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I. CHEMICAL REACTION DYNAMICS AND ENERGY REDISTRIBUTION Chemical Reaction Dynamics and Marcus’ Contributionst Richard B. Bernsteint and Ahmed H. Zewail* Arthur Amos Noyes Laboratory of Chemical Physics,f California Institute of Technology, Pasadena, California 91 I25 (Received: April 30, 1986) R. A. Marcus has made seminal contributions to our understanding of unimolecular rate processes. His 1951 paper with 0. K. Rice3 and his definitive, independent paper in 19526 represent the birth of the modern theory of unimolecular reactions. The RRKM theory is widely used by experimentalists to interpret measurements of all kinds of unimolecular rate phenomena, ranging from thermal decomposition and isomerization reactions initiated by energizing molecular collisions to chemical activation and photoexcitation. In this sense the RRKM theory has proven to be of near universality. The earlier RRK treatment (of Rice, H. C. Ramsperger, and L. S. Kassel), although qualitatively descriptive, neglected, among other things, the important influence of zero-point energies [of reactant, product(s), and transition state]. The *new” modification by Marcus (the M in RRKM) had “built-in” a credible quantum mechanical structural foundation and, moreover, made contact with the standard transition state theory (TST) of H. Eyring. The first-order rate constant for the decay of a molecule energized to some value, E, say k(E), is given by the expression obtained by Marcus (for chemical reactions): k(E) = N*(E - Eo)/hp(E) where N* is the number of vibrational states of the activated complex with internal energies up to E - Eo, where Eo is the zero-point level of the transition state measured from that of the reactant, and p ( E ) is the density of vibrational states of the energized molecule at E. (An analogous expression was obtained earlier in 1939 by N. Bohr and J. A. Wheeler for nuclear fission.) Integrating over E at a given temperature, T, yields the unimolecular rate constant
in accord with TST. Thus, RRKM theory is a statistical result. It is based on the assumption that, regardless of the *site” of energy deposition, the rate of intramolecular energy transfer is so rapid that the dynamical details are unimportant, and thus all energetically accessible quantum states will be populated equally. Accordingly, state counting (simplified by G. Whitten and B. S. Rabinovitch in.the early 1960s) suffices to predict the decay rate of the energized molecule. This fundamental assumption and deviations from it-non-RRKM behavior-is crucial to the description of energy flow in molecules and to mode-selective laser chemistry. To evaluate the state density and thus the state count a knowledge of the structure and normal-mode frequencies of the energized molecule at the transition state is required. If one knew ‘The few numbered references refer to the publication list of R. A. Marcus in this issue. In our somewhat subjective article we attempt to identify and comment upon some of the main issues in chemical reaction dynamics with particular relationship to Marcus’ contributions. The subject is not treated exhaustively and there is no attempt at literature coverage. *Sherman Fairchild Distinguished Scholar (1986). Permanent address: Department of Chemistry and Biochemistry, University of California, Los An eles, CA 90024. !Contribution 74 14.
the potential energy surface (PES), the actual molecular parameters in the transition state (TS) could be determined. However, the PES of large molecules is nontrivial to obtain and one is forced to “guess” the structure of the TS. One often assumes either a ”tight” or “loose” TS whose properties are estimated by various criteria and approximations, as prescribed by a number of researchers in the field, including Marcus, s. W. Benson, D. M. Golden, Rabinovitch, M. Quack, J. Troe, W. H. Miller, W. L. Hase, D. G. Truhlar, and others. One such criterion that is often used is the correspondence of k(E), when Boltzmann averaged, with the high-pressure rate constant k,(T). It has been found that this constraint makes the k(E) relatively insensitive to the choice of parameters. In general, these approximate frequencies of the TS, as well as Eo, must be empirically adjusted until k(E) agrees with experiment and Eo agrees satisfactorily with the appropriate 1/ T derivative. It is this parameterization procedure, normally required in fitting experimental data with RRKM calculations, which makes it difficult to ascertain possible nonstatistical behavior of a particular reaction. The low-pressure falloff of k( T ) is also well-represented by the RRKM formulation, which includes collisional excitation, but empirical adjustments of the gas collision frequency are usually required to obtain a fit. [Inclusion of collisional energy transfer involves the solution of the so-called master equation (recognized in the mid-1960s by J. Keck, G. Carrier, E. E. Nikitin, Troe, Rabinovitch, and others), now an active field of research (by J. Barker, Troe, and their co-workers)]. The same is true in connection with the application of the theory to the results of chemical activation experiments, starting from the classic work of Rabinovitch and co-workers in the 1960s. Thus, in practical application the RRKM theory has been mainly corroborative rather than fully predictive. Yet, we all recognize that the theory has captured the underlying physics of the phenomenon under those conditions in which statistical behavior prevails. Excellent books, e.g., by P. J. Robinson and K. A. Holbrook (1972) and W. Forst (1973) and many review articles on RRKM theory and its applications testify to the importance of Marcus’ contribution. The success in applying RRKM theory to reactions involving chemical activation as well as gaseous ion chemistry (by J. I. Brauman, T. Baer, M. T. Bowers, and others) is due in part to the fact that under the conditions of the experiments energy redistribution is (nearly) complete and, of course, the initial excitation spans a wide energy range. In the 1970s, R. Atkinson and B. A. Thrush, and Troe and co-workers measured the lifetimes of excited molecules falling apart. The results were analyzed by RRKM theory or a modified version called the adiabatic channel model (Quack and Troe). The molecules (sometimes a homologous series of compounds) ,are optically excited to an upper electronic state, which undergoes a relatively fast internal conversion to very high vibrational levels of the ground state, with E > Eo, and the lifetime (vis-a-vis isomerization or dissociation) directly measured. Later, through the work of M. J. Berry and co-workers, the technique of high overtone excitation (via the intracavity laser technique) made possible more rigorously specified excitation levels of the energized molecule which then undergoes isomerization, e.g., the case of
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methyl isocyanide. The k(E) data could be fitted within experimental uncertainty by RRKM theory with adjustments of the key low frequencies of the transition state relative to those of the product. In recent years, this overtone excitation technique has been similarly utilized by the groups of F. F. Crim, R. N. Zare, C. B. Moore, and others to test the validity of the RRKM theory. With the advent of molecular beams and lasers providing more controlled initial conditions, the RRKM theory has been subjected to more stringent tests. Modern beam conditions allow one to minimize internal thermal energy in unimolecular reactions and to initiate bimolecular reactions with well-specified relative translational energies. The laser makes it possible to excite a well-defined state in the reactant and probe a well-defined state in the product. Thus state-to-state reaction dynamics has become a reality. The direct measurement of k(E) vs. E using picosecond laser/molecular beam techniques by A. H. Zewail’s group is an example of the advances made in the early 1980s in this area of research. For isomerization reactions, the measured k ( E ) , spanning many orders of magnitude, from a given excited vibrational state was compared with RRKM theory. The qualitative trend is well accounted for by the calculations; however, there are quantitative deviations of experimental rates from RRKM theory, as shown by the groups of J. Jortner, Troe, and Zewail. These deviations can be removed by invoking modifications of the transition state, potential energy surface, or the nature of energy redistribution. Obviously care must be taken in treating the RRKM parameterization problem. RRKM theory has been applied successfully to the process of infrared laser-induced multiple-photon dissociation (IRMPD) of polyatomics, first studied by R. V. Ambartzumian and V. S. Lekokhov in the mid-l970s, not only at the level of estimating rates (e.g., from the work of Quack and others) but also at the more detailed level of product state distributions (e.g., by G. Hancock, C. Wittig, and others). The theory also explains recoil angular and translational energy distributions of decay products from crossed IR laser-molecular beam experiments (by Y . T. Lee and co-workers in the early 1980s). Once again, however, until detailed information on the PES becomes available it is necessary to make certain ad hoc assumptions about the transition state (“loose”, etc.) in order to fit the data. The theory is more useful in explaining than predicting experimental results (see below). Reaction dynamics experiments at the microscopic level became possible in the late 1950s and early 1960s, when the molecular beam scattering technique was introduced, by the groups of S. Datz and E. H. Taylor, R. B. Bernstein, E. F. Greene and J. Ross, and D. R. Herschbach and co-workers yielding direct measurements of angular distributions of reactive scattering. At the same time, infrared chemiluminescence experiments on nascent products of fast reactions, by J. C. Polanyi and co-workers, yielded detailed rovibrational product state distributions. Infrared emission techniques have been applied successfully to many important elementary reactions by D. W. Setser, J. D. McDonald, and S. R. Leone, and others, providing much chemical dynamical information. The early reactions studied were all of the so-called “directmode”, but experiments on the bimolecular formation and unimolecular decay of collision complexes in crossed molecular beam reactions were eventually carried out by Herschbach and COworkers in 1967. The results have been well-represented by RRKM modified somewhat to take into account special restrictions due to angular momentum conservation. The modified statistical theory of Herschbach and co-workers made it possible to account for measured angular and velocity distributions of product flux as well as branching ratios for the decay of collision complexes formed with known energies. More recently, important advances were made by Lee and co-workers in the study of products’ translational energy distributions, P(E’). The experiments showed (in certain cases) direct evidence of a barrier in the exit channel, with P(E’) rising abruptly from zero at somefinite E‘(close1y related to the barrier energy), maximizing, and falling off as predicted.
Bernstein and Zewail Deviations from the statistical forward-backward symmetry in the angular distribution of the products of the decay of the transient collision complexes have been observed and interpreted (by Herschbach and co-workers, J. L. Kinsey and others) quantitatively, assuming lifetimes of the complexes to be shorter than the classical rotational periods (in the picosecond range) expected for such complexes. Subtle differences have been observed (by S. Stoke, Bernstein, and co-workers) in the effect of rotational vs. translational energy upon the decay of a complex (at constant total energy). Of course, for “fast” direct-mode reactions (occurring in times
