A molecular theory of electron transfer reactions in polar liquids

Reorganization energy of electron transfer

The theory of electron transfer reactions establishes the conceptual foundation for redox solution chemistry, electrochemistry, and bioenergetics. Electron and proton transfer across the cellular membrane provide all energy of life gained through natural photosynthesis and mitochondrial respiration. Rates of biological charge transfer set kinetic bottlenecks for biological energy storage. The main system-specific parameter determining the activation barrier for a single electron-transfer hop is the reorganization energy of the medium. Both harvesting of light energy in natural and artificial photosynthesis and efficient electron transport in biological energy chains require reduction of the reorganization energy to allow fast transitions. This review article discusses mechanisms by which small values of the reorganization energy are achieved in protein electron transfer and how similar mechanisms can operate in other media, such as nonpolar and ionic liquids. One of the major mechanisms of reorganization energy reduction is through non-Gibbsian (nonergodic) sampling of the medium configurations on the reaction time. A number of alternative mechanisms, such as electrowetting of active sites of proteins, give rise to non-parabolic free energy surfaces of electron transfer. These mechanisms, and nonequilibrium population of donor-acceptor vibrations, lead to a universal phenomenology of separation between the Stokes shift and variance reorganization energies of electron transfer.

Electron transfer in nonpolar media

Electron transfer in nonpolar media violates the temperature scaling predicted by the fluctuation–dissipation theorem.

Dynamical transition, hydrophobic interface, and the temperature dependence of electrostatic fluctuations in proteins

Molecular dynamics simulations have revealed a dramatic increase, with increasing temperature, of the amplitude of electrostatic fluctuations caused by water at the active site of metalloprotein plastocyanin. The increased breadth of electrostatic fluctuations, expressed in terms of the reorganization energy of changing the redox state of the protein, is related to the formation of the hydrophobic protein-water interface, allowing large-amplitude collective fluctuations of the water density in the protein's first solvation shell. On top of the monotonic increase of the reorganization energy with increasing temperature, we have observed a spike at approximately 220 K also accompanied by a significant slowing of the exponential collective Stokes shift dynamics. In contrast to the local density fluctuations of the hydration-shell waters, these spikes might be related to the global property of the water solvent crossing the Widom line or undergoing a weak first-order transition.

Chemical approaches to artificial photosynthesis. 2

The goal of artificial photosynthesis is to use the energy of the sun to make high-energy chemicals for energy production. One approach, described here, is to use light absorption and excited-state electron transfer to create oxidative and reductive equivalents for driving relevant fuel-forming half-reactions such as the oxidation of water to O2 and its reduction to H2. In this "integrated modular assembly" approach, separate components for light absorption, energy transfer, and long-range electron transfer by use of free-energy gradients are integrated with oxidative and reductive catalysts into single molecular assemblies or on separate electrodes in photelectrochemical cells. Derivatized porphyrins and metalloporphyrins and metal polypyridyl complexes have been most commonly used in these assemblies, with the latter the focus of the current account. The underlying physical principles--light absorption, energy transfer, radiative and nonradiative excited-state decay, electron transfer, proton-coupled electron transfer, and catalysis--are outlined with an eye toward their roles in molecular assemblies for energy conversion. Synthetic approaches based on sequential covalent bond formation, derivatization of preformed polymers, and stepwise polypeptide synthesis have been used to prepare molecular assemblies. A higher level hierarchial "assembly of assemblies" strategy is required for a working device, and progress has been made for metal polypyridyl complex assemblies based on sol-gels, electropolymerized thin films, and chemical adsorption to thin films of metal oxide nanoparticles.

Theory of solvation in polar nematics

We develop a linear response theory of solvation of ionic and dipolar solutes in anisotropic, axially symmetric polar solvents. The theory is applied to solvation in polar nematic liquid crystals. The formal theory constructs the solvation response function from projections of the solvent dipolar susceptibility on rotational invariants. These projections are obtained from Monte Carlo simulations of a fluid of dipolar spherocylinders which can exist both in the isotropic and nematic phases. Based on the properties of the solvent susceptibility from simulations and the formal solution, we have obtained a formula for the solvation free energy which incorporates the experimentally available properties of nematics and the length of correlation between the dipoles in the liquid crystal. The theory provides a quantitative framework for analyzing the steady-state and time-resolved optical spectra and makes several experimentally testable predictions. The equilibrium free energy of solvation, anisotropic in the nematic phase, is given by a quadratic function of cosine of the angle between the solute dipole and the solvent nematic director. The sign of solvation anisotropy is determined by the sign of dielectric anisotropy of the solvent: solvation anisotropy is negative in solvents with positive dielectric anisotropy and vice versa. The solvation free energy is discontinuous at the point of isotropic-nematic phase transition. The amplitude of this discontinuity is strongly affected by the size of the solute becoming less pronounced for larger solutes. The discontinuity itself and the magnitude of the splitting of the solvation free energy in the nematic phase are mostly affected by microscopic dipolar correlations in the nematic solvent. Illustrative calculations are presented for the equilibrium Stokes shift and the Stokes shift time correlation function of coumarin-153 in 4-n-pentyl-4'-cyanobiphenyl and 4,4-n-heptyl-cyanopiphenyl solvents as a function of temperature in both the nematic and isotropic phases.

Salt and Solvent Effects on the Kinetics of the Oxidation of the Excited State of the [Ru(bpy)3]2+ Complex by S2O82-

The title reaction was studied in different reaction media: aqueous salt solutions (NaNO3) and water-cosolvent (methanol) mixtures. The observed rate constants, k(obs), show normal behavior in the solutions containing the electrolyte, that is, a negative salt effect. However, the solvent effect is abnormal, because a decrease of the rate constant is observed when the dielectric constant of the reaction medium decreases. These effects (the normal and the abnormal) can be explained using the Marcus-Hush treatment for electron transfer reactions. To apply this treatment, the true, unimolecular, electron-transfer rate constants, k(et), have been obtained from k(obs) after calculation of the rate constants corresponding to the formation of the encounter complex from the separate reactants, k(D), and the dissociation of this complex, k(-D). This calculation has been carried out using an exponential mean spherical approach (EMSA).

Solvent Friction Effect on Intramolecular Electron Transfer

U-shaped donor-bridge-acceptor molecules with different electronic couplings have been investigated as a function of temperature in solvents with slow polarization relaxation, in particular, N-methylacetamide (NMA) and N-methylpropionamide (NMP). At high temperature, the electron-transfer rate is well described by a nonadiabatic model; however, the rate at low temperature is controlled by the solvent friction. The change of the electron-transfer mechanism is discussed and compared with theoretical models.

Equilibrium solvation in quadrupolar solvents

We present a microscopic theory of equilibrium solvation in solvents with zero dipole moment and nonzero quadrupole moment (quadrupolar solvents). The theory is formulated in terms of autocorrelation functions of the quadrupolar polarization (structure factors). It can be therefore applied to an arbitrary dense quadrupolar solvent for which the structure factors are defined. We formulate a simple analytical perturbation treatment for the structure factors. The solute is described by coordinates, radii, and partial charges of constituent atoms. The theory is tested on Monte Carlo simulations of solvation in model quadrupolar solvents. It is also applied to the calculation of the activation barrier of electron transfer reactions in a cleft-shaped donor-bridge-acceptor complex dissolved in benzene with the structure factors of quadrupolar polarization obtained from molecular-dynamics simulations.

“Abnormal” Salt and Solvent Effects on Anion/Cation Electron-Transfer Reactions: An Interpretation Based on Marcus−Hush Treatment

Salt and solvent effects on the kinetics of the reactions [Fe(CN)6]3- + [Ru(NH3)5pz](2+) right arrow over left arrow [Fe(CN)6]4- + [Ru(NH3)5pz]3+ (pz = pyrazine) have been studied through T-jump measurements. The forward and reverse reactions show different behaviors: "abnormal" salt and solvent effects in the first case and normal effects in the second one. These facts imply an asymmetric behavior of anion/cation reactions depending on the charge of the oxidant. The results can be rationalized by using the Marcus-Hush treatment for electron-transfer reactions.

Theory of torsional non-Condon electron transfer: A generalized spin-boson Hamiltonian and its nonadiabatic limit solution

The paper develops a theory of electron transfer with torsionally induced non-Condon (NC) effects. The starting point of the theory is a generalized spin-boson Hamiltonian, where an additional torsional oscillator bilinearly coupled to other bath modes causes a sinusoidal NC modulation. We derive closed form time dependent nonadiabatic rate expressions for both sudden and relaxed initial conditions, which are applicable for general spectral densities and energetic condition. Under the assumption that the torsional motion is not correlated with the polaronic shift of the bath, simple stationary limit rate expression is obtained. Model calculations of this rate expression illustrate the effects of torsional quantization and gating on the driving force and temperature dependences of the electron transfer rate. The classical limit of the rate expression consists of three Marcus-type terms, and is shown to agree very well with the exact numerical result.

Solvent reorganization energy of electron-transfer reactions in polar solvents

A microscopic theory of solvent reorganization energy in polar molecular solvents is developed. The theory represents the solvent response as a combination of the density and polarization fluctuations of the solvent given in terms of the density and polarization structure factors. A fully analytical formulation of the theory is provided for a solute of arbitrary shape with an arbitrary distribution of charge. A good agreement between the analytical procedure and the results of Monte Carlo simulations of model systems is achieved. The reorganization energy splits into the contributions from density fluctuations and polarization fluctuations. The polarization part is dominated by longitudinal polarization response. The density part is inversely proportional to temperature. The dependence of the solvent reorganization energy on the solvent dipole moment and refractive index is discussed.

Solvent effects on charge transfer bands of nitrogen-centered intervalence compounds

Electron transfer parameters are extracted from the optical spectra of intervalence bis(hydrazine) radical cations. Compounds with 2-tert-butyl-3-phenyl-2,3-diazabicyclo[2.2.2]octyl-containing charge-bearing units that are doubly linked by 4-sigma-bond and by 6-sigma-bond saturated bridges are compared with ones having tert-butylisopropyl- and diphenyl-substituted charge bearing units and others having the aromatic units functioning as the bridge. Solvent effect studies show that the optical transition energy (E(op)) does not behave as dielectric continuum theory predicts but that solvent reorganization energy may be usefully separated from the vibrational reorganization energy by including linear terms in both the Pekar factor (gamma) and the Gutmann donor number (DN) in correlating the solvent effect. Solvation of the bridge for these compounds is too large to ignore, which makes dielectric continuum theory fail to properly predict solvent effects on either E(op) or the free energy for comproportionation.