Ensemble micro-canonical

According to (XII) it is first of all plausible that in general the average (Eq. 62) over the canonical ensemble will be very nearly identical with the average value taken over a microcanonical or even ergodic ensemble with E=E0> In fact, in that case also Eq. (57), for example, coincides with a relationship derived by Boltzmann (1871) for ergodic ensembles.182 Furthermore, the micro-canonical ensemble is very nearly equivalent to an ensemble that is distributed (cf. Section 12c) with constant density over the shell in T-space belonging to... 

Micro-canonical ensemble fiCE (each system has constant N, V, and U the walls between systems are rigid, impermeable, and adiabatic each system keeps its number of particles, volume, and energy, and it trades nothing with neighboring systems). The relevant partition function is the microcanonical partition function Cl ( N, V, U) ... 

TST22.23 also makes the statistical approximation and invokes an equihbrium between reactant and TS. TST invokes constant temperature instead of a micro-canonical ensemble as in RRKM theory. Using statistical mechanics, the reaction rate is given by the familiar equation... 

By choosing the initial conditions for an ensemble of trajectories to represent a quantum mechanical state, trajectories may be used to investigate state-specific dynamics and some of the early studies actually probed the possibility of state specificity in unimolecular decay [330]. However, an initial condition studied by many classical trajectory simulations, but not realized in any experiment is that of a micro-canonical ensemble [331] which assumes each state of the energized reactant is populated statistically with an equal probability. The classical dynamics of this ensemble is of fundamental interest, because RRKM unimolecular rate theory assumes this ensemble is maintained for the reactant [6,332] as it decomposes. As a result, RRKM theory rules-out the possibility of state-specific unimolecular decomposition. The relationship between the classical dynamics of a micro-canonical ensemble and RRKM theory is the first topic considered here. 

Classical dynamics of a micro-canonical ensemble intrinsic RRKM and non-RRKM behavior... 

The unimolecular rate constant k E), for a micro-canonical ensemble of reactant states, is identical with the RRKM rate constant. If N 0) is the number of reactant molecules excited at t = 0 in accord with a micro-... 

The fundamental assumption of RRKM theory is that the classical motion of the reactant is sufficiently chaotic so that a micro-canonical ensemble of states is maintained as the reactant decomposes [6,324]. This assumption is often referred to as one of a rapid intramolecular vibrational energy redistribution (IVR) [12]. By making this assumption, at any time k E) is given by Eq. (62). As a result of the fixed time-independent rate constant k(E), N(t) decays exponentially, i.e.. 

A RRKM unimolecular system obeys the ergodic principle of statistical mechanics [337]. A quantity of more utility than N t), for analyzing the classical dynamics of a micro-canonical ensemble, is the lifetime distribution Pc t), which is defined by... 

It is not immediately obvious, by simply looking at a molecule s Hamiltonian and/or its PES, whether the unimolecular dynamics will be intrinsic RRKM or not and computer simulations as outlined here are required. Intrinsic non-RRKM dynamics is indicative of mode-specific decomposition, since different regions of phase space are not strongly coupled and a micro-canonical ensemble is not maintained during the fragmentation. The phase space structures, which give rise to intrinsic RRKM or non-RRKM behavior, are discussed in the next section. 

The simplest approach [338] to describe a non-ergodic unimolecular system is to assume that the reactant s phase space only consists of quasi-periodic and chaotic trajectories, whose numbers are ATqp and Nch- If a micro-canonical ensemble is prepared at t = 0 and if it is assumed that a restricted micro-canonical ensemble is maintained within the chaotic region, while no trajectory dissociates from the quasi-periodic region, the number of reactant molecules versus time is... 

If a molecule decays in a mode-specific way, the assessment of the accuracy of classical calculations is much more complicated and depends, we believe, sensitively on the initially prepared resonance state. Considering a micro-canonical ensemble certainly will not be appropriate. The initial conditions of the ensemble of trajectories should mimic the quantum mechanical distribution function of coordinates and/or momenta as closely as possible [20,385]. The gross features of the final state distributions, e.g. the peaking of the CO vibrational distribution in the dissociation of HCO close to the maximum allowed state (Fig. 36), may be qualitatively reproduced. However, more subtle structures are unlikely to be described well, because they often reflect details of the quantum wave function (reflection principle [20]). More work to explore this question is certainly needed. 

If an intrinsically-RRKM molecule with many atoms is excited non-randomly, its initial classical non-RRKM dynamics may agree with the quantum dynamics for the reasons described above. But at longer times, after a micro-canonical ensemble is created, the classical unimolecular rate constant is much larger than the quantum value, because of the zero-point energy problem. Thus, the short-time unimolecular dynamics of a large molecule will often agree quite well with experiment if the molecule is excited non-randomly. The following is a brief review of two representative... 

The microcanonical ensemble is characterized by fixed values of the thermodynamic variables N, the total particle number, V, the volume of the system, and E, the total energy. The phase space distribution function for the micro-canonical ensemble is... 

To use the master equation, one needs a general formula for the rate constant, kj, out of minimum j through transition state f. In the micro-canonical ensemble this relation is provided by Rice-Ramsperger-Kassel-Marcus (RRKM) theory [166] ... 

For many of the model molecules studied by the trajectory simulations, the decay of P t) was exponential with a decay constant equal to the RRKM rate constant. However, for some models with widely disparate vibrational frequencies and/or masses, decay was either nonexponential or exponential with a decay constant larger than k E) determined from the intercept of P(f). This behavior occurs when some of the molecule s vibrational states are inaccessible or only weakly coupled. Thus, a micro-canonical ensemble is not maintained during the molecule s decomposition. These studies were a harbinger for what is known now regarding inelficient intramolecular vibrational energy redistribution (IVR) in weakly coupled systems such as van der Waals molecules and mode-specific unimolecular dynamics. 

As will be discussed in chapter 6, of fundamental importance in the theory of unimolecular reactions is the concept of a microcanonical ensemble, for which every zero-order state within an energy interval AE is populated with an equal probability. Thus, it is relevant to know the time required for an initially prepared zero-order state j) to relax to a microcanonical ensemble. Because of low resolution and/or a large number of states coupled to i), an experimental absorption spectrum may have a Lorentzian-like band envelope. However, as discussed in the preceding sections, this does not necessarily mean that all zero-order states are coupled to r) within the time scale given by the line width. Thus, it is somewhat unfortunate that the observation of a Lorentzian band envelope is called the statistical limit. In general, one expects a hierarchy of couplings between the zero-order states and it may be exceedingly difficult to identify from an absorption spectrum the time required for IVR to form a micro-canonical ensemble. 

To conclude this section, for many reactant molecules it is expected that a micro-canonical ensemble of resonance states will contain states which exhibit mode-specific decay and can be identified by patterns (i.e., progressions) in the spectrum, as well as unassignable states with random i and, thus, state-specific rate constants with random fluctuations. In general, it is not expected that the ij , which form a microcanonical ensemble, will have identical k which equal the RRKM k(E). 

Thus, for state-specific decay and the most statistical (or nonseparable) case, a micro-canonical ensemble does not decay exponentially as predicted by RRKM theory. It is worthwhile noting that when v/2 becomes very large, the right-hand side of Eq. (8.24) approaches exp -kt) (Miller, 1988), since lim (1 + xln) " = exp (-x), when n-> °o. Other distributions for P(k), besides the Porter-Thomas distribution, have been considered and all give M(f, E) expressions which are nonexponential (Lu and Hase, 1989b). 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

Figure 4 Variations of translational kinetic energy (a) micro-canonical ensemble (b) tight thermostating (frequency about four times higher than the natural one) (c) optimum thermostating (response time about half again of the resonance value).

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