Transition state flux correlation function

Consider second the frequency-dependent flux Jt(co). This is an unusual concept, but can be readily defined. The zero-frequency flux is defined from the time integral of the transition state flux correlation function ... 

The transmission coefficient k was calculated from a series of transition-state trajectories by monitoring the recrossings ( = St) that occur as a function of time until each trajectory is finally trapped in the product or reactant well. The normalized reactive flux-correlation function x(t) defined in Eq. 19 was constructed from this set of trajectories 131 the result is shown in Fig. 29. From its initial value, equal to the. transition-state result, ic(<) decreases rapidly until it becomes approximately constant for an extended period. The ra-... 

Figure 29. Value of the reactive flux-correlation function (/) versus time the function is normalized to the transition-state-theory value (see the text).

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. 

To deal with the ET rate in such a case, our strategy is to combine the generalized nonadiabatic transition state theory (NA-TST) and the Zhu-Nakamura (ZN) nonadiabatic transition probability.The generalized NA-TST is formulated based on Miller s reactive flux-flux correlation function approach. The ZN theory, on the other hand, is practically free from the drawbacks of the LZ theory mentioned above. Numerical tests have also confirmed that it is essential for accurate evaluation of the thermal rate constant to take into account the multi-dimensional topography of the seam surface and to treat the nonadiabatic electronic transition and nuclear tunneling effects properly. 

Summary. Rate constants of chemical reactions can be calculated directly from dynamical simulations. Employing flux correlation functions, no scattering calculations are required. These calculations provide a rigorous quantum description of the reaction process based on first principles. In addition, flux correlation functions are the conceptual basis of important approximate theories. Changing from quantum to classical mechanics and employing a short time approximation, one can derive transition state theory and variational transition state theory. This article reviews the theory of flux correlation functions and discusses their relation to transition state theory. Basic concepts which facilitate the calculation and interpretation of accurate rate constants are introduced and efficient methods for the description of larger systems are described. Applications are presented for several systems highlighting different aspects of reaction rate calculations. For these examples, different types of approximations are described and discussed. 

Since transition state theory is derived from a classical flux correlation function, it has all shortcomings of a classical description of the reaction process. Neither tunneling, which is especially important for H-atom transfer processes or low temperature reactions, nor zero point energy effects are included in the description. Thus, the idea to develop a quantum transition state theory (QTST) which accounts for quantum effects but retains the computational advantages of the transition state approximation has been very attractive (for examples see Refs.[5, 6] and references therein). The computation of these QTST rate constants does not require the calculation of real-time dynamics and is therefore feasible for large molecular systems. 

Recently, a QUAPI procedure was developed suitable for evaluating the full flux correlation function in the case of a one-dimensional quantum system coupled to a dissipative harmonic bath and applied to obtain accurate quantum mechanical reaction rates for a symmetric double well potential coupled to a generic environment. These calculations confirmed the ability of analytical approximations to provide a nearly quantitative picture of such processes in the activated regime, where the reaction rate displays a Kramers turnover as a function of solvent friction and quantum corrections are small or moderate, They also emphasized the significance of dynamical effects not captured in quantum transition state models, in particular under small dissipation conditions where imaginary time calculations can overestimate or even underestimate the reaction rate. These behaviors are summarized in Figure 7. 

The activation free energy AA can be used to compute the TST approximation of the rate constant = Ce, where C is the preexponential factor. Because not every trajectory that reaches the transition state ends up as products, the actual rate is reduced by a factor k (the transmission coefficient) as described earlier. The transmission coefficient can be calculated using the reactive flux correlation function method. " " " Starting from an equilibrated ensemble of the solute molecules constrained to the transition state ( = 0), random velocities in the direction of the reaction coordinate are assigned from a flux-weighted Maxwell-Boltzmann distribution, and the constraint is released. The value of the reaction coordinate is followed dynamically until the solvent-induced recrossings of the transition state cease (in less than 0.1 ps). The normalized flux correlation function can be calculated using " ... 

Another important experimental observable is the rate of a chemical reaction. With a validated PES at hand, such rates can be determined in a variety of ways, ranging from transition state theory (TST) to computationally demanding fully QM rates based on flux correlation functions [99]. As MMPT is a dissociable force field, it can be used in all such formalisms. Furthermore, because the environmental modes are described with a conventional force field and therefore computationally inexpensive to evaluate, and because only the motion along the PT motif is more highly parameterized, MMPT is a particularly efficient empirical representation of the intermolecular interactions. 

There are two distinct contributions to the flux. The initial 3-correlated contribution, which gives rise to the transition state rate, and a retarded backflow j t) associated with third-body collisions. The temporal characteristics of the flux can be determined from the phase space distribution function R, t R 0), R(0)) which, for the inverted parabolic potential, is ... 

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