Three-dimensional simulations of reversible bimolecular reactions. II. The excited-state target problem with different lifetimes

Theory of Diffusion-Influenced Reaction Networks

A formalism is developed to describe how diffusion alters the kinetics of coupled reversible association-dissociation reactions in the presence of conformational changes that can modify the reactivity. The major difficulty in constructing a general theory is that, even to the lowest order, diffusion can change the structure of the rate equations of chemical kinetics by introducing new reaction channels (i.e., modifies the kinetic scheme). Therefore, the right formalism must be found that allows the influence of diffusion to be described in a concise and elegant way for networks of arbitrary complexity. Our key result is a set of non-Markovian rate equations involving stoichiometric matrices and net reaction rates (fluxes), in which these rates are coupled by a time-dependent pair association flux matrix, whose elements have a simple physical interpretation. Specifically, each element is the probability density that an isolated pair of reactants irreversibly associates at time t via one reaction channel on the condition that it started out with the dissociation products of another (or the same) channel. In the Markovian limit, the coupling of the chemical rates is described by committors (or splitting/capture probabilities). The committor is the probability that an isolated pair of reactants formed by dissociation at one site will irreversibly associate at another site rather than diffuse apart. We illustrate the use of our formalism by considering three reversible reaction schemes: (1) binding to a single site, (2) binding to two inequivalent sites, and (3) binding to a site whose reactivity fluctuates. In the first example, we recover the results published earlier, while in the second one we show that a new reaction channel appears, which directly connects the two bound states. The third example is particularly interesting because all species become coupled and an exchange-type bimolecular reaction appears. In the Markovian limit, some of the diffusion-modified rate constants that describe new transitions become negative, indicating that memory effects cannot be ignored.

Diffusional effects on the reversible excited-state proton transfer. From experiments to Brownian dynamics simulations

We have studied an excited state proton transfer (ESPT) from the cationic "super" photoacid N-methyl 6-hydroxyquinolinium perfluorobutane sulfonate to non-aqueous solvents using picosecond and nanosecond time-resolved fluorescence spectroscopy. Upon the photoinduced adiabatic deprotonation from the hydroxyl moiety, a quinolinium zwitterion with a highly anisotropic charge distribution is formed. Due to the complexity of the resultant photodissociated system, the typical description of the reversible ESPT within the framework of the Spherically Symmetric Diffusion Problem (SSDP) is not possible. Additional complications are caused by the presence of a counteranion particle which affects the proton mobility. To better understand the ESPT process, we have performed extensive Brownian dynamics (BD) simulations of this three-body system as a tool to reveal the nature of the nonstationary interaction potentials and to elucidate the role of a counterion in the diffusion and reactive properties of the proton. Moreover, our results demonstrated that the anisotropy of the potential force can be taken into account after adapting this force for use in the SSDP. The results of both BD simulations and SSDP calculation with the adapted force field were used to fit the experimental kinetics of this three-body problem adequately.

Excited-state reversible geminate recombination in two dimensions

Excited-state reversible geminate recombination with two different lifetimes and quenching is investigated in two dimensions. From the exact Green function in the Laplace domain, analytic expressions of two-dimensional survival and binding probabilities are obtained at short and long times. We find that a new pattern of kinetic transition occurs in two dimensions. The long-time effective survival probabilities show a pattern of (ln t)(-1)-->constant-->e(t) depending on the rate constants while the effective binding probabilities show t(-1)(ln t)(-2)-->t(-1)-->e(t).

Diffusion-influenced excited-state reversible geminate ABCD reaction in the presence of an external field

We obtained the exact Green functions, in the Laplace domain, for a diffusion-influenced excited-state reversible geminate ABCD reaction with two different lifetimes and quenching processes under a constant external field in one dimension. Analytic expressions for the survival probabilities of the initial and final states are obtained in the time domain at short and long times, respectively. The short-time approximations obtained in this work are valid for t</K/(-1), where K depends on several parameters of the system. The analysis of the long-time asymptotic behaviors reveals rather complex kinetic transitions dependent upon the field and lifetimes. We also find a destructive interplay leading to the reduction in the number of kinetic transitions similar to that found for the excited-state geminate ABC reaction with an external field in one dimension.

Molecular-dynamics simulations for nonclassical kinetics of diffusion-controlled bimolecular reactions

Molecular-dynamics simulations are presented for the diffusion-controlled bimolecular reaction A+B<==>C in two and three dimensions. The reactants and solvent molecules are modeled as spheres interacting via continuous potential-energy functions. The interaction potential between two reactants contains a deep well that results in a reaction. When the solvent concentration is low and the reactant dynamics is essentially ballistic, the system reaches equilibrium rapidly, and the reaction follows classical kinetics with exponential decay to the equilibrium. When the solvent concentration is high the particles enter the normal diffusion regime quickly and nonclassical behavior is observed, i.e., the reactant concentrations approach equilibrium as t(-d/2) where d is the dimensionality of space. When the reaction well depth is large, however, the reaction becomes irreversible within the simulation time. In this case the reactant concentrations decay as t(-d/4). Interestingly this behavior is also observed at intermediate times for reversible reactions.

Diffusion-influenced excited-state reversible transfer reactions, A*+B⇌C*+D, with two different lifetimes: Theories and simulations

We report accurate Brownian simulation results for the kinetics of the pseudo-first-order diffusion-influenced excited-state reversible transfer reaction A(*) + Bright harpoon over left harpoonC(*) + D with two different lifetimes using two different propagation algorithms. The results are used to test approximate solutions for this many-particle problem. Available theories fail when one of the two reactions or (decay) rate constants is large. To remedy this situation, we develop two uniform approximations, which are based on introducing a generalized Smoluchowski term into the relaxation-time approximation. The best of these is the extended unified theory of reversible target reactions, which reduces correctly in all limits and exhibits superior agreement with simulations.

Elementary Steps in Excited-State Proton Transfer

The absorption of a photon by a hydroxy-aromatic photoacid triggers a cascade of events contributing to the overall phenomenon of intermolecular excited-state proton transfer. The fundamental steps involved were studied over the last 20 years using a combination of theoretical and experimental techniques. They are surveyed in this sequel in sequential order, from fast to slow. The excitation triggers an intramolecular charge transfer to the ring system, which is more prominent for the anionic base than the acid. The charge redistribution, in turn, triggers changes in hydrogen-bond strengths that set the stage for the proton-transfer step itself. This step is strongly influenced by the solvent, resulting in unusual dependence of the dissociation rate coefficient on water content, temperature, and isotopic substitution. The photolyzed proton can diffuse in the aqueous solution in a mechanism that involves collective changes in hydrogen-bonding. On longer times, it may recombine adiabatically with the excited base or quench it. The theory for these diffusion-influenced geminate reactions has been developed, showing nice agreement with experiment. Finally, the effect of inert salts, bases, and acids on these reactions is analyzed.

Exact solution of the excited-state geminate A*+B⇄C*+D reaction with two different lifetimes and quenching

The authors obtain, in the Laplace transform space, the exact analytic solution for the Green function and survival probabilities for the excited-state diffusion-influenced reversible geminate reaction, A*+B <==> C*+D, with two different lifetimes and in the presence of an added quenching process. This extends a previous investigation by Popov and Agmon [J. Chem. Phys. 117, 5770 (2002)] of the ground-state reaction without quenching. The long-time asymptotic behavior of the survival probabilities is obtained in the time domain. It is found to be different from the equal-lifetime case. This paper also provides a useful short-time approximation for the kinetics.

Diffusion-influenced reversible geminate recombination in one dimension. III. Field effect on the excited-state reaction

We obtain exact analytic solutions of the diffusion-influenced excited-state reversible geminate recombination reaction, A* + B<-->(AB)*, with two different lifetimes and quenching under the influence of a constant external field in one dimension. These fundamental solutions generalize two previous results [Kim et al., J. Chem. Phys. 111, 3791 (1999); 114, 3905 (2001)] and provide us with the insight necessary to analyze their specific relations and asymptotic kinetic transition behaviors. We find that the number of kinetic transitions can be changed due to interplay between the field strength and lifetimes. Unlike the previous works, the number of lifetime dependent transitions is found to be one or zero. On the other hand, the number of the field dependent transitions becomes two, one, or zero. We find a new pattern of kinetic transition e(t)-->t(-1/2)-->e(t) when there is only one field dependent transition.

Influence of diffusion on the kinetics of excited-state association–dissociation reactions: Comparison of theory and simulation

Several recent theories of the kinetics of diffusion influenced excited-state association--dissociation reactions are tested against accurate Brownian dynamics simulation results for a wide range of parameters. The theories include the relaxation time approximation (RTA), multiparticle kernel decoupling approximations and the so-called kinetic theory. In the irreversible limit, none of these theories reduce to the Smoluchowski result. For the pseudo-first-order target problem, we show how the RTA can be modified so that the resulting formalism does reduce correctly in the irreversible limit. We call this the unified Smoluchowski approximation, because it unites modern theories of reversible reactions with Smoluchowski's theory of irreversible reactions.