Gluing Potential Energy Surfaces with Rare Event Simulations

Transcription

1 This is an open aess artile published under an ACS AuthorChoie Liense, whih permits opying and redistribution of the artile or any adaptations for non-ommerial purposes. pubs.as.org/jctc Gluing Potential Energy Surfaes with Rare Event Simulations Anders Lervik and Titus S. van Erp* Department of Chemistry, Norwegian University of Siene and Tehnology, 7491 Trondheim, Norway ABSTRACT: We develop a new method ombining replia exhange transition interfae sampling with two distint potential energy surfaes. The method an be used to ombine different levels of theory in a simulation of a moleular proess (e.g., a hemial reation), and it an serve as a dynamial version of QM-MM, onneting lassial dynamis with Ab Initio dynamis in the time domain. This new method, whih we oin QuanTIS, ould be applied to use aurate but expensive density funtional theory based moleular dynamis for the breaking and making of hemial bonds, while the diffusion of reatants in the solvent are treated with lassial fore fields. We exemplify the method by applying it to two simple model systems (an ion dissoiation reation and a lassial hydrogen model), and we disuss a possible extension of the method in whih lassial fore field parameters for hemial reations an be optimized on the fly. 1. INTRODUCTION Many proesses in nature arise from the spontaneous transition between stable states separated by an ativation barrier. If the ativation barrier is suffiiently high, a transition between the stable states an be onsidered a rare event as the probability of a flutuation driving the transition dereases exponentially with the ativation barrier. The time sale for the overall transition (e.g., milliseonds) an be many orders of magnitude larger than the moleular time sale (e.g., femtoseonds), and this poses a hallenge for simulation tehniques whih aim to aurately model rare events. In priniple, moleular dynamis (MD) ould be used as a omputational tool to investigate suh transitions, but for rare events this is not feasible: The time sale required to aurately simulate the transition is far greater than what is urrently attainable by MD simulations. A brute-fore MD simulation beomes ineffiient as the traetory mainly samples the stable states, and only a tiny fration of it orresponds to the transition itself. Further, in order to orretly model the reative event, an aurate desription of the energy landsape is needed. For this purpose, one ould make use of a reative fore field; but this is often not aurate enough, and alulations using a more detailed desription, e.g. on the quantum mehanial (QM) level, are required. This would signifiantly inrease the ost of a MDbased approah, and we are thus faed with two interonneted hallenges: reahing the relevant time sale while maintaining an aeptable auray. In order to takle these hallenges, several methods have been proposed over the last deades. Hyperdynamis 1 aims at lowering the energy differene between the top of the barrier and the initial basin, the parallel replia method 2 exploits the power of parallel proessing to extend the moleular simulation time, and temperature-aelerated dynamis 3 speeds up the event by raising the temperature. The idea of driving energy into the system to esape the basin of the energy minimum in whih the system is initially prepared is also at the basis of onformational flooding 4 and the metadynamis method. 5 Other important methods are thermodynami integration 6 and umbrella sampling, 7 eigenvetor following, 8 the ativationrelaxation tehnique, 9 nudged elasti band, 10 string method, 11 and disrete path sampling. 12 These methods have in ommon that they enhane reative events by perturbing the atual dynamis. Using reweighting shemes, based on the laws of statistial physis, one an usually get exat results on (stati) statistis suh as free energy barriers, but important information on the spontaneous dynamial proess is usually lost. Moreover, free energies expressed as a low dimensional funtion of a set of reation oordinates an be very misleading. The height of free energy barriers depend sensitively on the set of reation oordinates that is hosen. As a result, Transition State Theory (TST) is insuffiient to make preditions about reation rates in omplex systems. The reative flux method 13 is the standard approah to orret the TST expression. It omplements the free energy alulation along a single reation oordinate with the alulation of a dynamial transmission oeffiient, by starting short traetories from the maximum of the free energy barrier. However, in omplex systems the orret reation oordinate an be exeedingly diffiult to find. If the reation oordinate does not apture the moleular mehanism, the biased sampling methods will suffer from substantial hysteresis when following the system over the barrier. Moreover, even if the free energy profile is obtained orretly for this partiular (but wrong) reation oordinate, the orresponding transmission oeffiient will be very low, making an aurate evaluation problemati. Reeived: January 7, 2015 Published: April 24, Amerian Chemial Soiety 2440

2 In order to overome these diffiulties, Chandler and oworkers developed the transition path sampling (TPS) 17,18 method whih does not require detailed knowledge of the transition state (or the reation oordinate/mehanism). In fat, the transition state and the reation mehanism an be inferred from the results of a TPS simulation. The TPS method uses a Monte Carlo (MC) approah to sample reative traetories onneting an initial and final state. This is ombined with the probability of reahing the final state from the initial state (obtained by an umbrella sampling approah) in order to estimate the rate onstant for the transition. Transition interfae sampling (TIS) 19 builds on the TPS approah and improves the effiieny by allowing flexible path lengths, it is less sensitive to rerossings, and the umbrella sampling tehnique is replaed by interfae ensemble averages. 19 In addition, the MC tehnique in the TIS method is simpler sine it only involves the so-alled shooting move and not both shooting and shifting moves as in the TPS method. The TIS method has been further developed into the replia exhange TIS (RETIS) 20 approah, where replia exhange moves are arried out between the different path simulations. This represents a signifiant improvement in effiieny ompared with the standard TIS method. The RETIS method is also better suited to takle omplex reation mehanisms with many degrees of freedom and multiple reation hannels. 21,22 For ompleteness, we also mention the reent ombined methods in whih TPS/TIS is ombined with thermodynami integration type of approahes or metadynamis. 26 Both the TIS and the RETIS methods introdue a set of interfaes between the reatant and the produt state and reasts the overall rossing probability as a produt of rossing probabilities for the intermediate interfaes. In order to estimate the rate onstant for the transition, the TIS method ombines this probability with a MD-simulation where the esape flux through the first interfae is alulated. In the RETIS approah, the esape flux is alulated using the average path lengths in the two ensembles assoiated with the first interfae. These two ensembles an be interpreted as orresponding to different proesses: nonreative exploration of the stable reatant state and the initial progress of the reative event. It is this interpretation that forms the basis of the new method we introdue in this artile. Even though the path-ensemble methods an be many times more effiient than a MD-based approah, they are still dependent on an aurate desription of the energy landsape. This will often require expensive alulations (e.g., on the QM level). One way to redue the ost is to limit the QM treatment to ertain spatial regions of the system and treat the rest with a lassial method. Suh quantum mehanis-moleular mehanis (QM-MM) 27,28 methods have suessfully been used together with TPS to boost the auray and attainable length sale of the system under onsideration. This requires that one an, a priori, selet whih regions require a QM or MM treatment. In this ontext, the interpretation of the two ensembles assoiated with the first interfae in the RETIS approah points to an alternative strategy for dereasing the omputational ost: The ensemble orresponding to nonreative exploration an be treated with a less detailed theory (e.g., MM) than the other ensembles (e.g., QM). Effetively, this orresponds to a dynamial QM-MM approah, where different auraies are used in different path ensembles and the hoie of employing QM or MM is based on the temporal rather than the spatial oordinate. In this new approah, whih we oin QuanTIS, the system is either treated fully with QM or fully with MM at a given time. This is advantageous sine the path lengths orresponding to the nonreative exploration an be made onsiderably longer than the path lengths in the other ensembles by positioning the first interfae at the barrier region. A signifiant redution in the omputational ost an be attained by applying the more (omputationally) expensive theory only to the reation itself. The aim of this artile is to introdue this new QuanTIS method, exemplify it, and disuss the prospets of the approah. We begin by disussing the RETIS approah in more detail sine this is the theoretial basis of the QuanTIS method. As a proof of onept, we investigate two different systems: (I) a model of a hypothetial ion transfer system where a solvent may redue the transfer barrier and (II) a realisti model for the dissoiation of hydrogen moleules. These models are defined before we disuss the omputational details in setion 4 and our results in setion 5. We provide a perspetive on the outlook for the new QuanTIS approah, and we propose further appliations before we end with the final onlusion. 2. THEORY The RETIS method is a very effiient strategy for performing path simulations. In this setion, we give a brief summary of the approah, before we disuss the new strategy of applying different levels of theory for the path ensembles onneted to the first interfae. We then disuss the two systems we onsider for testing the QuanTIS method Replia Exhange Transition Interfae Sampling. For the disussion of the RETIS approah, we onsider a transition between two stable states, labeled A (the reatant) and B (the produt), and we assume that the transition is a rare event. In the RETIS approah we first selet a reation oordinate, λ(r N, p N ), whih in general depends on the positions (R N ) and momenta (p N ) of all the N partiles, and we define a set of interfaes λ 0, λ 1,..., λ n, suh that the first interfae is the boundary of state A (i.e., the system is in state A for λ λ 0 = λ A ) and the last interfae is the boundary of the produt (i.e., the system is in state B for λ λ n = λ B ). The other interfaes are positioned to maximize the effiieny of the method and the rate onstant for the transition from A to B, k AB, an be obtained by kab = f ( λ λ ) A A B A (1) where f A is the flux through the first interfae and A (λ B λ A ) is the probability of rossing λ B before λ A given that λ A was rossed in the past. In pratie, it is more onvenient to estimate the rate onstant by expanding the probability as n 1 k = f ( λ λ ) = f A ( λi+ 1 λi) AB A A B A A i= 0 where A (λ i+1 λ i ) is the probability of a path rossing λ i+1 given that it originated in λ A, ended in λ A or λ B, and had at least one rossing with λ i. In order to alulate these probabilities, we define [i + ] as the olletion of paths that start at λ A and has at least one rossing with λ i before revisiting λ A or ending at λ B. The probability A (λ i+1 λ i ) an then be obtained as the fration of paths in the [i + ] ensemble that ross λ i+1 as well. The approah disussed so far orresponds to the onventional TIS method, 19 and f A is then obtained by performing a MD simulation. For the RETIS approah, replia exhange moves, or swapping moves, are attempted between paths orresponding to different path ensembles. In order to have full flexibility in the swapping moves the MD simulation in the TIS method is (2) 2441

3 replaed by another path ensemble [0 ] whih orresponds to all paths that start at λ A, then go in the negative diretion, and end bak at λ A.Defining t [k] path as the average path length in the [k] path ensemble, the flux through the first interfae is then obtained from ( path path ) [0 = ] [0 f t + t + ] A 1 The algorithm for the RETIS method is as follows: 20 At eah step it is (with equal probability) deided whether to perform shooting or swapping moves. In the first ase, all the path simulations are updated sequentially with a shooting move. In the latter ase, swaps will be attempted between the different path simulations, and it is again deided with equal probability if swaps between the ensembles [0 ] [0 + ], [1 + ] [2 + ], et. or between the ensembles [0 + ] [1 + ], [2 + ] [3 + ], et. are attempted. If the swapped paths are not valid, the move is reeted, and the old paths are reounted in the two ensembles involved in the swap. For the swapping between ensembles [0 ] and [0 + ] new traetories are generated in the following way: The last step of the old path in [0 ] is used as the initial point for generating a new traetory for [0 + ] by integrating forward in time, while the initial point in the old path in [0 + ] is used to generate a new path for [0 ] by integrating bakward in time. The desription so far orresponds to the ase where we treat all path ensembles with the same level of theory. In the new QuanTIS approah, we will use an approximate theory in the [0 ] ensemble and a higher level of theory in [0 + ] and all other ensembles. In order to ensure detailed balane in the MC sampling within the two levels of theories the following aeptane rule must be fulfilled before the swap [0 ] [0 + ]is aepted 1 Pa( rlo, rhi) = MIN 1, exp [ kt V hi ( r lo ) V lo ( r lo ) B + Vlo( rhi) Vhi( rhi)] where V hi and V lo denote the potential energy alulated with the higher level of theory and with the more approximate theory, respetively. We note that a similar expression appears in Li and Yang 33 where it is used to enhane onfigurational sampling in QM spae using a MM fore field. These energies are evaluated at the λ A interfae using r lo, the last point from the path in [0 ], and r hi, the initial point from the path in [0 + ]. Another tehniality is that, in the QuanTIS method, one needs to ensure that the interfae is still rossed in a single time step for the phase points that are being swapped. In pratie, this is done by swapping the very first phase point of the [0 + ] ensemble whih is at the left side of λ A and the one but last phase point of the [0 ] ensemble that is also still at the left side of this interfae. Then, in addition of the aeptane rule 4 we require that the next forward MD step starting from these points based on the new potentials will both reate a point that is at the right side of λ A. Otherwise, the move is reeted. The aeptane rule ensures that the rossing probability remains exat at the high level desription of theory. However, sine the aeptane rule 4 depends exponentially on the energy differenes between potentials V lo and V hi for onfiguration points at the first interfae, the aeptane an be low and the swapping move beomes ineffiient if the two levels of theory are too different. This point is further disussed in setion 6. (3) (4) The flux fator f A, on the other hand, beomes an average of the low and high level fluxes via eq 3. In pratie, the interfaes will be [0 hosen suh that t ] [0 path t +] path implying that the flux fator in the QuanTIS approah will be mostly resembling the flux of the low level system. This is not neessarily affeting the auray of the result in a negative manner. Classial fore fields have been fitted extensively to reprodue solvent dynamis and are, therefore, often more aurate for that aspet than Ab Initio MD Models. In order to exemplify our new method and as a proof of onept, we onsider two model reations: (I) A hypothetial ion transfer between two idential partiles where solvent partiles may redue the barrier to the transfer (i.e., a ooperative effet); (II) the spontaneous dissoiation of hydrogen using fore field from literature. 35 For the first example, the high level of theory orresponds to the full interation potential, while the approximate theory ignores the ooperative effet. For the seond example, a manybody effet is inluded in the high level of theory (via a threebody interation) whih is negleted in the approximate theory. This three-body term effetively forbids the formation of hydrogen lusters, H n with n > 2, and this behavior is preserved by adding a simple repulsive term to the approximate potential. We will in the following desribe these models in more detail Cooperative Ion Potential. The ooperative ion potential models an ion transfer aording to Ax + A A+ Ax, where the presene of a solvent (partiles of type B) may redue the barrier for the transfer. All partiles, A, B, and x, are treated as two-dimensional point partiles with idential mass. In the following, we will assume that we only have one partile of type x and two partiles of type A. The total interation energy, V tot, is given by N 1 V ( r, r,..., r ) = V ( r ) + V ( r, r,..., r ) tot 1 2 n i= 1 N =+ i 1 i i oop 1 2 n where V i desribes the pair interation between atom i and, r i = r i r, and V oop desribes the ooperative effet. The pair interation V i is based on the Lennard-Jones potential with parameters ϵ i, σ i, and utoff r i. Interations between partiles of type A and x are modeled by a shifted Lennard-Jones potential, while all other interations are modeled by the Weeks Chandler Andersen (WCA) potential with ut-offs given by r i =2 1/6 σ i (see eqs in the Appendix). The parameters for the pair interations are given in Table 1 where we also show the potential energies for the different pairs as a funtion of the distane in the inset figure. The ooperative part of the potential is modeled as V = F( n) ( V +ϵ ) (5) oop luster xa (6) where F(n) is a funtion of the oordination number (n,defined below) for partiles of type B around x, and V luster is the total pair interation energy of the x and A partiles (see eq 23 in the Appendix). Here, ϵ xa is the Lennard-Jones parameter for the x A pair. The oordination number for solvent partiles of type B around x is n = Hr ( ; R, N ) {type B} x oop where R oop and N d are parameters, and H is a smooth step funtion defined by d (7) 2442

4 Table 1. Parameters (in Redued Units) for the Pair Interations in the Ion Potential a when r i < r, and it is zero otherwise. The parameters are 35,36 α = kal/mol, β = Å 4, γ 2 = Å, and r = 2.8 Å, and the minimum of the pair interation for this set of parameters is approximately at a distane 0.74 Å, orresponding to the bond distane of H 2. The three-body term is given by V ( r, r, r ) = h( r, r, θ ) + h( r, r, θ ) ik i k i k ik i ki ik + hr (, r, θ ) where the funtion h (r i, r k, θ ik )=h ik is ik k ik (12) pair (i) ϵ i σ i r i potential AB, AA, BB /6 WCA xa /6 Shifted LJ xb /6 WCA a The interations are modeled using the Weeks Chandler Andersen (WCA) potential or a shifted 6-12 Lennard-Jones potential (Shifted LJ) as desribed in the text. In the inset figure, the pair interation potential energies are shown as a funtion of the partile partile distane (r i ) in redued units. The solid line is the x A pair interation, the dashed line is the x B pair interation, and the dashdotted line is the A B, A A, and B B pair interations. The two vertial dotted lines are the ut-offs, /6 and 2 1/6, and as the figure shows, only the x A pair interation has an attrative part. 1 H( r; R, M) = 1 + exp[ Mr ( R)] (8) Finally, the funtion F is the defined as Fn ( ) = f [1 Hn ( ; N, N )] n (9) where f 0, and N and N n are parameters. The funtion H(n; N, N n ) interpolates smoothly between 0 for large oordination numbers and 1 for small oordination numbers orresponding to V oop = f (V luster +ϵ xa ) 0 and V oop = 0, respetively. In the ground state, where x is bound to one A and the other A partile is at a distane where it is only interating with B partiles, V luster ϵ xa, whih also implies that V oop 0; the ooperative effet lowers the potential energy of the barrier without effeting the ground state energy (see Figure 4 in the Appendix), effetively reduing the barrier for ion transport. For the ooperative part of the potential, we set N d = 20, N n = 20, and N = 1.1. The remaining two parameters, R and f, an be used to tune the strength of the ooperative effet, and we will onsider these two parameters in more detail when we desribe our test ases in setion Dissoiation of Hydrogen. The hydrogen interation is modeled using the potential desribed by Kohen et al., 35 whih onsists of two- and three-body interations. For a olletion of hydrogen atoms, the potential energy is given by V( r, r,..., r ) = V ( r ) + V ( r, r, r ) 1 2 n i i i> i> > k ik i k (10) where r i is the position of hydrogen atom i, and V i and V ik denote the interation between pairs (i) and triplets (ik) of hydrogen atoms, respetively. The two-body term is here ompletely determined by the relative distane, r i = r i r, and it is given by 4 Vi( ri) = αβ ( ri 1) exp[ γ /( ri r )] 2 (11) 2 ik ik ik 3 i h = ξ(1 + μos( θ ) + νos ( θ )) exp( γ /( r r) + γ /( rk r)) 3 (13) if both r i < r and r k < r, and zero otherwise. The middle index for eah triplet refers to the enter atoms for the angle (see also Figure 5 in the Appendix) ri θ = rk os ik rr i k (14) and the parameters are 35,36 ξ = kal/mol, μ = , ν = , and γ 3 = 1.5 Å. The three-body term is always nonnegative and prevents the formation of hydrogen lusters H n for n > 2. For the study of the hydrogen dissoiation reation, all hydrogen atoms are initially bound in H 2 moleules, and the final state is reahed when at least one bond is broken. A simple twobody repulsive potential an then be used to mimi the repulsive ontribution from the three-body term, and, in the approximate model, the interation onsists of an intramoleular part (whih is idential to the two-body term given in eq 11) and an intermoleular part, V inter i (r i, r ), whih is evaluated between atoms in different moleules and modeled as a repulsive potential inter Vi ( ri) = ξ2[exp( γ ( r ri)) 1] 4 (15) if r i < r, and zero otherwise. We have obtained parameters (γ 4 = Å and ξ 2 = kal/mol) for this potential by fitting the fores obtained in the approximate desription to the orresponding fores from the more detailed potential as desribed in the Appendix. 3. TEST CASES We have onsidered several test ases for the two models presented in the previous setion. In this setion we briefly desribe these ases, and we present results from RETIS simulations we arried out in order to selet them. We postpone the desription of the omputational details to the next setion where we present the results from using the new method on the test ases presented here. As previously desribed, the strength of the solvent effet in the ion transfer potential an be tuned by hanging f and R oop.a larger f gives a stronger effet by inreasing the ontribution from the ooperative potential (see eqs 6 and 9), and a larger R oop inreases the oordination number whih leads to a redution in H(n; N, N n ) and gives a stronger effet as well. We arried out RETIS simulations (without gluing) for a ombination of parameters (R oop and f ), and the obtained rate onstants are given (together with the parameters) in Table 2. These results show that the rate onstant obtained by not inluding the ooperative effet an be several orders of magnitude lower than the rate onstants obtained when the 2443

5 Table 2. Rate Constant for the Ion Transfer Reation a ase R oop f k AB 0 (without ooperativity) (1.00 ± 0.06) /3 (1.92 ± 0.07) /3 (4.89 ± 0.16) /3 (9.47 ± 0.16) /20 (3.40 ± 0.04) 10 9 a All the rate onstants are alulated relative to the rate onstant for the ase where the ooperative effet is ignored, k AB = (2.6 ± 0.16) ooperative effet is inluded. Based on these RETIS simulations, ases 1 3 were seleted for further investigation. The last ase, 4, was not inluded as the rate onstant here is relatively high. For eah of these ases we onsider the following five variations whih we label a) e): a) No gluing, orresponding to the usual RETIS approah; b) Gluing, orresponding to the new QuanTIS method; ) RETIS with a swap frequeny set to zero (whih approximately orresponds to the TIS method); d) Fewer interfaes whih orresponds to method b) with a redued number of interfaes; and finally e) Aept all, orresponding to artifiially ignoring the detailed balane ondition in eq 4 and aepting all swapping moves [0 ] [0 + ]. In ase d), the first interfae was removed whih implies that the new first interfae λ 0 is shifted more into the barrier region so that the average path length of the low-level [0 ] system beomes even longer with respet to the high-level desription [0 + ] traetories. These variations will allow us to disuss the new method and ompare it with the RETIS and TIS approahes. For the hydrogen potential, we arried out RETIS simulations for the dissoiation at temperatures 1500, 2000, and 2500 K and a density of 0.07 g/m 3. For these onditions, the stable phase of hydrogen is a moleular liquid and the dissoiation is low. 37 At 1500 K, we estimate the thermal de Broglie wavelength to be approximately 0.45 Å whih is omparable to the hydrogen bond length (0.74 Å) used in the model. To assess the impat of possible quantum mehanial effets, we have ompared the radial distribution funtion for hydrogen at 600, 1000, and 1500 K using both the lassial model and path integral simulations (see Figure 7 and the desription in the Appendix), and we find that we an neglet suh effet for the temperatures we onsider. The rate onstants we obtain by RETIS simulations are given in Table 3. For an effiieny analysis it is required to have well Table 3. Rate Constant for the Hydrogen Dissoiation Reation a ase temp/k k AB (2.91 ± 0.15) (1.00 ± 0.03) (1.84 ± 0.04) 10 2 a All rate onstants are relative to the rate onstant at 2000 K, k AB = (2.96 ± 0.09) 10 6 ns 1. Simulation details are given in the next setion. onverged results for all methods, inluding the less effiient ones. We therefore fous on the 2000 and 2500 K ases whih are more aessible than the 1500 K ase. For this potential, we also onsider the same variations a) e) as for the ion exhange potential, and, in addition, we onsider a sixth ase: f) No gluing and using the approximate potential everywhere COMPUTATIONAL DETAILS The equations of motion were propagated in time under NVE dynamis using the Veloity Verlet integrator 38 with periodi boundaries (in two dimensions for the ion potential and three dimensions for the hydrogen potential). In the RETIS simulations, yles were performed and the swapping probability was set to 50%, while the probabilities of performing time reversal moves or shooting moves were both set to 25%. We applied the simplified version 20 of aimless shooting 39 whih implies that the veloities at the shooting point were generated from a Maxwellian distribution at the given temperature (0.1 in redued units for the ion transfer; 2000 or 2500 K for the hydrogen dissoiation), and the shooting points were piked with an equal probability along the path. The positions of the interfaes used in the RETIS alulation were plaed to inrease the effiieny of the method. 20,21 For the ion potential, the time step was δt = in redued units, and the partiles were assumed to have idential masses (of 1). Initially the ion, x, is bound to one of the A partiles (labeled A 1 ), and it is transferred to the other A partile (labeled A 2 ) over the ourse of the reation. The order parameter, λ, isdefined using the distane between x and A 2 (r xa2 ), as λ = r xa 2 (16) where the minus sign is inluded ust for onveniene: The order parameters hanges from a low value to a high value over the ourse of the ion transfer. The two stable states were defined as orresponding to λ < 0.7 and λ > 0.4, respetively, and interfaes were plaed at positions 0.7, 0.64, 0.62, 0.60, 0.59, 0.58, 0.57, 0.56, 0.55, 0.54, 0.53, 0.52, 0.51, 0.50, and Initially, the A and B partiles were positioned on a regular 2D grid suh that eah partile was surrounded by 4 neighbors, all at a distane of 1.2 from eah other. The unit ell was quadrati with a side length of redued units. For the hydrogen potential, we define redued units by the mass of the hydrogen atom, a length equal to 1 Å, and the binding energy of hydrogen ( kal/mol). The time unit is then τ = 4.81 fs, and the NVE simulations were performed with a time step of 0.1 τ. We onsidered 4 H 2 moleules in a ubi box with an edge length equal to Å (orresponding to a density of 0.07 g/m 3 ). Initially, we plaed the moleules (i.e., their enter of mass) on sites orresponding to a fae-entered ubi lattie and oriented the moleules randomly. The initial veloities of the moleules were drawn from a Maxwellian distribution orresponding to the target temperature for the simulation (2000 or 2500 K). The order parameter for the hydrogen reation was obtained in the following way: For eah pair of hydrogen atoms that were initially bound together in a H 2 moleule, we alulated the distane between the atoms. We then defined the order parameter as the maximum of these bond lengths. This unonventional reation oordinate is one of the advantages of the TIS-based algorithms: It is muh more flexible than for instane thermodynami integration with respet to the range of reation oordinates that an be hosen. Taking the maximum value of all intramoleular distanes allows for dissoiation of any H 2 moleule in the system whih ensures a faster deorrelation than for the ase that a single target moleule has to be seleted beforehand. Sine the method now alulates the rate onstant that any H 2 moleules dissoiates, the alulated rate onstants for the hydrogen potential have been normalized with the number of hydrogen moleules to obtain the dissoiation rate

6 [0 Table 4. Initial Flux (f A ), Crossing Probability ( A (λ B λ A )), Rate Constant (k AB ), and Ratio of Path Lengths ( t ] [0 path / t +] path ) for the Three Cases of the Ion Transfer Reation: Case 1 with R oop = 1.1 and f = 1/3; Case 2 with R oop = 1.3 and f = 1/3; and Case 3 with R oop = 1.3 and f = 2/3 a ase 1 f A A (λ B λ A ) 10 9 k AB [0 t [0 path / t path aeptane (%) a) ± ± ± ± 0.05 b) ± ± ± ± ) ± ± ± ± 0.07 d) ± ± ± ± ase 2 f A A (λ B λ A ) 10 8 k AB 10 8 [0 t [0 path / t path aeptane (%) a) ± ± ± ± 0.06 b) ± ± ± ± ) ± ± ± ± 0.16 d) ± ± ± ± e) ± ± ± ± ase 3 f A A (λ B λ A ) 10 4 k AB 10 5 [0 t [0 path / t path aeptane (%) a) ± ± ± ± 0.06 b) ± ± ± ± ) ± ± ± ± 0.06 d) ± ± ± ± e) ± ± ± ± a The different variations of the simulation methods are a) the usual RETIS approah; b) the new QuanTIS method; ) RETIS with a swap frequeny set to zero; d) QuanTIS with the first interfae removed; e) RETIS where all swapping moves are artifiially aepted (the detailed balane ondition in eq 4 is ignored). Sine the aeptane is relatively high in ase 1b), we did not perform the aept all simulations ase 1e) here. In eah olumn, we have italiized the different quantities that should be idential within statistial unertainties. All quantities are given in redued units based on the ion transfer potential. [0 Table 5. Initial Flux f A, Crossing Probability A (λ B λ A ), Rate Constant k AB, and Ratio of Path Lengths t ] [0 path / t +] path for the Hydrogen Dissoiation at 2000 and 2500 K a 2000 K f A 10 4 /ns 1 A (λ B λ A ) k AB 10 6 /ns 1 [0 t [0 path / t path aeptane (%) a) ± ± ± ± 0.07 b) ± ± ± ± ) ± ± ± ± 0.09 d) ± ± ± ± e) ± ± ± ± f) ± ± ± ± K f A 10 4 /ns 1 A (λ B λ A ) 10 8 k AB 10 4 /ns 1 [0 t [0 path / t path aeptane (%) a) ± ± ± ± 0.03 b) ± ± ± ± ) ± ± ± ± 0.04 d) ± ± ± ± e) ± ± ± ± f) ± ± ± ± 0.04 a The different variations of the simulation methods a) e) are the same as defined in Table 4, while the additional ase f) orresponds to RETIS where the approximate potential is used everywhere. In eah olumn, we have italiized the different quantities that should be idential within statistial unertainties. onstant for a single moleule. The initial and final states for the hydrogen reation orresponds to λ < 0.9 Å and λ > 2.4 Å, respetively. We onsidered interfaes plaed at positions (in units of Å): 0.95, 1.05, 1.15, 1.25, 1.30, 1.35, 1.40, 1.50, 1.60, 1.75, 1.90, 2.10, 2.20, 2.30, and RESULTS AND DISCUSSION We have alulated the initial flux (f A ), the overall rossing probability ( A (λ B λ A )), the rate onstant (k AB ), and the ratio of the average path lengths in the [0 ] and [0 + [0 ] ensembles ( t ] path / [ t path 0 + ] ) for the different ases we onsidered for the two potentials. The results for the ion transfer are presented in Table 4 and for the hydrogen dissoiation in Table 5. The individual rossing probabilities are shown for ase 3 of the ion potential in Figure 1 and for the hydrogen dissoiation at 2500 K in Figure 2. Overall we find good agreement for the obtained rate onstants when we ompare the results for the RETIS method with and without gluing. The results from the RETIS simulations where the swapping frequeny is set to zero are in lose agreement with the simulations where the swapping frequeny is 50%, as expeted. We do find variations up to about 50% in the rate onstant for the ion potential when we ompare RETIS with QuanTIS, see for instane ase 2a) and b) and ase 3a) and b) in Table 4. This is aused by an inreased initial flux when we apply the approximate potential in the [0 ] ensemble. The onditional rossing probabilities (see Figure 1) remain unhanged within the unertainty, as long as the positions of the interfaes are unhanged, and as long as the detailed balane ondition is respeted. Ignoring the detailed balane ondition, eq 4, and 2445

7 Figure 1. Crossing probabilities as a funtion of the order parameter for ase 3 of the ion potential (R oop = 1.3 and f = 2/3). The positions of the interfaes are indiated with vertial dotted lines. The different ases inluded here are a) the RETIS approah without gluing (irles, olor red); b) the QuanTIS method (squares, olor blue); ) RETIS with a swap frequeny set to zero (triangles, olor green); d) QuanTIS with the first interfae removed (diamonds, olor purple); e) RETIS where all swapping moves are artifiially aepted (down-pointing triangle, olor orange). Figure 2. Crossing probabilities as a funtion of the order parameter for the hydrogen dissoiation at 2500 K. The positions of the interfaes are indiated with vertial dotted lines. The different ases inluded here are a) the RETIS approah without gluing (irles, olor red); b) the new QuanTIS method (squares, olor blue); ) RETIS with a swap frequeny set to zero (triangles, olor green); d) QuanTIS with the first interfae removed (diamonds, olor purple); e) RETIS where all swapping moves are artifiially aepted (down-pointing triangle, olor orange); f) RETIS where the approximate potential is used everywhere (rosses, olor yan). aepting all swapping moves [0 ] [0 + ] leads to an overestimation of the rossing probability (and rate onstant): see ases 2e) and 3e) in Table 4. Inspetion of Figure 1 shows, however, that for i 3 all onditional rossing probabilities A (λ i+1 λ i ) of the aept-all approah give the orret values as follows from the total rossing probability whih is essentially running parallel to the other results in the logarithmi plot after λ 3 = 0.6. We note that although the initial flux and the overall rossing probability seems to differ signifiantly when we hange the positions of the interfaes, the rate onstants should still be idential. This is due to the fat that a rate onstant is obtained as the produt of the initial flux and the overall rossing probability. This an for instane be seen in Table 4 for ase 3b) and 3d) where the smaller initial flux in ase 3d) gives a proportionally larger rossing probability and the rate onstant remains approximately onstant. Case 3d) also show that we an modify the ratio of the path lengths by plaing the interfaes differently. Here, it is simply done by removing the first interfae. This is benefiial for two reasons: (1) the number of interfaes is dereased, and (2) the path lengths in the [0 ] ensemble are inreased, whih in turn will inrease the gain of using the QuanTIS method. For the hydrogen dissoiation, we do not see a large effet on the initial flux when using the approximate potential, ompare ases a) and b) in Table 5. All the path lengths in the first two ensembles are relatively short in ases a) and b) (on average approximately 50 steps) for the onditions we have onsidered, and there is not muh room for variation. By removing the first interfae, we obtain longer path lengths, and we see a derease in the initial flux and a orresponding inrease in the rossing probability. However, in this ase we see a larger disrepany when omparing the d) ases with ases b) and ): The rate onstant for the d) ases appear to be overestimated. In the urrent implementation of the method, we store all timeslies of the paths, and we restrit the length of these paths to a fixed value (20000 timeslies). When we remove the first interfae, the path lengths beome larger on average and may more often exeed the preset maximum length. This will effetively lead to an overestimation of the initial flux f A and subsequently of k AB, sine very long paths in the [0 ] ensemble may artifiially be reeted. For ase d) at 2000 K, the maximum allowed path lengths were inreased to timeslies in order to inrease the auray. We are urrently developing a method to irumvent this potential problem. To further investigate the effiieny of the new method, we [i have also alulated the effiieny times, τ +] eff, given by 21 1 = p τ p [ i + ] i eff i i i ξln i (17) [i where p i = A (λ i+1 λ i ), L i = t +] path /δt, N i is the effetive orrelation between traetories, and ξ i is the ratio between the average ost of the simulation yle and L i. The effiieny times an be interpreted as the minimum omputational ost (e.g., fore alulations needed) to obtain an overall relative error equal to For the ion potential, setting the swap frequeny to zero has a dramati effet on the effiieny times, see Figure 3. The overall effiieny time for ase 3) is approximately / times the overall effiieny time for ase 3b). This is aused by a muh faster deorrelation (as measured by N i ) for the ase when the swapping moves are used. It is also refleted in ξ i whih is onsistently larger for ase 3). The improved effiieny of QuanTIS ompared to RETIS is not refleted in these data sine effiieny times are expressed in the number of fore evaluations. Hene, this analysis does not take into aount that the fore evaluations in the lower level desription an in pratie be a few thousand times faster than in the higher level desription, as in the ombination of lassial MD with DFTbased MD. For the hydrogen potential, the effet of setting the swapping frequeny to zero is not so pronouned. In this ase ξ i is larger for the ase with a zero swapping frequeny; however, the effetive orrelation between the traetories is relatively low, and the all 2446

8 Figure 3. Calulated effiienies for the RETIS/QuanTIS simulations for ase 3 of the ion transfer potential (R oop = 1.3 and f = 2/3). From top to bottom: The onditional rossing probability p i = A (λ i+1 λ i ), with the average path length being L i, the ratio between the average ost of the simulation yle and the average path length being ξ i, the effetive orrelation between traetories being N i, and the effiieny times being [i τ +] eff. The effiieny times were alulated as desribed in the text. The different ases inluded here are a) the RETIS approah without gluing (olor red); b) the new QuanTIS method (olor blue); ) RETIS with a swap frequeny set to zero (olor green); d) QuanTIS with the first interfae removed (olor purple); e) RETIS where all swapping moves are artifiially aepted (olor orange). path lengths are short, giving a smaller effet. In fat, the ratio of the effiieny times is approximately 6780/ when omparing the ase with swap frequeny zero to the ase with a swap frequeny equal to 50%. This differene is mainly aused by the larger ξ i in the former ase. 6. PROSPECTS Sine the energy is an extensive quantity we an expet that the aeptane in eq 4 will drop for large systems. There are several possibilities to irumvent this problem. One possibility is to mix the two potentials using e.g. an alternative potential V lo at the lower level Vlo ( λ) = H( λ, Λ, Λ n) Vlo + ((1 H( λ, Λ, Λn)) Vhi (18) where H is the smooth step funtion that goes from 1 to 0 as previously defined in eq 8, and Λ and Λ n are parameters determining the position and the steepness of this infletion, respetively. Λ should be plaed somewhere around the first interfae. For λ values far lower than Λ the potential is simply V lo and expensive quantum alulations are not needed, but eah time that λ approahes the first interfae some fore field evaluations at the V hi will be required. However, one the exhange move is attempted, the aeptane probability should be muh higher. The gradual inrease of the high level desription ould in priniple also be extended to the other path ensembles, or the QuanTIS method ould be ombined with standard QM-MM whih redues the region in whih there is a mismath. An exiting idea that we plan to investigate is to adapt parameters of a lassial potential during the simulation in a similar fashion as fore-mathing 41 or learning-on-the-fly. 42 Sine the aeptane probability of eq 4 is maximized when V lo and V hi are similar at the first interfae, the average aeptane probability provides a natural ost funtion that an be optimized. This approah has two benefits: first of all it will enhane the effiieny of the QuanTIS method, and, seond, it will provide an aurate fore field whih ould be used in a multisale sheme. The algorithm that we plan to test is to update a database of onfigurations {r lo, r hi } eah time that the replia exhange move is attempted. Then, after a ertain number of replia exhange moves, the fore field parameters are adusted in order to maximize following ost funtion k Q wp ( r, r ) k, a lo hi N w swaps (19) where N swaps are the number of replia exhange moves performed so far, and, k are indexes of the different onfiguration points generated via V lo and V hi, respetively. w are weights orresponding to the {r lo } points whih we will disuss later on. As P a depends on V lo, Q depends on fore field parameters whih determine this potential, and optimization routines in parameter spae an be applied to maximize Q. The optimization of this ost funtion has speifi advantages ompared to standard parameter fitting shemes. One advantage already mentioned is that it is diretly linked to the effiieny of the QuanTIS method. Another advantage is that it is welldefined; there is no need to tune predefined required auraies of dissimilar properties like bond distanes, energies, angles, et. Moreover, the onfigurations that are sampled are relevant for the reation under study. Manually reated benhmark data using quantum alulations often fous too strongly on minimum energy onfigurations and transition states whih might not be relevant for the temperature that is onsidered in the atual simulation. Finally, the Q funtion tests the auray of the V lo fore field from both perspetives; not only should the fore field mimi the energies of the geometries from the quantum data, the points generated by V lo are also analyzed. Reative fore fields, fitted on high level quantum mehanial data, an simulate spurious reations whih naturally were never onsidered during the design of the Ab Initio database. In this sheme suh unphysial behavior would diretly be penalized. However, suppose the lassial fore field generates a point r lo with a orresponding unrealistially high energy V hi (r lo ). In that ase, it would be suffiient to hange the parameters of V lo suh that this point will never be generated again. Requiring that V lo (r lo ) V hi (r lo ) would put a too strong restrition on the parameter set sine this point r lo has beome irrelevant in the next update of the V lo potential. It is therefore that we put a weight w in the expression whih should be proportional to the relevane of the r lo points. There are several pratial solutions how to define appropriate weights, but it is yet far from trivial to predit what is the most effetive way to do so. One natural requirement is that the weights should not hange, at least not relatively, when adding a trivial onstant to the potential V lo. We shall report on these tehnialities in forth-oming publiations. 7. CONCLUSION We have introdued the new method QuanTIS that glues different potential energy surfaes in order to model rare transitions suh as hemial reations. In this way, QuanTIS allows using different levels of theory to desribe distint subproesses of this transition. One interesting appliation is to use QuanTIS to study hemial reations in a solvent in whih the breaking and making of hemial bonds is done by aurate Ab Initio MD, while the diffusion of reatants is treated by 2447

9 lassial fore fields. The new method is based on the previously developed RETIS 20 method whih already gives a dramati inrease in effiieny ompared to brute-fore MD. By adusting the aeptane rule of the replia exhange move between the different path simulations, we an maintain detailed balane and inrease the effiieny even further in the QuanTIS method. Both in the RETIS and in the QuanTIS sheme the rate onstant of the transition is alulated from a flux fator f A times the overall rossing probability A (λ B λ A ). In the QuanTIS sheme the flux fator will be mainly determined by the potential energy surfae V lo that is based on the lower level theory. The rossing probability will be purely determined on the higher level potential V hi. Hene, A (λ B λ A ) is not affeted by V lo, but the effiieny of its omputation ertainly will. The replia exhange moves between the two systems redue signifiantly the orrelation between traetories in all path ensembles. By positioning the first interfae far into the barrier region, the path on the heap V lo potential an be made very long, and an exhange move to the V hi potential, if aepted, is likely to generate a path that is ompletely unorrelated to the previous path in that ensemble. In this artile we give the first proof-of-onept of this new approah by applying it on two model systems. The first system desribes a hypothetial ion transport model in whih we exlude or inlude ooperative effets in the V lo and V hi potentials, respetively. In a seond stage, we also applied it to the dissoiation of hydrogen based on a reative fore field from the literature. 35 In this model we ignored the three-body term of the hydrogen potential in the lower desription but added a simple repulsive steep potential to obstrut trimer formations. Our results show that the rossing probability A (λ B λ A ) is indeed onverging to the same results as in the RETIS simulations using the high-level desription on all path ensembles. We note that the aeptane of the swapping between the two levels of theory an drop for large systems, and we have disussed possible solutions to this problem (see e.g. eq 18). For the systems that we studied this was not the ase. The realisti hydrogen model even showed aeptane lose to 100% after a simple optimizing proedure for positioning the repulsive potential. The results for the detailed-balane violating aeptall approah shows that only the very first rossing probabilities are affeted. This suggests that the aept-all approah might be a pratial method if an approximate value for the reation rate (within 1 order of magnitude) is suffiient. In onlusion, we believe that the QuanTIS method is a promising method to onnet different levels of theory for studying rare transitions. In addition, it also provides interesting prospetives regarding fore field optimization whih are presently under investigation. A. APPENDIX A.1. Detailed Desription of the Interation Potentials. In the ion potential, the pair interation is based on the Lennardi Jones potential, V LJ 12 6 σ σ i = ϵ i i V LJ( ri) 4 i ri ri (20) where ϵ i and σ i are the parameters for the potential. The form of the pair interation between partiles of type A and x is given by a shifted Lennard-Jones potential xa xa VLJ ( rxa) VLJ ( rxa), if rxa < rxa VAx( rax) = VxA( rxa) = 0, if rxa rxa (21) and the other interations are modeled by the Weeks Chandler Andersen (WCA) potential i V ( r ) +ϵ, if r < r WCA LJ i i i i Vi ( ri) = 0, if ri ri (22) with ut-offs given by r i =2 1/6 σ i. The luster energy is given by the interations between the two A partiles labeled A 1 and A 2 and the x partile WCA luster xa1 xa1 xa 2 xa 2 A1A 2 AA 1 2 (23) V = V ( r ) + V ( r ) + V ( r ) Here, V xa1 and V xa2 are modeled with the shifted Lennard-Jones potential, while V A1 A 2 is given by the WCA potential. In Figure 4 we illustrate the effet of ooperation on the interation. Figure 4. Illustration of the ooperative effet on the x A interation. The figure shows the potential energy as a funtion of the distane (both in redued units) between the x partile and one A partile (labeled A 1 ) in the presene of B partiles. The seond A partile is assumed to be at a distane where the interation with x an be negleted. The solid line orresponds to the ase where the oordination number is zero, the dashed line to a ase where the oordination number is intermediate (here: equal to N ), and the dotted line orresponds to the ase where the oordination number is high. The inset shows the variation of F(n) with respet to the oordination number for N = 1.1, N n = 20, and f = 1/3. In Figure 5 we illustrate the hydrogen potential and elaborate the triplet notation we have used in the main text. A.2. Parameters for the Approximate Hydrogen Potential. The two parameters for the approximate hydrogen potential were obtained by fitting the simplified potential to the more detailed potential in the following way: First, the more detailed potential was used to run a short simulation (10 6 steps with a time step of 0.1 in redued units) of 32 partiles in a ubi box (box length Å) with NVT dynamis and a temperature orresponding to 1500, 2000, and 2500 K. The positions (r i ) and fores (f i ) on all the atoms in the system were stored every 10th step. We then re-evaluated the fores using the approximate potential and used the Levenberg Marquardt algorithm 43,44 to optimize the parameters γ 4 and ξ 2. This was done by minimizing the funtion g defined as 2448