Adiabatic Electron Transfer Reactions Research Articles

Effects of electron correlations in a surface-molecule model for the adiabatic electrochemical electron transfer reactions with allowance for the electrostatic repulsion of electrons on an effective orbital of metal

Within the framework of a surface-molecule model for the adiabatic electrochemical electron transfer reactions, exact expressions for the adiabatic free energy surfaces are obtained and the diagrams of kinetic modes are constructed with allowance made for the electrostatic repulsion between electrons with the opposite spin projection both on the valence orbital of the reactant and on the effective electron orbital of the metal. It is shown that taking into account the electrostatic repulsion on the effective orbital of the metal and the correlation effects connected with it is very substantial for a number of electrochemical electron-transfer reactions and leads not only to an alteration of the activation free energies but also to qualitatively different forms of adiabatic free energy surfaces in some regions of values of the model’s parameters.

Fully adiabatic dissociative electrochemical electron transfer reactions induced by scanning tunneling microscopy

A theory of fully adiabatic dissociative electrochemical processes of the electron transfer that are induced by scanning tunneling microscopy is constructed. Adiabatic free energy surfaces are calculated and properties of their symmetry are examined under various conditions. Diagrams of kinetic regimes, which characterize possible kinetic processes, which may proceed in the system under consideration, are constructed in the space of model parameters. Dependence of activation free energy on the bias voltage, overvoltage, physical properties of a molecule, and intensity of interaction of a molecule with an electrode and the tip of the scanning tunneling microscope is explored.

The effect of the electron–electron interaction on the pre-exponential factor of the rate constant of the adiabatic electrochemical electron transfer reaction

It is shown within the simple model of the polar solvent and the classical multidimensional transition state method in the harmonic approximation that the pre-exponential factor of the rate constant of the adiabatic electrochemical electron transfer reaction depends on the energy of the Coulomb repulsion of electrons occupying the valence orbital of the reactant. The exact expressions for the effective frequency ωeff determining the pre-exponential factor are obtained using two exactly solvable limits of the Anderson–Newns model: the surface-molecule model and the wide-bands approximation for the electronic structure of an electrode. The list of expressions for ωeff for different particular cases is presented. It is shown that, due to the non-zero Coulomb repulsion, the transition state method in the harmonic approximation breaks down in some regions of values of physical parameters of the system. The stochastic approach permits the proper calculations of ωeff in this case. The results of calculations of ωeff are presented and the effect of the Coulomb repulsion on the dependencies of ωeff on the electronic matrix element and the overvoltage are studied. It is shown that this effect can be rather significant.

The effect of ‘hot’ electrons on the heterogeneous adiabatic charge transfer reactions

The spin-less version of the Anderson Hamiltonian is extended to describe the adiabatic electron transfer reactions at a metal electrode involving ‘hot’ electrons which can be reached by the subpicosecond laser pump. The latter presumes high temperatures of the electron gas in the metal substrate, significantly exceeding the temperature of solvent bath. Some analytical formulas are derived and employed to construct the adiabatic free energy surfaces along the solvent coordinate as a function of the electrode overpotential. The model calculations show a noticeable decrease of the reaction barrier resulted from the effect of ‘hot’ electrons. A difference in the kinetics of adiabatic and diabatic electron transfer induced by ‘hot’ electrons was found to be noticeable. The dependencies of the symmetry coefficient vs. the overpotential are discussed, as well as the contribution from vibrationally excited states.

The protein-folding speed limit: intrachain diffusion times set by electron-transfer rates in denatured Ru(NH3)5(His-33)-Zn-cytochrome c.

The kinetics of electron transfer from the triplet-excited Zn-porphyrin to a Ru(NH(3))(5)(His-33)(3+) complex have been measured in Zn-substituted ruthenium-modified cytochrome c under denaturing conditions. In the folded protein, the electron-tunneling rate constant is 7.5 x 10(5) s(-1). As the protein is denatured with guanidine hydrochloride, a faster adiabatic electron-transfer reaction appears (4.0 x 10(6) s(-1), [guanidine hydrochloride] = 5.4 M) that is limited by the rate of intrachain diffusion to bring the Zn-porphyrin and Ru complex into contact. The 250-ns contact time for formation of a 15-residue loop in denatured cytochrome c is in accord with a statistical model developed by Camacho and Thirumalai [Camacho, C. J. & Thirumalai, D. (1995) Proc. Natl. Acad. Sci. USA 92, 1277-1281] that predicts that the most probable transient loops formed in denatured proteins are comprised of 10 amino acids. Extrapolation of the cytochrome c contact time to a 10-residue loop sets the folding speed limit at approximately 10(7) s(-1).

An infinite bandwidth limit model for adiabatic electrochemical electron transfer including electron correlation effects: the exact solution

A theory of the adiabatic interfacial electron transfer reactions is presented based on the exactly solvable infinite bandwidth limit of the Anderson–Newns model which can be applied to the s–p metals. Unlike the other papers on this subject, the electron correlation effects are taken into account from the very beginning and exactly. It is shown that the electron correlation effects play an important role in the region of parameters describing adiabatic electron transfer and lead not only to quantitative corrections to the results obtained earlier in the spin-less model or in the Hartree–Fock approximation but also to new qualitative effects. The critical regions which correspond to different types of electron transfer processes are obtained within the space of parameters of the model and classified. The corresponding kinetic regime diagram is constructed and presented. Examples of the adiabatic Gibbs energy surfaces for a number of characteristic electrochemical electron transfer reactions are given and discussed. The results are compared with those for the surface molecule model.

Electron Correlations for Adiabatic Electrochemical Electron Transfer Reactions in a Model for an Electrode with an Infinitely Wide Conduction Band

Fresh general relationships for adiabatic free-energy surfaces (AFES) and corresponding diagrams of kinetic modes for adiabatic electrochemical electron transfer reactions are derived in the framework of an exactly solvable model for a metallic electrode with an infinitely wide conduction band. The model is a limiting case of the Anderson model applicable to the sp metals. In contrast to earlier studies of adiabatic reactions in a model for an electrode with an infinitely wide conduction band, this work accounts for the electron–electron correlation effects exactly. As an illustration, an AFES is calculated and a diagram of kinetic modes is constructed for a special case corresponding to the equilibrium electrode potential of a two-electron reaction. The exact AFES is compared with the AFES computed in the Hartree–Fock approximation and a spinless model. The correlation effects are shown to play a substantial role and lead to a considerable decrease in the activation free energy.

Crossover from nonadiabatic to adiabatic electron transfer reactions: Multilevel blocking Monte Carlo simulations

The crossover from nonadiabatic to adiabatic electron transfer has been theoretically studied under a spin-boson model (dissipative two-state system) description. We present numerically exact data for the thermal transfer rate and the time-dependent occupation probabilities in largely unexplored regions of parameter space, using real-time path-integral Monte Carlo simulations. The dynamical sign problem is relieved by employing a variant of the recently proposed multilevel blocking algorithm. We identify the crossover regime between nonadiabatic and adiabatic electron transfer, both in the classical (high-temperature) and the quantum (low-temperature) limit. The electron transfer dynamics displays rich behaviors, including multi-exponential decay and the breakdown of a rate description due to vibrational coherence.

Real-Time Observation of Photoinduced Adiabatic Electron Transfer in Strongly Coupled Dye/Semiconductor Colloidal Systems with a 6 fs Time Constant

Electron transfer from organic dye molecules to semiconductor−colloidal systems is among the fastest reported charge-separation reactions. We present investigations on alizarin complexing the surface of TiO2 semiconductor colloids in solution. Because of the very strong electronic coupling between the sensitizer and the semiconductor in the alizarin/TiO2 system, very fast electron injection from the photoexcited dye to the conduction band of TiO2 occurs. The real-time observation of the injection process is achieved by transient absorption spectroscopy using a 19-fs excitation pulse provided by a pump pulse from a noncollinear optical parametric amplifier and a probe pulse from a quasi-chirp-free supercontinuum. An injection time τinj of 6 fs can be unambiguously derived in three different ways from the experimental data: (i) analysis of individual transients at spectral positions without contributions from subsequent reactions (relaxation, recombination); (ii) global fitting procedure for 31 wavelengths over a wide spectral range; and (iii) calculation of the S* state and comparison to the “nonreactive” system alizarin/ZrO2. The spectral signature of the 6-fs kinetic component can be assigned to electron transfer from the excited dye molecule to the TiO2 colloid. Even for this strongly coupled system, we propose a localized excitation with a subsequent adiabatic electron transfer reaction that is, to our knowledge, the fastest electron-transfer reaction that has been directly measured by transient spectroscopy.

Simulations of adiabatic bond-breaking electron transfer reactions on metal electrodes

Adiabatic electron transfer at metal electrodes accompanied by bond breaking is studied by molecular dynamic simulations in a two-dimensional model potential. In the simulations the redox system is coupled to a thermal bath that ensures proper statistical mechanics and provides the energy fluctuations required for the system to cross the activation barrier. The rate constants and transmission factors obtained from the simulations are analysed for various system parameters such as the applied overpotential and the strength of the interaction between the reactants and the electrode. A dependency of the rate constant on the friction parameter is also tested considering both isotropy and anisotropy effects. The Kramers turnover region is found in both cases, but the rate constant is much lower than predicted by the Kramers theory. In all simulations a strong saddle-point avoidance was observed.

Potential-Barrier Profile for an Adiabatic Electron Transfer Reaction in a Condon Approximation with Allowance for Nonsphericity of Electron Wavefunctions

The free-energy surface profile for an adiabatic electron transfer reaction in a polar solvent is calculated. The calculation allows for the dependence of electron wavefunctions of reactants on the environment polarization and for nonsphericity of electron wavefunctions. Allowance for the nonsphericity reduces significantly the activation barrier and increases the reaction rate.

A model of the surface molecule for adiabatic electrochemical electron transfer including electron correlation effects: the exact solution

Abstract A theory of the adiabatic interfacial electron transfer reactions is presented based on the exactly solvable surface molecule limit of the Anderson–Newns model. Unlike the other papers on this subject, the electron correlation effects are taken into account from the very beginning and exactly. It is shown that the electron correlation effects play an important role in the region of parameters describing adiabatic electron transfer and lead not only to quantitative corrections to the results obtained earlier in the spinless model or in the Hartree–Fock approximation but also to new qualitative effects. The critical regions, which correspond to different types of electron transfer processes, are obtained within the space of parameters of the model and classified. The corresponding kinetic regime diagram is constructed and presented. Examples of the adiabatic Gibbs energy surfaces for a number of characteristic, electrochemical electron transfer reactions are given and discussed.

A semiclassical theory of electron transfer reactions in Condon approximation and beyond

The effect of the modulation of the electronic wave functions by configurational fluctuations of the molecular environment on the kinetic parameters of electron transfer reactions is discussed. A self-consistent algorithm for the calculation of the potential profile along the reaction coordinate of adiabatic electron transfer reactions is elaborated. A new formula for the transition probability of non-adiabatic electron transfer reactions is obtained in an improved Condon approximation. A regular method for the calculation of non-Condon corrections is suggested. The importance of these effects for some specific biological and electrochemical electron transfer systems is discussed.

Potential-Barrier Profile for the Adiabatic Electron Transfer Reaction in the Condon Approximation

An effective reaction coordinate is introduced and a self-consistent algorithm is proposed for the calculation of the potential barrier profile of the adiabatic electron transfer reaction in a polar solvent, taking into account electron wave functions of reagents with respect to the media polarization. A change in the functions when moving along the reaction coordinate reduces significantly the activation barrier and increases the reaction rate.

A simulation of an electrochemical adiabatic electron-transfer reaction

Adiabatic electron exchange between an electroactive species and a metal electrode is investigated by molecular dynamics simulation. The system is assumed to be in contact with a thermal bath, which provides the activation energy required for an electron transfer. The method allows the explicit calculation of transmission factors.

An advanced dielectric continuum approach for treating solvation effects: Time correlation functions. I. Local treatment

A local continuum solvation theory, exactly treating electrostatic matching conditions on the boundary of a cavity occupied by a solute particle, is extended to cover time-dependent solvation phenomena. The corresponding integral equation is solved with a complex-valued frequency-dependent dielectric function ε(ω), resulting in a complex-valued ω-dependent reaction field. The inverse Fourier transform then produces the real-valued solvation energy, presented in the form of a time correlation function (TCF). We applied this technique to describe the solvation TCF for a benzophenone anion in Debye (acetonitrile) and two-mode Debye (dimethylformamide) solvents. For the Debye solvent the TCF is described by two exponential components, for the two-mode Debye solvent, by three. The overall dynamics in each case is longer than that given by the simple continuum model. We also consider a steady-state kinetic regime and the corresponding rate constant for adiabatic electron-transfer reactions. Here the boundary effect introduced within a frequency-dependent theory generates only a small effect in comparison with calculations made within the static continuum model.

Electron transfer reactions in electrolyte solutions: effects of ion atmosphere and solvent relaxation

A molecular theory of the effects of solution dynamics on the rates of adiabatic outer-sphere electron transfer reactions in electrolyte solutions is presented. The theory correctly includes the dynamics of both ions and solvent molecules through frequency dependent ion and solvent susceptibilities. The rate of electron transfer is calculated by employing the well-known Grote-Hynes theory. The barrier-crossing frequency of electron transfer is found to depend rather strongly on the salt concentration dependent solvent longitudinal relaxation time and on the ionic conductivity. Numerical results are presented for some common aqueous and non-aqueous electrolyte solutions.

Electrolyte dynamics effect on adiabatic outersphere electron transfer reactions

A theoretical study of the effects of electrolyte solution dynamics on adiabatic outersphere electron transfer reactions is presented. The theory is self-contained and it properly includes the diffusive and non-diffusive modes of relaxation of ions and solvent molecules. Numerical results are obtained for the rate of electron transfer in a model electrolyte solution consisting of spherical ions in Stockmayer solvent and are compared with the rate of electron transfer in the pure Stockmayer solvent. It is found that the deviation from the prediction of transition state theory is higher for electron transfer in the electrolyte solution and this is because of the extra frictional contribution from slow ion atmosphere relaxation.

Adiabatic and nonadiabatic outersphere electron transfer reactions in methanol: Effects of the ultrafast solvent polarization modes

Recent studies have demonstrated that the solvation dynamics in common dipolar liquids like water and acetonitrile is dominated by an initial ultrafast Gaussian component which seems to account for about 60%–70% of the total energy relaxation. Methanol, on the other hand, exhibits a rather different behavior with a much smaller amplitude of the initial Gaussian component and the relaxation is primarily caused by a much slower exponential decay. In the present study, we have investigated the role of these solvent modes on both adiabatic and nonadiabatic outersphere electron transfer reactions in methanol. It is found that the rate of the adiabatic barrier crossing is greatly enhanced due to the ultrafast solvation. For nonadiabatic reactions, the relative importance of the solvent dynamic modes increases enormously compared to the situation when only the slow, overdamped modes are included. Another important conclusion is that because of the dominance of the inertial modes, the rate of electron transfer reaction is almost independent of the longitudinal relaxation time, τL, of the solvent. The results of the present study are compared with those obtained earlier by us for water and acetonitrile to elucidate the underlying difference in the high frequency polar response of these liquids.

Inorganic solution chemistry at elevated pressures

Recent interest in pressure effects on inorganic systems in solution has been centred upon the use of volumes of activation ΔV‡ as criteria of reaction mechanism. Work in our laboratory has sought to determine whether ΔV‡ is indeed a useful parameter in this respect, i.e., whether it is substantially independent of pressure and reaction conditions and whether it can be quantitatively predicted for suitably "simple" reactions. For solvent exchange on metal ions (the simplest conceivable substitution process), a semi-empirical model predicts ΔV‡ for limiting dissociative (bond breaking) and associative (bond making) mechanisms in water, but experimental values lie between these extremes, vindicating the Langford–Gray "interchange" model in which associative and dissociative processes are viewed as being concerted. For adiabatic electron transfer reactions of the outer-sphere type (the simplest conceivable oxidation–reduction process) in water, an adaptation of Marcus theory accounts for the essentially pressure-independent ΔV‡ satisfactorily, and failures of such predictions can be ascribed to complications such as nonadiabaticity or the incursion of inner-sphere (ligand-bridged) reaction pathways. The theory is less successful in nonaqueous solvents. Experimental methods used for these and related studies include high pressure adaptations of nuclear magnetic resonance, UV–visible spectrophotometry, stopped-flow techniques, cyclic voltammetry, and sampling under pressure.