Reorganization Energy of Electron Transfer in Ionic Liquids.

Bandshape analysis of charge-transfer optical bands in room-temperature ionic liquids (ILs) was performed to extract the reorganization energy of electron transfer. Remarkably, the reorganization energies in ILs are close to those in cyclohexane. This result runs against common wisdom in the field since conducting ILs, which are characterized by an infinite static dielectric constant, and nonpolar cyclohexane fall to the opposite ends of the polarity scale based on their dielectric constants. Theoretical calculations employing structure factors of ILs from molecular dynamics simulations support the low values of the reorganization energy. Standard dielectric arguments do not apply to solvation in ILs, and nonergodic reorganization energies are required for a quantitative analysis.

Electron-Induced Proton Transfer.

The pathway of activationless proton transfer induced by an electron-transfer reaction is studied theoretically. Long-range electron transfer produces highly nonequilibrium medium polarization that can drive proton transfer through an activationless transition during the process of thermalization, dynamically altering the screening of the electron-proton Coulomb interaction by the medium. The cross electron-proton reorganization energy is the main energy parameter of the theory, which exceeds in magnitude the proton-transfer reorganization energy roughly by the ratio of the electron-transfer to proton-transfer distance. This parameter, which can be either positive or negative, is related to the difference in pKa values in two electron-transfer states. The relaxation time of the medium is on the (sub)picosecond time scale, which establishes the characteristic time for activationless proton transfer. Microscopic calculations predict substantial retardation of the collective relaxation dynamics compared to the continuum estimates due to the phenomenology analogous to de Gennes narrowing. Nonequilibrium medium configuration promoting proton transfer can be induced by either thermal or photoinduced charge transfer.

Optical Absorption by Charge-Transfer Molecules.

An analytical formalism is developed for the calculation of absorption of radiation by charge-transfer molecules with electronic density delocalized between the donor and acceptor parts. The theory consistently incorporates both the vibronic coupling to quantum intramolecular vibrations and electrostatic interactions with a classical polarizable medium. The formulation operates in terms of basis-invariant parameters and can be used for calculations based on both the localized diabatic states and delocalized adiabatic wave function produced by standard quantum-chemistry algorithms. The basis-invariant reorganization energy is particularly important, since it determines observable spectroscopic parameters, in contrast to the adiabatic reorganization energy based on the molecular charge distributions in the adiabatic vacuum states. Analytical formulas are derived for the charge-transfer absorption band and first two spectral moments connecting spectroscopy to properties of charge-transfer molecules. Electronic delocalization requires a new definition of the Huang-Rhys factor traditionally used to construct vibronic band shapes. Theory's performance is illustrated by application to experimental data for self-exchange charge-transfer optical absorption and for metal-to-ligand emission transitions.

Thermodynamics of Reactions Affected by Medium Reorganization.

We present a thermodynamic analysis of the activation barrier for reactions which can be monitored through the difference in the energies of reactants and products defined as the reaction coordinate (electron and atom transfer, enzyme catalysis, etc.). The free-energy surfaces along the reaction coordinate are separated into the enthalpy and entropy surfaces. For the Gaussian statistics of the reaction coordinate, the free-energy surfaces are parabolas, and the entropy surface is an inverted parabola. Its maximum coincides with the transition state for reactions with zero value of the reaction free energy. Maximum entropic depression of the activation barrier, anticipated by the concept of transition-state ensembles, can be achieved for such reactions. From Onsager's reversibility, the entropy of equilibrium fluctuations encodes the entropic component of the activation barrier. The reorganization entropy thus becomes the critical parameter of the theory reducing the problem of activation entropy to the problem of reorganization entropy. Standard solvation theories do not allow reorganization entropy sufficient for the barrier depression. Complex media, characterized by many relaxation processes, need to be involved. Proteins provide several routes for achieving large entropic effects through incomplete (nonergodic) sampling of the complex energy landscape and by facilitating an active role of water in the reaction mechanism.

Q-model of electrode reactions: altering force constants of intramolecular vibrations.

A theory of redox reactions involving electron transfer between a metal electrode and a molecule in solution is formulated in terms of two types of nuclear coordinates of the thermal bath: electrostatic polarization of the medium and local low-frequency vibrations. The polarization fluctuations follow Gaussian statistics. In contrast, the vibrational coordinate is allowed to change its force constant between two oxidation states of the reactant, which is projected onto non-Gaussian fluctuations of the reactant's electronic states. A closed-form analytical theory for the electrode redox reactions is formulated in terms of three reorganization energies: the reorganization energy for the electrostatic polarization of the medium and two internal (vibrational) reorganization energies for the reduced and oxidized states of the reactant. The theory predicts asymmetry between the cathodic and anodic branches of the electrode current driven by the corresponding difference in the vibrational force constants.

Electrode reactions in slowly relaxing media.

Standard models of reaction kinetics in condensed materials rely on the Boltzmann-Gibbs distribution for the population of reactants at the top of the free energy barrier separating them from the products. While energy dissipation and quantum effects at the barrier top can potentially affect the transmission coefficient entering the rate pre-exponential factor, much stronger dynamical effects on the reaction barrier are caused by the breakdown of ergodicity for populating the reaction barrier (violation of the Boltzmann-Gibbs statistics). When the spectrum of medium modes coupled to the reaction coordinate includes fluctuations slower than the reaction rate, such nuclear motions dynamically freeze on the reaction time scale and do not contribute to the activation barrier. Here we consider the consequences of this scenario for electrode reactions in slowly relaxing media. Changing the electrode overpotential speeds the electrode electron transfer up, potentially cutting through the spectrum of nuclear modes coupled to the reaction coordinate. The reorganization energy of electrochemical electron transfer becomes a function of the electrode overpotential, switching between the thermodynamic value at low rates to the nonergodic limit at higher rates. The sharpness of this transition depends on the relaxation spectrum of the medium. The reorganization energy experiences a sudden drop with increasing overpotential for a medium with a Debye relaxation but becomes a much shallower function of the overpotential for media with stretched exponential dynamics. The latter scenario characterizes the electron transfer in ionic liquids. The analysis of electrode reactions in room-temperature ionic liquids shows that the magnitude of the free energy of nuclear solvation is significantly below its thermodynamic limit. This result applies to reaction times faster than microseconds and is currently limited by the available dielectric relaxation data.

Impact of Temperature and Non-Gaussian Statistics on Electron Transfer in Donor-Bridge-Acceptor Molecules.

A combination of experimental data and theoretical analysis provides evidence of a bell-shaped kinetics of electron transfer in the Arrhenius coordinates ln k vs 1/T. This kinetic law is a temperature analogue of the familiar Marcus bell-shaped dependence based on ln k vs the reaction free energy. These results were obtained for reactions of intramolecular charge shift between the donor and acceptor separated by a rigid spacer studied experimentally by Miller and co-workers. The non-Arrhenius kinetic law is a direct consequence of the solvent reorganization energy and reaction driving force changing approximately as hyperbolic functions with temperature. The reorganization energy decreases and the driving force increases when temperature is increased. The point of equality between them marks the maximum of the activationless reaction rate. Reaching the consistency between the kinetic and thermodynamic experimental data requires the non-Gaussian statistics of the donor-acceptor energy gap described by the Q-model of electron transfer. The theoretical formalism combines the vibrational envelope of quantum vibronic transitions with the Q-model describing the classical component of the Franck-Condon factor and a microscopic solvation model of the solvent reorganization energy and the reaction free energy.

Solvent-Induced Shift of Spectral Lines in Polar-Polarizable Solvents.

Solvent-induced shift of optical transition lines is traditionally described by the Lippert-McRae equation given in terms of the Onsager theory for dipole solvation. It splits the overall shift into the equilibrium solvation by induced dipoles and the reaction field by the permanent dipoles in equilibrium with the chromophore in the ground state. We have reconsidered this classical problem from the perspective of microscopic solvation theories. A microscopic solvation functional is derived, and continuum solvation is consistently introduced by taking the limit of zero wavevector in the reciprocal-space solvation susceptibility functions. We show that the phenomenological expression for the reaction field of permanent dipoles in the Lippert-McRae equation is not consistent with the microscopic theory. The main deficiency of the Lippert-McRae equation is the use of additivity of the response by permanent and induced dipoles of the liquid. An alternative closed-form equation for the spectral shift is derived. Its continuum limit allows a new, nonadditive functionality for the solvent-induced shift in terms of the high-frequency and static dielectric constants. The main qualitative outcome of the theory is a significantly weaker dependence of the spectral shift on the polarizability of the solvent than predicted by the Lippert-McRae formula.

Free energy functionals for polarization fluctuations: Pekar factor revisited.

The separation of slow nuclear and fast electronic polarization in problems related to electron mobility in polarizable media was considered by Pekar 70 years ago. Within dielectric continuum models, this separation leads to the Pekar factor in the free energy of solvation by the nuclear degrees of freedom. The main qualitative prediction of Pekar's perspective is a significant, by about a factor of two, drop of the nuclear solvation free energy compared to the total (electronic plus nuclear) free energy of solvation. The Pekar factor enters the solvent reorganization energy of electron transfer reactions and is a significant mechanistic parameter accounting for the solvent effect on electron transfer. Here, we study the separation of the fast and slow polarization modes in polar molecular liquids (polarizable dipolar liquids and polarizable water force fields) without relying on the continuum approximation. We derive the nonlocal free energy functional and use atomistic numerical simulations to obtain nonlocal, reciprocal space electronic and nuclear susceptibilities. A consistent transition to the continuum limit is introduced by extrapolating the results of finite-size numerical simulation to zero wavevector. The continuum nuclear susceptibility extracted from the simulations is numerically close to the Pekar factor. However, we derive a new functionality involving the static and high-frequency dielectric constants. The main distinction of our approach from the traditional theories is found in the solvation free energy due to the nuclear polarization: the anticipated significant drop of its magnitude with increasing liquid polarizability does not occur. The reorganization energy of electron transfer is either nearly constant with increasing the solvent polarizability and the corresponding high-frequency dielectric constant (polarizable dipolar liquids) or actually noticeably increases (polarizable force fields of water).

Extension of Hopfield's Electron Transfer Model To Accommodate Site-Site Correlation.

Extension of the Förster analogue for the ET rate constant (based on virtual intermediate electron detachment or attachment states) with inclusion of site-site correlation due to coulomb terms associated with solvent reorganization energy and the driving force, has been developed and illustrated for a simple three-state, two-mode model. The model is applicable to charge separation (CS), recombination (CR), and shift (CSh) ET processes, with or without an intervening bridge. The model provides a unified perspective on the role of virtual intermediate states in accounting for the thermal Franck-Condon weighted density of states (FCWD), the gaps controlling superexchange coupling, and mean absolute redox potentials, with full accommodation of site-site coulomb interactions. Two types of correlation have been analyzed: aside from the site-site correlation due to coulomb interactions, we have emphasized the intrinsic "nonorthogonality" which generally pertains to reaction coordinates (RCs) for different ET processes involving multiple electronic states, as may be expressed by suitably defined direction cosines (cos(θ)). A pair of RCs may be nonorthogonal even when the site-site coulomb correlations are absent. While different RCs are linearly independent in the mathematical sense for all θ ≠ 0°, they are independent in the sense of being "uncorrelated" only in the limit of orthogonality (θ = 90°). Application to more than two coordinates is straightforward and may include both discrete and continuum contributions.

Electron Localization of Anions Probed by Nitrile Vibrations.

Localization and delocalization of electrons is a key concept in chemistry, and is one of the important factors determining the efficiency of electron transport through organic conjugated molecules, which have potential to act as "molecular wires". This, in turn, substantially influences the efficiencies of organic solar cells and other molecular electronic devices. It is also necessary to understand the electronic energy landscape and the dynamics that govern electron transport capabilities in one-dimensional conjugated chains so that we can better define the design principles for conjugated molecules for their applications. We show that nitrile ν(C≡N) vibrations respond to the degree of electron localization in nitrile-substituted organic anions by utilizing time-resolved infrared detection combined with pulse radiolysis. Measurements of a series of aryl nitrile anions allow us to construct a semiempirical calibration curve between the changes in the ν(C≡N) infrared (IR) shifts and the changes in the electronic charges from the neutral to the anion states in the nitriles; more electron localization in the nitrile anion results in larger IR shifts. Furthermore, the IR line width in anions can report a structural change accompanying changes in the electronic density distribution. Probing the shift of the nitrile ν(C≡N) IR vibrational bands enables us to determine how the electron is localized in anions of nitrile-functionalized oligofluorenes, considered as organic mixed-valence compounds. We estimate the diabatic electron transfer distance, electronic coupling strengths, and energy barriers in these mixed-valence compounds. The analysis reveals a dynamic picture, showing that the electron is moving back and forth within the oligomers with a small activation energy of ≤kBT, likely controlled by the movement of dihedral angles between monomer units. Implications for the electron transport capability in molecular wires are discussed.

Multistate treatments of the electronic coupling in donor-bridge-acceptor systems: insights and caveats from a simple model.

We use a simple one-dimensional delta function electronic structure model (dfm) to investigate the results of a pair of multistate diabatization techniques (i.e., based on n states, with n ≥ 2) for linear DBA and DBBA (donor-bridge-acceptor) electron-transfer systems. In particular, we focus on the physical meaning of the couplings obtained from multistate methods and their relationship to two-state (n = 2) coupling elements. On the basis of the simple dfm approach, which allows exact as well as finite basis set treatment and has no many-electron effects, we conclude that for orthogonal diabatic states, it is difficult to assign clear physical significance to multistate matrix elements for coupling beyond nearest-neighbor contacts. The implications of these results for more complex multistate many-electron treatments are discussed. It is emphasized that physically meaningful coupling elements must involve states that are orthogonal, either explicitly or implicitly.

Reduced electronic spaces for modeling donor/acceptor interactions.

Diabatic states for donor (D) and acceptor (A) interactions in electron transfer (ET) processes are formulated and evaluated, along with coupling elements (H(DA)) and effective D/A separation distances (r(DA)), for reduced electronic spaces of variable size, using the generalized Mulliken Hush model (GMH), applicable to an arbitrary state space and nuclear configuration, and encompassing Robin-Day class III and as well as class II situations. Once the electronic state space is selected (a set of n ≥ 2 adiabatic states approximated by an orbital space based on an effective 1-electron (1-e) Hamiltonian), the charge-localized GMH diabatic states are obtained as the eigenstates of the dipole moment operator, with rotations to yield locally adiabatic states for sites with multiple states. The 1-e states and energies are expressed in terms of Kohn-Sham orbitals and orbital energies. Addressing questions as to whether the estimate of H(DA) "improves" as one increases n, and in what sense the GMH approach "converges" with n, we carry out calculations for three mixed-valence binuclear Ru complexes, from which we conclude that the 2-state (2-st) model gives the most appropriate estimate of the effective coupling, similar (to within a rms deviation of ≤15%) to coupling elements obtained by superexchange correction of H(DA) values based on larger spaces (n = 3-6), and thus yielding a quasi-invariant value for H(DA) over the range explored in the calculations (n = 2-6). An analysis of the coupling and associated D and A states shows that the 2-st coupling involves crucial mixing with intervening bridge states (D and A "tails"), while increasingly larger state spaces for the same system yield increasingly more localized D and A states (and weaker coupling), with H(DA) tending to approach the limit of "bare" or "through space" coupling. These results help to reconcile seemingly contradictory assertions in the recent literature regarding the proper role of multistate frameworks in the formulation of coupling for both intra- and intermolecular ET systems.The present results are compared in detail with other reported results.

The spectral elucidation versus the X-ray structure of the critical precursor complex in bimolecular electron transfers: application of experimental/theoretical solvent probes to ion-radical (redox) dyads.

The mechanistic conundrum is commonly posed by the intrinsic structural disconnect between a bimolecular (reactive) intermediate that is fleetingly detected spectroscopically in solution versus that rigorously defined by isolation and X-ray crystallography. We resolve this ambiguity by the combined experimental and theoretical application of the solvent media probe to the transient (1:1) precursor complex in the simplest chemical reaction involving direct adiabatic electron transfer (ET) among various donor/acceptor pairs. Of particular help in our resolution of such an important ET problem is the characterization of the bimolecular precursor complex as Robin-Day class II (localized) or class III (delocalized) from either the solvent-dependent or the solvent-independent response of the diagnostic intervalence absorption bands for the quantitative evaluation of the electronic coupling elements. The magnitudes of these intracomplex bindings are confirmed by theoretical (ab initio and DFT) computations that derive from X-ray structures and Marcus-Hush theories. Most importantly, the experimental solvent-induced ET barriers evaluated from the intervalence absorption bands are also quantitatively verified by the calculated outer-shell reorganization energies to establish unambiguously the intimate interconnection between the loosely bound bimolecular intermediate identified concurrently in solution and in the solid state.

Structure, energetics, and electronic coupling in the (TCNE2)-* encounter complex in solution: a polarizable continuum study.

For the prototypical dyad (TCNE2)-*, previous in vacuo calculations indicate that sizable distortion of the equilibrium gas-phase structure may be required to reduce the donor/acceptor electronic coupling element (HDA) to the solution-phase experimental estimates. We employ the polarizable continuum model (PCM) to simulate the solvation environment for several polar solvents, finding noticeable structure change associated with the promotion of charge localization due to solvation. We have extended the counterpoise (CP) correction procedure so as to include fragment relaxation energies within the PCM model, and it would be of interest to incorporate this approach into schemes for optimizing coordinates on CP-corrected energy surfaces. The calculations include face-to-face encounter geometries as well as several lateral and twist distortions of the face-to-face structures. In proceeding from vacuum to solution, the calculated stabilization energy is reduced from -18 to -3 kcal/mol, and the calculated energy surface becomes flatter, with a somewhat larger minimum-energy separation of the monomer units (rDA). The corresponding minimum-energy structures are, respectively, delocalized and charge-localized. Using TD-DFT, spin-projected MP2 (PUMP2), and state-averaged two-configuration SCF (SA-TCSCF) calculations to evaluate HDA for symmetric encounter complex geometries (models for transition-state structures) indicates that HDA has comparable magnitude in the gas phase and in solution for a given dimer structure. SA-TCSCF calculations comparing HDA based on symmetric charge-delocalized structures and their asymmetric (minimum-energy) charge-localized counterparts (at a given rDA) yield very similar values. Even with account taken of the energetically accessible configurations probed by the PCM calculations, the HDA values remain significantly higher than the experimental estimates inferred from solution spectra and assumption of rDA based on crystal data. Clearly, additional calculations based on molecular-level solvent models would be of value in helping to characterize the intermolecular structures accessible to the encounter complex in polar solution.

Multichromophoric Förster resonance energy transfer from b800 to b850 in the light harvesting complex 2: evidence for subtle energetic optimization by purple bacteria.

This work provides a detailed account of the application of our multichromophoric Förster resonance energy transfer (MC-FRET) theory (Phys. Rev. Lett. 2004, 92, 218301) for the calculation of the energy transfer rate from the B800 unit to the B850 unit in the light harvesting complex 2 (LH2) of purple bacteria. The model Hamiltonian consists of the B800 unit represented by a single bacteriochlorophyll (BChl), the B850 unit represented by its entire set of BChls, the electronic coupling between the two units, and the bath terms representing all environmental degrees of freedom. The model parameters are determined, independent of the rate calculation, from the literature data and by a fitting to an ensemble line shape. Comparing our theoretical rate and a low-temperature experimental rate, we estimate the magnitude of the BChl-Qy transition dipole to be in the range of 6.5-7.5 D, assuming that the optical dielectric constant of the medium is in the range of 1.5-2. We examine how the bias of the average excitation energy of the B800-BChl relative to that of the B850-BChl affects the energy transfer time by calculating the transfer rates based on both our MC-FRET theory and the original FRET theory, varying the value of the bias. Within our model, we find that the value of bias 260 cm-1, which we determine from the fitting to an ensemble line shape, is very close to the value at which the ratio between MC-FRET and FRET rates is a maximum. This provides evidence that the bacterial system utilizes the quantum mechanical coherence among the multiple chromophores within the B850 in a constructive way so as to achieve efficient energy transfer from B800 to B850.

Interfacial bridge-mediated electron transfer: mechanistic analysis based on electrochemical kinetics and theoretical modelling.

Understanding the physical and chemical factors that control the kinetics of interfacial electron-transfer (ET) reactions is important for a large number of technological applications. The present article describes electrochemical kinetic studies of these factors, in which standard interfacial ET rate constants (k(0)(l)) have been measured for ET between substrate Au electrodes and various redox couples attached to the electrode surfaces by variable lengths (l) of oligomethylene (OM), oligophenylenevinylene (OPV) and oligophenyleneethynylene (OPE) bridges, which were constituents of mixed self-assembled monolayers (SAMs). The k(0)(l) measurements employed the indirect laser-induced temperature jump (ILIT) technique, which permits the measurement of interfacial ET rates that are orders of magnitude faster than those measurable by conventional techniques using the macroelectrodes that are the most convenient substrates for the mixed SAMs. The robustness of the measured rate constants (k(0)(l)), together with the Arrhenius activation energies (E(a)(l)) and preexponential factors (A(l)), is demonstrated by their invariance with respect to several experimental system parameters (including the chemical nature and length of the diluent component of the mixed SAM). Analysis of the kinetic results demonstrates that all of the observed interfacial ET processes proceed through a common type of transition state (predominantly associated with solvent reorganization around the redox moiety) and that the actual ET step involves direct electronic tunnelling between the Au electrode and the redox moiety. However, for the full range of l investigated, a global exponential decay of A(l) is not found for any of the three types of bridges. Possible reasons for this behavior, including the role of rate determining steps associated with adiabatic mechanisms within or beyond the transition state theory framework, are discussed, and comparisons with related conductance measurements are presented.