Approximate microcanonical ensemble

This sampling, with the normal-mode/rigid-rotor Hamiltonian, provides an exact microcanonical ensemble for this Hamiltonian, but an approximate microcanonical ensemble for the actual anharmonic and reactive Hamiltonian with vibrational-rotational coupling. 

This section deals with the question of how to approximate the essential features of the flow for given energy E. Recall that the flow conserves energy, i.e., it maps the energy surface Pq E) = x e P H x) = E onto itself. In the language of statistical physics, we want to approximate the microcanonical ensemble. However, even for a symplectic discretization, the discrete flow / = (i/i ) does not conserve energy exactly, but only on... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

If the statistical approximation were correct, one could estimate the rate constant for a microcanonical ensemble of reactant molecules by estimating the volume of... 

The RRKM theory is the most widely used of the microcanonical, statistical kinetic models It seeks to predict the rate constant with which a microcanonical ensemble of molecules, of energy E (which is greater than Eq, the energy of the barrier to reaction) will be converted to products. The theory explicitly invokes both the transition state hypothesis and the statistical approximation described above. Its result is summarized in Eq. 2... 

In order to remove the need for explicit trajectory analysis, one makes the statistical approximation. This approximation can be formulated in a number of equivalent ways. In the microcanonical ensemble, all states are equally probable. Another formulation is that the lifetime of reactant (or intermediate) is random and follows an exponential decay rate. But perhaps the simplest statement is that intramolecular vibrational energy redistribution (IVR) is faster than the reaction rate. IVR implies that if a reactant is prepared with some excited vibrational mode or modes, this excess energy will randomize into all of the vibrational modes prior to converting to product. 

RRKM theory assumes both the statistical approximation and the existence of the TS. It assumes a microcanonical ensemble, where all the molecules have equivalent energy E. This energy exceeds the energy of the TS (Eq), thanks to vibration, rotation, and/or translation energy. Invoking an equilibrium between the TS (the activated complex) and reactant, the rate of reaction is... 

The microcanonical ensemble may be depleted in the vicinity of the transition state by the absence of trajectories in the reverse direction. This assumption is often referred to as the ergodic approximation, that the microcanonical ensemble is rapidly randomized behind the reaction bottleneck faster that reactive loss can perturb the distribution. 

The key idea that supplements RRK theory is the transition state assumption. The transition state is assumed to be a point of no return. In other words, any trajectory that passes through the transition state in the forward direction will proceed to products without recrossing in the reverse direction. This assumption permits the identification of the reaction rate with the rate at which classical trajectories pass through the transition state. In combination with the ergodic approximation this means that the reaction rate coefficient can be calculated from the rate at which trajectories, sampled from a microcanonical ensemble in the reactants, cross the barrier, divided by the total number of states in the ensemble at the required energy. This quantity is conveniently formulated using the idea of phase space. 

The classical unimolecular dynamics is ergodic for molecules like NO2 and D2CO, whose resonance states are highly mixed and unassignable. As described above, their unimolecular dynamics is identified as statistical state specific. The classical dynamics for these molecules are intrinsically RRKM and a microcanonical ensemble of phase space points decays exponentially in accord with Eq. (3). The correspondence found between statistical state specific quantum dynamics and quantum RRKM theory is that the average of the N resonance rate constants fe,) in an energy window E + AE approximates the quantum RRKM rate constant k E) [27,90]. Because of the state specificity of the resonance rates, the decomposition of an ensemble of the A resonances is non-exponential, i.e. 

We now have established a framework for the thermodynamic properties of a model system. What would be desirable is that we could approximate the force terms arising in the bulk, without the need to simulate them directly. It should be apparent that constant-energy dynamics (designed to sample the microcanonical ensemble with constant energy E) will not sample the canonical distribution in the absence of the heat bath such Newtonian trajectories cannot access regions of the phase space where H z) H(zo), where zo is the initial condition. 

The partition function Z, which normalizes the density, is effectively a function of N, V and E it represents the number of microstates available under given conditions. As this ensemble is associated to constant particle number N, volume V and energy E, it is often referred to as the NVE-ensemble, and when we speak of NVE simulation, we mean simulation that is meant to preserve the microcanonical distribution this, most often, would be based on approximating Hamiltonian dynamics, e.g. using the Verlet method or another of the methods introduced in Chaps. 2 and 3, and assuming the ergodic property. For a discussion of alternative stochastic microcanonical methods see [126]. 

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