Diffusion-assisted long-range reactions in confined systems: Projection operator approach

Abstract

Articles you may be interested inAn accurate expression for the rates of diffusion-influenced bimolecular reactions with long-range reactivity J. Chem. Phys. 138, 164123 (2013); 10.1063/1.4802584Diffusion-assisted long-range reaction between the ends of a polymer: Effective sink approximation Long time behavior of reversible diffusion-influenced reaction perturbed by photolysis: Brownian dynamics simulationThe diffusion-assisted long-range reversible reaction equation is solved for the pair survival probability using a projection operator method in terms of the diffusion propagator in the absence of reaction. For a localized ͑delta function͒ reaction sink, the well-known analytical solution is immediately reproduced from the operator expression. It is emphasized that the mean reaction time approach, often used to approximate the overall reaction rate, is not adequate for a nonequilibrium initial condition. The general operator solution for a delocalized sink is shown to reduce to a closed matrix form, provided the propagator has a discrete spectrum of eigenmodes. The matrix solution is exact and applies for an arbitrary functional form and strength of the reaction sink. Although matrices of infinite dimensions are involved, they can be truncated at a certain finite dimension to attain any prescribed precision. Convergence of the truncated matrix solution is fast and often only a few of the lowest eigenmodes are sufficient to obtain quantitatively reasonable results. Several long-range reaction models are analyzed in detail revealing the breakdown of the widely used closure approximation obtained as a first-order Padé approximation of the operator solution.

Solvent Effects in Nonadiabatic Electron‐Transfer Reactions: Theoretical Aspects

“…The situation is different in the inverted regime, where back reaction has no effect on the ET rate, except when the fast mode is absolutely absent. In their influential paper, Sumi and Marcus [89] demonstrated and many others confirmed [90][91][92][93]161] that an irreversible stochastic model of ET with a wide sink (classical vibrational mode) predicts nonexponential kinetic behavior for large coupling and a considerable increase of the long-time rate constant over the usual solvent-control plateau, indicating the breakdown of the Pade´approximation, as shown in Figures 9.18 and 9.20. However, this increase is still rather weak, asymptotically logarithmic [161], as can be understood by converting the reaction-diffusion equation into a Schro¨din-ger-type equation with a self-adjoint operator and invoking a local harmonic approximation after locating the minimum of the effective potential [310].…”

“…However, in order to fit the data, a very high value of the electronic coupling had to be taken, V : 0.1-0.3 eV, which is far beyond the applicability condition of the local Golden Rule, which is at the heart of Zusman's model. Besides, exact solution of this nonequilib- rium model predicts strongly nonexponential kinetics of the ion pair population [105b, 106,161], in contradiction with experiments where simple exponential decays were observed [322].…”

“…Equation (9.121) should be solved with the uniform initial condition, (r, 0) : 1, and the reflecting boundary condition at encounter. An equivalent formulation defines P(t) in terms of the pair survival probability [335,340] [161,303,304] that although the closure approximation is exact for contact reactions (reducing to the Collins-Kimball model [334] in this limit) and works well for short-range reactions, it progressively deviates from exact results as the sink becomes wider and diffusion becomes slower. Therefore, in order to be able to extract reliable ET parameters involved in the Marcus expression from experimental kinetic data, one has to solve the whole problem numerically.…”

Corrections to the law of mass action and properties of the asymptotic t=∞ state for reversible diffusion-limited reactions

“…Strictly speaking, reaction events are not describable within the framework of a classical theory only; an elementary reaction act results from an interplay of many factors and is influenced by solvent structure, potential interactions, a variety of particles' energies and angular orientations, quantum processes of different origin and etc [38,39,40,41,42,43].…”

Theory of Spin Echo in Restricted Geometries under a Step-wise Gradient Pulse Sequence

“…We have been able to fulfill this task recently for the case of a steady gradient by using the projection operator technique (48). The method is described in full detail elsewhere for a physically different but mathematically equivalent problem of diffusion-assisted reaction kinetics (50). By using the standard eigenmode expansion of the Green's function (49),…”