Sampling ensembles of deterministic transition pathways

Faraday Discuss., 1998, 110, 421È436

Sampling ensembles of deterministic transition pathways Peter G. Bolhuis, Christoph Dellago and David Chandler Department of Chemistry, University of California, Berkeley, California 94720, USA

We extend the method of transition-path sampling to the case of deterministic dynamics. This method is a Monte Carlo procedure for sampling the ensemble of trajectories that carry a many-particle system from one set of stable or metastable states to another. It requires no preconceived notions of transition mechanisms or transition states. Rather, it is from the resulting set of suitably weighted dynamical transition paths that one identiÐes transition mechanisms, determines relevant transition states and calculates transition rate constants. In earlier work, transition-path sampling was considered in the context of stochastic dynamics. Here, the necessary modiÐcations that make it applicable to deterministic dynamics are discussed and the modiÐcations illustrated with microcanonical simulations of isomerization events in two-dimensional seven-atom Lennard-Jones clusters.

1 Introduction Molecular dynamics simulation is one of the most useful tools for the study of dynamical processes in condensed matter systems. But there is a general problem with the method when the dynamics of interest is dominated by rare events. For these cases, relevant time scales exceed those accessible to conventional molecular dynamics simulations. One might face this situation in the calculation of rate constants for transitions between stable states separated by a high energy barrier. A standard way to overcome this problem is to initiate many short trajectories from the relevant dynamical bottleneck or transition state.1h4 This approach is based upon transition-state theory. Rate constants are obtained as a product of the probability for observing the system at the transition states, and a transmission coefficient obtained from dynamical trajectories initiated from those states. In most high-dimensional systems, however, the transition state is not known a priori.5 Moreover, it might not even be possible to specify the transition state in terms of a small number of variables. In the present work, we consider a method to study rare events without any prior knowledge of transition states or underlying transition mechanisms. The basis of the method is the deÐnition of a joint probability distribution for initial and Ðnal conditions for molecular dynamics trajectories. Each member of the ensemble exhibits a transition from a stable (or metastable) reactant state to a stable (or metastable) product state. We sample this transition path ensemble using a Monte Carlo procedure. The method is suitable for sampling deterministic transition paths as obtained from a solution of NewtonÏs equation of motion. The algorithm is very similar to the “ shooting Ï algorithm we recently developed to sample an ensemble of stochastic transition paths.6 We demonstrate how the method is used to calculate rate constants, and to determine other quantities of interest, such as rates of energy dissipation. We also demonstrate how the method can be used to identify an ensemble of transition states. The approach is not limited to Newtonian dynamics but, rather, is capable of treating any dynamical evolution described by deterministic equations of motion. 421

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Sampling ensembles of deterministic transition pathways

The paper is organized as follows. In Section 2, we deÐne the ensemble of transition paths and develop an appropriate sampling algorithm. The calculation of rate constants is described in Section 3. Then, in Section 4, we apply the method to an isomerization process in a two-dimensional cluster of Lennard-Jones atoms. For this system, we calculate microcanonical rate constants and determine an ensemble of transition states. We compare our exact results for rate constants with those predicted from the classical limit of the statistical theory RRKM.7 We also calculate quantities that are beyond the scope of such statistical theories. Conclusions are drawn in Section 6.

2 Simulation methods The ensemble of deterministic transition paths Consider a classical many-body system speciÐed by the phase-space vector x \ Mr, pN and evolving in time according to HamiltonÏs equations of motion : r5 \

dH , dp

p5 \ [

dH dr

(1)

where H(r, p) \ K ] V is the Hamiltonian of the system and the dot denotes the time derivative. Integration of these equations of motion yields the trajectory x \ x (x ) (2) t t 0 where x speciÐes the state of the system at time t \ 0. To simplify notation, we omit the 0 dependence of x on the initial state x in what follows. However, it is understood that t 0 the whole trajectory x is completely determined by the corresponding initial condition t x . 0 We shall assume that the statistics of our system can be described by an ensemble, with the probability to Ðnd the system in a certain state x given by the distribution o(x). For example, the canonical distribution function o (x) \ Mexp[[bH(x)]N/Q (3) c c is the appropriate distribution function for a system in contact with a heat bath at temperature T . In the microcanonical ensemble, o (x) \ d[H(x) [ E]/Q (4) mc mc is the appropriate distribution of a mechanically and thermally isolated system with energy E. Here, Q and Q are the partition functions that normalize their respective c mc distributions. Di†erent ensembles and equations of motion can easily be incorporated into our scheme. Here, however, we limit the discussion to systems distributed according to either the canonical or microcanonical ensemble, and evolving according to HamiltonÏs equations of motion. Each point x belonging to the ensemble can be regarded as an initial point x for a trajectory x . We are interested in paths connecting the phase-space regions A 0and B described byt the characteristic functions h and h : A B 1, if x ½ A, B (5) h (x) \ A, B 0, if x ¾ A, B

G

Accordingly, we consider a restricted ensemble that includes only those paths that originate in region A and reach region B within a time T. The distribution function for this restricted ensemble can be thought of as a joint distribution for initial and Ðnal phasespace points. For deterministic dynamics, however, the distribution collapses to one for

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the initial phase-space points. This distribution is where Q AB

f (x ) 4 o(x )h (x )H (x )/Q AB 0 0 A 0 B 0 AB is the normalizing factor Q \ AB

P

(6)

dx o(x )h (x )H (x ) 0 0 A 0 B 0

(7)

The function H (x ) is unity if the system reaches B within time T and vanishes other1 0 wise :

G

1, H (x ) \ B 0 0,

if there is a t ½ [0, T] such that h (x ) \ 1 B t otherwise.

(8)

Each phase point for which f (x ) D 0 initiates a transition path connecting region A AB 0 with region B in a time T. We can use the restricted ensemble to study transition paths by analyzing the probability distribution (6). For consistency, the time evolution of the system should conserve the canonical or microcanonical distribution, as is the case for NewtonÏs equations of motion. Sampling the path ensemble by shooting An ensemble of correctly weighted transition paths can be obtained by sampling the distribution (6) with a Monte Carlo procedure. In the Monte Carlo procedure, a new path represented by the initial condition xn is created from an old path with the starting point xo . Then the new path is accepted 0or rejected according to the detailed balance 0 condition f (xo )Po?n Po?n \ f (xn)Pn?o Pn?o (9) AB 0 gen acc AB 0 gen acc where Po?n and Po?n are the probabilities to generate and accept a path, respectively. gen The superscripts o acc and n refer to the old and to the new path, respectively. This acceptance probability ensures that the paths are sampled with the correct weight given by f (x ). AB In0 the present work, we create new transition paths by what we call the “ shooting method.Ï A very similar method has been successfully applied to the sampling of stochastic transition paths.6 In the shooting algorithm, a new transition path is created by slightly changing an existing one. First, we randomly choose a time t on an existing path, as depicted in Fig. 1. Next, we change the momenta po by a small amount dp. With the new momenta pn , we then integrate NewtonÏs equationst of motion backward in time t in time to t \ T. The new path is then accepted or rejected to t \ 0 and forward according to the Metropolis criterion8

C

D

f (xn )Pn?o (10) Po?n \ min 1, AB 0 gen , acc f (xo )Po?n AB 0 gen where the min function returns the smaller of its arguments. Assuming a symmetric generation probability, i.e., Po?n \ Pn?o, we obtain gen gen o(xn) 0 h (xn)H (xn) (11) Po?n \ min 1, acc o(xo ) A 0 B 0 0 For a canonical equilibrium distribution, the momentum change dp can be drawn from a Gaussian distribution. The acceptance probability then becomes

C

D

Po?n \ min(1, expM[b[H(xn ) [ H(xo )]Nh (xn)H (xn)) acc 0 0 A 0 B 0

(12)

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Sampling ensembles of deterministic transition pathways

Fig. 1 Scheme of the shooting algorithm. A point ro is selected at random along an existing path t (ÈÈ) connecting A with B. The momenta po at time t are then changed by a small amount dp t creating the new momenta pn . Starting from Mqo , pnN the new path (----) is calculated by forward t t and backward integration oft the equations of motion. The new path is accepted according to a Metropolis criterion.

For the microcanonical ensemble, the total energy E of the system is Ðxed. This constraint must be taken into account in generating a new path from an old one. Because a new path is generated by changing only the momenta at a certain time slice, without changing the positions of the particles, the momentum displacement dp must be chosen such that the total kinetic energy of the system is not changed. This criterion may be met by drawing new momenta from a Gaussian distribution and then rescaling them. In this case the acceptance probability becomes Po?n \ h (xn)H (xn) (13) acc A 0 B 0 We have used the fact that HamiltonÏs equations of motion conserve the total energy of the system. Thus, any new path which connects region A with region B is accepted. For some systems, additional considerations are necessary. For example, one might be interested in systems with Ðxed total linear and angular momentum. Since the Newtonian equations of motion conserve the total momentum P \ ;p and, in the absence i of periodic boundary conditions, also the total angular momentum L \ ;r p , it is i natural to impose the constraints P \ P and L \ L . In the present work, we iillustrate 0 0 the path sampling method by studying a seven-particle Lennard-Jones cluster that neither moves nor rotates as a whole. Consequently, we require P \ 0 and L \ 0 and 0 0 the path distribution becomes f (x ) 4 o(x )h (x )H (x )d(L )d(P)/Q (14) AB 0 0 A 0 B 0 AB Clearly, these constraints must be taken into account in generating a new path from an old one. Owing to the chaotic nature of NewtonÏs equations, one might think that it is highly unlikely that a new path will satisfy the endpoint constraints. However, the change in dp can be made arbitrarily small, ensuring a reasonable acceptance ratio. Furthermore, because the stable states have, in general, a lower potential energy than the transition state, they act as an attractor for the trajectories. Summarizing : In the shooting algorithm, we choose a random time t, make a small change in the momenta, and then shoot o† a trajectory forward and backward in time. The new path is accepted with a certain probability if its endpoints are still in the initial and Ðnal stable-state regions. The average acceptance probability can be controlled by tuning the magnitude of the momentum displacement dp, allowing optimization of the accessible phase-space sampling rate. The computational cost of the method clearly scales linearly with the length of the path. Therefore, the cost to harvest N trajectories,

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each of length T, speciÐcally directed between A and B, is of the same order as that for a straightforward trajectory of length NT. Enhancement by shifting To enhance the efficiency of path sampling one can shift the paths in time. In a path shifting move a new starting point xn for a path is obtained by evolving the old initial 0 conditions xo in time by an interval t 0 xn \ x (xo ) (15) 0 t 0 Here, t may be either positive or negative. In other words, one chooses a point along an existing path, that already satisÐes the endpoint constraints, as the starting point of a new path. For t \ 0, the equations of motion must be integrated backwards to Ðnd xn \ x (xo ). For 0 \ t O T, the new starting point, x (xo ), is already known and one has 0 t 0 t 0 to extend the old path from T to T ] t. If we select the shifting interval t at random from a symmetrical distribution w(t), i.e. w(t) \ w([t), the acceptance probability of the new path is Po?n \ h (xn)H (xn) (16) acc A 0 B 0 Here, we have used the fact that xn and xo have the same energy, since the equations of 0 motion conserve the total energy.0 Eqn. (16) is true for both a canonical and microcanonical ensemble of transition paths. Thus, any new path obtained from translation of an old one by an interval t is accepted, provided the new path connects region A with region B. The average acceptance probability of a shifting move can be easily controlled by tuning the magnitude of the interval t. Because shifting only moves the path forward and backward in time, it cannot be used to sample the transition path ensemble ergodically. Used in combination with the shooting algorithm described above, however, shifting moves can increase the sampling efficiency considerably. Characterizing the stable regions The initial and Ðnal stable regions A and B are characterized by the functions h (x) and A h (x). In a practical application to a high-dimensional system, one desires to parametrize B these functions with a single order parameter (or at most a few). As noted in ref. 6, this order parameter must discriminate between the two stable states. If the order parameter fails to distinguish unambiguously between the initial and Ðnal region, the path simulation will tend to produce paths that remain in only one of the stable regions, thus failing to sample transition pathways. Careful consideration is required in selecting order parameters that characterize the high-dimensional coordinate space.

3 Rate constants Phenomenological reaction rate constants are related to microscopic time correlation functions.4, 9 In particular, for a time t past a molecular transient time t , mol Sh (x )h5 (x )T k(t) 4 A 0 B t B k exp([t/t ) (17) A?B rxn Sh (x )T A 0 where t \ (k ]k )~1 is the reaction time and k and k are the rate conrxn the reactions A?B B?A A?Bbrackets B?ASÉ É ÉT denote equistants for A ] B and B ] A, respectively. The librium averages over initial conditions. For a transition between stable states separated by a high free energy barrier, the transient time t required for the system actually to cross the barrier is much shorter than the reactionmol time t . Owing to this separation of rxn

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Sampling ensembles of deterministic transition pathways

timescales, exp([t/t ) D 1, and k(t) reaches a plateau during the intermediate time rxn regime, t \ t @ t . The plateau value of k(t) is then equal to the reaction rate conmol rxn stant k . A?B Eqn. (17) can be cast in a form that is convenient for calculations with a transitionpath ensemble. (To simplify notation, we omit the arguments of h , h , and H whenA B B ever they refer to the initial state x .) First, we factorize k(t) as 0 Sh h5 (x )T Sh h (x )T k(t) \ A B t ] A B C 4 l(t)P (18) Sh h (x )T Sh T A B C A The frequency factor, l(t), contains the full time dependence of k(t) ; the probability factor, P, depends only on the total duration T of the paths. To determine the reaction rate constant both factors must be calculated. In order to compute the frequency l(t) using the path ensemble, we perform a second factorization Sh h5 (x )H T Sh H T Sh5 (x )T A B A B t B \ B t AB (19) Sh H T Sh h (x )H T Sh (x )T A B A B C B B C AB where we have used the fact that H \ 0 only if h (x ) \ 0 at all times along the path. B t the average of the quantity O The notation SO(x )T 4 Sh O(x )HB T/Sh H T denotes t AB A t B A B at time t in the path ensemble. As eqn. (19) implies, the frequency factor, l(t), can be calculated as the ratio of two path averages, computed in a single transition-path simulation. Since l(t) contains the full time dependence of k(t), it must reach a plateau in the time regime t \ t @ t . Absence of this plateau would indicate that the total time T mol was not sufficiently longrxn to allow sampling of all relevant transition paths. For the calculation of the probability factor P, we use an approach10 very similar to the umbrella sampling technique for estimating free energy di†erences.9,11h13 To begin, we identify a low dimensional parameter j, such that the Ðnal region B can be identiÐed by some particular value of that parameter. The whole phase space should be identiÐed with some other particular value of j . Varying j corresponds to changing the size of region B. We divide phase space into slightly overlapping ranges of j and thus slightly overlapping regions B . For each region, we perform separate path simulations. In each simulation, we use thejsame initial region A. We construct a histogram of the parameter j for each simulation. By matching the histograms in the overlapping regions, we obtain the probability distribution f (j) of the parameter j. Finally, we obtain the probability factor P by integration over all js belonging to the Ðnal region B : l(t) \

/ f (j) dj P\ B / f (j) dj

(20)

Summarizing : our method for the calculation of the transition rate constant involves two steps. First, we calculate the frequency factor l(t) from a single path simulation. If l(t) does reach a plateau within time T, the paths are sufficiently long to capture all essential transitions. Second, we calculate the probability factor P by performing a series of path simulations with di†erent Ðnal regions. Since this step usually involves several independent path simulations, it is the more computationally expensive of the two. Multiplying P by the plateau value of l(t) gives the transition rate constant k . A?B

4 Illustrative example Lennard-Jones cluster Rare-gas clusters consisting of just a few atoms (N D 10 [ 100) show a remarkably complex behaviour despite the relatively small number of degrees of freedom.14 As is typical for complex systems, the potential-energy surface of such clusters has many local

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minima connected by saddle points. Each potential-energy minimum represents a stable (or metastable) isomer of the cluster. At low energies, such a cluster oscillates around a potential-energy minimum. At high energies, it quickly evaporates. In an intermediate energy range, the cluster is able to cross energy barriers separating neighbouring potential-energy minima.6,15 In this case, the time evolution of the system consists of long sojourns in potential energy minima, interrupted by transitions between them. Transition-path sampling methods can be used to study such isomerization events. As an illustration, we consider a cluster of N \ 7 identical particles of mass m in two dimensions. The particles interact via the Lennard-Jones potential

CA B A B D

p 12 p 6 [ (21) V \ ; 4v r r ij ij i:j where r \ o r [ r o is the distance between particle i and j. The cluster evolves at conij i j stant energy E \ K ] V , where K \ ; p2/m. Natural units are used throughout, and i i distances are measured in units of p, energies in units of v, time in units of q \ (mp2/v)1@2. Accordingly, transition rate constants are given in units of 1/q. The most stable state of the heptamer is a conÐguration in which one central particle is surrounded symmetrically by six others. If the total energy of the cluster exceeds some threshold energy, V , the central particle can escape to the surface of the cluster. In a 0 recent paper,6 we have studied this process in detail using stochastic transition-paths sampling. By quenching the path action, we were able to Ðnd the predominant paths leading to the transfer of the central particle to the surface of the cluster. Three of these paths are shown in Fig. 2. Typically, the system visits one or more metastable states before it eventually reaches the most stable conÐguration with a di†erent particle at the centre of the cluster. The life times of the metastable states are large compared with the time needed to cross the barrier separating adjacent potential-energy minima, or isomers. Transitions between these isomers can, therefore, be regarded as independent. Consequently, the transition rate constant for the centre-to-surface migration of a particle can be calculated from the rate constants for the transitions between adjacent potential-energy minima. In ref. 6, we calculated thermal rate constants for all the subprocesses shown in Fig. 2. Since, in the present work, we are mainly concerned with methodology, we focus on only one of several possible isomerization processes. In particular, we consider the isomerization shown in Fig. 3. State A is centered around the global minimum of the potential energy where V \ [12.53v ; state B is the region around the next lowest minimum A of the potential energy where V \ [11.47v. Quenched path simulations6 show that B

Fig. 2 Predominant pathways for the migration of a particle initially in the centre of the cluster to its surface (adapted from ref. 6). As described in detail in ref. 6, the paths were found by quenching stochastic transition paths. The shaded particle is the particle initially at the centre of the cluster.

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Fig. 3 Schematic representation of the transition from the stable state A with V \ [12.53v to the A metastable Ðnal state B with V \ [11.47v. A minimum energy of V \ [11.04v is required for B 0 transitions to occur. The conÐguration corresponding to the saddle point is found automatically by the algorithm.

transitions between A and B can only occur if the total energy of the system is larger than V \ [11.04v. This threshold energy is the potential energy of the saddle point 0 regions A and B. Prior knowledge of the saddle point is not required as an separating input of our simulation scheme, rather, it is automatically found by the method. The Ðnal and initial regions A and B consist of all conÐgurations close to the respective energy minima. A simple measure of the proximity of a speciÐc conÐguration to a stable state is the mean square displacement dr2 : N dr2 \ ; (Ur [ r0)2 (22) i i i/1 Here, N is the number of particles or atoms, and r0 is a reference conformation characterizing the stable state, for example the minimumi potential-energy conÐguration. U is the rotation matrix that minimizes the mean square displacement dr2. This minimum value is used in the deÐnition of the initial and Ðnal regions A and B. The functions h (x) and h (x) can then be written as A B 1, if d2 \ c min (23) h (x) \ A,B 0, if dr2 [ c min The constant c is chosen to accommodate the equilibrium Ñuctuations of the system around the potential-energy minima. In all our simulations regions A and B are deÐned by dr2 \ 0.1p2 and dr2 \ 0.1p2, respectively. A B

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Rate constants For di†erent values of the total energy, we calculated microcanonical rate constants for the reaction A ] B shown in Fig. 3. Transition paths were generated using the shooting and shifting algorithms, while maintaining constant total energy and vanishing total linear and angular momentum. In each shooting move, we selected a time slice t at random along the old path. Then we randomly selected two particles x and changed t a Gaussian one velocity component of the Ðrst particle by a small amount drawn from distribution. All other momentum components of the two particles were then determined from the constraints of vanishing total linear and total angular momentum. This

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procedure was performed N times. Finally, the momenta of all particles were rescaled to obtain the desired total kinetic energy. The new path was calculated by integrating NewtonÏs equations using the Verlet algorithm. The time duration of each path was T \ 5q. The time step used in the Verlet algorithm was dt \ 0.01q. The new path was accepted if it satisÐed the endpoint constraints. To reduce the amount of required computer memory, only every Ðfth integration step was saved. Thus, the discretized path consisted of 100 consecutive time slices, each separated by an interval of *t \ 0.05q. Those 100 instantaneous conÐgurations and momenta along the path were stored in memory and used as potential starting points for shooting moves and for calculations of path averages. The total energies considered in the microcanonical simulations were E \ [11.03v, E \ [11.02v, E \ [11.00v, E \ [10.98v, E \ [10.90v, and E \ [10.50v. The Ðrst energy is only slightly above the minimum energy required for a transition, V \ 0 [11.04025v. In Fig. 4, the frequency factor l(t) is plotted for the di†erent energies. The higher the energy the shorter the time for l(t) to reach a plateau value. This behavior is due to the fact that, for the higher energies, the system reaches region B more quickly on average than for the lower energies. Further, since l(t) is the time derivative of a function that rises from 0 to 1 in time T, more rapid crossings lead to lower l(t). Thus, the plateau value of l(t) decreases as the total energy is increased. Fig. 5 shows the normalized probability distribution f (j). The order parameter j denotes the minimum squared distance dr2 to the metastable state B. (j \ 0, for example, coincides with the system located exactly at the local potential energy minimum at the center of B.) The probability for the system to reach region B, after starting in region A decreases with decreasing energy. The corresponding rate constants are displayed in Table 1. The path-sampling method is very efficient as the computational cost scales linearly with the total time duration of the path. The shifting algorithm plays an important role in the efficiency of the path sampling method. To estimate the e†ect of the shifting moves on the total efficiency of the method we introduce the correlation function12,13 C(n ) \ c

Sdh (x0 )dh (xnc )T B C@2 B C@2 AB S[dh (x0 )]2T B C@2 AB

(24)

Fig. 4 Frequency factor l as a function of time for the total energies E : [11.03v, [11.02v, [11.00v, [10.98v, [10.90v and [10.50v. l is measured in units of (mp2/v)~1@2 and t is measured in units of (mp2/v)1@2. For all energies l(t) reaches a plateau within time T, indicating that T is large enough to capture all relevant transition paths.

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Fig. 5 Logarithm of the probability f (j) that j is the value of the order parameter for the last path point x . For small values of j, the system is in state B, whereas, for larger values of j, it is far T from B.

where n is the number of simulation cycles and dh (xi) \ h (xi) [ Sh (x )T is the deviB q Baverage q q AB ation ofcthe characteristic function h (x ) from its ensemble atB simulation cycle i. B q This function is a measure for how fast paths diverge in path space. Fig. 6 shows the correlation function C(n ) for a simulation using both shooting and shifting moves and one with only shooting. cThe di†erence between both simulations is very clear. Although shifting the path does not change its salient properties, it leads to a considerable improvement of the efficiency. Comparison with RRKM A common approximation of rate constants is the classical limit of the RRKM theory.7 In this microcanonical transition state theory, the rate constant is approximated by the Table 1 Energy E, plateau value l of the frequency plateau factor l(t), probability factor P(T), and transition rate constant k for the process A ] B depicted in Fig. 2 E/v [11.03 [11.02 [11.00 [10.98 [10.90 [10.50

l q plateau 0.38 0.36 0.34 0.32 0.30 0.22

P(T)

kq

1.07 ] 10~19 2.67 ] 10~17 3.15 ] 10~15 1.92 ] 10~13 2.89 ] 10~10 6.18 ] 10~6

4.07 ] 10~20 9.61 ] 10~18 1.07 ] 10~15 6.14 ] 10~14 8.67 ] 10~11 1.36 ] 10~6

The minimum energy for which a transition can occur is V \ [11.04v, such that the respective energy di†erences 0 are E [ V \ 0.01v, 0.02v, 0.04v, 0.06v, 0.14v, and 0.54v. 0 We note that l varies only slightly, whereas the probability factorplateau P(T) varies over more than 13 orders of magnitude in the energy range studied.

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Fig. 6 Correlation function C(n ) for path sampling with shooting and shifting (ÈÈ) and only c shooting (É É É É).

ratio of the density of states at the transition state and the density of states at the stable state. The density of states can be estimated by assuming the system behaves as a collection of harmonic oscillators both at the transition and the stable state. Within the classical RRKM theory the reaction rate constant in a system of N particles with a total energy E is given by

A

B

E [ V s~1