Microcanonical rate coefficients

This reaction was investigated by Klippenstein and Harding [57] using multireference configuration interaction quantum chemistry (CAS + 1 + 2) to define the PES, variable reaction coordinate TST to determine microcanonical rate coefficients, and a one-dimensional (ID) master equation to evaluate the temperature and pressure dependence of the reaction kinetics. There are no experimental investigations of pathway branching in this reaction. 

Although it would be possible to use the approximate microcanonical rate coefficient in the general framework for pressure-dependent rate coefficients, Eq. (24), it is an improvement to use the more detailed Slater model. Unfortunately the integrals required using Eq. (50) are too complicated, and so Slater developed a more tractable approximate form, which is valid when the critical value of the controlling coordinate is close to the maximum possible value for the given energy. 

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... 

The RRKM microcanonical rate coefficient is the rate at which states pass through the transition state (per unit energy) divided by the total density of states of the reactants. 

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 shape of the fall-off must then be corrected, both for the energy-dependence of the microcanonical rate coefficient in the strong collision approximation, and for the fact that the collisions are not, in fact, strong. 

The first term represents the rate of transitions to E from all other energies on collision. The second term is the rate at which collisions remove density from energy E and the third term is the rate at which density at energy E is removed by reaction, k E) being the microcanonical rate coefficient. 

The generation of pressure-dependent thermal rate coefficients from such microcanonical rate coefficients involves some averaging over a collision induced distribution. This averaging can once again reduce some of the errors due to the neglect of vibrational anharmonicities. In particular, at fairly high pressures, the distribution function is close to Boltzmann. In the calculation of thermal rate coefficients the errors in the distribution function then cancel with those in the microcanonical rate coefficients, just as in the high pressure limit. 

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). 

The ILT technique may be used to calculate microcanonical rate coefficients from experimental Arrhenius expressions of the high pressure canonical rate coefficient. The idea was first proposed by Slater [38] who observed that the standard expression that relates the microcanonical and canonical rate coefficients can be written in terms of a Laplace transform... 

Equation (2.30) can be solved analytically only for a few simple systems, for most other cases numerical solution is required. Solution is usually effected by using the graining technique whereby the energy is partitioned into a series of contiguous intervals or grains, each of which is characterized by a number of states, a mean energy and a mean microcanonical rate coefficient. Equation (2.30) is thus approximated by. 

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. 

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... 

Microcanonical VTST minimizes the microcanonical rate coefficients, k E), and takes into account that the dividing surface location is most likely energy dependent... 

Microcanonical rate coefficient versus energy above dissociation for the H00H 20H reaction with J=Q computed using three methods. The red curve is the semiclassical adiabatic model, the blue curve is the HO-RR method, and the purple curve shows the result of the separable hindered rotor approximation. 

Figure 6.14 Microcanonical rate coefficient versus energy above dissociation calculated using the semiclassical adiabatic model evaluated at five values of the total angular momentum.

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