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J. Phys. Chem. B 2008, 112, 206-212
Transition-State Theory Rate Calculations with a Recrossing-Free Moving Dividing Surface† Thomas Bartsch* Department of Mathematical Sciences, Loughborough UniVersity, Loughborough LE11 3TU, UK
T. Uzer Center for Nonlinear Science, Georgia Institute of Technology, Atlanta, Georgia 30332-0430
Jeremy M. Moix Chemical Physics Department, The Weizmann Institute of Science, 76100 RehoVot, Israel
Rigoberto Hernandez Center for Computational Molecular Sciences & Technology, School of Chemistry & Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400 ReceiVed: July 16, 2007; In Final Form: August 7, 2007
Two different methods for transition-state theory (TST) rate calculations are presented that use the recently developed notions of the moving dividing surface and the associated moving separatrices: one is based on the flux-over-population approach and the other on the calculation of the reactive flux. The flux-over-population rate can be calculated in two ways by averaging the flux first over the noise and then over the initial conditions or vice versa. The former entails the calculation of reaction probabilities and is closely related to previous TST rate derivations. The latter results in an expression for the transmission factor as the noise average of a stochastic variable that is given explicitly as a function of the moving separatrices. Both the reactive-flux and flux-over-population methods suggest possible new ways of calculating approximate rates in anharmonic systems. In particular, numerical simulations of harmonic and anharmonic systems have been used to calculate reaction rates based on the reactive flux calculation using the fixed and moving dividing surfaces so as to illustrate the computational advantages of the latter.
I. Introduction Transition-state theory (TST) has proven to be a powerful method for predicting the rate of activated processes. While originally formulated as a description of the chemical reaction rates for small molecules,1-3 it has been shown that any system that evolves from well-defined “reactant” to “product” states can be treated within this framework.4-10 TST is based on the observation that large barriers hindering reactive events lead to bottlenecks in phase space. It is then possible to define a surface that separates reactants from products. The (classical) reaction rate is given simply by the steady-state flux through that surface. This approach yields accurate approximations for isolated systems if the dividing surface is chosen well. However, coupling with the environment, as occurs in almost all chemical reactions in solution and many other systems of practical interest, leads to recrossings of the dividing surface. These deviations result in an overestimate of the true rate by traditional TST calculations. The generalized Langevin equation has been used extensively as a prototypical model for simple chemical and physical reaction processes that are coupled to the surrounding thermal environment.11-13 As mentioned above, the corresponding rate †
Part of the James T. (Casey) Hynes Festschrift. * Corresponding author.
for such systems must necessarily be modified from that of the bare gas-phase result predicted by traditional TST because of interactions with the external bath.14,15 An early attempt to formulate a rate theory in the Langevin setting was provided by Kramers’ seminal work.16 He was able to derive explicit solutions for the rate over a parabolic barrier separately in the limits of weak and strong damping based on a flux-overpopulation calculation. However, a general solution valid for the entire friction regime (the famous “Kramers’ Turnover” problem) was not found until much later17,18 by taking advantage of the equivalence19 of the generalized Langevin equation to a Hamiltonian model of a subsystem bilinearly coupled to a bath of harmonic oscillators. Here, results are presented for the strong coupling limit that lead to the well-known Kramers-GroteHynes correction to the TST rate constant.20-22 However, the present formalism remains within the Langevin representation in the spirit of Kramers’ original work without resorting to the Hamiltonian model. There are many advantages associated with this methodology. It offers an alternative geometrical interpretation of the reaction process, which is convenient and intuitive from both a theoretical and computational perspective, along with the ability to observe the contributing factors to a particular rate process at a trajectory level. In a recent series of papers, a new framework for rate theory calculations has been proposed.23-25 The Langevin equation
10.1021/jp0755600 CCC: $40.75 © 2008 American Chemical Society Published on Web 10/13/2007
TST Rate Calculations using a Moving Dividing Surface
J. Phys. Chem. B, Vol. 112, No. 2, 2008 207
gives rise to a distinguished trajectory that remains in the vicinity of the barrier for all time. This trajectory was termed the “transition-state (TS) trajectory” since it plays a similar role to the fixed saddle point in traditional TST. A key idea in the theory23-25 arises from the observation that Langevin dynamics in a moving coordinate system relative to the TS trajectory is noiseless. One can easily construct a TST dividing surface in the relative phase space. It gives rise to a moving dividing surface in the absolute frame that serves as an alternate TS in noisy environments and possesses the important property that it is free of the recrossings that plague traditional calculations. Building on these previous observations, this paper presents two equivalent, but conceptually different methods that take advantage of time-dependent TS structures in the calculation of the rate. Through both methods, the well-established Kramers-Grote-Hynes result for the transmission factor is obtained in the strong friction limit. One involves a traditional approach including averages over the probability that a trajectory resides in the product state at large times in close analogy with previous calculations.26 This probability was calculated in ref 27 from an explicit model of the heat bath (see also refs 28 and 29 for anharmonic corrections) and in ref 24 from moving TS structures. The other approach is afforded only by the moving dividing surface formalism and results in an average over realizations of the noise of a stochastic variable that is given explicitly as a function of the moving separatrices. Hence it requires only a sampling of the noise, which leads to a very efficient computational approach without the need for any trajectory simulations. Supporting numerical computations are also included, confirming the analytical results along with reactive flux calculations using both moving and traditional fixed dividing surfaces. As the former is free of recrossings for harmonic systems, it results in improved convergence properties and several computational advantages over the fixed TS. It is also demonstrated that these benefits persist even in the presence of quite substantial nonlinearities. II. Traditional Rate Formulas In this section we will briefly outline traditional approaches to rate calculations as described in, for example, refs 12, 30, and 31, which we will later adapt to incorporate the additional information that is made available by the moving TSs. TST is based on the assumption that a dividing surface in phase space separating reactants and products can be found that is crossed once and only once by every reactive trajectory. If the dividing surface is characterized by the value x ) x‡ of a generalized reaction coordinate x, so that the product region in phase space is given by x > x‡, the TST approximation to the reaction rate is given by
kTST )
〈δ(x - x‡)VxΘ (Vx)〉 〈Θ(x‡ - x)〉
(1)
where the angular brackets indicate the average over a thermal equilibrium ensemble, and
{
1 : x>0 Θ(x) ) 0 : x 0 2
where the subscript f indicates an average over initial conditions drawn from the distribution (eq 7), and the subscript R denotes an average over realizations of the noise. A similar procedure allows us to replace the reactive flux (eq 2) by
1 s ) - (γ + xγ2 + 4ωx2) < 0 2
JTST f
〈Vx(0)Θ(Vx(0))〉f
1 t1 ) - (γ - xγ2 - 4ωy2) 2
) 〈VxΘ(Vx)〉f ) )
)
1 t2 ) - (γ + xγ2 - 4ωy2) 2
∫dx∫dVx∫dy∫dVy f(x,y,Vx,Vy)VxΘ(Vx) 1
x2πkBT
∫
∞ 0
{ }
dVxVx exp -
Vx2 2kBT
x
kBT 2π
(9)
They correspond to an unstable and a stable direction in the reactive degree of freedom x and damped oscillations in the transverse degree of freedom y. We describe the dynamics in diagonal coordinates measured in these eigendirections. In the reactive degree of freedom, the position-momentum coordinates and the diagonal coordinates are related by
which differs in normalization. The transmission factor can then be obtained via
Jf ) κJTST f
(10)
from any improved estimate Jf of the reactive flux that was calculated with the barrier distribution (eq 7). The remainder of this paper will be devoted to the calculation of such improved estimates with the help of the moving TS structures. III. The Moving TS of a Paradigmatic Reaction A. A Langevin Model for an Anharmonic Chemical Reaction. The Langevin equation describing the reduced dynamics in an external bath is given by12
bR(t)) - Γq b˙R(t) + B ξ R(t) b q¨ R(t) ) -∇bq U(q
(11)
where b qR is an N dimensional vector describing the reaction coordinate with one unstable degree of freedom, Γ is an N × N matrix of time-independent damping constants, and B ξR(t) is the corresponding stochastic force, which is connected to the friction via a fluctuation-dissipation theorem:
〈ξR(t)ξTR(t′)〉R ) 2kBTΓδ(t - t′)
(12)
The subscript R describes a given realization of the noise to which all trajectories are subjected. For simplicity, we consider the case in which the friction is diagonal and isotropic, i.e., γx ) γy ) γ. The potential of mean force, U(q bR(t)), is chosen to be described by eq 6, as was also selected in the earlier work of ref 25, because it serves as a prototype for simple chemical reactions. B. Moving Transition States. The fundamental concepts of time-dependent TST were developed in refs 23-25. In this
(13)
x ) zu + zs, Vx ) uzu + szs
(14)
with the inverse
zu )
Vx - sx -Vx + ux ,z ) u - s s u - s
(15)
As was shown in ref 23, for every realization R of the noise, there is a unique trajectory b z‡R (the TS trajectory) that remains in the vicinity of the saddle point for all time. Its diagonal coordinates are given by ‡ zRj (t) )
{
0 e- τ ξRj(t + τ)dτ ∫-∞ -∫∞0 e- τ ξRj(t + τ)dτ j
j
if j is such that Re j < 0 if j is such that Re j > 0
}
(16)
The variances of the components of the TS trajectory as noted in ref 24 are provided by the expressions
〈z‡2 u 〉R )
kBTγ
; 〈z‡2 s 〉R )
(u - s) u 2
kBTγ (u - s)2s
(17)
It is very useful to consider the relative coordinate ∆zj ) zRj - z‡Rj. In the case of a harmonic barrier (k ) 0), the dynamics of the relative coordinate is noiseless. For the reactive degree of freedom, it is illustrated in Figure 1. The moving TST offers a rich geometrical structure: it gives rise to a moving dividing surface that is strictly free of recrossings and that can play the role of the traditional fixed dividing surface in a rate calculation. It also gives rise to moving separatrices allowing one to identify reactive trajectories. Either
TST Rate Calculations using a Moving Dividing Surface
J. Phys. Chem. B, Vol. 112, No. 2, 2008 209 and the two ways lead to two different interpretations of the resulting rate expression. A. Approach 1: Averaging over the Noise. If we average over the noise first, we find
Jf ) 〈VxP+ (Vx)〉f
(23)
P+ ) 〈Θ(Vx > V+ x )〉R
(24)
where
Figure 1. Phase portrait of the relative dynamics in the reactive degree of freedom. Dashed lines indicate the stable and unstable manifolds of the equilibrium point, which act as separatrices. Solid curves illustrate typical trajectories. White dots indicate two possibilities for the position and velocity of the TS trajectory at time t ) 0; the vertical lines represent the corresponding barrier ensembles. The probability density is given by the line widths. Also shown (shifted) are the coordinate axes for the diagonal coordinates ∆zu and ∆zs.
is the probability that a trajectory with a given initial condition will end as a product. We calculated this probability in ref 24 to be
( )
zu 1 P+ ) erfc 2 x2σu where
σu2 )
of these structures can be used in a rate calculation, as will be demonstrated below. IV. Rates Obtained from the Moving Separatrices While the no-recrossing assumption in conventional TST is an approximation, the moving TS gives us the possibility of identifying reactive trajectories exactly. They are easiest to identify in the moving relative coordinate system. From the phase space plot in Figure 1, it is clear that a trajectory will be on the product side of the barrier in the distant future if
∆zu ) zu - z‡uR > 0
(18)
(19)
For trajectories that are sampled from the barrier ensemble, the initial position is x ) 0, so that the reactivity condition reads Vx > V+ R . We will henceforth study only such trajectories that start on the saddle. It is useful to introduce a dimensionless version of the critical velocity by
V+ R )
V+ R
xkBT
)
u - s
xkBT
z‡uR
(20)
γ u
(21)
If the TST approximation to the reactivity condition in eq 9 is replaced with the exact condition in eq 18, one obtains the flux
Jf+ ) 〈VxΘ(Vx > V+ x )〉Rf
σ+2
(u - s)2
(26)
zu ) and
Vx u - s
(
1 P+ ) erfc 2
(27)
)
Vx
(28)
x2kBTσ+
We thus obtain the flux
Jf ) 〈VxP+(Vx)〉f )
∫dVx Vx
)
x
1
2 1 e-Vx /2kBT erfc 2 x2πkBT
kBT 2π
∞ du u e-u ∫-∞
2
(
Vx
x2kBTσ+
)
( )
erfc -
u , with u ) σ+
) JfTSTκ
Vx
x2kBT (29)
This approach yields the transmission coefficient
It follows from eqs 20 and 17 that V+ R is a Gaussian random variable with zero mean and variance:
σ2+ :) 〈V+2 R 〉)
kBT
is the variance (eq 17) of the unstable component of the TS trajectory, and zu is the unstable coordinate of the initial phase point. Using eq 15 and x ) 0 for points in the barrier ensemble, we find
Using the transformation formulas (eq 15), we can rewrite this as
Vx - sx > (u - s)z‡uR ): V+ R
(25)
(22)
This expression contains averages over the realizations of the noise that are labeled by R and over the ensemble f of initial conditions. These averages can be carried out in arbitrary order,
κ) )
∞ du u e-u ∫-∞
2
( )
erfc -
1
u σ+
(30)
x1 + σ+2
This result can be further rewritten as
κ+ )
1
x1 + γ/u
)
u ωx
(31)
which is identical to the famous Kramers prefactor in the spacediffusion limited regime.
210 J. Phys. Chem. B, Vol. 112, No. 2, 2008
Bartsch et al.
B. Approach 2: Averaging over the Subsystem Ensemble. It is instructive to carry out the calculation of the transmission factor in a different way by averaging over the initial conditions first. We then find
Jf )
)
〈
1
x2πkBT
{ }〉
Vx2 exp 2kBT
∫
∞ VR+ dVxVx
〈x { }〉 〈 { }〉 2 V+ kBT R exp 2π 2kBT
) JTST exp f
R
R
and thus +2
(32)
2 Because the distribution of V+ R is Gaussian with variance σ+ , this result gives
κ+ )
∫dV e-V /2 e-V /2σ 2
1
2
2
x2πσ+
2
+
)
1
x1 + σ+2
which is the Kramers prefactor (eq 31). Of course, the final result is the same as before. However, the form of eq 32 is very suggestive: the transmission factor is represented as an average over a random variable that is independent of the initial conditions. All dynamics are hidden in the parameter V+ R, which in turn encodes the location of the moving separatrix. We therefore expect a similar formula to hold in a nonlinear system. Equation 32 uses the full power of the time-dependent TST to express the reaction rate in terms of the geometric structures in phase space. The geometric structures of time-dependent TST can be constructed23 even in the more general case of the generalized Langevin equation in which the particle is subject to colored noise.12 There still exists a single unstable mode, but the eigenvalue u that describes the dynamics in this mode is replaced by the Kramers-Grote-Hynes reaction frequency.20-22 The reactivity condition (eq 18) remains unchanged. It thereby leads to the expression (eq 32) for the transmission factor, and thereby the celebrated Kramers-Grote-Hynes result. C. Connection to a Fokker-Planck-Based Rate. The result found earlier in eq 31 can be obtained from a reactive-flux calculation similar to that of Tannor and Kohen26 starting from the traditional formula30
e-E /kBT Q ‡
k)
∞ Vxe-V /2k T χ(Vx) dVx ∫-∞ 2
x
B
(
1 ∞ dVx Vxe-V /2k T erfc ∫-∞ 2 2
x
e-E /kBT ) kBT Q ‡
) kTST
R
κ ) 〈e-VR /2〉R
‡
B
Vx
)
x2kBTσ+
1
x1 + σ+2
2 V+ R
2kBT
e-E /kBT k) Q
(33)
where Q is the partition function of the well and the characteristic function χ(Vx) gives the probability that a trajectory starting from x ) 0 with velocity Vx will end up in the reactant well in the distant future. Tannor and Kohen calculate it from the Fokker-Planck equation. In our terminology, it is the reaction probability P+, which indeed agrees with their eq 34 if the notation is adapted appropriately. We can therefore substitute our P+ into eq 33, which yields
1
x1 + σ+2
The calculation is exactly the same as that in eq 30, except that all normalization factors for the rate constant are in place from the outset. The comparison of the result in the earlier section to the established reactive-flux formula (eq 33) makes it clearer that the requisite assumptions are those that establish the barrier ensemble. All trajectories in this ensemble are assumed to come up from the reactant side, and friction must be so strong that a trajectory crosses the dividing surface sufficiently often to forget about its past so as to establish an ensemble that is symmetric in Vx. The derivation of the Kramers rate that is presented in ref 12 also uses an exact solution of the Fokker-Planck equation on a parabolic barrier. The distribution function found in this way reduces at x ) x‡ to the Boltzmann distribution times P+, as was also pointed out by Tannor and Kohen.26 Their derivation uses the Fokker-Planck equation for a fixed initial condition on the barrier top. In some sense, it is halfway between Kramers’ original approach and ours. V. Rates Obtained from the Moving Dividing Surface A. Harmonic Systems. The review of the past literature and the central results of the previous sections suggest three methods for evaluating the transmission coefficient. The first is based on the traditional evaluation of eq 5 in trajectory simulations of the barrier ensemble (eq 7) using the fixed dividing surface to identify reactive trajectories. Alternatively, one can replace the fixed dividing surface in the step function in the numerator of eq 5 with the moving diving surface. This simple substitution removes recrossings in harmonic systems and greatly reduces the number in anharmonic systems while requiring no additional computational effort once the TS trajectory has been constructed. An efficient numerical approach for this task was proposed in ref 24. Finally, the semianalytic result of eq 32 can be evaluated numerically in a straightforward manner. This third approach requires only a sampling of the noise and therefore is potentially much more efficient than the first two options. However, it is restricted to systems that are approximately harmonic, whereas the other two approaches are valid in general. We will now compare the reliability and efficiency of these three methods for the model potential (eq 6). As in our previous work,25 we will restrict the calculations to a single realization of the noise in order to illustrate the level of microscopic detail that the time-dependent TST can provide. Carrying out a subsequent noise average to obtain a macroscopic reaction rate is, of course, straightforward and will require the same computational effort for all three schemes. The results for each of the three approaches evaluated for the same realization of the noise are displayed in Figure 2. All three methods converge to the same result, although the convergence properties of the two trajectory-based methods are very different. The transmission coefficient evaluated with the standard fixed surface displays the expected behavior. It starts from unity and shows erratic fluctuations at intermediate times
TST Rate Calculations using a Moving Dividing Surface
Figure 2. Transmission coefficients evaluated for a given realization of the noise with 50 000 ensemble trajectories. The dotted lines represent simulations using eq 5 with the fixed dividing surface, and the dashed lines are the corresponding results using the moving dividing surface. The analytic estimate from eq 32 is displayed as the solid line. For this case, the temperature kBT ) 1, the friction γ ) 1.5, the transverse frequency ωy ) 1.5, and the barrier frequency ωx ) 0.75.
as trajectories recross the dividing surface, possibly repeatedly, before finally settling to the equilibrium value. The moving dividing surface, in contrast, is initially located at some distance from the ensemble of trajectories. Thus, the results for the transmission coefficient start from zero instead of one. In the example shown in Figure 2, the surface is initially on the product side. As the members of the ensemble that have large positive velocities quickly cross the moving surface and proceed to products, the transmission coefficient increases steeply. At a later time, some trajectories that were initially started with negative velocities and were identified as nonreactive by the TST approximation will turn around and proceed to products. They contribute to a decrease in the transmission coefficient. While these trajectories have necessarily recrossed the fixed dividing surface, which leads to the irregular fluctuations seen in Figure 2, they cannot recross the moving dividing surface by definition, and the curve obtained from the moving surface is smooth. More importantly, the moving dividing surface requires trajectory simulations that need only be run for 1/4 of the time needed to obtain converged results using the traditional fixed surface. Thus, the moving surface offers the same reduction in the computational effort needed to calculate a rate, as was found in ref 25 for the identification of reactive trajectories. The generation of the moving dividing surface itself requires approximately as much effort as the simulation of a trajectory.24 This effort is negligible if one simulates several trajectories under the influence of the same noise sequence, as has been shown to be advantageous in ref 25. The most efficient approach is provided by the direct evaluation of the quantity +2 e(-VR /2) in eq 32 (when it is directly available), as it requires no trajectory simulations at all, although one must still sample over the noise. B. Anharmonic Systems. When the potential is not purely harmonic, the transmission coefficient calculated from the analytic expressions given in Section IV are no longer exact, and recrossings of the chosen dividing surface are inevitable. The advantage of using the moving dividing surface over the traditional fixed surface is that it remains approximately free of recrossings and so will still offer many of the benefits described above when the anharmonicity is small enough.25 The transmission coefficient for several values of the anharmonicity are displayed in Figure 3. The TS trajectory in each case is calculated assuming that the nonlinearity is absent. In all cases, the fixed and moving dividing surface converge to the same result as should be expected since the TS trajectory remains in
J. Phys. Chem. B, Vol. 112, No. 2, 2008 211
Figure 3. The transmission coefficient, κ(t), calculated for various values of the anharmonicity using the fixed surface (solid lines) and the moving surface (dashed lines). The parameters for each case are the same as those in Figure 2.
the vicinity of the barrier for all times, and any arbitrary trajectory must eventually escape into one of the product or reactant wells. As can be seen in every case, the transmission coefficient based on the moving surface provides a better approximation to the true transmission coefficient at all times than that based on the fixed surface. For small values of the anharmonicity, eq 32 still provides a reasonable estimate. VI. Concluding Remarks The calculation of the rate of a chemical reaction (or any other finite-dimensional process) using TS approaches and its ensuing corrections has been a primary objective of the scientific community for some time. Such calculations are even more difficult when they occur in solvents (or large-dimensional environments) because one no longer has the ability to observe or record the dynamics in the complete space. The seminal contribution of the Kramers-Grote-Hynes rate formula20-22 involved their discovery of the correction of the reduceddimensional TST rate due to the reorientation of the dividing surface caused by the solvent.33 This work represents a new attempt to correct TST rate formulas in noisy or time-dependent environments using the recently developed geometric perspective of the reactive dynamics. Two distinct derivations of the rate formula for reactive events described by the Langevin equation have been presented in this work. They, in turn, lead to new algorithms for the calculation of rates for systems in which the barrier regions may or may not be harmonic. In particular, the numerical results for the anharmonic rates show high accuracy and fast convergence even for large anharmonicities. Acknowledgment. This work was partly supported by the U.S. National Science Foundation and by the Alexander von Humboldt Foundation. The computational facilities at the CCMST have been supported under NSF Grant CHE 04-43564. J.M. would like to thank Prof. Eli Pollak and the Feinberg Graduate School of the Weizmann Institute of Science for support during the preparation of this manuscript. We dedicate our paper to Casey Hynes, whose erudition, insight, and scientific rigor is an example for all of us. References and Notes (1) Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 15, 2664. (2) Miller, W. H. Faraday Discuss. Chem. Soc. 1998, 110, 1. (3) Truhlar, D. G.; Garrett, B. C.; Klippenstein, S. J. J. Phys. Chem. 1996, 100, 12771. (4) Toller, M.; Jacucci, G.; DeLorenzi, G.; Flynn, C. P. Phys. ReV. B 1985, 32, 2082. (5) Eckhardt, B. J. Phys. A 1995, 28, 3469.
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