Microcanonical rate constant, unimolecular

RRKM theory, an approach to the calculation of the rate constant of indirect reactions that, essentially, is equivalent to transition-state theory. The reaction coordinate is identified as being the coordinate associated with the decay of an activated complex. It is a statistical theory based on the assumption that every state, within a narrow energy range of the activated complex, is populated with the same probability prior to the unimolecular reaction. The microcanonical rate constant k(E) is given by an expression that contains the ratio of the sum of states for the activated complex (with the reaction coordinate omitted) and the total density of states of the reactant. The canonical k(T) unimolecular rate constant is given by an expression that is similar to the transition-state theory expression of bimolecular reactions. 

In dynamical theories, one solves the equation of motion for the individual nuclei, subject to the potential energy surface. This is the exact approach, provided one starts with the Schrodinger equation. The aim is to calculate k(E) and kn(hi/), the microcanonical rate constants associated with, respectively, indirect (apparent or true) unimolecular reactions and true (photo-activated) unimolecular reactions. 

This is an approach for the calculation of the microcanonical rate constant k(E) for indirect unimolecular reactions that is based on several approximations. The molecule is represented by a collection of s uncoupled harmonic oscillators. According to Appendix E, such a representation is exact close to a stationary point on the potential energy surface. Furthermore, the dynamics is described by classical mechanics. 

For reactions that are unimolecular in one or both directions, the reaction rate is expected to be pressure dependent, as discussed in detail in an earlier chapter of this text. In the high-pressure limit, conventional transition state theory as described in the previous section can be applied to estimate the rate constant. The only change in equation (20) is that only a single reactant partition function appears in the denominator. The pressure dependence can then be described at various levels of sophistication, from QRRK theory to RRKM theory, to full master equation treatments using microcanonical rate constants from RRKM theory, as described in the chapter by Carstensen and Dean. Because these approaches have been described in detail there, they are not treated in the present chapter. 

RRKM theory assumes a microcanonical ensemble of A vibrational/rotational states within the energy interval E E + dE, so that each of these states is populated statistically with an equal probability [4]. This assumption of a microcanonical distribution means that the unimolecular rate constant for A only depends on energy, and not on the maimer in which A is energized. If N(0) is the number of A molecules excited at / =... 

In deriving the RRKM rate constant in section A3.12.3.1. it is assumed that the rate at which reactant molecules cross the transition state, in the direction of products, is the same rate at which the reactants fonn products. Thus, if any of the trajectories which cross the transition state in the product direction return to the reactant phase space, i.e. recross the transition state, the actual unimolecular rate constant will be smaller than that predicted by RRKM theory. This one-way crossing of the transition state, witii no recrossmg, is a fiindamental assumption of transition state theory [21]. Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

Song K and Hase W L 1999 Fitting classical microcanonical unimolecular rate constants to a modified RRK expression anharmonic and variational effects J. Chem. Phys. 110 6198-207... 

How thermal activation can take place following the Lindemann and the Lindemann-Hinshelwood mechanisms. An effective rate constant is found that shows the interplay between collision activation and unimolecular reaction. In the high-pressure limit, the effective rate constant approaches the microcanonical rate... 

RRKM theory, developed from RRK theory by Marcus and others [20-23], is the most commonly applied theory for microcanonical rate coefficients, and is essentially the formulation of transition state theory for isolated molecules. An isolated molecule has two important conserved quantities, constants of the motion , namely its energy and its angular momentum. The RRKM rate coefficient for a unimolecular reaction may depend on both of these. For the sake... 

These remarks apply equally to the complementary unimolecular reaction and it is helpful to look at the unimolecular reaction to begin with, always bearing in mind that association and dissociation are connected via the equilibrium constant. In Section 2.4.4 it was shown that for the RRKM model, the microcanonical rate coefficient is proportional to the sum of states, G, at the transition state, which is a function of the energy, E. Application of the minimum flux criterion means that G must be altered... 

Bunker [37] surmised that the unimolecular rate constant k(E) should correspond to that for a microcanonical ensemble of reactant molecules, for which every state in the energy... 

The intercept of Eq. (3) is the unimolecular rate constant k(E) for a microcanonical ensemble of reactant states. Bunker found that k(E) is well represented by the RRKM expression in Eq. (1) if anharmonicity effects are included for N E) and p( ) and if variational effects are included in identifying the transition state for reactions which do not have a barrier for the reverse association reaction [37-40]. Each of these two findings motivated extensive future studies [41-47]. 

The irregular trajectories in Fig. 15.6 display the type of motion expected by RRKM theory. These trajectories moves chaotically throughout the coordinate space, not restricted to any particular type of motion. RRKM theory requires this type of irregular motion for all of the trajectories so that the intramolecular dynamics is ergodic [1]. In addition, for RRKM behavior the rate of intramolecular relaxation associated with the ergodicity must be sufficiently rapid so that a microcanonical ensemble is maintained as the molecule decomposes [1]. This assures the RRKM rate constant k E) for each time interval f —> f + df. If the ergodic intramolecular relaxation is slower than l/k(E), the unimolecular dynamics will be intrinsically non-RRKM. 

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. 

The classical anharmonic RRKM rate constant for a fluxional molecule may be calculated from classical trajectories by following the initial decay of a microcanonical ensemble of states for the unimolecular reactant, as given by equation 1A3.12.41. Such a calculation has been performed for dissociation of the Alg and A1j3 clusters using a model analytic potential energy function written as a sum of Lennard-Jones and Axelrod-Teller potentials [30]. Stmctures of some of the Alg minima, for the potential function, are shown in figure A3.12.6. The deepest potential minimum has... 

The trajectories for this statistical ensemble will give the TS recrossing correction for the microcanonical unimolecular rate constant fc(E) in Eq. 

The Lindemann mechanism consists of three reaction steps. Reactions (1.4) and (1.5) are bimolecular reactions so that the true unimolecular step is reaction (1.6). Because the system described by Eqs. (1.4)-(l. 6) is at some equilibrium temperature, the high-pressure unimolecular rate constant is the canonical k T). This can be derived by transition state theory in terms of partition functions. However, in order to illustrate the connection between microcanonical and canonical systems, we consider here the case of k(E) and use Eq.(1.3) to convert to k(T). 

The random lifetime assumption is perhaps most easily tested by classical trajectory calculations (Bunker, 1962 1964 Bunker and Hase, 1973). Initial momenta and coordinates for the Hamiltonian of an excited molecule can be selected randomly, so that a microcanonical ensemble of states is selected. Solving Hamilton s equations of motion, Eq. (2.9), for an initial condition gives the time required for the system to reach the transition state. If the unimolecular dynamics of the molecule are in accord with RRKM theory, the decomposition probability of the molecule versus time, determined on the basis of many initial conditions, will be exponential with the RRKM rate constant. That is, the decay is proportional to exp[-k( )t]. The observation of such an exponential distribution of lifetimes has been identified as intrinsic RRKM behavior. If a microcanonical ensemble is not maintained during the unimolecular decomposition (i.e., IVR is slower than decomposition), the decomposition probability will be nonexponential, or exponential with a rate constant that differs from that predicted by RRKM theory. The implication of such trajectory studies to experiments and their relationship to quantum dynamics is discussed in detail in chapter 8. 

Anharmonic corrections have also been determined for unimolecular rate constants using classical mechanics. In a classical trajectory (Bunker, 1962, 1964) or a classical Monte Carlo simulation (Nyman et al., 1990 Schranz et al., 1991) of the unimolecular decomposition of a microcanonical ensemble of states for an energized molecule, the initial decomposition rate constant is that of RRKM theory, regardless of the molecule s intramolecular dynamics (Bunker, 1962 Bunker, 1964). This is because a... 

Analytic potential energy functions for unimolecular reactions without reverse activation energies can be obtained by semi-empirical methods or by ab initio calculations, and enhanced by experimental information such as vibrational frequencies, bond energies, etc. To determine microcanonical VTST rate constants from such a potential function, the minimum in the sum of states along the reaction path must be determined. Two approaches have been used to calculate this sum of states. 

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... 

Microcanonical TST has found wide application in the case of unimolecular reactions at the limit of high pressure (see below). In this situation, the translational partition functions for the reactants and transition state species are identical, so that eqn (1.14) for the canonical rate constant simplifies to ... 

The discussion so far has concentrated on the calculation of bimolecular rate constants for gas-phase reactions under thermal conditions. Many extensions, e.g., unimolecular reactions [10], microcanonical ensembles [8j,81,10,29], and state-selected reactions [19c,30] are described elsewhere but are not reviewed here. 

When TST is applied to a unimolecular reaction, A C, it is often called RRKM theory. One simply replaces 4> (T) by Q iT). For unimolecular reactions it is customary to consider the rate constant as a function of total energy E rather than temperature. This is called microcanonical TST since an equilibrium ensemble with a fixed total energy is called a microcanonical ensemble. The microcanonical generalized TST rate is... 

E = total energy h = Planck constant kg = Boltzmeinn constant k(B) = unimolecular micro canonical rate constant k(7) = canonical rate con stant Nq = initial number of ions formed M E - E< = number of states of the transition state up to - Elj above the critical energy E P E) = distribution of internal energies R(E,tj = rate of dissociation T = temperature Af = activation enthalpy A5 = activation entropy A5j = microcanonical entropy of activation Vr = reaction coordinate frequency p E) = density of states of the parent ion at internal energy E o= degeneracy of reaction path. 

The coefficient k( , J) is named the microscopic rate constant of unimolecular reaction. Sometimes this rate constant is named microcanonical because all states with equal e and J values are assumed to the equiprobable. If active molecules A ( e) are formed with some distribution fi[e, J) over the states with the energy e and angular moment J, the averaged rate constant < k(8, J)> is described by the equation... 

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