RRKM method

Ion-molecule radiative association reactions have been studied in the laboratory using an assortment of trapping and beam techniques.30,31,90 Many more radiative association rate coefficients have been deduced from studies of three-body association reactions plus estimates of the collisional and radiative stabilization rates.91 Radiative association rates have been studied theoretically via an assortment of statistical methods.31,90,96 Some theoretical approaches use the RRKM method to determine complex lifetimes others are based on microscopic reversibility between formation and destruction of the complex. The latter methods can be subdivided according to how rigorously they conserve angular momentum without such conservation the method reduces to a thermal approximation—with rigorous conservation, the term phase space is utilized. 

Powerful formalisms such as the Rice-Rampsperger-Kassel-Marcus (RRKM) method exist to analyze simple energy-transfer-limited uni-molecular reactions in detail (see, for example, Robinson and Holbrook,... 

The most recent determination, made using relative rate methods [101], gives a value approximately a factor of two higher than either the extrapolation of the low pressure data of Plumb and Ryan [102] or the atmospheric pressure measurement of Munk et al. [98]. However, it is close to the extrapolation of the low pressure data of Wagner et al. [108] made using variational RRKM methods. 

The most realistic description of the kinetics of a unimolecular reaction is given by the RRKM method,3,16 which has been successfully used in the investigations of a wide variety of reaction systems. However, the general equation of the RRKM method is quite complex, and values of the rate constant at a given temperature and pressure are obtained by numerical evaluation of a complicated integral expression. This is an important limitation of the RRKM method because kinetic modeling studies require a simple expression, best in an analytical form, convenient for estimating the rate constant under any experimental conditions of pressure and temperature. 

Another more direct approach has recently been used to define the excitation function for a reaction product from a recoil reaction (21). Instead of using an RRKM method for a single competitive unimolecular process, the sequential decomposition of excited CF4 was studied. The fragments formed were scavenged with CI2 to give direct yield data for each path as indicated below ... 

Vibrational frequencies for various normal modes must be estimated and active as well as inactive energies should be decided. Numerical methods may be used to calculate rate constant k at various concentrations obtained by RRKM theory. The rate constant has been found to be same as given by conventional transition state theory, i.e. 

However, central to any truly accurate determination of the radiative rate are the integrated absorption (emission) intensities, A", which for gaseous ions are almost completely unknown as are, usually, the vibrational frequencies. Fortunately, however, ab initio and density functional methods have recently been shown to be quite accurate in their predictions of vibrational spectra for a wide variety of systems, and there is no reason to suspect that this accuracy would not carry over to comparable data for gaseous ions. The one caveat must be that the low-frequency modes that are common in cluster ions will be decidedly anharmonic, and prediction of both these frequencies and their intensities may be suspect. However, these modes are not generally expected to be dominant contributors to the overall radiative rate. In addition, standard RRKM procedures can be applied to the unimolecular dissociation of the same adduct ions and, in principle therefore, the overall kinetics of formation of stabilized association complexes can be accurately modeled. 

Transition state theory yields rate coefficients at the high-pressure limit (i.e., statistical equilibrium). For reactions that are pressure-dependent, more sophisticated methods such as RRKM rate calculations coupled with master equation calculations (to estimate collisional energy transfer) allow for estimation of low-pressure rates. Rate coefficients obtained over a range of temperatures can be used to obtain two- and three-parameter Arrhenius expressions ... 

R. A. Marcus It certainly is a good point that transition state theory, and hence RRKM, provides an upper bound to the reactive flux (apart from nuclear tunneling) as Wigner has noted. Steve Klippenstein [1] in recent papers has explored the question of the best reaction coordinate, e.g., in the case of a unimolecular reaction ABC — AB + C, where A, B, C can be any combination of atoms and groups, whether the BC distance is the best choice for defining the transition state, or the distance between C and the center of mass of AB, or some other combination. The best combination is the one which yields the minimum flux. In recent articles Steve Klippenstein has provided a method of determining the best (in coordinate space) transition state [1]. 

The thermodynamic stability of unsubstituted silacyclopropane to fragmentation has been studied by ab initio quantum mechanical methods and the enthalpy of decomposition to H2Si + CH2=CH2 was predicted to be 44.878 and 43.279 kcalmol-1. There is indirect experimental support for these theoretical estimates. When these values were employed in RRKM calculations on silirane decomposition, the pressure dependence of the bimolecular rate constant for addition of H2Si to ethylene could be accurately modeled80. 

One example of non-IRC trajectory was reported for the photoisomerization of cA-stilbene.36,37 In this study trajectory calculations were started at stilbene in its first excited state. The initial stilbene structure was obtained at CIS/6-31G, and 2744 argon atoms were used as a model solvent with periodic boundary conditions. In order to save computational time, finite element interpolation method was used, in which all degrees of freedom were frozen except the central ethylenic torsional angle and the two adjacent phenyl torsional angles. The solvent was equilibrated around a fully rigid m-stilbene for 20 ps, and initial configurations were taken every 1 ps intervals from subsequent equilibration. The results of 800 trajectories revealed that, because of the excessive internal potential energy, the trajectories did not cross the barrier at the saddle point. Thus, the prerequisites for common concepts of reaction dynamics such TST or RRKM theory were not satisfied. 

The lowest-lying potential energy surfaces for the 0(3P) + CH2=C=CH2 reaction were theoretically characterized using CBS-QB3, RRKM statistical rate theory, and weak-collision master equation analysis using the exact stochastic simulation method. The results predicted that the electrophilic O-addition pathways on the central and terminal carbon atom are dominant up to combustion temperatures. Major predicted end-products are in agreement with experimental evidence. New H-abstraction pathways, resulting in OH and propargyl radicals, have been identified.254... 

Rice et al. [99] developed a global potential energy surface based on the Mowrey et al. [103] results and performed extensive classical trajectory calculations to study the dynamics of the CH2NN02 dissociation reactions. They calculated rates for reactions (III) and (IV) with classical barriers of 35 and 37 kcal/mol, respectively. They found that N-N bond fission dominates at low energy but that HONO elimination is competitive. Chakraborty and Lin [104] predict the opposite on the basis of their ab initio barriers and RRKM theory calculations. The two dissociations channels are closely competitive and it is not clear that ab initio methods are sufficiently reliable to distinguish between two reactions that have such similar energy requirements. Also, the Zhao et al. results [33] are not in accord with the theoretical predictions. 

It must be emphasized that such phenomena are to be expected for a statistical system only in the regime of low level densities. Theories like RRKM and phase space theory (PST) (Pechukas and Light 1965) are applicable when such quantum fluctuations are absent for example, due to a large density of states and/or averaging over experimental parameter such as parent rotational levels in the case of incomplete expansion-cooling and/or the laser linewidth in ultrafast experiments. However, in the present case, it is unlikely that such phenomena can be invoked to explain why different rates are obtained when using ultrafast pump-probe methods that differ only in experimental detail. 

A more general discussion of the dependence of the decomposition rate on internal energy was developed by Marcus and Rice [4] and further refined and applied by Marcus [5] (RRKM). Their method is to obtain the reaction rate by summing over each of the accessible quantum states of the transition complex. The first-order rate coefficient for decomposition of an energised molecule is shown to be proportional to the ratio of the total internal quantum states of the transition complex divided by the density of states (states per unit energy) of the excited molecule. It is a great advance over previous theory because it can be applied to real molecules, counting the states from the known vibrational frequencies. 

In general, the structure and frequencies of the transition complex are not known for unimolecular reactions and, consequently, neither transition state theory nor detailed RRKM calculations can be tested. However, provided a physically plausible choice is made which will match the koc over the range of measured temperatures, the derived ft (e) are only slightly dependent on the particular model selected. Details of these procedures are available [11—13] and an excellent discussion is given by Robinson and Holbrook [11]. Readers should also refer to the detailed methods used by Schneider and Rabinovitch [14] for the CH3NC isomerisation. The following brief comments are intended to complete this introductory outline of the basic theory and to show how it may be applied. 

Statistical methods represent a background for, e.g., the theory of the activated complex (239), the RRKM theory of unimolecular decay (240), the quasi-equilibrium theory of mass spectra (241), and the phase space theory of reaction kinetics (242). These theories yield results in terms of the total reaction cross-sections or detailed macroscopic rate constants. The RRKM and the phase space theory can be obtained as special cases of the single adiabatic channel model (SACM) developed by Quack and Troe (243). The SACM of unimolecular decay provides information on the distribution of the relative kinetic energy of the products released as well as on their angular distributions. 

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