Reactive flux method

Den Otter, W.K., Briels, W.J. The reactive flux method applied to complex reactions using the unstable normal mode as a reaction coordinate. J. Chem. Phys. 106 (1997) 1-15. 

From a practical point of view, integrating trajectories for times which are of the order of eP is very expensive. When the reduced barrier height is sufficiently large, then solution of the Fokker-Planck equation also becomes numerically very difficult. It is for this reason, that the reactive flux method, described below has become an invaluable computational tool. 

The major advantage of the reactive flux method is that it enables one to initiate trajectories at the barrier top. instead of at reactants or products. Computer time is not wasted by waiting for the particle to escape from the well to the barrier. The method is based on the validity of Onsager s regression hypothesis/ which assures that fluctuations about the equilibrium state decay on the average with the same rate as macroscopic deviations from equilibrimn. It is sufficient to know the decay rate of equilibrimn correlation fimctions. There isn t any need to determine the decay rate of the macroscopic population as in the previous subsection. 

In this central result the choice of the point q(0) is arbitrary. This means that at time t = 0 one can initiate trajectories anywhere and after a short induction time the reactive flux will reach a plateau value, which relaxes exponentially, but at a very slow rate, It is this independence on the initial location which makes the reactive flux method an important nmnerical tool. 

In the very short time limit, q (t) will be in the reactants region if its velocity at time t = 0 is negative. Therefore the zero time limit of the reactive flux expression is just the one dimensional transition state theory estimate for the rate. This means that if one wants to study corrections to TST, all one needs to do munerically is compute the transmission coefficient k defined as the ratio of the numerator of Eq. 14 and its zero time limit. The reactive flux transmission coefficient is then just the plateau value of the average of a unidirectional thermal flux. Numerically it may be actually easier to compute the transmission coefficient than the magnitude of the one dimensional TST rate. Further refinements of the reactive flux method have been devised recently in Refs. 31,32 these allow for even more efficient determination of the reaction rate. 

To summarize, the reactive flux method is a great help but it is predicated on a time scale separation, which results from the fact that the reaction time (1/T) is very long compared to all other times. This time scale separation is valid, only if the reduced barrier height is large. In this limit, the reactive flux method, the population decay method and the lowest nonzero eigenvalue of the Fokker-Planck equation all give the same result up to exponentially small corrections of the order of For small reduced barriers, there may be noticeable differences between the different definitions and as aheady mentioned each case must be handled with care. 

As shown by TalkneP there is a direct connection between the Rayleigh quotient method and the reactive flux method. Two conditions must be met. The first is that phase space regions of products must be absorbing. In different terms, the trial function must decay to zero in the products region. The second condition is that the reduced barrier height pyl" 1. As already mentioned above, differences between the two methods will be of the order e P. ... 

Drozdov and Tucker have recently criticized the VTST method claiming that it does not bound the exact rate constant. Their argument was that the reactive flux method in the low barrier limit, is not identical to the lowest nonzero eigenvalue of the corresponding Fokker-Planck operator, hence an upper bound to the reactive flux is not an upper bound to the true rate. As aheady discussed above, when the barrier is low, the definition of the rate becomes problematic. All that can be said is that VTST bounds the reactive flux. Whenever the reactive flux method fails, VTST will not succeed either. 

The main advantage of the VTST method is that it can be applied also to realistic simulations of reactions in condensed phases.The optimal planar coordinate is determined by the matrix of the thermally averaged second derivatives of the potential at the barrier top. VTST has been applied to various models of the CP-i-CHsCl Sn2 exchange reaction in water, a system which was previously studied extensively by Wilson, Hynes and coworkers.Excellent agreement was found between the VTST predictions for the rate constant and the numerically exact results based on the reactive flux method. The VTST method also allows one to determine the dynamical source of the friction and its range, since it identifies a collective mode which has varying contributions from differ-... 

The first poly chalcogenide complex, K4USeg, was obtained by a solid-state reaction. It has a molecular sfructure with a distorted dodecahedral anion, [U(Se2)4] , which is isosfructural with the known peroxoanions [M(02)4]" , where M = V, Nb, Ta, Cr (n = 3) or Mo, W (n = 2). Recently, two additional uranium selenides have been synthesized, MU2Se6 (M = K+, Cs+), using a reactive flux method. The oxidation state of the uranium in these compounds was found to be tetravalent. The selenium has two distinct oxidation states, Se and one similar to a polyselenide network. 

A brief review of some of the basic concepts and numerical methods used for the study of classical reaction dynamics in condensed phases is presented in Sec. II. Of central importance is the reactive flux method (5,10,37,41) without which it would be... 

The major advantage of the reactive flux method is that it enables one to initiate trajectories at the barrier top instead of at reactants or products. In this way, one saves all the wasted computer time waiting for the particle to escape from the well to the barrier. 

The VTST study of this same system has been described in detail in Refs. 82 and 83. The main results are that there is excellent agreement between the VTST predictions for the rate constants and the numerically exact results based on the reactive flux method. Convergence of the iteration method described in this section is very fast, at worst one needed five iterations, typically, two or three were sufficient. The friction function generated by the VTST method was in excellent agreement with the ad hoc procedure used in Refs. 48 and 49, this was attributed to the relatively fast time scale of the reaction, such that only the near vicinity of the barrier really contributes to the friction function. 

The reactive flux method is also useful in calculating rate constants in quantum systems. The path integral formulation of the reactive flux together with the use of the centroid distribution function has proved very useful for the calculation of quantum transition-state rate constants [7]. In addition new methods, such as the Meyer-Miller method [8] for semiclassical dynamics, have been used to calculate the flux-flux correlation function and the reactive flux. 

The reactive flux method is widely used to compute rate constants. Several reviews [9, 10] illustrate its utility. Because of this utility, the title paper represents an important milestone in theoretical chemical kinetics. 

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