Microcanonical process

The formation of the transition state from the excited molecule is referred to as a microcanonical process, while the formation of the transition state in conventional TST in Chapter4 and in VTST in Chapter 6 is referred to as canonical process. The terms microcanonical and canonical in statistical mechanics refer respectively to processes at constant energy and processes at constant temperature. 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

At the start of the production phase all counters are set to zero and the system is permitted t< evolve. In a microcanonical ensemble no velocity scaling is performed during the produc tion phase and so the temperature becomes a calculated property of the system. Varioui properties are routinely calculated and stored during the production phase for subsequen analysis and processing. Careful monitoring of these properties during the simulation car show whether the simulation is well behaved or not it may be necessary to restart i simulation if problems are encountered. It is also usual to store the positions, energie ... 

Once the boundary conditions have been implemented, the calculation of solution molecular dynamics proceeds in essentially the same manner as do vacuum calculations. While the total energy and volume in a microcanonical ensemble calculation remain constant, the temperature and pressure need not remain fixed. A variant of the periodic boundary condition calculation method keeps the system pressure constant by adjusting the box length of the primary box at each step by the amount necessary to keep the pressure calculated from the system second virial at a fixed value (46). Such a procedure may be necessary in simulations of processes which involve large volume changes or fluctuations. Techniques are also available, by coupling the system to a Brownian heat bath, for performing simulations directly in the canonical, or constant T,N, and V, ensemble (2,46). 

When the total energy was decreased in these experiments, the decay became slower than 30 ps. This energy dependence of the process could be understood considering the time scale for 1VR and the microcanonical rates, k(E) at a given energy, in a statistical RRKM description. Such dynamics of consecutive bond breakage are common to many systems and are also relevant to the mechanism in different classes of reactions discussed in the organic literature. 

In this chapter we consider the problem of reaction rates in clusters (micro-canonical) modified by solvent dynamics. The field is a relatively new one, both experimentally and theoretically, and stems from recent work on well-defined clusters [1, 2]. We first review some theories and results for the solvent dynamics of reactions in constant-temperature condensed-phase systems and then describe two papers from our recent work on the adaptation to microcanonical systems. In the process we comment on a number of questions in the constant-temperature studies and consider the relation of those studies to corresponding future studies of clusters. 

Elementary processes in chemical dynamics are universally important, besides their own virtues, in that they can link statistical mechanics to deterministic dynamics based on quantum and classical mechanics. The linear surprisal is one of the most outstanding discoveries in this aspect (we only refer to review articles [2-7]), the theoretical foundation of which is not yet well established. In view of our findings in the previous section, it is worth studying a possible origin of the linear surprisal theory in terms of variational statistical theory for microcanonical ensemble. 

Defining the microcanonical temperature as a kinetic energy that maximizes a phase-space distribution when projected onto the potential energy coordinate, we have shown that this temperature can characterize a time scale of structural isomerization dynamics in the liquid-like phase. In particular, it has been found that the local microcanonical temperature bears an Arrhenius-like relation to the inverse of the average lifetime in isomerization of M7 clusters. Thus, with this temperature one can extract critical information hidden behind the stepwise fluctuation of the kinetic energy of a trajectory in an isomerization process [33]. We have explored a possible origin of the Arrhenius-like relation. 

These phenomena lead us to a rather complicated situation. The first phenomenon reminds us of ergodicity, the realization of microcanonical distribution in systems with many degrees of freedom and the validity of statistical mechanics. We know that KAM tori cannot divide the phase space (or energy surface) for systems with many degrees of freedom, and the first phenomenon tells us that two neighborhoods in different parts of the phase space are connected not only topologically but also dynamically. In this sense the phenomenon can be considered as an elementary process of relaxation in systems with many degrees of freedom. 

The physical process of protein folding involves a phase transition from a statistical coiled state to a uniquely compact native state. A powerful approach to define these systems is to make use of the microcanonical entropy function [18-27]. The entropy function S(E) is related to the... 

An exact knowledge of the microcanonical entropy, or the density of the states, of a protein model in both the native and nonnative states is crucial for a precise characterization of the folding process of the model, such as whether the folding is first-order or gradual, whether the model can fold uniquely to the native structure, whether there is a discontinuity in the order parameter of the conformation in the folding transition, and so on. Once the microscopic entropy function is accurately determined, the statistical mechanics of the protein folding problem is solved. The accurate determination of the entropy function of protein models by the ESMC method requires a proper treatment of the computational problems discussed in Section IV. 

Classical trajectory studies of unimolecular decomposition have helped define what is meant by RRKM and non-RRKM behavior (Bunker, 1962, 1964 Bunker and Hase, 1973 Hase, 1976, 1981). RRKM theory assumes that the phase space density of a decomposing molecule is uniform. A microcanonical ensemble exists at t = 0 and rapid intramolecular processes maintain its existence during the decomposition [fig. 8.9(a), (b)]. The lifetime distribution, Eq. (8.35a), is then... 

It is also interesting to consider the classical/quantal correspondence in the number of energized molecules versus time N(/, E), Eq. (8.22), for a microcanonical ensemble of chaotic trajectories. Because of the above zero-point energy effect and the improper treatment of resonances by chaotic classical trajectories, the classical and quantal I l( , t) are not expected to agree. For example, if the classical motion is sufficiently chaotic so that a microcanonical ensemble is maintained during the decomposition process, the classical N(/, E) will be exponential with a rate constant equal to the classical (not quantal) RRKM value. However, the quantal decay is expected to be statistical state specific, where the random 4i s give rise to statistical fluctuations in the k and a nonexponential N(r, E). This distinction between classical and quantum mechanics for Hamiltonians, with classical f (/, E) which agree with classical RRKM theory, is expected to be evident for numerous systems. 

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