The full reaction dynamics

At the most fundamental level one follows the time development of the system in detail. The reactants are started in a specific initial (quantum) state and the equation of motion are propagated to give the final state. The equation of motion of the system is the time dependent Schroinger equation, or, if the atoms involved are heavy enough (not H or Li) Newtons equation. The starting point is the adiabatic potential energy surface on which the process takes place. For some reactions electronic excitations during the reaction are important and must be included in addition to the electronically adiabatic dynamics. 

The adiabatic potential energy surface is the ground state electronic energy of the system as a function of all the degrees of freedom of the system. 

Such a detailed description could in principle be made of every elementary step in a given reaction, while a dynamical simulation of a whole chemical process including several elementary steps is usually impossible. Typically intermediates in a reaction can have lifetimes which are many orders of magnitude larger than typical times for a dynamical simulation (a few picoseconds). Another point is that if the aim is to get an elementary reaction rate for a system at a given temperature, the full dynamical approach may be too detailed. 

In this description only average properties are considered. The rate of a given elementary step involving adsorbates A and B are assumed given by 

In the case where there is one slow step in the reaction mechanism, the solution for the rate of the catalytic reaction is straightforward. 

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The methods we have described in this section so far all work best in one or two epochs of the reaction dynamics. We shall now consider several methods that can be used to model the full reaction dynamics in all the epochs. In doing so, we shall, among other things, show how this division of the dynamics into epochs arises naturally from these models. 

The reaction dynamics of the internal quantum states of molecules is also relevant to understanding the kinetics of reactions such as the temperature dependence of rate constants. Furthermore, reactions such as those considered here between diatomic free radicals often have very large rate coefficients with unusual temperature dependences that are not explained using simple theories that do not treat the full reaction dynamics of the chemical reaction [7]. 

Abstract. This paper presents results from quantum molecular dynamics Simula tions applied to catalytic reactions, focusing on ethylene polymerization by metallocene catalysts. The entire reaction path could be monitored, showing the full molecular dynamics of the reaction. Detailed information on, e.g., the importance of the so-called agostic interaction could be obtained. Also presented are results of static simulations of the Car-Parrinello type, applied to orthorhombic crystalline polyethylene. These simulations for the first time led to a first principles value for the ultimate Young s modulus of a synthetic polymer with demonstrated basis set convergence, taking into account the full three-dimensional structure of the crystal. 

As a first case study to consider the impact of advection processes on reaction-diffusion dynamics (thus leading to the full reaction-diffusion-advection problem) we address here the case of a binary reaction. The limit of a fast reaction (as compared with diffusion) becomes simple to analyze, and the filament or lamellar approach developed in Sect. 2.7.1 is a very appropriate tool to understand the dynamics in simple geometries and to interpret observations in more complex ones. 

In the case of the fast binary reaction we could eliminate the reaction term from the reaction-diffusion-advection equation. But in general this is not possible. In this chapter we consider another class of chemical and biological activity for which some explicit analysis is still feasible. We consider the case in which the local-reaction dynamics has a unique stable steady state at every point in space. If this steady state concentration was the same everywhere, then it would be a trivial spatially uniform solution of the full reaction-diffusion-advection problem. However, when the local chemical equilibrium is not uniform in space, due to an imposed external inhomogeneity, the competition between the chemical and transport dynamics may lead to a complex spatial structure of the concentration field. As we will see in this chapter, for this class of chemical or biological systems the dominant processes that determine the main characteristics of the solutions are the advection and the reaction dynamics, while diffusion does not play a major role in the large Peclet number limit considered here. Thus diffusion can be neglected in a first approximation. 

Here we also assume that the reaction term does not depend explicitly on the spatial coordinate, therefore the dynamics of the medium is uniform in space. It is easy to see that the spatially uniform time-periodic oscillation is a trivial solution of the full reaction-diffusion-advection system, so the question is whether this uniform solution is stable to small non-uniform perturbations and more generally, if there are any persistent spatially non-uniform solutions in which the spatial structure does not decay in time. 

The applications of this stochastic picture to the trans-gauche isomerization of -alkanes and the boat-chair isomerization of cyclohexane have demonstrated the usefulness of this approach. In both cases, the transmission coefficients calculated from stochastic dynamics agreed quite well with those from the (later) molecular dynamics calculations, given that there can be an uncertainty in the correct value of the collision frequency to use in comparing with the full molecular dynamics in solution. Stochastic dynamics therefore can allow the rapid calculation of reaction dynamics over a wide range of solvent densities and/or viscosities. 

Such an approach results in a description of the system in terms of a reaction coordinate and a solvent coordinate. The value of such a reduced description has been shown by Gertner et al. ° for the Sn2 reaction. Their two-dimensional model reproduced many of the results of the full molecular dynamics. 

In most circumstances the spatiotemporal dynamics of reacting systems constrained to lie far from equilibrium can be described adequately by reaction-diffusion equations. These equations are valid provided the phenomena of interest occur on distance and time scales that are sufficiently long compared to molecular scales. Naturally, the complete microscopic description of the reacting medium, whether near to or far from equilibrium, must be based on the full molecular dynamics of the system, as embodied in Newton s or Schrodinger s equations of motion. 

Substitution of this reaction probability matrix in the automaton mean-field equations (7) yields the Willamowski-Rbssler rate law (24). Since the full automaton dynamics is not mean field, we can now use the automaton to investigate the mesoscopic dynamics of this reacting system. In the simulations presented below we take the diffusion coefficients of all of the species to be the same thus, henceforth we dispense with the species label and refer to this common diffusion coefficient as D. 

The automaton dynamics provides an ideal way to investigate such a possible breakdown since the mean-field limit of the automaton dynamics is the mass-action rate law and the full automaton dynamics incorporates correlations and fluctuations thus, the automaton dynamics can be compared with the mean field-limit to assess its range of validity. Such a comparison is very difficult to make in real systems since any real system is subject to both external and internal noise. Also, in physical systems the reaction mechanism is usually imperfectly known, which in turn can lead to uncertainties in the form of the rate law. In the automaton one can control the interplay between internal and external noise as well as noise arising from spatial inhomogeneities and reaction kinetics. 

Many of the species involved in the endogenous metabolism can undergo a multitude of transformations, have many reaction channels open, and by the same token, can be produced in many reactions. In other words, biochemical pathways represent a multi-dimensional space that has to be explored with novel techniques to appreciate and elucidate the full scope of this dynamic reaction system. 

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples, THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG and STABIL. 

In addition to the fact that MPC dynamics is both simple and efficient to simulate, one of its main advantages is that the transport properties that characterize the behavior of the macroscopic laws may be computed. Furthermore, the macroscopic evolution equations can be derived from the full phase space Markov chain formulation. Such derivations have been carried out to obtain the full set of hydrodynamic equations for a one-component fluid [15, 18] and the reaction-diffusion equation for a reacting mixture [17]. In order to simplify the presentation and yet illustrate the methods that are used to carry out such derivations, we restrict our considerations to the simpler case of the derivation of the diffusion equation for a test particle in the fluid. The methods used to derive this equation and obtain the autocorrelation function expression for the diffusion coefficient are easily generalized to the full set of hydrodynamic equations. 

Based on this physical view of the reaction dynamics, a very broad class of models can be constructed that yield qualitatively similar oscillations of the reaction probabilities. As shown in Fig. 40(b), a model based on Eckart barriers and constant non-adiabatic coupling to mimic H + D2, yields out-of-phase oscillations in Pr(0,0 — 0,j E) analogous to those observed in the full quantum scattering calculation. Note, however, that if the recoupling in the exit-channel is omitted (as shown in Fig. 40(b) with dashed lines) then oscillations disappear and Pr exhibits simple steps at the QBS energies. As the occurrence of the oscillation is quite insensitive to the details of the model, the interference of pathways through the network of QBS seems to provide a robust mechanism for the oscillating reaction probabilities. 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

An overview of the time-dependent wavepacket propagation approach for four-atom reactions together with the construction of ab initio potential energy surfaces sufficiently accurate for quantum dynamics calculations has been presented. Today, we are able to perform the full-dimensional (six degrees-of-freedom) quantum dynamics calculations for four-atom reactions. With the most accurate YZCL2 surface for the benchmark four-atom reaction H2 + OH <-> H+H2O and its isotopic analogs, we were able to show the following ... 

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