol steps. The magenta horizontal line is a guide\nto the eye near the outer turning point of HBr vibration at v = 0. Note that the other coordinates also change during the course of the reaction;\nthus, the section of the PES along the r(O−H)−r(H−Br) plane in reality also varies slightly between successive time steps. Panel (d) is a magnified\nview of the same cut of the PES, with energy spacing 0.2 kJ/mol. The Journal of Physical Chemistry A\npubs.acs.org/JPCA\nArticle The Journal of Physical Chemistry A\npubs.acs.org/JPCA\nArticle The Journal of Physical Chemistry A Article pubs.acs.org/JPCA pubs.acs.org/JPCA Figure 4. (a−c) Representative trajectories for the HO + HBr (ν = 0) reaction, together with the contour representation of the potential energy\nsurface plotted as a function of the lengths of the forming and breaking bonds. The rest of the geometrical parameters are set to their value at the\nsaddle point of the PES. The collision energy is 0.05 kJ/mol. The contour lines are drawn in 1 kJ/mol steps. The magenta horizontal line is a guide\nto the eye near the outer turning point of HBr vibration at v = 0. Note that the other coordinates also change during the course of the reaction;\nthus, the section of the PES along the r(O−H)−r(H−Br) plane in reality also varies slightly between successive time steps. Panel (d) is a magnified\nview of the same cut of the PES, with energy spacing 0.2 kJ/mol. Figure 5. Sections of the potential energy surfaces as functions of the lengths of the forming and breaking bonds for the (a) CH3 + HBr and (b)\nHO + HBr reactions. The colored horizontal lines represent the bond length of the breaking bond at the vibrational outer turning point of HBr at\nvarious vibrational quantum numbers. In each panel, the lowest line corresponds to the ground state of HBr, followed by ν = 1, 2. Energies in\nkJ/mol. Figure 5. https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 ■RESULTS Thus,\nthe attraction arising when the zero-point vibration is at the\nouter turning point effectively supplements the weak non-\nbonding attraction between the reactants with equilibrium\nbond lengths. This is the reason the excitation functions for\nthese reactions diverge so quickly when the reactant molecule\nis vibrationally unexcited. It is reasonable to consider the\nattraction arising when the breaking bond is significantly\nstretched as originating from the dynamics of the reacting\nsystem, compared with the “static” van der Waals-type\ninteraction that brings together the components of the pre-\nreaction complex. We shall refer to the attraction due to the\nlarge amplitude of vibration as “vibrationally induced”. Since\nthis kind of attraction is strong already in the vibrational\nground state of the HBr reactant, it is not surprising that when\nthe vibrational energy available for the reactant is increased\nabove the ground-state level by one or two quanta, the cross\nsections do not increase as much as one might expect. The\nvibrational excitation certainly increases the amplitude of the\nHBr bond length oscillation above that of the vibrational\nground state. However, the reactivity enhancement caused by\nthe zero-point oscillation is so large that increasing the\namplitude further will not induce as large a change as\nvibrational excitation of the reactant would in the absence of\nthe extra, dynamically induced attraction. Figure 6. Cuts of the potential energy surfaces of reactions R1 and R2\nalong the horizontal lines in Figure 5: the potential energy as a\nfunction of the length of the forming C−H (left panel) and O−H\n(right panel) bond at fixed values of the length of the breaking H−Br\nbond. The top three lines correspond to the outer turning point of the\n(from top downward) v = 2, 1, 0 vibrational states of HBr. Figure 6. Cuts of the potential energy surfaces of reactions R1 and R2\nalong the horizontal lines in Figure 5: the potential energy as a\nfunction of the length of the forming C−H (left panel) and O−H\n(right panel) bond at fixed values of the length of the breaking H−Br\nbond. The top three lines correspond to the outer turning point of the\n(from top downward) v = 2, 1, 0 vibrational states of HBr. These arguments seem to provide (one of) the reasons why\nthe reactivity enhancement by vibrational excitation of HBr is\nlarger for reaction R1 than for R2. ■RESULTS Sections of the potential energy surfaces as functions of the lengths of the forming and breaking bonds for the (a) CH3 + HBr and (b)\nHO + HBr reactions. The colored horizontal lines represent the bond length of the breaking bond at the vibrational outer turning point of HBr at\nvarious vibrational quantum numbers. In each panel, the lowest line corresponds to the ground state of HBr, followed by ν = 1, 2. Energies in\nkJ/mol. possible vibrational energy content of the reactant, i.e., the\nzero-point energy. visible in the projection shown in the figure, such as the\nrotation of the reactants. The energy gained in this coupling\nwas lost by the relative translation, and the trajectory was\ntrapped for about 15 H−Br vibrational periods, because the\nremaining translational energy was not enough to get out of\nthe well either across the barrier or back to reactants, until the\ncoupling with bending channeled some energy back into this\nmode. The trajectory in Figure 4b allows a glimpse at the role of\nthe pre-reaction potential well. In this specific collision, the\ntrajectory spent a long time in the well. Temporarily, the\namplitude of the vibration of HBr increased, due to the\ncoupling with the relative translation. This coupling was\ninduced by the motion along the degrees of freedom not https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 8390 The Journal of Physical Chemistry A pubs.acs.org/JPCA Article Article ■DISCUSSION less efficient in promoting early barrier reactions than\ntranslational energy, as it is familiar from Polanyi’s rules. Role of the Shape of the PES for Reactions with\nExtreme Reactivity Enhancement by Vibrational Ex-\ncitation. The trajectory calculations indicate that in reactions\nR1 and R2, the trajectories leading to reaction follow a path\nsimilar to what was seen for some other reactions when the\nvibration of the breaking bond was highly excited. The earlier\nobservations of the appearance of capture-type excitation\nfunctions include some highly endothermic hydrogen-\nabstraction reactions (H + H2O, H + HF)35−39 with reactant\nvibrational quantum number above 3, and a close to\nthermoneutral reaction (H + HCl).40 For the latter, the\nbarrier is comparable in height to a vibrational quantum, and\nthe extreme reactivity can be seen already at ν(HCl) = 2. Reactions R1 and R2 are exothermic, and the reactivity\nenhancement seems to arise already when the reactant HBr is\nin the vibrational ground state. ■RESULTS While the location of the\nbarrier is distinctly different for the highly endothermic, almost\nthermoneutral, and exothermic reactions, the condition for\nhigh reactivity seems to be common for all cases: The\namplitude of the vibration of the breaking bond is large enough\nto ensure that the outer turning point is displaced far from the\nMEP, to the region of configuration space where the\ninteraction of the reactants is attractive. The attraction is\nmanifested in the shape of the contour lines of the potential\nenergy plotted against the lengths of the forming and breaking\nbonds: The equipotentials turn away from the horizontal axis\nwhen followed right to left (see Figures 4, 5, and S3). Another\nmanifestation of the attraction critical for reactivity enhance-\nment is that the cuts of the PES as functions of the length of\nthe forming bond plotted at fixed values of the length of the\nextended breaking bond are monotonously attractive (Figure\n6). Every time the breaking bond vibration approaches the Figure 4d and Figure 5 show that for reactions R1 and R2\ncharacterized by a submerged barrier, the repulsive contour\nlines (those that turn downward when traced from right to left\nin the figures) are at very low energies, below the reactant level. The reason for this is that the barrier is low even with respect\nto the well of the prereaction complex. This feature of the PES\nof reactions R1 and R2 (and very probably many other\nreactions with submerged, early potential barriers) are\nexceptional in the sense that the enhancement of the reactivity\nby vibrational excitation in fact occurs already when the\nreactants are in the vibrational ground state. Figure 5 shows\nthat the vibrational energy of the vibrationally unexcited HBr\nbond is so large that for both reactions, on the cut of the PES\nalong the r(X−H)−r(H−Br) plane (X = C or O), the contour\nline corresponding to it turns upward with decreasing r(X−H). In Figure 6, the cut of the PESs for both reactions are plotted\nas a function of the length of the forming bond at fixed values\nof the breaking bond. When the H−Br bond is long, the\npotential curves monotonously decrease when the length of the\nforming bond shortens toward its equilibrium distance. https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 The Journal of Physical Chemistry A In the other set of\ncalculations, we reduced the vibrational quantum number of\nHBr from 0 to −0.4 and −0.49, i.e., decreased the vibrational\nenergy, respectively, to 20% and 2% of the semiclassical zero-\npoint energy. For some cases, we combined the two methods. Figure 7 shows the change of the excitation functions due to\nthe variation of the mass of the abstracted H atom for reaction\nR1 and the influence of the initial vibrational quantum number\nof HBr for reaction R2. The reactivity drastically drops both\nwhen one increases the mass of the H atom and when the\nvibration is frozen. y\ny\nThe calculations with artificially reduced vibrational\namplitude indicate that the lack of vibrational energy inhibits\nthe reaction. In reality, complete quenching of reactant\nvibration is not possible, because it is quantized and the\nlowest energy level, the zero-point energy, is finite. In reactions\nR1 and R2 the corresponding amplitude is large enough to\nallow the system to visit the region of the PES where attraction\narises between the reactants. This means that the large\nreactivity in these systems is made possible by the existence of\nzero-point energy, which is the manifestation of an interesting\nquantum effect. q\nIt is instructive to look at the effect of changing the mass of\nthe transferred atom or the vibrational quantum number of the\nHBr reactant on the opacity functions underlying the reaction\ncross sections. Figure 8a shows that when the mass of the\nreactive H atom increases, the opacity functions shrink both in\nheight and in width, and their character changes: the large-\nimpact-parameter range becomes completely nonreactive when\nthe mass of the atom to be transferred is large. At low collision\nenergy (left panels), the shape of the opacity functions for\nreaction R1 reflects capture-type behavior when the mass of\nthe reactive H atom is small (1 or 2 g/mol): the reaction\nprobabilities are very large up to large impact parameters, with\na sudden falloff. In contrast, at large masses the probabilities\nare drastically smaller at small impact parameters and decrease\nquickly and monotonously with increasing impact parameter. At larger collision energies, where the reactant’s attraction has\na smaller influence on the dynamics, the opacity functions\ndecrease roughly linearly with increasing impact parameter,\nand no qualitative difference can be seen between the small\nand large masses of the transferred atom. The Journal of Physical Chemistry A Article pubs.acs.org/JPCA pubs.acs.org/JPCA Figure 7. Excitation functions for (a) reaction R1 with the masses of the transferred H atom set to 1, 2, 15, and 80 g/mol (left panel) with ν(HBr)\n= 0 and with ν(HBr) = 4 for mass 15 g/mol; (b) for reaction R2 with vibrational quantum numbers −0.49, −0.4, 0, 1, and 2 with protium mass as\nwell as with mass for the H atom set to 80 g/mol, combined with ν(HBr) = 0. is added. When the vibrational excitation is increased further,\nthe reactivity will not increase significantly for either reaction,\nbecause the vibrationally induced attraction almost fully\noperates already at v = 1. probably negligible effect. The mass effect is excluded when\nthe vibrational amplitude is decreased by diminishing the\nrespective quantum number. In Figure 7b, one can see that the\nreaction can be frozen this way too. In both sets of calculations,\nthe excitation functions turn upward at very low collision\nenergies even when the mass of the transferred H atom is set to\nvery large values or when the vibrational “quantum number” is\nmade very small. This remaining capture-type reactivity shows\nthe effect of the “static” attraction, which is rather small when\nit is not assisted by the vibrationally induced attraction. p\ny\nDynamics at Small Vibrational Amplitudes. In QCT\nsimulations, by directly controlling the vibrational amplitude of\nHBr, one can test whether really the large amplitude of the\nground-state HBr vibration is responsible for the large\nreactivity. One possibility is that one increases the mass of\nthe H atom that can be abstracted from the HBr molecule. Experimentally, this can be done by deuterium substitution,\nbut since the amplitude decreases roughly as the 1/4th power\nof the ratio of the mass of protium to that of the heavy isotope,\nthe observable consequences are not spectacular. In\nsimulations of collision dynamics in QCT calculations, one\nhas more freedom and can increase the mass arbitrarily. Another way to reduce the amplitude of the HBr vibration that\nis not available in experiments is that one reduces the\nvibrational action below the 1/2 hν semiclassical prescription. We used both methods. In one set of simulations, we increased\nthe mass of the reactive H atom to 2, 15, and 80 g/mol (the\nlatter two values match those of the group/atom between\nwhich the H atom is transferred). https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 ■RESULTS Figure 5 shows that for\nreaction R2 the contour lines turn away from the horizontal\naxis starting from about the HBr bond length corresponding to\nthe OTP of the zero-point vibration, while for reaction R1, the\neffect occurs at significantly lower energy. This suggests that\nfor the reaction of HO with HBr, the zero-point vibrational\namplitude is at the borderline where the vibrationally induced\nattraction arises, while for the reaction of CH3, the zero-point\namplitude is well above the limit. One can expect that for\nreaction R1 the zero-point vibration induces a reactivity\nenhancement that approaches an upper limit, and by exciting\nthe HBr vibration, the reactivity cannot increase very much. In\nreaction R2, the reactivity enhancement generated by the zero-\npoint vibration is still not as close to the upper limit, and there\nis room for further enhancement when a vibrational quantum OTP, a force component arises that is parallel to the direction\nof approach of the reactants and pulls the reactants toward\neach other. For early barrier reactions, this is not typical\nbehavior. When the barrier is relatively high and is located at\nthe same early position as for reactions R1 and R2, the\ntopography of a saddle requires that the equipotentials turn\ntoward the horizontal axis, similarly to the three lowest contour\nlines above the barrier in Figure 4d. The force component\narising from the curvature of the contour lines is repulsive and\nslows down the approach of the reactants. This force\ncomponent is one of the factors that make vibrational energy 8391 Figure 7. Excitation functions for (a) reaction R1 with the masses of the transferred H atom set to 1, 2, 15, and 80 g/mol (left panel) with ν(HBr)\n= 0 and with ν(HBr) = 4 for mass 15 g/mol; (b) for reaction R2 with vibrational quantum numbers −0.49, −0.4, 0, 1, and 2 with protium mass as\nwell as with mass for the H atom set to 80 g/mol, combined with ν(HBr) = 0. The Journal of Physical Chemistry A\npubs.acs.org/JPCA\nArticle The Journal of Physical Chemistry A In contrast,\nthe probability increases approximately by only a factor of 1.15\nwhen v(HBr) grows from 0 to 1, and the reactivity\nenhancement is negligible when one more vibrational quantum\nbecomes available. These trends show that the vibrational\nground state of HBr is close to some “large vibrational\namplitude limit” for reaction R2 in the capture regime. As\ndiscussed in connection with Figures 5 and 6, this is connected\nto the shape of the potential energy surface: At small\nvibrational amplitude, the OTP of the vibration remains\nclose to the minimum energy path, and the system cannot\nreach the region where the vibrationally induced attraction is\neffective (see Figures 4d and 5). Under such conditions, the\n“static” attraction does bring together the reactants, but it is\nnot strong enough to effectively guide them through the\nbarrier. Once the vibrational amplitude of HBr is large enough\nto allow the reactants to visit the region of the PES where the\nvibrationally induced attraction arises, significant reactivity\nenhancement can be seen. Figure 8b shows from another\nperspective that for reaction R2 this happens somewhere close\nto v(HBr) = 0, and the change is quite sudden. y\nVirtual Violation of Polanyi’s Rule. At first glance, the\nenhancement of the reactivity of an early-barrier reaction by\nvibrational excitation seems to violate one of Polanyi’s rules. The latter states that when the potential barrier is shifted to the\nreactant valley, if one provides the same energy in the form of\ntranslation or vibration, the former is more favorable in\npromoting the reaction. The reason is that the collision energy\nis deposited in the degree of freedom that is directed along the\nuphill valley toward the crest of the barrier, so it can be fully\nutilized by the reactive system to surmount the potential\nbarrier. Vibration, in contrast, corresponds to motion in the\nperpendicular direction, and is de facto useless in climbing the\nbarrier. It can be utilized for surmounting the potential barrier\nonly if it is channeled to the right, uphill direction. If the\npotential barrier is early, the coupling to translation is rather\nlimited, and the vibration remains completely passive. The Journal of Physical Chemistry A pubs.acs.org/JPCA Article Article barrier of the PES, even when the vibrational amplitude is\nsmall. The large vibrational amplitude in this regime cannot\nincrease the reactivity effectively, because the ratio of the time\nscales of vibration and translation becomes less favorable than\nwhen the reactants approach each other slowly. As the relative\nvelocity of the reactants increases, fewer and fewer vibrational\nperiods are completed while the reactants are close to each\nother, and the oscillation of the length of the breaking bond\nrarely reaches the outer turning point where the vibrationally\ninduced attraction is effective. Connection to Non-IRC Dynamics. For the vibrationally\ninduced attraction to operate, the vibrational amplitude needs\nto be large in comparison with the width of the reactant and\nproduct valleys and of the saddle point region. Thus, an\nintrinsic property of such reacting systems is that they do not\nclosely follow the minimum energy path. There is a group of\nphenomena known in the literature where the reaction does\nnot visit certain regions of the PES that are close to the MEP,\nfor example, some van der Waals-type wells. These processes\nare said to be characterized by non-IRC dynamics53−55 (IRC,\nor intrinsic reaction coordinate in this context is equivalent to\npathways involving multiple minima and saddle points\ncontiguously connected by the respective minimum energy\npaths). Non-IRC dynamics is generally considered to take\nplace when the reactants pass a potential barrier, and for\ndynamical reasons, the trajectories leave the neighborhood of\nthe product side of the MEP. In a sense, the behavior of\nreactions where reactivity enhancement is caused by vibra-\ntionally induced attraction can be considered a kind of non-\nIRC dynamics. Here, however, the large deviation from the\nIRC acts before the reactants pass the potential barrier, instead\nof arising after that. As in reactions R1 and R2 the barrier\nregion is often crossed far from the saddle point itself, there is a\nlarge chance of post-transition state non-IRC dynamics in the\nconventional sense. This can be manifested in the product\nenergy distribution among the translational, vibrational, and\nrotational degrees of freedom, but can have chemical\nconsequences such as roaming. In fact, roaming was found\nto take place in the H + HF and H + HCl reactions.39,40 We\nplan to study the possible manifestations of post-transition-\nstate non-IRC dynamics for reactions R1 and R2. Figure 8. The Journal of Physical Chemistry A Opacity functions of (a) reaction R1with different masses of\nthe H atom (denoted as M) of the HBr molecule and (b) reaction R2\nwith various vibrational quantum numbers of HBr. All other degrees\nof freedom are in the ground state. R2 is much larger when v(HBr) increases from −0.4 to 0 than\nwhen it changes from 0 to 1 or 1 to 2, especially when\nconsidering that in the former case the increase of the\nvibrational energy is less than one-half of that in the latter two\ncases. When v(HBr) changes from −0.49 to 0 and −0.4 to 0,\nthe enhancement of the reaction probability at, for example, b\n= 2 Å is roughly a factor of 10 and 5, respectively. In contrast,\nthe probability increases approximately by only a factor of 1.15\nwhen v(HBr) grows from 0 to 1, and the reactivity\nenhancement is negligible when one more vibrational quantum\nbecomes available. These trends show that the vibrational\nground state of HBr is close to some “large vibrational\namplitude limit” for reaction R2 in the capture regime. As\ndiscussed in connection with Figures 5 and 6, this is connected\nto the shape of the potential energy surface: At small\nvibrational amplitude, the OTP of the vibration remains\nclose to the minimum energy path, and the system cannot\nreach the region where the vibrationally induced attraction is\neffective (see Figures 4d and 5). Under such conditions, the\n“static” attraction does bring together the reactants, but it is\nnot strong enough to effectively guide them through the\nbarrier. Once the vibrational amplitude of HBr is large enough\nto allow the reactants to visit the region of the PES where the\nvibrationally induced attraction arises, significant reactivity\nenhancement can be seen. Figure 8b shows from another\nperspective that for reaction R2 this happens somewhere close\nto v(HBr) = 0, and the change is quite sudden. R2 is much larger when v(HBr) increases from −0.4 to 0 than\nwhen it changes from 0 to 1 or 1 to 2, especially when\nconsidering that in the former case the increase of the\nvibrational energy is less than one-half of that in the latter two\ncases. When v(HBr) changes from −0.49 to 0 and −0.4 to 0,\nthe enhancement of the reaction probability at, for example, b\n= 2 Å is roughly a factor of 10 and 5, respectively. The Journal of Physical Chemistry A In Figure 8b, the\neffect of the variation of v(HBr) is displayed. Reduction of the\nreactant’s vibrational quantum number below 0 induces a\nqualitative change of the opacity functions, similarly to the\nincrease of the mass of the transferred atom. Remarkably, at\nlow collision energy the difference of the reactivity in reaction To test the whether the larger inertia of the reactant that\naccompanies the increase of the mass of the H atom of HBr\ncan influence the reactivity beyond reducing the vibrational\namplitude, we calculated the excitation functions with\ncombined variation of mass and vibrational quantum number. For reaction R1, increasing the mass of the H atom to 15 g/\nmol at νHBr = 0 reduces the cross section at Ecoll = 2 kJ/mol\nfrom 40.9 to 4.3 Å2 (see Figure 7). The reactivity, however, can\nbe recovered by vibrational excitation of HBr by increasing the\nvibrational quantum number to 4 (the respective cross section\nbecomes 25.3 Å2). This proves that the mass change\npredominantly affects the reactivity via reducing the vibrational\namplitude, and the lower velocity of approach has a small, 8392 https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 Figure 8. Opacity functions of (a) reaction R1with different masses of\nthe H atom (denoted as M) of the HBr molecule and (b) reaction R2\nwith various vibrational quantum numbers of HBr. All other degrees\nof freedom are in the ground state. The Journal of Physical Chemistry A The Journal of Physical Chemistry A In terms\nof the potential energy surface, for reactions for which\nPolanyi’s rules are designed, the early potential barrier is\nrelatively high or at least positive, and the contour lines on the\nreactant side of the barrier remain repulsive up to large\ndistances from the minimum energy path (in the cuts the\ncontour lines turn toward the forming bond axis). Even for this It is worth noting that at large collision energies, where the\nattraction between the reactants does not affect the reactivity\nas much as at low collision energies, the reaction probabilities\nfor reaction R2 are not very sensitive to the reactant’s\nvibrational quantum number. This agrees with the expectation\nthat the translational energy can be utilized for climbing the 8393 https://doi.org/10.1021/acs.jpca.1c05839\nJ. Phys. Chem. A 2021, 125, 8386−8396 The Journal of Physical Chemistry A pubs.acs.org/JPCA Article Article reducing the vibrational action below the zero-point value,\nwhich resulted in enormous reduction of the reactivity. The\nexcitation functions are capture-type even in this case, which is\ndue to the attractive potential leading into the prereaction\npotential well, but the magnitude of the residual reactivity is\nmuch smaller than what the “vibrationally enhanced attraction”\ninduces. This means that the zero-point energy of the reactant\nvibration is necessary to make these reactions as fast as it is\nmeasurable in experiments. The reactions would be much\nslower without z.p.e., which is a unique quantum effect. kind of reaction, the “vibrationally induced” attraction does\narise when the amplitude of the vibration of the breaking bond\nis large. The attraction facilitates the reaction and even induces\nenormous rate enhancement, such as that observed exper-\nimentally and explained by theory for the reaction of\nvibrationally highly excited water with H atoms.36−40 In the\nprevious sections we have seen that the same factors determine\nthe reactivity when the potential barrier to reaction is early and\nis submerged so that it forms a reef next to a shallow\nprereaction potential well. Vibrational excitationin fact,\nalready when it is relatively lowin such cases does promote\nthe reaction, which might sound like a violation of Polanyi’s\nrule. However, the rule also includes a comparison with the\nefficiency of collisional energy. The translational energy in the\ncases of vibrationally induced attraction, instead of promoting\nthe reaction, is not favorable; instead, the reactivity decreases\nwith increasing collision energy. Notes The authors declare no competing financial interest. The authors declare no competing financial interest. ■AUTHOR INFORMATION The effect of vibrational excitation on the reactivity of systems\nwhose potential energy surfaces are characterized by a\npotential well corresponding to a prereaction complex and a\nsubmerged barrier acting as a reef in the formed “lagoon” were\nstudied. For reactions of CH3 and HO radicals with HBr, the\nQCT calculations revealed unexpectedly large reactivity: the\nreaction cross sections diverge swiftly when the collision\nenergy is reduced. In addition, both reactions are accelerated\nwhen the HBr reactant is vibrationally excited. The capture-\ntype excitation functions observed for this reaction repeat what\nhas been seen for the HO + HBr reaction. The large reactivity\nhas been traced back to a static component determined by the\nshape of the potential energy surface and a dynamical factor\noriginating in the dynamics of the motion of atoms. The static\nfactor is that the potential energy decreases on the path leading\ninto the prereaction potential well when the distance between\nthe reactants diminishes. The dynamical factor comes from the\nbond length oscillation of the breaking H−Br bond, the\namplitude of which is large already in the vibrational ground\nstate of HBr. The oscillation generates extra attraction between\nthe reactants, essentially independent of whether there is a\nbarrier to the reaction. The additional attraction arises when\nthe reacting system leaves the close neighborhood of the\nminimum energy path (MEP). In particular, when the HBr\nmolecule is fully stretched, the trajectories visit the region of\nthe PES where, as a function of the length of the forming bond,\nthe potential is attractive. The larger the vibrational excitation,\nthe larger the attraction in the stretched phase of the reactant\nvibration is, and the larger the reactivity. The curiosity of the\nreactions studied here is that the potential energy surface\nallows large-amplitude vibration even in the vibrational ground\nstate of the reactants. As a consequence, the divergence of the\nexcitation function is enhanced more by “vibrationally\nenhanced attraction” than by static attraction. In test\ncalculations, the reactant vibrational amplitude was artificially\ndecreased by increasing the mass of the abstracted atom or by Corresponding Author György Lendvay −Institute of Materials and Environmental\nChemistry, Research Centre for Natural Sciences, H-1117\nBudapest, Hungary; Center for Natural Sciences, Faculty of\nEngineering, University of Pannonia, Veszprém 8200,\nHungary;\norcid.org/0000-0002-2150-0376;\nEmail: lendvay.gyorgy@ttk.mta.hu The Journal of Physical Chemistry A Thus, the question, “in which\nmode should one provide the same energy to achieve larger\nreactivity?” is meaningless. Polanyi’s rules apply to systems\nwhich more-or-less follow the minimum energy path. This is\nnot the case when the vibrational amplitude is large, so one\nshould not expect the rules to apply to this kind of reactivity\nenhancement. It remains to be seen whether the extension to\nPolanyi’s rules, the sudden vector projection model of Guo and\nco-workers56,57 can explain this phenomenon. The conditions inducing the vibrational enhancement of the\nreactivity of this kind of early barrier reactions are different\nfrom those that govern reactions that more closely follow the\nMEP, for which Polanyi’s rules have been designed. Thus,\nthese reactions are beyond the sphere of applicability of these\nrules. Authors Benjámin Csorba −Institute of Materials and Environmental\nChemistry, Research Centre for Natural Sciences, H-1117\nBudapest, Hungary Budapest, Hungary\nPéter Szabó −Institute of Materials and Environmental\nChemistry, Research Centre for Natural Sciences, H-1117\nBudapest, Hungary; Present Address: Department of\nPhysics and Materials Science, University of Luxembourg,\nL-1511 Luxembourg, Luxembourg\nSzabolcs Góger −Institute of Materials and Environmental\nChemistry, Research Centre for Natural Sciences, H-1117\nBudapest, Hungary; Present Address: Department of\nPhysics and Materials Science, University of Luxembourg,\nL-1511 Luxembourg, Luxembourg Péter Szabó −Institute of Materials and Environmental\nChemistry, Research Centre for Natural Sciences, H-1117\nBudapest, Hungary; Present Address: Department of\nPhysics and Materials Science, University of Luxembourg,\nL-1511 Luxembourg, Luxembourg Péter Szabó −Institute of Materials and Environmental\nChemistry, Research Centre for Natural Sciences, H-1117\nBudapest, Hungary; Present Address: Department of\nPhysics and Materials Science, University of Luxembourg,\nL-1511 Luxembourg, Luxembourg g\ng\nSzabolcs Góger −Institute of Materials and Environmental\nChemistry, Research Centre for Natural Sciences, H-1117\nBudapest, Hungary; Present Address: Department of\nPhysics and Materials Science, University of Luxembourg,\nL-1511 Luxembourg, Luxembourg Complete contact information is available at:\nhttps://pubs.acs.org/10.1021/acs.jpca.1c05839 *\nsı Supporting Information Plots of potential energy surfaces for the H+HF reaction\nand the vibrational enhancement factors for reactions R1\nand R2 (PDF) ■ASSOCIATED CONTENT *\nsı Supporting Information\nThe Supporting Information is available free of charge at\nhttps://pubs.acs.org/doi/10.1021/acs.jpca.1c05839. ■REFERENCES 2014, 53, 1122. 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