RRKM rate constant

The RRKM rate constant is often expressed as an average classical flux tlirough the transition state [18,19 and 20]. To show that this is the case, first recall that the density of states p( ) for the reactant may be expressed as... 

The RRKM rate constant written this way is seen to be an average flux tlirough the transition state. 

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. 

Regardless of the nature of the intramolecular dynamics of the reactant A, there are two constants of the motion in a nnimolecular reaction, i.e. the energy E and the total angular momentum j. The latter ensures the rotational quantum number J is fixed during the nnimolecular reaction and the quantum RRKM rate constant is specified as k E, J). 

If K is adiabatic, a molecule containing total vibrational-rotational energy E and, in a particular J, K level, has a vibrational density of states p[E - EjiJ,K). Similarly, the transition state s sum of states for the same E,J, and Kis [ -Eq-Ef(J,K)]. The RRKM rate constant for the Kadiabatic model is... 

In the above section a hannonic model is described for calculating RRKM rate constants with hamionic sums and densities of states. This rate constant, denoted by Ic iE, J), is related to the actual anhamionic RRKM rate constant by... 

The bulk of the infomiation about anhannonicity has come from classical mechanical calculations. As described above, the aidiannonic RRKM rate constant for an analytic potential energy fiinction may be detemiined from either equation (A3.12.4) [13] or equation (A3.12.24) [46] by sampling a microcanonical ensemble. This rate constant and the one calculated from the hamionic frequencies for the analytic potential give the aidiannonic correctiony j ( , J) in equation (A3.12.41). The transition state s aidiannonic classical sum of states is found from the phase space integral... 

The above expressions are empirical approaches, with m and D. as parameters, for including an anliamionic correction in the RRKM rate constant. The utility of these equations is that they provide an analytic fomi for the anliamionic correction. Clearly, other analytic fomis are possible and may be more appropriate. For example, classical sums of states for Fl-C-C, F1-C=C, and F1-C=C hydrocarbon fragments with Morse stretching and bend-stretch coupling anhamionicity [M ] are fit accurately by the exponential... 

Modifying equation (A3.12.45) to represent the transition state s sum of states, the aniiamionic correction to the RRKM rate constant becomes... 

It is of interest to detennine when the linewidth F( ) associated with the RRKM rate constant lc(E) equals the average distance p( ) between the reactant energy levels. From equation (A3.12.54) F( ) = Dk( ) and from the RRKM rate constant expression equation (A3.12.15) p(Ef = hl% K( )/M( - q). Equating these two... 

The analysis is performed for the calculations with rrot=0 K for the CH3C1 reactant, so that the angular momentum distribution for the complex P(j) is the distribution of orbital angular momentum for complex formation P(i). This latter distribution is given in ref. 37. Jm , the quantum number for j, varies from 282 for Enl = 0.5 kcal/mol to 357 for rel = 3.0 kcal/mol. The term k iEJ) in equation 24 is written as k (.EJ)=k Ejyf E), where k EJ) is the classical RRKM rate constant with the CH3C1 intramolecular modes inactive and / ( ) is treated as a fitting factor. 

For an electronic system, how do you define a transition state necessary for calculating an RRKM rate constant ... 

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-... 

Suppose we have a simple unimolecular dissociation embedded in a microcano-nical ensemble in phase space, in which only one dissociating channel is available. The Rice-Ramsperger-Kassel-Marcus (RRKM) rate constant is given as [9]... 

Intrinsic RRKM behavior is defined by Eq. (3), where an initial microcanonical ensemble of states decomposes exponentially with the RRKM rate constant [56]. Such dynamics can be investigated by computational chemical dynamics simulations. Therefore, an intrinsic non-RRKM molecule is one for which the intercept in P(t) is k(E), as a result of the initial microcanonical ensemble, but whose decomposition probability versus time is not described by k E). For such a molecule there is a bottleneck (or bottlenecks) restricting energy flow into the dissociating coordinate. Intrinsic RRKM and non-RRKM dynamics are illustrated in Fig. 15.3(a), (b), and (e). 

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. 

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