RRKM theory microcanonical

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 / =... 

The rapid IVR assumption of RRKM theory means that a microcanonical ensemble is maintained as the A molecnles decompose so that, at any time t, k(E) is given by... 

Figure A3,12.2(a) illnstrates the lifetime distribution of RRKM theory and shows random transitions among all states at some energy high enongh for eventual reaction (toward the right). In reality, transitions between quantum states (though coupled) are not equally probable some are more likely than others. Therefore, transitions between states mnst be snfficiently rapid and disorderly for the RRKM assumption to be mimicked, as qualitatively depicted in figure A3.12.2(b). The situation depicted in these figures, where a microcanonical ensemble exists at t = 0 and rapid IVR maintains its existence during the decomposition, is called intrinsic RRKM behaviour [9].

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. 

As a result of possible recrossings of the transition state, the classical RRKM lc(E) is an upper bound to the correct classical microcanonical rate constant. The transition state should serve as a bottleneck between reactants and products, and in variational RRKM theory [22] the position of the transition state along q is varied to minimize k E). This minimum k E) is expected to be the closest to the truth. The quantity actually minimized is N (E - E ) in equation (A3.12.15). so the operational equation in variational RRKM theory is... 

In the above discussion it was assumed that the barriers are low for transitions between the different confonnations of the fluxional molecule, as depicted in figure A3.12.5 and therefore the transitions occur on a timescale much shorter than the RRKM lifetime. This is the rapid IVR assumption of RRKM theory discussed in section A3.12.2. Accordingly, an initial microcanonical ensemble over all the confonnations decays exponentially. However, for some fluxional molecules, transitions between the different confonnations may be slower than the RRKM rate, giving rise to bottlenecks in the unimolecular dissociation [4, ]. The ensuing lifetime distribution, equation (A3.12.7), will be non-exponential, as is the case for intrinsic non-RRKM dynamics, for an mitial microcanonical ensemble of molecular states. 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

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... 

II. Microcanonical Solvent Dynamics Modified RRKM Theory... 

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. 

First, we want to derive an expression for the microcanonical rate constant k(E) when the total internal energy of the reactant is in the range E to E + dE. From Eq. (7.43), the rate of reaction is given by the rate of disappearance of A or, equivalently, by the rate at which activated complexes A pass over the barrier, i.e., the flow through the saddle-point region. The essential assumptions of RRKM theory are equivalent to the assumptions underlying transition-state theory. 

The dimension of the factor IIf=1z/j/IIlz v is that of a frequency. If the frequencies of the reactant and the activated complex are not too different, this frequency is roughly a typical vibrational frequency vr (typically in the range 1013 to 1014 s 4). Since the energy-dependent factor is less than one, we have that the microcanonical rate constant k(E) < i/r, i.e., it is less than a typical vibrational frequency. The energy dependence as a function of the number of vibrational degrees of freedom was illustrated in Fig. 7.3.2, and as shown previously in Eq. (7.38) it can be interpreted as the probability that the energy in one out of s vibrational modes exceeds the energy threshold Eq for the reaction. Note that if we make the identification vr n =1z/i/n 11i/ , we have recovered RRK theory, Eq. (7.39), from RRKM theory. 

In the RRKM theory, the microcanonical rate constant k(E, J) at a given E and total angular momentum quantum number / is given by [62, 68],... 

The microcanonical rate constant at energy E can be expressed using the RRKM theory [48],... 

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... 

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... 

Variational transition state theory was suggested by Keck [36] and developed by Truhlar and others [37,38]. Although this method was originally applied to canonical transition state theory, for which there is a unique optimal transition state, it can be applied in a much more detailed way to RRKM theory, in which the transition state can be separately optimized for each energy and angular momentum [37,39,40]. This form of variational microcanonical transition state theory is discussed at length in Chapter 2, where there is also a discussion of the variational optimization of the reaction coordinate. 

Chapter 3 deals with an even more fundamental difficulty of RRKM theory. The theory expresses the microcanonical rate coefficient in terms of a sum and... 

The determination of the microcanonical rate coefficient k E) is the subject of active research. A number of techniques have been proposed, and include RRKM theory (discussed in more detail in Section 2.4.4) and the derivatives of this such as Flexible Transition State theory. Phase Space Theory and the Statistical Adiabatic Channel Model. All of these techniques require a detailed knowledge of the potential energy surface (PES) on which the reaction takes place, which for most reactions is not known. As a consequence much effort has been devoted to more approximate techniques which depend only on specific PES features such as reaction threshold energies. These techniques often have a number of parameters whose values are determined by calibration with experimental data. Thus the analysis of the experimental data then becomes an exercise in the optimization of these parameters so as to reproduce the experimental data as closely as possible. One such technique is based on Inverse Laplace Transforms (ILT). 

Modern unimolecular theory has its origins in the work of Rice, Ramsberger and Kassel [44] who investigated the rate of dissociation of a molecule as a function of energy. Marcus and Rice [44] subsequently extended the theory to take account of quantum mechanical features. This extended theory, referred to as RRKM theory, is currently the most widely used approach and is usually the point of departure for more sophisticated treatments of unimolecular reactions. The key result of RRKM theory is that the microcanonical rate coefficient can be expressed as... 

State specific experiments can now test unimolecular rate theories by probing microcanonical rate coefficients. Moore and coworkers [45] have studied the dissociation of ketene close to the reaction threshold in an attempt to test RRKM theory. 

For some unimolecular reactions involving large molecules in supersonic beams, the microcanonical RRKM theory apjjears to overestimate (by about one order of magnitude) the rate. This has been reported for t-stilbene and for where A is anthracene and is N, N-... 

Intramolecular hydrogen transfer is another important class of chemical reactions that has been widely studied using transition state theory. Unimolecular gas-phase reactions are most often treated using RRKM theory [60], which combines a microcanonical transition state theory treatment of the unimolecular reaction step with models for energy redistribution within the molecule. In this presentation we will focus on the unimolecular reaction step and assume that energy redistribution is rapid, which is equivalent to the high-pressure limit of RRKM theory. 

Trajectory calculations have been used to study the intrinsic RRKM and apparent non-RRKM dynamics of ethyl radical dissociation, i.e. C2H5 — H - - C2H4 [61,62]. When C2H5 is excited randomly, with a microcanonical distribution of states, it dissociates with the exponential P t) of RRKM theory [61], i.e. it is an intrinsic RRKM molecule. However, apparent non-RRKM behavior is present in a trajectory simulation of C2H5... 

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. 

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