Intramolecular reactions inside macromolecules (e.g., binding of ligands to iron inside heme proteins) may often be coupled to slow random fluctuations in the reaction center geometry. This motion is ‘‘perpendicular’’ to the reaction coordinate. It can be described as bounded diffusion in the presence of a binding potential field and an intramolecular rate constant which depends on the perpendicular degree of freedom. The diffusion equation is solved under the appropriate reflective boundary conditions. The transient decay of the total population is multiexponential (power law) for small diffusivity, changing to monoexponential kinetics for large diffusivity. For large times or large diffusivity, direct integration is very tedious, but an eigenvalue expansion converges rapidly. It also allows the calculation of the ‘‘average survival time’’ (an extension of the ‘‘first passage time’’) a natural candidate for replacing the reciprocal rate constant in multiexponential kinetics. An example is given for electron transfer between two loosely bound sites in a macromolecule. The average survival time shows a non‐Kramers dependence on diffusivity, of the type found in the binding kinetics in heme proteins.

Transient kinetics of chemical reactions with bounded diffusion perpendicular to the reaction coordinate: Intramolecular processes with slow conformational changes

We extend Tachiya’s method for treating totally diffusion‐controlled recombination of a pair of particles to partially diffusion‐controlled recombination. The differential equation satisfied by the survival probability of the pair is derived, and solved for several cases of practical interest. Recent experiments on the solvated electron yield in liquid ammonia are analyzed by use of the present theory. It is confirmed that the rather high yield of solvated electrons in irratiated liquid ammonia is due to the slowness of the recombination reaction e−s+NH+4.

Partially diffusion‐controlled recombination

A unified theory of reversible diffusion‐influenced geminate and pseudo‐ first‐order reactions is developed. Explicit results are presented for the time dependence of the fraction of molecules that are dissociated at time t for a variety of initial conditions. To introduce the basic ideas of our approach, an elementary and rather complete treatment of the irreversible reaction between a pair of interacting, spherically symmetric particles is presented. The focus is on deriving relations among survival probabilities and bimolecular time‐dependent rate coefficients for the radiation and absorbing boundary conditions and the asymptotic behavior of these quantities. These relations are then generalized to reversible geminate reactions. For example, it is shown that the separation probability for an initially bound pair satisfies a simple convolution relation involving the survival probability of an irreversibly reacting geminate pair initially at contact. An analytic expression is obtained for this separation probability that is exact for free diffusion and is an accurate approximation for interacting particles. Finally, the Smoluchowski approach to irreversible pseudo‐first‐order reactions is extended to reversible reactions. The analysis is based on the generalization of the convolution relations that are rigorously valid for isolated pairs.

Theory of reversible diffusion‐influenced reactions

There is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences. In this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the National Institutes of Health on May 6–8, 1985. This review will focus on (1) the Smoluchowski model for reaction rates together with its extension by Onsager, (2) first passage time formalism for discrete and continuous master equations and Fokker-Planck equations, (3) the Kramers model and its extensions, (4) diffusion in the presence of trapping centers.

Overview of theoretical models for reaction rates

Rate coefficients of diffusion‐limited fast reactions of charged reactants are predicted to be time dependent or time independent in different theoretical approaches to solve the Debye–Smoluchowski equation. Different rate coefficient expressions are also predicted in different time domains. Fluorescence quenching experiments using two cationic dyes (rhodamine B and cresyl violet) and four efficient ionic quenchers (iron complexes) in aqueous solutions were carried out to verify the theoretical prediction. A detailed data analysis of the quenched fluorescence decay over a range of concentration of the quenchers supports the prediction that the ‘‘long’’ time rate coefficient is k(t)∼a+bt−1/2. Exact agreement with the theoretical predictions is not obtained in any of the fluorophore–quencher systems described in this work. Cresyl violet–ferrocyanide and cresyl violet–ferricyanide are two systems for which the experimental values of diffusion coefficient D and distance parameter RHN are justifiably close to theoretical expectation.

Diffusion controlled reactions: fluorescence quenching of cationic dyes by charged quenchers

Reactions of isolated ion pairs in solution have been modelled using the Debye–Smoluchowski equation for diffusion and conduction An activation step was incorporated using a partially reflecting boundary condition The method of matched expansions and the Abelian theorem of Laplace transforms was used to give an approximate solution of the Debye–Smoluchowski equation Numerical integrations based on the finite‐difference method confirmed these approximate analytic formulas

A solution of the Debye–Smoluchowski equation for the rate of reaction of ions in dilute solution

Fick's second diffusion equation, with an added exponential sink term, is integrated numerically to simulate the decay of electrons at short times in the presence of scavengers The time dependence of the scavenger concentration profile, the scavenging rate constant, and the electron concentration are illustrated graphically Using the experimental results of Buxton et al and Jonah et al It is shown that the Smoluchowski equation is valid within their experimental time ranges provided the cage encounter distance is replaced by Reff, where Reff can be evaluated explicitly in terms of reaction parameters It is also shown that tunnelling from relaxed traps may make a significant contribution to ultra-short time electron scavenging

Numerical investigation of tunnelling contributions to electron scavenging reactions in liquids at short times

Application of the Smoluchowski–Debye equation results in a uniform treatment of the kinetics of either of the bulk reactions e− +M+→M or e−+A→A−, and the corresponding photoionization reaction, either M⇄hνM++e− or A−⇄hνA+e−. For simplicity of terminology, the bulk reactions are referred to as class I and the photoionizations as class II. Included in the theory are the following effects: (1) solvent screening of the interaction potential between e− and either of the acceptors M+ or A, (2) spatial variation of electron transport coefficients (mobility and diffusion tensor) due to the internal electric field associated with the interaction potential, (3) electron tunneling, and (4) chemical activation processes. By use of Green’s function methods, we arrive at the result n1(r)exp[−ψ (r)]=NsP (r), where n1(r) is the pair density at a relative separation r in a reaction of class I, and P (r) is the escape probability for a pair initially at a separation r in the corresponding reaction of class II. The quantit...

Uniform theory of electron kinetics in nonpolar liquids

The field dependence of the electron mobility in dielectric liquids is discussed.(AIP)The field dependence of the electron mobility in dielectric liquids is discussed.(AIP)

Comment on ’’Probability of escaping neutralization when the mobility is field dependent’’

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A solution of the Debye–Smoluchowski equation for the rate of reaction of ions in dilute solution

Numerical investigation of tunnelling contributions to electron scavenging reactions in liquids at short times

Uniform theory of electron kinetics in nonpolar liquids

Comment on ’’Probability of escaping neutralization when the mobility is field dependent’’