Statistical unimolecular rate theory

The results of the rate constant calculations by d Anna et al,156 seem to confirm this reaction mechanism. In Fig. 25 is shown the temperature dependence of the observed and calculated rate constants. The rate constant k describes the rate of formation of the post-reaction adduct under the assumption that the pre-reactive adducts are not stabilized by collisions, whereas kadd describes the kinetics of formation of the stable pre-reactive complexes at a total pressure of 1 bar. Thus the overall rate constant for the decay of reactants (denoted in the figure by a solid line) is given by the sum k + k. The values of k predicted by d Anna et al.156 distinctly underestimate the reaction rate at low temperatures, but they approach the results of measurements at temperatures above 700 K. The limiting rate constants kadd, and kadd,0 for the addition channels were analyzed in terms of statistical unimolecular rate theory. Results of the calculations show a fall-off behavior of the reaction kinetics under typical atmospheric conditions corresponding to a total pressure of 1 bar. Therefore, all kadd values were derived from the... 

The information from such experiments, of which there are now many examples(2,3), is important to our understanding of both dynamics and structure. First, since the dynamics of the initially excited state are reflected in spectral line widths, factors which control the rate of vibrational energy flow within molecules can be studied by recording spectra as a function of molecular structure. Fast randomization of this energy is a postulate of statistical unimolecular rate theories. On the... 

Shown in Fig. 2 are examples of two kinetic traces of transient absorbance difference which reflect formation and decarboxylation of carbonyloxy radicals after photo-induced decomposition of DINPO and TBNC. A detailed understanding of the kinetics is obtained from modeling the data. Within the experimental time resolution (150 fs), peroxide primary dissociation produces carbonyloxy radical intermediates, which decay either directly from an electronically excited state within about 500 fs or in a statistical unimolecular reaction on a ps to ps time-scale in the electronic ground state (see Fig. 3). In the case of DINPO photodissociation at 266 nm, the excited state of the 1-naphthylcarbonyloxy radical is too high energetically to be populated to any relevant extent.The reaction on the ground state PES can be treated by statistical unimolecular rate theory. 

Finally, there is the situation (figure 3.13(d)) in which the molecule in an excited electronic state can either be stabilized by fluorescence (rate kf) or undergo a nonradia-tive transition to the ground electronic state with a rate, Once the molecule is in the ground electronic state, it can dissociate with a rate constant (k ) that is often treated in terms of statistical unimolecular rate theory. If the excited electronic state is the first excited single state (5,), one has the mechanism... 

RRKM theory is a microcanonical transition state theory and as such, it gives the connection between statistical unimolecular rate theory and the transition state theory of thermal chemical reaction rates. Isomerization or dissociation of an energized molecule A is assumed in RRKM theory to occur via the mechanism... 

From this discussion it becomes quite clear that the fundamental expression of statistical unimolecular rate theory in equation (46) is an average quantity. When considered as an energy average over many isolated long-lived resonance scattering states in the interval A , one has the inequality (48) ... 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

By choosing the initial conditions for an ensemble of trajectories to represent a quantum mechanical state, trajectories may be used to investigate state-specific dynamics and some of the early studies actually probed the possibility of state specificity in unimolecular decay [330]. However, an initial condition studied by many classical trajectory simulations, but not realized in any experiment is that of a micro-canonical ensemble [331] which assumes each state of the energized reactant is populated statistically with an equal probability. The classical dynamics of this ensemble is of fundamental interest, because RRKM unimolecular rate theory assumes this ensemble is maintained for the reactant [6,332] as it decomposes. As a result, RRKM theory rules-out the possibility of state-specific unimolecular decomposition. The relationship between the classical dynamics of a micro-canonical ensemble and RRKM theory is the first topic considered here. 

Theoretical chemistry is the discipline that uses quantum mechanics, classical mechanics, and statistical mechanics to explain the structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of (1) electronic structure, potential energy surfaces, and force fields (2) vibrational-rotational motion and (3) equilibrium properties of condensed-phase systems and macromolecules. Chemical dynamics includes (1) bimolecular kinetics and the collision theory of reactions and energy transfer (2) unimolecular rate theory and metastable states and (3) condensed-phase and macromolecular aspects of dynamics. 

Detailed Cross-sections and Rates.—The RRKM version of transition-state theory for unimolecular reactions, as developed 25 years ago and sununarized in its useful practical form in recent books, has continued to find wide applications in unimolecular rate theory. As has been pointed out by Marcus in the 1973 Faraday Discussion on molecular beams, it is both a weakness and a strength of transition-state theory that it does not make very detailed statements on specific cross-sections and rates. With such information becoming accessible experimentally, more detailed statistical dynamical theories were to come. We have now four such detailed statistical approaches ... 

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 frequency with which the transition state is transformed into products, iT, can be thought of as a typical unimolecular rate constant no barrier is associated with this step. Various points of view have been used to calculate this frequency, and all rely on the assumption that the internal motions of the transition state are governed by thermally equilibrated motions. Thus, the motion along the reaction coordinate is treated as thermal translational motion between the product fragments (or as a vibrational motion along an unstable potential). Statistical theories (such as those used to derive the Maxwell-Boltzmann distribution of velocities) lead to the expression ... 

Experimental rate constants, kinetic isotope effects and chemical branching ratios for the CF2CFCICH3-do, -d, -d2, and -d2 molecules have been experimentally measured and interpreted using statistical unimolecular reaction rate theory.52 The structural properties of the transition states needed for the theory have been calculated by DFT at the B3PW91 /6-31 G(d,p/) level. 

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. 

The most important conclusions of these dynamical studies is that van der Waals clusters behave in a statistical manner and that IVR/VP kinetics are given by standard vibrational relaxation theories (Beswick and Jortner 1981 Jortner et al. 1988 Lin 1980 Mukamel and Jortner 1977) and unimolecular dissociation theories (Forst 1973 Gilbert and Smith 1990 Kelley and Bernstein 1986 Levine and Bernstein 1987 Pritchard 1984 Robinson and Holbrook 1972 Steinfeld et al. 1989). One can even arrive at a prediction for final chromophore product state distributions based on low energy chromophore modes. If rIVR tvp [4EA(Ar)i], a statistical distribution of cluster states is not achieved and vibrational population of the cluster does not reflect an internal equilibrium distribution of vibrational energy between vdW and chromophore states. If tvp rIVR, and internal vibrational equilibrium between the vibrational modes is established, and the relative intensities of the Ar = 0 torsional sequence bands of the bare chromophore following IVR/VP can be accurately calculated. A statisticsl sequential IVR/VP model readily explains the data set (i.e., rates, intensities, final product state distributions) for these clusters. 

As shown above, classical unimolecular reaction rate theory is based upon our knowledge of the qualitative nature of the classical dynamics. For example, it is essential to examine the rate of energy transport between different DOFs compared with the rate of crossing the intermolecular separatrix. This is also the case if one attempts to develop a quantum statistical theory of unimolecular reaction rate to replace exact quantum dynamics calculations that are usually too demanding, such as the quantum wave packet dynamics approach, the flux-flux autocorrelation formalism, and others. As such, understanding quantum dynamics in classically chaotic systems in general and quantization effects on chaotic transport in particular is extremely important. 

In this chapter we have reviewed the development of unimolecular reaction rate theory for systems that exhibit deterministic chaos. Our attention is focused on a number of classical statistical theories developed in our group. These theories, applicable to two- or three-dimensional systems, have predicted reaction rate constants that are in good agreement with experimental data. We have also introduced some quantum and semiclassical approaches to unimolecular reaction rate theory and presented some interesting results on the quantum-classical difference in energy transport in classically chaotic systems. There exist numerous other studies that are not considered in this chapter but are of general interest to unimolecular reaction rate theory. 

In the first part, our aim is to discuss how we can apply concepts drawn from dynamical systems theory to reaction processes, especially unimolecular reactions of few-body systems. In conventional reaction rate theory, dynamical aspects are replaced by equilibrium statistical concepts. However, from the standpoint of chaos, the applicability of statistical concepts itself is problematic. The contribution of Rice s group gives us detailed analyses of this problem from the standpoint of chaos, and it presents a new approach toward unimolecular reaction rate theory. 

Several conclusions can be drawn from Eqs. (76) and (77). First, the influence of fluctuations is the largest when the number of open channels u is of the order of unity, because then the distribution Q k) is the broadest. Second, the effect of a broad distribution of widths is to decrease the observed pressure dependent rate constant as compared to the delta function-like distribution, assumed by statistical theories [288]. The reason is that broad distributions favor small decay rates and the overall dissociation slows down. This trend, pronounced in the fall-of region, was clearly seen in a recent study of thermal rate constants in the unimolecular dissociation of HOCl [399]. The extremely broad distribution of resonances in HOCl caused a decrease by a factor of two in the pressure-dependent rate, as compared to the RRKM predictions. The best chances to see the influence of the quantum mechanical fluctuations on unimolecular rate constants certainly have studies performed close to the dissociation threshold, i.e. at low collision temperatures, because there the distribution of rates is the broadest. 

From the measurement of the time of flight of the cations their dissociation rates can be calculated and, using statistical theory, it was possible to conclude that the unimolecular rate constant for dissociation (of the order of 10 s " at / = 10-10.5 eV) was too small by a factor of 10 for the possibility that the bicyclobutane cation itself could be the immediate precursor of the fragments. The results were suggestive that a complete isomerization to the cation of 1,3-butadiene occurs before dissociation. 

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