A general theoretical model for electron transfer reactions in complex systems

Abstract

In this paper we present a general theoretical-computational model for treating electron transfer reactions in complex atomic-molecular systems. The underlying idea of the approach, based on unbiased first-principles calculations at the atomistic level, utilizes the definition and the construction of the Diabatic Perturbed states of the involved reactive partners (i.e. the quantum centres in our perturbation approach) as provided by the interaction with their environment, including their mutual interaction. In this way we reconstruct the true Adiabatic states of the reactive partners characterizing the electron transfer process as the fluctuation of the electronic density due to the fluctuating perturbation. Results obtained by using a combination of Molecular Dynamics simulation and the Perturbed Matrix Method on a prototypical intramolecular electron transfer (from 2-(9,9'-dimethyl)fluorene to the 2-naphthalene group separated by a steroidal 5-α-androstane skeleton) well illustrate the accuracy of the method in reproducing both the thermodynamics and the kinetics of the process.

In silico characterization of bimolecular electron transfer reactions: The ferrocene–ferrocenium reaction as a test case

“…The present study is entirely based on the theoretical model for ET processes in complex systems described in Amadei et al The model refers to reactions occurring with no covalent reorganization, for which the changes of the properties of the QC and leading to the electron transfer process, are provided by the fluctuations of the perturbing electric potential and field exerted by the atomic‐molecular environment and the semiclassical internal motions of the QC, assuming instantaneous relaxation to the Hamiltonian eigenstate (ie, Adiabatic approximation).…”

“…To model the unimolecular ET rate constant k (ie, the rate constant for the last reaction step in Figure ), we use the approach described in a previous paper based on the adiabatic approximation, that is, virtually instantaneous relaxation of the QC dynamical quantum‐state on the Born–Oppenheimer Hamiltonian eigenstate (the adiabatic state). We also assume that at virtually every reactants (R) and environment configuration, except for a very limited number of tiny configurational regions where the charge transition occurs (ie, the transition regions TR), the true vibronic eigenstate (adiabatic state) can be well approximated by a combination of the A and D perturbed vibronic eigenstates, that is, the product of the corresponding eigenfunctions with the exchanging charge fully located in one of the chemical species involved in the reaction (diabatic state).…”

“…We also assume that at virtually every reactants (R) and environment configuration, except for a very limited number of tiny configurational regions where the charge transition occurs (ie, the transition regions TR), the true vibronic eigenstate (adiabatic state) can be well approximated by a combination of the A and D perturbed vibronic eigenstates, that is, the product of the corresponding eigenfunctions with the exchanging charge fully located in one of the chemical species involved in the reaction (diabatic state). Making use of PMM, after computing via Quantum‐chemical (QM) calculations the unperturbed properties of the quantum states of the isolated A and D as obtained by fixing the exchanging negative charge alternatively on each of the partner of the reaction (unperturbed diabatic states), we reconstruct the reactive adiabatic process through the evaluation of the perturbed diabatic states, that is, the A and D quantum states when including in the Hamiltonian their mutual interaction and the environment perturbation. Whitin the adiabatic approximation, a productive ET process occurs every time the reactant and product diabatic perturbed energy surfaces cross (ie, when where and are the diabatic perturbed energy of the reactants and products state, respectively), as described in our previous paper .…”

“…For a fixed donor–acceptor distance, we can use the following reaction scheme for the reaction (the chemical state transition corresponding to the Fc–Fc+ ET reaction, see Figure ) where we assumed a TR negligible size with hence forbidden inversion of the TR crossing velocity (ie, we do not assume anymore equilibrium within TR, implying for the tiny transition region virtually identical outward fluxes, as in typical ET reactions the mean crossing time can be rather shorter than the autocorrelation mean time of the crossing velocity) and hence …”

“…The basic idea of PMM, somewhat related to the most popular Quantum Mechanics/Molecular Mechanics (QM/MM) techniques, is the use of the configurational sampling produced by a MD or MC simulation for evaluating the quantum states of a pre‐selected subportion of the system (the Quantum Center, QC) perturbed by its atomistic environment (eg, the solvent). Such a method has been applied for a number of processes ranging from ligand‐exchange to charge‐transfer reactions in complex atomistic environments . Our strategy to model the kinetics and thermodynamics of electron‐transfer (ET) reactions, is based on the use of MD simulations providing the reactants and products ensembles (ie, each of these simulations is performed with the proper corresponding diabatic state force‐field).…”