Stochastic Transition State Theory - The Journal of Physical Chemistry

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Stochastic Transition State Theory Eli Pollak J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02712 • Publication Date (Web): 02 Oct 2018 Downloaded from http://pubs.acs.org on October 2, 2018

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The Journal of Physical Chemistry Letters

Stochastic Transition State Theory Eli Pollak∗ Chemical and Biological Physics Department, Weizmann Institute of Science, 76100, Rehovot, Israel E-mail: [email protected]

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Abstract Kramers’ original paper on the diffusion model of chemical reactions was based on the consideration that only the barrier region determines the outcome of transmission over a barrier. Subsequently it became understood that Kramers’ approach was identical to variational transition state theory (VTST), and as such used only thermodynamic information. Here, using Kramers’ philosophy in conjunction with perturbation theory and the realization that the dynamics which is rate determining usually occurs in the vicinity of the transition state leads to a novel stochastic rate theory in which the momentum change induced by the medium is the stochastic variable. A first successful application of the theory is to the old and challenging problem of motion over a cusped barrier. This has implications for the study of transition path time distributions as well as the theory of tunneling via nonadiabatic coupling.

Table of Content Graphic

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Transition State Theory (TST) is a fundament of rate theory. 1,2 In classical mechanics, one notes that in order to react, the system under study must cross a dividing surface with a given sign of the velocity perpendicular to it. This is due to the fact that by definition, the dividing surface divides the space between reactants and products such that any trajectory leading from reactants to products must cross it. 3 This unidirectional velocity leads to an upper bound estimate for the thermal or microcanonical rate of reaction. Since TST gives an upper bound to the reaction rate, it is only natural to vary the dividing surface such that the upper bound is minimized, this is known as Variational TST (VTST). 4 In 1940 Kramers derived a rate expression, based on a stochastic description of the dynamics, in which the system’s equation of motion is a Langevin equation. 5 The system moves under the influence of a potential and a friction force which is balanced by an external Gaussian random force. Kramers used the Fokker-Planck equation in phase space equivalent of the Langevin equation and solved it by considering only the motion in the vicinity of the barrier top separating reactants and products. Although seemingly Kramers’ approach was very different from TST, it was shown in 1986 that Kramers’ theory and VTST are identical. 6 The route leading towards this identity was based on the realization that the Langevin equation may be derived from the continuum limit of a Hamiltonian in which the system is coupled bilinearly to a harmonic bath. This meant that if the potential is quadratic, as considered by Kramers, then the Hamiltonian equivalent is quadratic and may be diagonalized by a normal mode transformation giving one unstable normal mode while the rest were stable. The dividing surface perpendicular to the unstable mode was then identified as the variational transition state. The unidirectional flux through this surface gave the exact reactive flux since within the quadratic model the unstable mode is separable from all other modes. It was identical to Kramers’ result for the Langevin equation and to the Grote-Hynes estimate for the rate across the parabolic barrier 7 when the dynamics was described by the Generalized Langevin Equation (GLE) in which the frictional force includes memory. In ret-

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rospect, Kramers’ theory which was reformulated as a straightforward application of VTST, is an equilibrium theory in the sense that the rate coefficient is expressed only in terms of thermal equilibrium properties of the Hamiltonian underlying the GLE. The model of a system bilinearly coupled to a harmonic bath is arguably the ”standard” model for thermal reactions in liquids. One computes a (classical) potential of mean force along some reasonably chosen reaction coordinate. 8 The liquid is modeled by a force autocorrelation function which is used to approximate the full dynamics in terms of a GLE. One expressly assumes that in the long time limit, everything relaxes to the thermal equilibrium distribution. This assures, among others, that detailed balance is obeyed. 1 One may then use TST to estimate the rate. 9 The heart of the theory remains in the observation that the fate of the reacting system is determined by the location of the optimal dividing surface, typically in the vicinity of the free energy barrier dividing between reactants and products. The central advantage of VTST is that there is no need to obtain dynamical information, but this is also its limitation. Especially when the temperature is not very low, that is when the barrier height V ‡ is not much larger than the thermal energy (kB T ) at temperature T , ( barring special circumstances such as a very flat or cusped barrier) one may expect that the actual nonlinear dynamics becomes important.

This is of special interest in view of

the recent experiments on the transition path time distribution for the folding unfolding transition of hairpin DNA 10 where the low barrier (in terms of kB T ) seems to be ubiquitous. This led to the derivation of finite barrier corrections to the VTST rate, 11 in which perturbation theory was used to estimate the leading order correction to the VTST rate estimate whose magnitude was of the order of kB T /V ‡ . In this approach, the difference between the full potential and the parabolic part (termed as the nonlinear part of the potential) was considered to be the ”small parameter” in the sense that it leads to small corrections to the rate estimate which are of the order of (kB T )/V ‡ . The main theme of this letter is a generalization of this perturbation theory in the form of a stochastic transition state theory in which one takes into account the stochastic dynamics induced by the thermal surround-

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ings. Although the resulting theory no longer leads to an upper bound to the transmission coefficient it may still be considered to be a transition state theory since it takes into consideration the dynamics only in the vicinity of the transition state. This is in the spirit of the original theory of Kramers who also considered the dynamics of the system coordinate only in the vicinity of the transition state. We shall show that this Stochastic TST (STST) not only improves upon VTST it also serves as a guide to the accuracy of the variational upper bound estimate to the rate. As a practical but important example, we consider the thermal flux across symmetric 12,13 and asymmetric cusped barriers. 14–16 This problem which is notoriously nonlinear, has posed a challenge to theory and defied solution for many years. Yet it plays an essential role in understanding nonadiabatic rate theory. 17,18 Our point of origin is the Hamiltonian equivalent which leads in the continuum limit to the GLE, whereby a particle with mass M moves along a reaction coordinate q with momentum pq under the influence of a potential V (q) whose motion is coupled bilinearly to a harmonic bath: " 2 # X p2xj ωj2 p2q cj √ + V (q) + + xj − 2 M q . H= 2M 2 2 ω j j

(1)

xj and pxj are the mass weighted coordinate and momentum of the j-th oscillator whose frequency is ωj which is coupled to the system via the coupling constant cj . It is well known 19 that one may recast the equation of motion for the system as a GLE:

0

Z

M q¨ + V (q) + M

t

dt0 γ (t − t0 ) q˙ (t0 ) = F (t)

(2)

with the identification of the friction function

γ (t) =

X c2j cos (ωj t) 2 ω j j

(3)

and the Gaussian random force F (t) whose mean vanishes and whose correlation function

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is proportional to the friction function (fluctuation dissipation relation 1 ). Without loss of generality we assume that the potential V (q) has a barrier located at q = 0 with V (0) = 0. The formally exact flux side expression 20 for the thermal reactive flux (with β = 1/ (kB T )) originating from the left of the barrier and crossing it to the right at the dividing surface located at q = 0 is i h pq FR = lim Tr exp (−βH) δ (q) θ (qt ) t→∞ M

(4)

where the Tr operation is a classical integral over all phase space variables of the system and the bath, δ (q) denotes the Dirac ”delta” function which localizes one to the dividing surface. All the dynamics hides in the Heaviside function θ (qt ) which is unity when the trajectory is to the right of the barrier and zero otherwise. The standard estimate for the TST flux (FT ST ) is obtained by replacing limt→∞ θ (qt ) with θ (pq ) in Eq. 4.

It is then straightforward to carry out the integration over the Q 2π . We then define a system and bath phase space variables to find that βFT ST = j βω j transmission probability, which is the central property considered in this letter as: Y FR PR = = FT ST j

βωj 2π

βFR .

(5)

This transmission probability reflects the effect of the bath on the ”standard” TST rate. (By defining a transmission probability we were able to leave out the normalization factor for the flux in Eq. 4.) The potential may always be rewritten as

V (q) = −

2 M ω ‡2 q − q ‡ + V1 (q) 2

(6)

where the barrier frequency ω ‡ and barrier location q ‡ will be considered as variational parameters. V1 (q) is termed the nonlinear part of the potential. If the potential is not

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quadratic at the barrier top then this ”nonlinearity” may include even linear terms in the coordinate. If one sets V1 (q) = 0 the Hamiltonian is quadratic in the system and bath phase space variables and may be diagonalized, using a normal mode transformation, the details of which may be found for example in Ref. 21 . Shifting the coordinate q to q − q ‡ , the transformed Hamiltonian in the normal modes is # X" 2 2 2 p λ y p2ρ λ‡2 ρ2 1 yj j j − + V1 q ‡ + √ [u00 ρ + σ] + + H= 2 2 2 2 M j

(7)

where ρ and pρ are the mass weighted unstable mode coordinate and momentum whose barrier frequency is λ‡ and yj , pyj are the mass weighted phase space variables of the j-th stable bath mode whose frequency is λj . A central object in the stochastic theory presented here is the collective bath mode

σ=

X

uj0 yj

(8)

j

where the uj0 ’s are the matrix elements projecting the mass weighted system coordinate √ M q on the normal modes. Choosing the dividing surface to be located at the top of the normal mode barrier (ρ = 0) and performing the integration over the stable bath mode phase space variables and the unstable momentum and coordinate one finds 11 that the VTST estimate for the transmission probability of Eq. 5 is

‡

PV T ST q , ω

‡

λ‡ = ‡ ω

r

βΩ2 2π

∞

βΩ2 σ 2 σ ‡ dσ exp − − βV1 q + √ 2 M −∞

Z

(9)

where Ω is the collective bath mode frequency defined by

Ω2 =

X u2j0 j

λ2j

!−1 .

(10)

The transmission probability depends on the two variables ω ‡ and q ‡ which are then varied

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to give the least upper bound to the transmission probability. The frequencies λ‡ and Ω as well as the matrix element u00 may be expressed in terms of Laplace transforms of the time dependent friction, the expressions are given in Ref. 21 . From Eq. 9 one notes that already in the VTST formulation, the collective mode σ (Eq. 8) may be considered as a stochastic Gaussian random variable. The present variational theory differs from a previous version. 14 Here, we consider the location and frequency of the parabolic barrier to be variational parameters. In Ref. 14 the dividing surface was represented as a planar dividing surface, 22 that is a linear combination of the system coordinate q and bath modes xj whose location and linear combination coefficients were considered as variational parameters. It turns out though that the two approaches give identical results, this is in retrospect not surprising, since the unstable mode is a linear combination of all degrees of freedom. The real point of departure from previous theory is the introduction below of the effect of the stochastic dynamics on the transmission probability. The central idea is to solve explicitly for the motion of the unstable mode to account correctly for multiple recrossing of the dividing surface. From the normal mode Hamiltonian one sees that the exact equation of motion for the unstable mode is (the prime denotes the derivative with respect to the argument) 1 u00 0 ‡ ρ¨ − λ ρ = − √ V1 q + √ [u00 ρ + σ] . M M ‡2

(11)

Following Kramers, we may assume that only the dynamics in the vicinity of the dividing surface is important. Since it is located at the top of a parabolic barrier, where deviations are typically at least quadratic in the distance from the barrier we may assume that the nonlinear part of the potential in this region is in some sense small. It is then possible to solve for the unstable mode motion (Eq. 11) perturbatively 11 . In the absence of the nonlinearity, the time dependence of the collective bath mode is just the sum of the separate

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harmonic oscillator motions of the stable modes:

σ0 (t) =

X j

uj0

py yj cos (λj t) + j sin (λj t) λj

(12)

and it may be considered as a Gaussian random variable with zero mean. The unperturbed motion of the unstable mode is pρ ρ0 (t) = ρ cosh λ‡ t + ‡ sinh λ‡ t . λ

(13)

The first order equation of motion for the unstable mode, given that initially it was at the barrier top (ρ = 0) is obtained by inserting the zero-th order solutions of Eqs. 12 and 13 into the right hand side of the exact equation of motion (Eq. 11). This is then a forced oscillator equation of motion whose analytic solution is well known. One finds, as also detailed in Ref. 11 , that for long times, to first order in the nonlinearity

ρ1 (t)t→∞

Z ∞ 0 ‡ σ0 (t0 ) exp λ‡ t u00 0 ‡ 0 dt exp −λ t V1 q + √ pρ − √ = 2λ‡ M 0 M ‡ exp λ t ≡ [pρ − ∆pρ ] . 2λ‡

(14)

The momentum shift ∆pρ is a stochastic random variable, since it depends on the Gaussian variable σ0 (t0 ). In the variational theory, the dividing surface is perpendicular to the unstable mode coordinate ρ instead of the system coordinate q. Therefore, in Eq. 4 the Heaviside function limt→∞ θ(qt ) which limits the phase space integration to all the trajectories that ultimately end up on the right side of the dividing surface, must be replaced by limt→∞ θ(ρt ). From the second line of Eq. 14, this means that the initial unstable mode momentum at the dividing surface pρ ≥ ∆pρ . It remains to estimate the transmission probability by suitable averaging over the stochastic momentum shift ∆pρ . First we note that the mean value of the stochastic momentum is obtained by averaging q βΩ2 1 2 2 the derivative of the nonlinearity over the Gaussian distribution exp − βΩ σ (t) 0 2π 2 9



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which already appeared in Eq. 9. This mean is however independent of the time since σ0 (t) involves only the unperturbed stable oscillator motion. The mean value of the stochastic momentum is thus a Gaussian average over the force emanating from the nonlinear part of the potential: u00 h∆pρ i = √ λ‡ M

r

βΩ2 2π

∞

βΩ2 σ 2 σ 0 ‡ dσ exp − V1 q + √ . 2 M −∞

Z

(15)

It remains to obtain an expression for the variance of the stochastic momentum. This has already been carried out in Eq. 3.13 of Ref. 11 (here we used some of the symmetries of the variables to simplify the expression somewhat):

∆p2ρ

u200 β = πM λ‡ · (V10

Z

∞

dt exp −λ‡ t

∞

Z

Z du

0

0

[σ+ (u, v, t)] V10

0

∞

β dv exp − u2 + v 2 2

[σ− (u, v, t)] + V10 [−σ+ (u, v, t)] V10 [−σ− (u, v, t)]) (16)

where we used the notation ζ (t) σ± (u, v, t) = √ [u ± η (t) v] , 2Ω ζ 2 (t) = 1 + K (t) , η 2 (t) =

(17) 1 − K (t) . 1 + K (t)

(18)

and the so called ”stable mode normalized friction function” is

2

K (t) = Ω

X u2j0 j

λ2j

cos (λj t) .

(19)

The function K(t) is similar in form to the friction function (Eq. 3) and this is the source of its name. We are now at the point where we can derive the stochastic TST expression for the rate.

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From Eqs. 4 and 5 and the condition that pρ ≥ ∆pρ we have that:

PST ST =

Y βωj j

2π

βTr [exp (−βH) pρ δ (ρ) θ (pρ − ∆pρ )]

= PV T ST q ‡ , ω

‡

β h∆pρ i2 q

exp − 2 1 + β ∆p2 ρ 1 + β ∆p2ρ 1

! (20)

and as in Eq. 4 the Tr operation implies an integration over all phase space variables. The second equality in Eq. 20 is the central formal result of this paper. It has been derived by using perturbation theory in the vicinity of the variational dividing surface. Trajectories that move far from the dividing surface have little chance of returning to it. Therefore it should suffice to consider the actual motion only in the vicinity of the dividing surface and it is here that the ”nonlinear part of the potential” is truly small. This allows for a perturbation theory estimate of the fate of the trajectories. Due to the bath of stable modes, this fate is stochastic, hence this leads to the stochastic transition state theory derived here. If the original potential V (q) is symmetric in q then the optimal value of the shift parameter q ‡ = 0 and V10 (q) is antisymmetric in q. This means that h∆pρ i = 0 and the q

−1

1 + β ∆p2ρ ' 1−β ∆p2ρ /2, regaining VTST transmission factor is reduced only by the previous result considered as dynamic TST in Ref. 11 . As seen from the expression, the STST result is always smaller than the VTST result, as it should be. The asymmetry in the 2

h∆pρ i potential is seen as adding a temperature dependent activation energy ( 2 1+β ) to the ( h∆p2ρ i) rate expression. The resulting STST theory is valid for memory friction, all expressions may

be expressed in terms of the time dependent friction function (Eq. 3) in its continuum limit form. To exemplify the utility of this stochastic TST estimate for the rate, we will consider the old but as yet unsolved problem which lies at the heart of nonadiabatic transitions in condensed phases, namely the transmission probability for motion over a (not necessarily symmetric) cusped barrier in the presence of Ohmic friction (γ (t) = 2γδ (t)). The system

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potential is:

V (q) = M k− qθ (−q) − M k+ qθ (q) = M kq [θ (−q) − θ (q)] +

M ∆k q 2

and in the second line we used the symmetrized notation k =

k+ +k− , ∆k 2

(21)

= k− − k+ for

the slopes of the potential. In this case, due to the cusped form the nonlinear part of the potential does not strictly vanish around the barrier top. Nevertheless, its deviation from the parabolic barrier potential is sufficiently small, especially in the case of a symmetric cusped potential as may be inferred from the magnitude of the variance of the stochastic momentum, so that it may be treated perturbatively. The detailed solution for VTST and STST in this case is given in the supporting information to this letter. Here we summarise the main results. For this purpose we use some reduced notations. The shift reduced energy Q‡ is defined as :

Q‡ =

βM ω ‡2 q ‡2 2

(22)

and is the variational parameter which determines the location q ‡ of the parabolic barrier. The (inverse) barrier frequency variational parameter is taken to be

α=

γ 2ω ‡

(23)

where γ is the friction coefficient. In other words, the two independent parameters needed to minimize the VTST estimate for the rate, at a fixed value of the Ohmic friction coefficient γ, are α and Q‡ . It is also useful to define a reduced friction strength parameter γ γ˜ = √ . k βM

12


(24)

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Table 1: Transmission probabilities for a symmetric cusped potential. The numerically exact results (PR ) are adapted from Table 1 of Ref. 15 γ˜ 0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.8 1 1.5 2 3 4 5 6 8 10

α 0.003925 0.0088244 0.015671 0.035148 0.062173 0.096443 0.13749 0.23723 0.35467 0.68230 1.01544 1.65903 2.28162 2.89361 3.49996 4.70336 5.90069

hβ∆p2ρ i 0.7979e-3 0.1796e-2 0.3193e-2 0.7183e-2 0.1274e-1 0.1981e-1 0.2822e-1 0.48016e-1 0.6922e-1 .11123 .13122 .14047 .13967 .13779 .13617 .13400 .13277

PR 0.9966 0.9929 0.9875 0.9760 0.9527 0.9271 0.9003 0.8379 0.7769 0.6300 0.5229 0.3816 0.2965 0.2399 0.2028 0.1538 0.1223

PST ST .99764 .99470 .99060 .97902 .96313 .94326 .91990 .86519 .80461 .65836 .54267 .39215 .30380 .24698 .20771 .15726 .12637

PV T ST .99804 .99560 .99219 .98253 .96924 .95255 .93270 .88571 .83199 .69401 .57717 .41879 .32432 .26345 .22140 .16746 .13450

In Table I, for the symmetric cusped potential, we give the optimized values (we first obtain the value of α that minimizes the VTST transmission factor and then use it to obtain the momentum variance and the STST transmission factor) of the frequency parameter α as a function of the reduced friction strength γ˜ , the resulting VTST and STST estimates and the numerically exact result adapted from Table 1 of Ref. 15 In panel a of Figure 1 we compare the VTST and STST estimates for the transmission probability with the numerically determined values as functions of the magnitude of the reduced friction. The error bars on the numerical data reflect the accuracy of these computations, as reported in Ref. 15 . The enhanced accuracy of the STST estimate is reflected more clearly in panel b of the figure where the relative error (∆j = (Pj − Pex ) /Pex ,

j =VTST,STST) of both estimates is

compared. In both panels, the solid (red) line and the dashed (blue) line represent the STST and VTST data respectively. We note that with STST the relative error for the whole range of friction is less than 4% and considerably reduces the error as compared to the VTST estimate. 13



Figure 1: Stochastic TST for the symmetric cusped barrier. The solid (red) and dashed (blue) lines shown in the left panel of the figure are the transmission probabilities as obtained from the STST and VTST theories respectively, shown as functions of the reduced friction parameter γ˜ as defined in Eq. 24. The asterisks are the numerically exact results as reported in Table 1 of Ref. 15 . The right panel shows the relative error of the two theories (closed circles, blue for VTST and open circles, red, for STST) as compared with the numerically exact results of Ref. 15 The error bars result from the noise in the numerically exact results. Note that STST is significantly more accurate than VTST. In this context we also note, as shown in the supplementary material, that the error in the STST estimate for the rate of the symmetric cusped barrier in the Smoluchowski strong friction limit is just 1.67% when compared with Kramers’ analytic result, while the VTST error is ca. 5 times higher. For the asymmetric cusped potential in the Smoluchowski limit, STST provides an estimate which is within ∼ 10% or less of the exact result as long as the asymmetry parameter (κas = ∆k/(2k)) is less than 1/3. In this range the momentum variance is smaller than unity and the perturbation theory is valid. For larger values of the asymmetry, the momentum variance grows considerably and the STST estimate which was based on the validity of perturbation theory is no longer reliable. However, the same holds true in this parameter region for the VTST estimate. In other words, the range in which 14


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the perturbation theory fails provides valuable information, it indicates that also the VTST estimate is no longer accurate. In summary, we have presented a novel stochastic transition state theory which is valid for dissipative systems whose dynamics is described by a generalized Langevin equation. The utility of this theory has been exemplified by providing a solution to the elusive problem of the transmission probability through cusped barriers. The result is very accurate for the symmetric potential. It should be noted that the cusped barrier is on the face of it an especially difficult case, since the parabolic barrier potential which lies at the heart of the perturbation theory approach is never exact in this case. STST is successful even here since the relevant dynamics is really limited to the barrier region and in this region, especially when the asymmetry is not too strong, the deviation of the cusped potential from the parabolic barrier potential remains small.

The same is even more so when considering ”normal”

potentials which are quadratic around the barrier top. Here, STST is expected to be even more accurate. The sceptic reader might ask, well ok, why do we need such an analytic theory at all? Present day computers can solve for the classical dynamics of very large systems and in addition the interest is if anything in the quantum mechanics rather than the classical mechanics theory considered here. To answer we make a few observations. The parabolic barrier has been a paradigm in solving for the transition path time distribution in the experimentally measured folding and unfolding of DNA hairpins. 10 The reason for this is that the parabolic problem is amenable to analytic solution. 23–25 The present STST implies that we can now solve for a different model problem, the symmetric cusped potential, to obtain also in this case analytic expressions for the transition path time distribution. Given the intense discussion regarding the transition times for the folding-unfolding transition it will be of quite some interest to see whether the experimental data is sufficiently accurate to distinguish between the two paradigms. The STST approach has also important implications in the quantum mechanical context.

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The theory of tunneling in nonadiabatic systems presents an ongoing challenge. 26–28 The present STST approach can be used to identify the extent within which the unstable mode separable dynamics is sufficient for solution of the nonadiabatic quantum tunneling problem. It may also be used to do the same in the context of adiabatic tunneling. Finally, we note that although the present theory has been given in the context of dissipative systems whereby the reaction coordinate is bilinearly coupled to a harmonic bath, it is possible to identify the unstable normal mode in a general molecular dynamics simulation. 29 This raises the possibility of formulation of STST in a much more general context.

Acknowledgements I thank I. Rips for discussions and P. Talkner and A. M. Berezhkovskii for insight on the dynamics of the asymmetric cusped potential and for their constructive critical comments on an early version of this paper. This work was generously supported by grants of the Israel Science Foundation, the Minerva Foundation, Munich and the German Israel Foundation for Basic Research. Supporting Information Available The supporting information for this letter provides the detailed working VTST and STST expressions used to obtain the numerical results for the symmetric and asymmetric cusped potentials.

References (1) H¨anggi, P.; Talkner, P.; Borkovec, M. Reaction-rate Theory: Fifty Years After Kramers. Rev. Mod. Phys. 1990, 62 251-341 (2) Pollak E.; Talkner P.; Reaction Rate Theory: What it Was, Where is it Today, and Where is it Going? Chaos 2005 15 026116(1-11) (3) Pechukas P.; Transition State Theory Ann. Rev. Phys. Chem. 1981 32 159-177 16


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Stochastic Transition State Theory - The Journal of Physical Chemistry

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