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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.

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.

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.

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.

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.

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.

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.

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 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.

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).

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.

A common strategy to calculate electronic coupling matrix elements for charge or energy transfer is to take the adiabatic states generated by electronic structure computations and rotate them to form localized diabatic states. In this paper, we show that, for intermolecular transfer of singlet electronic excitation, usually we cannot fully localize the electronic excitations in this way. Instead, we calculate putative initial and final states with small excitation tails caused by weak interactions with high energy excited states in the electronic manifold. These tails do not lead to substantial changes in the total diabatic coupling between states, but they do lead to a different partitioning of the total coupling between Coulomb (Forster), exchange (Dexter), and one-electron components. The tails may be reduced by using a multistate diabatic model or eliminated entirely by truncation (denoted as “chopping”). Without more information, we are unable to conclude with certainty whether the observed diabatic tails are a physical reality or a computational artifact. This research suggests that decomposition of the diabatic coupling between chromophores into Coulomb, exchange, and one-electron components may depend strongly on the number of states considered, and such results should be treated with caution.

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.

Intermolecular potential energy curves are presented and analyzed for three different water dimer structures: the normal hydrogen-bonded structure (C/sub s/); a weakly hydrogen-bonded structure (C/sub 1/) thought to be important in encounters of hydrated ions in aqueous solution; and a repulsive configuration (C/sub 2h/) characterized by a head-on approach of hydrogen atoms (OH...HO). Results are based on ab initio calculations (SCF and CI (at the MP2 level), with and without counterpoise (CP) correction for basis set superposition error (BSSE)) and also a recent semiempirical effective pair potential (RWK2). The CP corrections have a significant effect on the location and depth of the potential wells for the hydrogen-bonded structures, even when CI and extended basis sets are employed.

Formalisms suitable for calculating the rate of electron exchange between transition metal complexes in aqueous solution are reviewed and implemented in conjunction with ab initio quantum chemical calculations which provide crucial off-diagonal Hamiltonian matrix elements as well as other relevant electronic structural data. Rate constants and activation parameters are calculated for the hex-aquo Fe2 +-Fe3+ system, using a simple activated complex theory, a non-adiabatic semi-classical model which includes nuclear tunnelling, and a more detailed quantum mechanical method based on the Golden Rule. Comparisons are made between calculated results and those obtained by extrapolating experimental data to zero ionic strength. All methods yield similar values for the overall rate constant (a 0.1 L/(mol-sec)). The experimental activation parameters (δHd and δSd) are in somewhat better agreement with the semi classical and quantum mechanical results than with those from the simple activated complex theory, thereby providing some indication that non-adiabaticity and nuclear tunnelling may be important in the Fe2+/3+ exchange reaction. It is concluded that a model based on direct metal-metal overlap can account for the observed reaction kinetics provided the reactants are allowed to approach well within the traditional contact distance of 6.9 a. 6 figures, 7 tables.

A method for approximating hybrid and exchange integrals over atomic orbitals is discussed and examined in detail for the case of Slater-type orbitals (STO'S). The crux of the approximation is the least-squares projection of 2-center charge distributions onto suitably chosen 1-center charge distributions. The approximate integrals are thus linear combinations of Coulomb-type integrals, and are generated at a rate of ∼150/sec on a CDC 1604A computer. The method is employed in an SCF MO study of benzene and several of its ions.

Design of visible-light-driven photocatalytic materials has attracted intense interest in recent years in an attempt to enhance photonic efficiencies of hydrogen production via water splitting. Recent studies have shown that a solid solution of GaN and ZnO, (Ga1-xZnx)(N1-xOx), can oxidize water under visible light in the presence of a sacrificial electron acceptor. Here we present a systematic study of the structural and electronic properties of this (Ga1-xZnx)(N1-xOx) solid solution as a function of zinc (oxygen) concentration, x, using density-functional theory (DFT). The DFT+U approach has been adopted, and two different periodic supercells, the 16-atom (Ga8-nZnn)(N8-nOn) and 32-atom (Ga16-nZnn)(N16-nOn), have been used to model this solid solution. Results obtained from both supercells are in qualitative agreement with available experimental findings (in the lower concentration range x < 0.25), although overall the larger 32-atom supercell provides a better description of the (Ga1-xZnx)(N1-xOx) solid ...

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.

The main structural and electronic factors playing a role in intramolecular dissociative electron transfer of a simple donor-peptide-acceptor (D-peptide-A) model have been investigated by an integrated computational protocol based on the density functional theory, its time-dependent extension, and the polarizable continuum model. Our results allow us to elucidate the electronic states involved in the process and how they are perturbed by the orientation of the donor and the acceptor with respect to the peptide chain and by the presence of the solvent. We also report a semiquantitative estimation of the rate constant governing electron transfer obtained by a direct quantum mechanical evaluation of all the terms entering the kinetic expressions based on the Marcus theory and its extensions.

The paramagnetic [1:1] encounter complex ( TCNE ) 2 - is established as the important precursor in the kinetics and mechanism of electron-transfer for the self-exchange between tetracyanoethylene acceptor (TCNE) and its radical-anion as the donor. Spectroscopic observation of the dimeric complex ( TCNE ) 2 - by its intervalence absorption band at the solvent-dependent wavelength of λIV ∼ 1500 nm facilitates the application of Mulliken–Hush theory which reveals the significant electronic interaction extant between the pair of cofacial TCNE moieties with the sizable coupling of HDA = 1000 cm−1. The transient existence of such an encounter complex provides the critical link in the electron-transfer kinetics by lowering the classical Marcus reorganization barrier by the amount of HDA in this strongly adiabatic system. Ab initio quantum-mechanical methods as applied to independent theoretical computations of both the reorganization energy (λ) and the electronic coupling element (HDA) confirm the essential correctness of the Mulliken–Hush formalism for fast electron transfer via strongly coupled donor/acceptor encounter complexes.

We report microscopic calculations of free energies and entropies for intramolecular electron transfer reactions. The calculation algorithm combines the atomistic geometry and charge distribution of a molecular solute obtained from quantum calculations with the microscopic polarization response of a polar solvent expressed in terms of its polarization structure factors. The procedure is tested on a donor-acceptor complex in which ruthenium donor and cobalt acceptor sites are linked by a four-proline polypeptide. The reorganization energies and reaction energy gaps are calculated as a function of temperature by using structure factors obtained from our analytical procedure and from computer simulations. Good agreement between two procedures and with direct computer simulations of the reorganization energy is achieved. The microscopic algorithm is compared to the dielectric continuum calculations. We found that the strong dependence of the reorganization energy on the solvent refractive index predicted by continuum models is not supported by the microscopic theory. Also, the reorganization and overall solvation entropies are substantially larger in the microscopic theory compared to continuum models., Comment: 21 pages, 15 figures, typos corrected

A relationship is developed that facilitates a comparison between the standard electron-transfer rate constant (k0) of a redox couple covalently attached (as a constituent of a self-assembled monolayer (SAM)) to an electrode and the k0 of a similar redox couple dissolved in an electrolyte solution in contact with a similar (but not electroactive) electrode/SAM assembly. Such comparisons performed for ferrocene and ruthenium redox couples demonstrate that the (normalized) rate of electron-transfer (and, also, the electronic coupling) for a solution-dissolved couple through an alkanethiol SAM/aqueous electrolyte solution interface is, within one order of magnitude, equal to that for a covalently attached redox species. A related comparison involving the Marcus theory limiting rate constants for solution-dissolved couples and the Arrhenius preexponential factors for attached couples supplies additional evidence for this surprising result.

Definitive X-ray structures of "separated" versus "contact" ion pairs, together with their spectral (UV-NIR, ESR) characterizations, provide the quantitative basis for evaluating the complex equilibria and intrinsic (self-exchange) electron-transfer rates for the potassium salts of p-dinitrobenzene radical anion (DNB(-)). Three principal types of ion pairs, K(L)(+)DNB(-), are designated as Classes S, M, and C via the specific ligation of K(+) with different macrocyclic polyether ligands (L). For Class S, the self-exchange rate constant for the separated ion pair (SIP) is essentially the same as that of the "free" anion, and we conclude that dinitrobenzenide reactivity is unaffected when the interionic distance in the separated ion pair is r(SIP)or =6 Angstroms. For Class M, the dynamic equilibrium between the contact ion pair (with r(CIP) = 2.7 Angstroms) and its separated ion pair is quantitatively evaluated, and the rather minor fraction of SIP is nonetheless the principal contributor to the overall electron-transfer kinetics. For Class C, the SIP rate is limited by the slow rate of CIP right arrow over left arrow SIP interconversion, and the self-exchange proceeds via the contact ion pair by default. Theoretically, the electron-transfer rate constant for the separated ion pair is well-accommodated by the Marcus/Sutin two-state formulation when the precursor in Scheme 2 is identified as the "separated" inner-sphere complex (IS(SIP)) of cofacial DNB(-)/DNB dyads. By contrast, the significantly slower rate of self-exchange via the contact ion pair requires an associative mechanism (Scheme 3) in which the electron-transfer rate is strongly governed by cationic mobility of K(L)(+) within the "contact" precursor complex (IS(CIP)) according to the kinetics in Scheme 4.

The standard heterogeneous electron-transfer rate constants (k(n)0) between substrate gold electrodes and the ferrocene redox couple attached to the electrode surface by variable lengths of substituted or unsubstituted oligophenyleneethynylene (OPE) bridges as constituents of mixed self-assembled monolayers were measured as a function of temperature. The distance dependences of the unsubstituted OPE standard rate constants and of the preexponential factors (An) obtained from an Arrhenius analysis of the unsubstituted OPE k(n)0 versus temperature data are not monotonic. This surprising result, together with the distance dependence of the substituted OPE preexponential factors, may be assessed in terms of the likely conformational variability of the OPE bridges (as a result of the low intrinsic barrier to rotation of the phenylene rings in these bridges) and the associated sensitivity of the rate of electron transfer (and, hence, the single-molecule conductance which may be estimated using An) through these bridges to the conformation of the bridge. Additionally, the measured standard rate constants were independent of the identity of the diluent component of the mixed monolayer, and using an unsaturated OPE diluent has no effect on the rate of electron transfer through a long-chain alkanethiol bridge. These observations indicate that the diluent does not participate in the electron-transfer event.

In the case of superexchange tunneling, sensitivity of the tunneling gap (A) to progress along the thermal et reaction coordinate (including molecular as well as solvent contributions) has been formulated in terms of a general linear response framework, which is used to express the correlation between linearly independent reaction coordinates for distinct et steps. When the tunneling gap is relatively small (i.e., of the order of magnitude of reorganization energies), the corresponding sensitivity of the donor (D)/acceptor (A) coupling (H D A ) and its distance dependence (decay coefficient β) may be significant. The magnitude of Δ at the thermal et transition state has been compared with other reference values obtainable from optical or thermodynamic data. The initial state at equilibrium yields an upper bound for Δ, and so also does the equilibrium free energy change for charge injection onto the bridge in cases of et processes which are only moderately exothermic. The in vacuo value of A may constitute either an upper or lower bound, depending on the nature of the et process ( charge separation or charge shift) and the details of the relevant reorganization energies.

Contemporary continuum-based models of solvation in polar media are surveyed and assessed, with special focus on non-equilibrium solvation. A new hybrid approach combining molecular-level treatment of inertial solvent response, and inclusion of inertialess solvent response at the continuum level, is presented and illustrated in terms of calculated equilibrium solvation free energies for small molecular ions and reorganization free energies for model dumbbell systems.

Long-range electronic coupling of local donor and acceptor sites is formulated in the context of thermal and optical electron transfer and then illustrated with examples based on electronic structure calculations. The relationship of the calculated results to available experimental kinetic and optical data is discussed in detail. The influence of nuclear modes on the magnitude of the coupling (i.e., departures from the Condon approximation) is investigated in terms of both discrete molecular modes and solvent modes, and a general expression is presented for the modulation of the superexchange tunneling gap by motion along the electron transfer reaction coordinate.

The hybrid molecular–continuum model for polar solvation considered in this paper combines the dielectric continuum approximation for treating fast electronic (inertialess) polarization effects and a molecular dynamics (MD) simulation for the slow (inertial) polarization component, including orientational and translational solvent modes. The inertial polarization is generated by average charge distributions of solvent particles, composed of permanent and induced (electronic) components. MD simulations are performed in a manner consistent with the choice of solvent and solute charges such that all electrostatic interactions are scaled by the factor 1/e∞, where e∞ is the optical dielectric permittivity. This approach yields an ensemble of equilibrium solvent configurations adjusted to the electric field created by a charged or strongly polar solute. The electrostatic solvent response field is found as the solution of the Poisson equation including both solute and explicit solvent charges, with accurate acco...

This is the report of a DOE-sponsored workshop organized to discuss the status of our understanding of charge-transfer processes on the nanoscale and to identify research and other needs for progress in nanoscience and nanotechnology. The current status of basic electron-transfer research, both theoretical and experimental, is addressed, with emphasis on the distance-dependent measurements, and we have attempted to integrate terminology and notation of solution electron-transfer kinetics with that of conductance analysis. The interface between molecules or nanoparticles and bulk metals is examined, and new research tools that advance description and understanding of the interface are presented. The present state-of-the-art in molecular electronics efforts is summarized along with future research needs. Finally, novel strategies that exploit nanoscale architectures are presented for enhancing the efficiences of energy conversion based on photochemistry, catalysis, and electrocatalysis principles.

The charge-transfer transition energies and the electronic-coupling matrix element, |H(DA)|, for electron transfer from aminopyridine (ap) to the 4-carbonyl-2,2'-bipyridine (cbpy) in cbpy-(gly)(n)-ap (gly = glycine, n = 0-6) molecules were calculated using the Zerner's INDO/S, together with the Cave and Newton methods. The oligopeptide linkages used were those of the idealized protein secondary structures, the alpha-helix, 3(10)-helix, beta-strand, and polyproline I- and II-helices. The charge-transfer transition energies are influenced by the magnitude and direction of the dipole generated by the peptide secondary structure. The electronic coupling |H(DA)| between (cbpy) and (ap) is also dependent on the nature of the secondary structure of the peptide. A plot of 2.ln|H(DA)| versus the charge-transfer distance (assumed to be the dipole moment change between the ground state and the charge-transfer states) showed that the polyproline II structure is a more efficient bridge for long-distance electron-transfer reactions (beta = 0.7 A(-1)) than the other secondary structures (beta approximately 1.3 A(-1)). Similar calculations on charged dipeptide derivatives, [CH(3)CONHCH(2)CONHCH(3)](+/)(-), showed that peptide-peptide interaction is more dependent on conformation in the cationic than in the anionic dipeptides. The alpha-helix and polyproline II-helix both have large peptide-peptide interactions (|H(DA)|800 cm(-1)) which arise from the angular dependence of their pi-orbitals. Such an interaction is much weaker than in the beta-strand peptides. These combined results were found to be consistent with electron-transfer rates experimentally observed across short peptide bridges in polyproline II (n = 1-3). These results can also account for directional electron transfer observed in an alpha-helical structure (different ET rates versus the direction of the molecular dipole).

Current theories of electron transfer (ET) kinetics in conjunction with techniques of computational quantum chemistry are reviewed and applied to the analysis of thermal and optical ET processes involving transition metal complexes (TMCs). The roles of geometrical and electronic structure in controlling the relevant energetics and electronic coupling are discussed, and a number of limiting cases of kinetic behavior are displayed. Particular attention is paid to the formulation of the initial and final states involved in the ET, including the consequences of near-degeneracy and spin–orbit coupling. The roles of metal/ligand mixing and other types of ‘mediation’ of donor/acceptor coupling by molecular spacers are noted and illustrated with calculated results. An example of a detailed computation of activation parameters is also presented, including entropic effects and nuclear tunnelling.

The linearized MD technique is developed in order to treat systematically free energies of simple charged and dipolar solutes in water. The solvent electrostatic response field in the solute region is modeled by averaging instantaneous fields found in a MD computation for solvent configurations confined within a cavity that conforms to the real shape of the solute particle. At this stage, all electrostatic interactions are explicitly treated inside the cavity. The solvent in the external region (outside the cavity) is modeled in terms of a standard continuum theory. For nonspherical cavities, the present approach is more accurate than the field computation employed at the preceding MD stage, where spherically truncated Coulomb potentials are modified by the reaction field corrections. We considered two different linearization schemes based on a computation of either the average response field or of its fluctuations. Only the first algorithm proved to be successful. For a series of single-charged monatomic...

Fluorescence of fac-AlQ3 (Q=8-quinolinolato), mer-AlQ3, mer-Al(mQ)3 (mQ=4-methyl-8-quinolinolato), and BeQ2 were investigated with electronic structure calculations. The molecular structure of the first singlet excited state (the emission state) was optimized with the ab initio “configuration interaction with single excitations” (CIS) method. Ab initio CIS and semiempirical “Zerner’s intermediate neglect of differential overlap” (ZINDO) methods were used to calculate the emission energies (ΔE) and also the corresponding absorption energies. Although the ab initio CIS method overestimated the experimental value of ΔE by 1.09–1.16 eV, the ZINDO method reproduced it to a reasonable accuracy (within 0.26 eV). The optimized excited-state structure has an interesting feature in that one of the equivalent ligands distorts appreciably, while the thers keep their ground-state structures. As a result the highest occupied and the lowest unoccupied molecular orbitals (HOMO and LUMO) are localized on the distorted lig...

We have developed a band-shape analysis of optical transitions in polarizable chromophores characterized by large magnitudes of the transition dipole (intense transitions). The model is tested on steady-state spectra of the coumarin-153 optical dye, employing an explicit solvent description accounting for dipole moment, quadrupole moment, and polarizability of the solvent molecules. The calculations are performed for solvents ranging from nondipolar to strongly dipolar. The solvent dependence of both the experimental Stokes shift and the spectral width is satisfactorily reproduced over the whole polarity range. The optical width is shown to demonstrate a qualitatively different solvent dependence for absorption and emission. The solvent-induced absorption width increases with solvent polarity, whereas the solvent-induced emission width passes through a maximum. This is a result of non-Gaussian statistics of the energy gap fluctuations in polarizable/electronically delocalized chromophores. The total (i.e....

Theoretical relationships are developed to describe the open-circuit responses associated with the indirect laser-induced temperature-jump (ILIT) method, a method for measuring fast electron-transfer rate constants of surface-attached redox species. The analysis is also applicable to data obtained using the coulostatic charge-injection method. The unique relationship between km, the relaxation rate constant for the ILIT (or coulostatic) response, and Ei, the potential at which the system is initially poised, exhibits a surprising sensitivity to the values of k0, Ei0′ (the standard rate constant and formal potential for the redox couple), α (the transfer coefficient in the Butler–Volmer equation) and γ (a dimensionless parameter which is directly proportional to the total surface concentration of the redox moiety). ILIT data for several examples of surface-attached ferrocene moieties confirm the theoretically predicted km vs Ei behavior. Values of Ei0′ and γ extracted from the ILIT data agree well with the values obtained from cyclic voltammetric data thereby confirming that the ILIT and cyclic voltammogram (CV) experiments are sampling the same ferrocene population. Keywords: Temperature-jump, Laser, ILIT, Coulostatic, Surface-attached redox species, Redox capacitance, Electron transfer resistance

We examine a simple approximate method for calculating the electron transfer (ET) distance suitable for extracting the off-diagonal electronic coupling element (Hab) of Marcus−Hush theory from the ...

An extended superexchange model for electron transfer (ET) matrix elements (H{sub DA}) has been formulated as a superposition of McConnell-type pathways and implemented by combined use of configuration interaction wave functions (obtained using the INDO/s model of Zerner and co-workers) and the generalized Muliken-Hush formulation of charge-localized diabatic states. Applications are made for et (and hold transfer) in several donor/bridge/acceptor radical anion (and cation) systems, (DBA){sup {+-}}, allowing detailed comparison with experimental H{sub DA} estimates. For the case of oligo phenylene ethynylene (OPE) bridges, the role of {pi} and {sigma} electronic manifolds for different distributions of phenylene torsion angles is analyzed in detail.

Quantum rate theory, molecular dynamics simulations, and semiempirical electronic structure calculations are used to fully investigate electron transfer mediated by a solvated polypeptide for the first time. Using a stationary-phase approximation, the nonadiabatic electron-transfer rate constant is calculated from the nuclear free energies and the electronic coupling between the initial and final states. The former are obtained from quantum path integral and classical molecular dynamics simulations; the latter are calculated using semiempirical electronic structure calculations and the generalized Mulliken−Hush method. Importantly, no parameters are fit to kinetic data. The simulated system consists of a solvated four-proline polypeptide with a tris(bipyridine)ruthenium donor group and an oxypentamminecobalt acceptor group. From the simulation data entropy and energy contributions to the free energies are distinguished. Quantum suppression of the barrier, including important solvent contributions, is demo...

The spectral parameters for the optically induced intervalence charge transfer and the rates of thermal electron transfer as a function of temperature have been measured for a rigid, triply linked mixed-valence dinuclear tris(2,2‘-bipyridine)iron complex. The total reorganizational energy associated with the intramolecular electron exchange in this complex is almost exclusively outer-sphere in nature and comes from thermal fluctuations of the solvent. Thus, the system can be treated rigorously at the classical level, where in this context classical refers to treatments of the nuclear modes. The theories developed to describe the optical electron transfer and the thermal electron transfer are evaluated by analysis of the spectral and rate data, respectively. The quantities common to both theories are the donor−acceptor coupling matrix element, H12, and the total reorganizational energy. Applying the respective theories to the appropriate corresponding sets of data yields reorganizational energies that are ...

The Born–Oppenheimer (BO) formulation of polar solvation is developed and implemented at the semiemperical (PM3) configuration interaction (CI) level, yielding estimates of electron transfer (ET) coupling elements ( V 0 ) for intramolecular ET in several families of radical ion systems. In contrast to the traditional treatment based on a single solvent coordinate and a fixed gas-phase coupling element, the present treatment yields a self-consistent characterization of kinetic parameters in a 2-dimensional solvent framework which includes an exchange coordinate. The dependence of V 0 on inertial solvent contributions and on donor/acceptor separation ( r DA ) is discussed.

The frequency-resolved cavity model (FRCM), a generalized continuum reaction field model, which allows for distinct effective solute cavities pertaining to optical (op) and inertial (in) solvent response, has been implemented and applied to the evaluation of solvent reorganization energy ( E s ) for a number of intramolecular electron transfer (ET) processes in polar media. Specifically, effective radii are defined for the solute atoms: r ∞ = κ · r vdW (where κ is taken as a universal scale factor) and r in = r ∞ + δ (where δ is specific to a particular solvent). Optimal values of κ and δ are determined through the use of solvation free energy data for small atomic and molecular ions, together with the experimental estimates of solvation reorganization energy ( E s ) for intramolecular ET in the steroid-based radical ions studied by Closs, Miller and co-workers [G.L. Closs, L.T. Calcaterra, N.J. Green, K.W. Penfield, J.R. Miller, J. Phys. Chem. 90 (1986) 3673; M.D. Johnson, J.R. Miller, N.S. Green, G.L. Closs, J. Phys. Chem. 93 (1989) 1173; J.R. Miller, B.P. Paulson, R. Bal, G.L. Closs, J. Phys. Chem. 99 (1995) 6923]. With these optimal parameters, E s is then evaluated for a number of other intramolecular ET processes, yielding results which are in generally good agreement with experimentally based estimates, and which give support for some of the assumptions employed in the analysis of the experimental data. Calculations with conventional solute atom radii ( r ∞ = r in , with κ =1.2 and δ =0) fitted to equilibrium solvation data yield E s values exceeding the FRCM results by factors of ≥2.

Two quantities crucial in controlling long-range electron transfer (ET) kinetics are the medium (or solvent) reorganization energy E r and the donor/acceptor (D/A) electronic coupling integral H DA . Detailed theoretical models for E r and H DA are exploited in a variety of calculations, which include either continuum or molecular-level representations of solvent. The ability of different types of interactions (covalent, hydrogen-bonded and non-bonded) in mediating effective D/A coupling are evaluated, and new procedures for defining diabatic states, including a generalization of the Mulliken-Hush model, allow applications to optical and excited state as well as ground state ET in a many-state framework.

Two independent methods are presented for the nonperturbative calculation of the electronic coupling matrix element (Hab) for electron transfer reactions using ab initio electronic structure theory. The first is based on the generalized Mulliken–Hush (GMH) model, a multistate generalization of the Mulliken Hush formalism for the electronic coupling. The second is based on the block diagonalization (BD) approach of Cederbaum, Domcke, and co-workers. Detailed quantitative comparisons of the two methods are carried out based on results for (a) several states of the system Zn2OH2+ and (b) the low-lying states of the benzene–Cl atom complex and its contact ion pair. Generally good agreement between the two methods is obtained over a range of geometries. Either method can be applied at an arbitrary nuclear geometry and, as a result, may be used to test the validity of the Condon approximation. Examples of nonmonotonic behavior of the electronic coupling as a function of nuclear coordinates are observed for Zn2OH2+. Both methods also yield a natural definition of the effective distance (rDA) between donor (D) and acceptor (A) sites, in contrast to earlier approaches which required independent estimates of rDA, generally based on molecular structure data.

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.

Important energy quantities governing electron transfer (ET) kinetics in polar solutions (reorganization energy, Er, and net free energy change, ΔU) are evaluated on the basis of quantum-chemical self-consistent reaction-field (SCRF) models. Either self-consistent field (SCF) or configuration interaction (CI) wavefunctions are used for the solute, which occupies a molecular cavity of realistic shape in a dielectric continuum. A classical SCRF model together with unrestricted Hartree-Fock SCF wavefunctions based on the semiempirical PM3 Hamiltonian is applied to the calculation of the solvent portion of Er (denoted Es) for two different series of radical ion ET systems: radical cations and anions of biphenylyl/naphthyl donor/acceptor ( D A ) pairs linked by cyclohexane-based spacer groups and trans-staggered radical anions of the type (CH2)2m, m = 2–5. Results for Es based on two-configurational CI wavefunctions and an alternative reaction field (the so-called Born-Oppenheimer model, which recognizes the fast timescales of solvent electrons relative to those involved in ET) are also noted. Results for inner-sphere (i.e. intra-solute) reorganization, Ei, and for ΔU are also reported. The semiempirical Es results are quite similar to corresponding ab initio results and display the form of the two-sphere Marcus model for Es as a function of D A separation. Nevertheless, in the one case where direct comparison is possible, the calculated Es result is more than twice the magnitude of the estimate based on experimental ET kinetic data. To reconcile this situation, a generalized SCRF model is proposed, which assigns different effective solute cavity sizes to the optical and inertial components of the solvent response, using ideas based on non-local solvation models.