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J. Phys. Chem. 1994,98, 7300-7306
Scaling in a Model of Chemical Reaction Rates with Space-Dependent Friction Surjit Singht and G . Wilse R o b h n ' a Subpicosecond and Quantum Radiation Laboratory, Departments of Chemistry and Physics, Texas Tech University, Lubbock, Texas 79409 Received: January 21, 1994; In Final Form. March 29. I994@
We study a chemical reaction model with a one-dimensional reaction coordinate that moves in a parabolic barrier joined to a parabolic well. A different dynamic friction is present in the well and the barrier regions. It is found that, in the limit of large solvent response times, the reaction rate has singularities that can be interpreted as critical phenomena. The model is shown to have two main scaling regimes, governed by large barrier damping and large well damping, respectively. In either of these two regimes, there are nine subregimes, depending on whether the well or the barrier coupling is strong, intermediate, or weak. The behavior of the rate in these subregimes is governed by a multitude of critical exponents. However, in either of the main regimes, scaling reduces the number of independent exponents to just three, which result from the barrier friction, the well friction, and the barrier height.
I. Introduction It is well-known1 that standard transition state theory (TST) is inadequate for dealing with the complexities of solvation dynamics in chemical reactions. By assuming equilibrium solvation, TST attributes the reaction almost entirely to the reactants, the solvent merely providing an isothermal heat bath. Since Kramers's early work? it has been known that this idea is too simple. One must treat the solvent as a more complex system, even though, at the present time, this must be done phenomenologically. Kramersused a one-dimensional reaction coordinate moving in a parabolic mean potential, with Markovian friction for the solvent. This work has now been generalized in many directions. For our purposes, we note recent work involving friction with memory3s4 and space-dependent f r i ~ t i o n . ~ . ~ The memory friction kernel (MFK) is characterized by its strength and its relaxation time. In the limit of large relaxation times, the reaction rate has a singular behavior occurring near a certain critical strength. Because of this singularity, thebehavior of the rate is found to be described by a multitude of power laws in the various regimes. This has made the activated barrier crossing problem difficult to solve in general. In fact, an exact solution to this problem is not known except in the purely parabolic Grote-Hynes (GH) limit.3 For other potentials only approximate solutions are known. Recently, in a series of papers7-10 from our laboratory, it has been demonstrated that this singular behavior can be described as a critical phenomenon.ll We studied this phenomenon within the Pollak-Grabert-Htinggi (PGH) theory12 in refs 7 and 8, the canonical variational transition state theory13 in ref 9, and the van der Zwan-Hynes theory14 in ref 10. This has not helped solve the problem exactly but has certainly reduced the number of exponents and other poperties that need to be calculated in an exact solution. The reason for this simplification is as follows. The presence of a critical point causes the reaction rate to obey a scaling relation. This, in turn, leads to relations between the critical exponents. For example, in the GH limit, only one independent exponent is needed. When one goes beyond the GH theory, one needs to take into account the finiteness of the barrier height, which introducesjust one more exponent. In all three of the above-mentioned theories, we calculated the scalingfunctions and obtained the behavior of the rate in different regimes of the parameters. Department of Chemistry. *.Abstract Departments of Chemistry and Physics. published in Advunce ACS Absrrucrs, July 1, 1994. t
0022-3654/94/2098-7300S04.50/0
A question naturallyariseswhether models incorporating spacedependent friction will obey scaling and universality. This is the subject of the present paper. In order to study this question, we have chosen a models proposed and studied in our group. This model is the simplest possible one that reproduces the correct results in the appropriate limits; and in contrast with all other available models, it can be studied analytically. In section I1 we summarize the scaling results for the GH theory and the space-independent friction model in the PGH approximation. In section 111, we outline our space-dependent friction model and the relevant parts of its solution. This will help set the stage for the analysis and to introduce notation. In section IV, we calculate the rate in the scaling limit, Le. in the case when the well and the barrier memory friction times are very long. This divides the complex behavior of the model into two main regimes. The first of these, when the barrier damping is much greater than the well damping, is studied in section V, whereas the opposite case is the subject of section VI. The results arediscussedin section VII, and our concluding remarks constitute section VIII.
II. Summary of Scaling for the CH and the PGH Theories for the Space-Independent Friction Case The fact that thereactionratein theGH theory has a singularity at a critical coupling strength and an infinite correlation time was first discovered by van der Zwan and Hynes.14 They applied the GH theory to their model for the dipole isomerization rate in polar solvents. They found that, in the limit of long solvent response times and near a critical coupling, the rate could be described by power laws. The connection between this behavior and critical phenomena was discovered in our laboratory.718 In studying the GH theory from the viewpoint of critical phenomena:,* we used a general MFK of the form
where the static friction or the damping parameter y is defined as y = Jldt {(t), so that Jldu g(u) = 1. Here, T = cry is the memory relaxation time, where cr is inversely proportionalto the strength of the MFK. For fixed a,the memory correlation time is proportional toy. Fromvariousstudies in critical phenomena," it is found that the critical behavior occurs whenever the correlation length or time or both become very large. In our problem, the variable T is the correlationtime of the fluctuationsof the random force. Therefore, we expect critical phenomena to be present in Q 1994 American Chemical Society
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The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 7301
Scaling in a Model of Chemical Reaction Rates the limit
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03. Now 7 can go to infinity in two ways: fixed fixed a,y m. We have seen previously that the former possibility is somewhat t r i ~ i a l . ~Therefore, J in this paper, we focus on the case of fixed a and large y. The fact that this case may give results that aredifferent from theusual large friction limits was clearly recognized by Zwanzig.ls In the GH theory, the reduced rate, i.e. the rate divided by the TST rate, depends on a,y, the imaginary barrier frequency O b , and the function g(u) describing the shape of the MFK. The critical behavior occurs near the point a* = a W b 2 = g(o), l/y* - W b / y = 0. In order to study the critical region, which wcurs only for large memory correlation time, we need to consider the behavior of g(u) in eq 1 only for small u. We assume that
y, a
7
a;or
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g(u) g(0) - wlutp I4 > 1 and Me(u) = u for u
