Robert I. Cukier

Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. III. Analysis of the approximations

In Parts I and II of this series [J. Chem. Phys. 59, 3873, 3879 (1973)] we developed a new method of sensitivity analysis for large sets of coupled nonlinear equations with many parameters. In developing this theory and in carrying out the computer calculations involved in this analysis we made a number of approximations. We present here a quantitative analysis of these approximations and, where applicable, develop rigorous error bounds. Our analysis shows that we can specify the approximations which enter into our theory so as to obtain sensitivity measures of known accuracy. On this basis we feel that the techniques developed in this series of papers provide a useful and efficient method of sensitivity analysis of large systems with many parameters.

The transition from nonadiabatic to solvent controlled adiabatic electron transfer: solvent dynamical effects in the inverted regime

We analyze the effect of dynamical solvent effects on the rate of a nonadiabatic electron transfer (ET) reaction. Starting from a Hamiltonian for a reaction coordinate for motion along the potential surfaces of donor and acceptor species, and a bath representing the solvent dynamical effects, we obtain a system of four coupled reduced equations of motion for the elements of the density matrix of the donor/acceptor system. In this derivation the dynamics along the reaction coordinate are reduced to a classical Fokker–Planck operator since we assume the temperature is high compared with bath frequencies. At temperatures where the nuclear motion describing the transition between the surfaces can be treated classically we show that the ET processes may be viewed as a consecutive reaction scheme with rate constant k=kNA kD/(kD+kNA), the steps are diffusion along the reaction coordinate with rate constant kD followed by crossing between the donor and acceptor surfaces at the point of intersection of the surface...

Dynamics of diffusion‐controlled reactions among stationary sinks: Scaling expansion approach

Diffusion‐controlled reactions in a nondilute sink system are rigorously studied with the aid of a scaling expansion method. A space‐time coarse graining is carried out in a manner consistent with an expansion in sink concentration to obtain macroscopic transport equations from microscopic equations. It is shown that, beyond the lowest order in sink concentration, the macroscopic transport equation for the reaction–diffusion process cannot be written in a conventional local form in space and time since a nonlocal contribution in space becomes important. Properties of the fluctuations around the macroscopic motion are also explicitly explored, and they are shown to be small in comparison with the macroscopic motion for three‐dimensional systems with an appropriate choice for the size of a sink radius and obey a Gaussian process. An absorption process whose characteristic length is much longer than that of the reaction–diffusion process is also investigated, and a local damping equation in space and time is...

On the effects of solvent and intermolecular fluctuations in proton transfer reactions

We present a theory of proton transfer reactions which incorporate the modulation of the proton’s potential surface by intermolecular vibrations and the effect of coupling to solvent degrees of freedom. The proton tunnels between states corresponding to it being localized in the wells of a double minimum potential. The resulting tunnel splitting depends on the intermolecular separation. The solvent response to the proton’s charge is modeled as that of a damped oscillator, allowing the introduction of friction effects in the solvent dynamics. The rate of transfer is evaluated by perturbation theory in the level splitting. We find that typically the intermolecular and solvent contributions enhance the rate relative to the values obtained in their absence. This effect is evident at low temperature where friction can enhance the rate by increasing opportunity for solvent tunneling. At high temperature the intermolecular motion enhances the rate by sampling over a distribution of tunnel splittings.

Solvent effects on proton‐transfer reactions

We present a theory of solvent effects on the rate of intramolecular proton‐transfer (IPT) reactions. The proton tunnels between two vibrational levels of a double minimum potential. The proton’s coupling to the solvent is modeled with an oscillator bath, appropriate to reactions where a charge interacts with many solvent molecules. The rate is evaluated by use of the Golden Rule; the perturbation is the level splitting. The IPT rate constant has several limiting expressions, one of which has an activated form. The activation energy is related to the medium reorganization energy, and provides a mechanism to slow the IPT reaction. Since reorganization energies are small in nonpolar and large in polar solvents, the rate is expected to be smaller in the latter class of solvents. Isotopic substitution is predicted to only affect the prefactor of the rate expression. Another regime is obtained for smaller reorganization energies where the solvent dynamics, as described by a dielectric relaxation time, are important. Comparison is made with recent experimental studies of IPT in solution.

A quantum molecular dynamics simulation of an excess electron in methanol

The structure, energetics, and dynamics of a ground‐state, excess electron in the polar solvent methanol are simulated. Two pseudopotentials describing the interaction of the excess electron and the methanol molecules are developed. An adiabatic simulation method is used whereby the Schrodinger equation for the electron is solved in the presence of a fixed solvent configuration and the solvent configuration is advanced with the forces arising from the methanol interactions and the expectation value of the electron–methanol interaction. We find that the electron is localized with average radii of 3.1 and 2.6 A, depending on which pseudopotential is used, and both show a fairly strong solvation structure. The methanols are on average methoxyl bond‐dipole oriented toward the electron in one model and hydroxyl bond‐dipole ordered in the other. The binding energy (kinetic plus potential) of the electron fluctuates about the value −2.2 eV. The electron solvates on about a 400 fs time scale with a fast decay com...

Diffusion controlled processes among stationary reactive sinks: Effective medium approach

We formulate an effective medium theory for diffusion controlled processes among randomly distributed, stationary, reactive sinks. An implicit expression for the exact ‘‘self‐energy’’ of the combined reactive density field and sinks is obtained in terms of the exact medium propagator. The lowest order truncation of this expression can be solved by numerical iteration for the self‐energy. The rate and diffusion coefficients as a function of sink concentration are extracted from this self‐energy. We also consider the next order truncation which explicitly introduces the pair correlation function of the sinks.

The transition from nonadiabatic to solvent controlled adiabatic electron transfer kinetics: The role of quantum and solvent dynamics

We analyze the transition from nonadiabatic to solvent controlled adiabatic electron transfer kinetic behavior with emphasis on the inverted regime. By viewing the electron transfer process as dynamical motion on a donor surface followed by possible crossing to the acceptor surface, we are able to obtain a simple consecutive reaction expression for the overall rate in terms of rate constants for motion on the surfaces and crossing motion. When the crossing between surfaces is not localized to a point, we find that there are frictional effects on this crossing rate constant, in contrast to the localized case. Use of typical electron transfer parameters shows that these friction effects will only be in evidence for the inverted regime of electron transfer kinetics.

Generalized stochastic model for molecular rotational motion in dense media

We introduce a general stochastic model that encompasses all other commonly used stochastic models for the orientational motion of molecules in dense media. In this model the rotational motion is envisaged to proceed by an alternating sequence of collision and between collision events. These events are characterized by transition probabilities whose form can be chosen in accordance with the physical situation. The durations of the events are governed by probability density functions. These probability densities enable us to investigate the effects of finite durations of collisions and of time correlations between successive collisions. The effect of correlations between successive collisions is to produce nonexponential behavior in the time correlation functions of orientational variables. Indeed these time correlation functions can become negative and can have damped oscillatory behavior even in the complete absence of explicit inertial effects (free rotations). The finite duration of collisions insures ...

Effect of static correlations on the pair friction coefficient

A calculation of the spatial dependence of the pair friction tensors ζij(r12), (i, j=1,2) for two hard sphere solute particles in a hard sphere fluid is presented. The calculation is based on a kinetic theory expression for the pair friction tensor. The contribution from dynamically uncorrelated collisions is computed here. (We do not consider the contribution from correlated collisions, which are important at high density.) The friction tensor is found to have a significant dependence on the solute pair separation, which arises from static correlations among the bath and solute particles.