LONG RANGE ELECTRON TUNNELING IN BIOLOGICAL AND MODEL SYSTEMS

Abstract— This review paper reports the data on the role of long range electron tunneling in photosynthetic and model systems. The main papers concerned with elucidation mechanisms and kinetical peculiarities of electron transfer in reaction centers of bacteria and green plants are considered. The paper reviews the articles on long range electron transfer in reactions of metalloporphyrins—synthetic analogs of the main natural pigment of photosynthesis, chlorophyll. Liquid and solid phase redox reactions of electronically excited porphyrin molecules, intramolecular and photosensitized electron transfer processes are considered.     

Tunneling transfer of atomic particles in chemical and biological reactions: The role of intermolecular vibrations and media reorganization

Various types of chemical and biological tunneling reactions in a condensed phase are discussed. The analytical expressions for the rate constants in different temperature ranges are given. Experimental data on such low-temperature processes as hydrogen transfer from a molecule to a radical between two molecules and intramolecular transformations are considered. Data on the kinetic isotope effect upon the transfer of atomic particles in the solid phase and biological liquids are presented. The effect pressure has on different tunneling reactions is also considered; where possible, experimental results are compared with theory.     

Long-distance electron tunneling in proteins

Long-distance tunneling is the major mechanism of electron transfer (ET) in proteins. For a number of years, a major question has been whether specific electron tunneling pathways exist. This question is still debated in the literature, because the pathways are not observed directly, and interpretation of experimental results on ET rates involves ambiguities. The extremely small tunneling interactions are difficult to calculate accurately. Recently, there has been remarkable progress in the area; however, some problems still remain unresolved. The accurate prediction of the absolute rates of long-distance ET reactions and other biological charge-transfer reactions is a particularly pressing issue. The current theoretical calculations indicate that the specific paths do exist in static protein structures. However, the protein motions can result in significant averaging of the spatial tunneling patterns, and it is not clear how accurately subtle quantum interference effects are described by the present theories. The key to resolving these issues is to perform accurate, first-principles calculations of electron tunneling that include the dynamics of the protein. This paper reviews some of theoretical issues of electron tunneling dynamics in inhomogeneous organic media.     

Rate-promoting vibrations and coupled hydrogen-electron transfer reactions in the condensed phase: a model for enzymatic catalysis

A model is presented for coupled hydrogen-electron transfer reactions in condensed phase in the presence of a rate promoting vibration. Large kinetic isotope effects (KIEs) are found when the hydrogen is substituted with deuterium. While these KIEs are essentially temperature independent, reaction rates do exhibit temperature dependence. These findings agree with recent experimental data for various enzyme-catalyzed reactions, such as the amine dehydrogenases and soybean lipoxygenase. Consistent with earlier results, turning off the promoting vibration results in an increased KIE. Increasing the barrier height increases the KIE, while increasing the rate of electron transfer decreases it. These results are discussed in light of other views of vibrationally enhanced tunneling in enzymes.     

Electron tunneling in solid phase redox reactions. Experimental values of parameters characterizing the rate of tunneling and theoretical models for long range electron transfer

The problem of the quantitative analysis of the kinetic data for the tunnel electron transfer reactions in solids is discussed. Various procedures are considered with the help of which one can determine from the experimental data the parameters v and a in the equation w = v exp (?2r/a) describing the dependence of the electron tunneling frequency w on the distance r between the reacting particles. For numerous tunnel reactions studied in the literature these parameters are calculated and their values are analyzed in terms of theoretical models of electron tunneling. The physical background of these models is discussed in detail to make it more understandable for experimentalists. The limits of applications for these models are discussed. A novel model for nuclear motion in the process of electron tunneling is proposed to account for the unusually high values of the frequency factor found for some of the reactions.     

Hydrogen Evolution on a Mercury Electrode: The Effect of a Condensed Adsorption Layer

The electroreduction of hydrogen on a dropping mercury electrode with and without condensed adsorption layers (CAL) of organic compounds is studied by methods of classical polarography and laser photoemission. An analysis of effects CAL has on some reactions reveals that the CAL influence on the first-electron transfer and on reactions involving intermediates can retard or even completely block the process. The most profound changes occur in the processes that involve a proton donor. Possible reasons for different effects of CAL on the electron transfer processes are discussed.     

Anomalous temperature-isotope dependence in proton-coupled electron transfer

Motivated by the experiments of Hodgkiss et al. [J. Phys. Chem. (submitted)] on electron transfer (ET) through a H-bonding interface, we present a new theoretical model for proton-coupled electron transfer (PCET) in the condensed phase, that does not involve real proton transfer. These experiments, which directly probe the joint T-isotope effects in coupled charge transfer reactions, show anomalous T dependence in k(H)k(D), where k(H) and k(D) are the ET rates through the H-bonding interface with H-bonded protons and deuterons, respectively. We address the anomalous T dependence of the k(H)k(D) in our model by attributing the modulation of the electron tunneling dynamics to bath-induced fluctuations in the proton coordinate, so that the mechanism for coupled charge transfer might be better termed vibrationally assisted ET rather than PCET. We argue that such a mechanism may be relevant to understanding traditional PCET processes, i.e., those in which protons undergo a transfer from donor to acceptor during the course of ET, provided there is an appropriate time scale separating both coupled charge transfers. Likewise, it may also be useful in understanding long-range ET in proteins, where tunneling pathways between redox cofactors often pass through H-bonded amino acid residues, or other systems with sufficiently decoupled proton and electron donating functionalities.     

Extended spin-boson model for nonadiabatic hydrogen tunneling in the condensed phase

A nonadiabatic rate expression for hydrogen tunneling reactions in the condensed phase is derived for a model system described by a modified spin-boson Hamiltonian with a tunneling matrix element exponentially dependent on the hydrogen donor-acceptor distance. In this model, the two-level system representing the localized hydrogen vibrational states is linearly coupled to the donor-acceptor vibrational mode and the harmonic bath. The Hamiltonian also includes bilinear coupling between the donor-acceptor mode and the bath oscillators. This coupling provides a mechanism for energy exchange between the two-level system and the bath through the donor-acceptor mode, thereby facilitating convergence of the time integral of the probability flux correlation function for the case of weak coupling between the two-level system and the bath. The dependence of the rate constant on the model parameters and the temperature is analyzed in various regimes. Anomalous behavior of the rate constant is observed in the weak solvation regime for model systems that lack an effective mechanism for energy exchange between the two-level system and the bath. This theoretical formulation is applicable to a wide range of chemical and biological processes, including neutral hydrogen transfer reactions with small solvent reorganization energies.     

偶氮苯自组装单分子膜中长程电子转移机理

利用自组装技术在金电极表面构造了具有不同前端健长度偶氮苯功能化的单分子膜体系：Au／S（CH2）nNHCO-N＝N-OCH2CH3（n＝2，3，4，6）．研究结果表明，仍氮苯到金电极的表现电子转移速率随它们之间的距离长度的增加而呈指数性的下降趋势．基于Marcus电子隧穿理论，得到了此自组装膜体系的长程电子隧穿系数ρ＝（1．35±0．2）／CH2在和可逆电活性分子自组装膜体系及理论计算相比较的基础上，从偶氮苯分子自组装膜结构与电子转移过程的关系角度对这一结果进行了分析和说明．     

Electron tunneling in lithium-ammonia solutions probed by frequency-dependent electron spin relaxation studies

Electron transfer or quantum tunneling dynamics for excess or solvated electrons in dilute lithium-ammonia solutions have been studied by pulse electron paramagnetic resonance (EPR) spectroscopy at both X- (9.7 GHz) and W-band (94 GHz) frequencies. The electron spin-lattice (T(1)) and spin-spin (T(2)) relaxation data indicate an extremely fast transfer or quantum tunneling rate of the solvated electron in these solutions which serves to modulate the hyperfine (Fermi-contact) interaction with nitrogen nuclei in the solvation shells of ammonia molecules surrounding the localized, solvated electron. The donor and acceptor states of the solvated electron in these solutions are the initial and final electron solvation sites found before, and after, the transfer or tunneling process. To interpret and model our electron spin relaxation data from the two observation EPR frequencies requires a consideration of a multiexponential correlation function. The electron transfer or tunneling process that we monitor through the correlation time of the nitrogen Fermi-contact interaction has a time scale of (1-10) × 10(-12) s over a temperature range 230-290 K in our most dilute solution of lithium in ammonia. Two types of electron-solvent interaction mechanisms are proposed to account for our experimental findings. The dominant electron spin relaxation mechanism results from an electron tunneling process characterized by a variable donor-acceptor distance or range (consistent with such a rapidly fluctuating liquid structure) in which the solvent shell that ultimately accepts the transferring electron is formed from random, thermal fluctuations of the liquid structure in, and around, a natural hole or Bjerrum-like defect vacancy in the liquid. Following transfer and capture of the tunneling electron, further solvent-cage relaxation with a time scale of ～10(-13) s results in a minor contribution to the electron spin relaxation times. This investigation illustrates the great potential of multifrequency EPR measurements to interrogate the microscopic nature and dynamics of ultrafast electron transfer or quantum-tunneling processes in liquids. Our results also impact on the universal issue of the role of a host solvent (or host matrix, e.g. a semiconductor) in mediating long-range electron transfer processes and we discuss the implications of our results with a range of other materials and systems exhibiting the phenomenon of electron transfer.     

Analysis of kinetic models for the tunnel electron transfer reactions. Reaction kinetics for various radial and angular dependences of the tunneling probability

A new version of the “stepwise” approximation used in kinetic models for the long range electron transfer reactions is suggested. Using this approach the effects on the kinetics of these reactions of such factors are examined as a more complicated dependence (compared to the commonly used exponential one) of the tunneling probability, w, on the distance between the reagents as well as the angular dependence of w. For many practically important situations taking account of the above factors is shown to have no significant effect on the form of the reaction kinetics for both the pairwise and random spatial distributions of the reagents. However, the relations between the parameters a_ef and ν_cf in the expression for the tunneling distance R* = ₂¹a_ef In ν_cft, which can be determined from the experimental kinetics, and similar theoretical parameters ν_o and a that are involved in the expressions for w are found to be different for various types of the radial and the angular dependences of w. It is shown that when the orbitals between which the electron transfer takes place are located at sites removed from the geometric centers of the reactants the values of ν_ef can exceed by many orders of magnitude the characteristic frequency of the electron movement in molecules ν_e ≈ 10¹³ s^?1.     

Coherence in electron transfer pathways

Central to the view of electron-transfer reactions is the idea that nuclear motion generates a transition state geometry at which the electron/hole amplitude propagates coherently from the electron donor to the electron acceptor. In the weakly coupled or nonadiabatic regime, the electron amplitude tunnels through an electronic barrier between the donor and acceptor. The structure of the barrier is determined by the covalent and noncovalent interactions of the bridge. Because the tunneling barrier depends on the nuclear coordinates of the reactants (and on the surrounding medium), the tunneling barrier is highly anisotropic, and it is useful to identify particular routes, or pathways, along which the transmission amplitude propagates. Moreover, when more than one such pathway exists, and the paths give rise to comparable transmission amplitude magnitudes, one may expect to observe quantum interferences among pathways if the propagation remains coherent. Given that the effective tunneling barrier height and width are affected by the nuclear positions, the modulation of the nuclear coordinates will lead to a modulation of the tunneling barrier and hence of the electron flow. For long distance electron transfer in biological and biomimetic systems, nuclear fluctuations, arising from flexible protein moieties and mobile water bridges, can become quite significant. We discuss experimental and theoretical results that explore the quantum interferences among coupling pathways in electron-transfer kinetics; we emphasize recent data and theories associated with the signatures of chirality and inelastic processes, which are manifested in the tunneling pathway coherence (or absence of coherence).     

Nonadiabatic transition rates in a random motion model. Near adiabatic limit

The near adiabatic limit for nonadiabatic electronic transitions in condensed media is considered. The motion of a classical subsystem is approximated by a one-dimensional stochastic gaussian process. An analytical expression for the near adiabatic transition rate W is obtained in the limits: λ <a and λ ? a, where λ is the step size of classical stochastic motion and a is the size of the nonadiabaticity region. An approx interpolating expression for the intermediate case λ ≈ a is proposed. It is shown that W decreases rapidly with the increase of adiabatic s Application of the obtained expressions to the electron transfer processes in polar liquids is discussed briefly.     

Eyringpy: A program for computing rate constants in the gas phase and in solution

Eyringpy is a modular program for calculating thermochemical properties and rate constants for reactions in the gas phase and in solution. The code is written in Python and it has a user-friendly interface and a simple input format. Unimolecular and bimolecular reactions with one and two products are supported. Thermochemical properties are estimated through canonical ensemble and rate constants are computed according to the transition state theory. One-dimensional Wigner and Eckart tunneling corrections are also available. Rate constants of bimolecular reactions involving the formation of pre-reactive complexes are also estimated. To compute rate constants in solution, Eyringpy uses the Collins–Kimball theory to include the diffusion-limit, the Marcus theory for electron transfer processes, and the molar fractions to account for the solvent pH effect.     

Quantum effects in ab-initio calculations of rate constants for chemical reactions occuring in the condensed phase

An overview of recent advances in the development of methods designed to calculate rate constants for chemical reactions obeying mass action kinetic equations in condensed phases is presented. A general framework addressing mixed quantum-classical systems is elaborated that enables quantum features such as tunneling effects, zero-point vibrations, dynamic quantum coherence, and non-adiabatic effects to be calculated. An efficient Monte Carlo sampling method for performing ab-initio calculations of rate constants and isotope effects in chemical processes in condensed phases is outlined, and the connection of isotope effects to reaction mechanism is explored     

Separation of pinhole and tunneling electron transfer processes at self-assembled polymeric monolayers on gold electrodes

Self-assembled monolayers of poly(3-alkylthiophene) on gold electrodes are examined by cyclic voltammetry in solutions containing electroactive species. Two well-separated electron transfer processes, namely, electron tunneling through the monolayer and electron exchange at pinholes (defects) of the monolayer are observed. The voltammetric responses of the pinhole electron transfer process take place around the standard potential of the electroactive species and resemble those of a nanoelectrode ensemble of independent individual nanoelectrodes. The voltammetric characteristics of the electron tunneling agree well with predictions of the Marcus theory. Satisfactory values of tunneling coefficient, standard rate constant and organization energy are derived from the voltammetric data.     

One-dimensional description of multidimensional electron transfer reactions in condensed phase

We derive a one-dimensional energy diffusion equation for describing the dynamics of multidimensional electron transfer reactions in condensed phase, which is conceptually simpler and computationally more economic than the conventional approaches. We also obtain an analytical expression for the rate of electron transfer reactions for a general one-dimensional effective potential as well as an energy dependent diffusitivity. As an illustrative example, we consider application to electron transfer in a contact ion pair system modeled through harmonic potentials consisting of two slow classical modes and a high frequency vibrational mode for which the numerical results calculated using the proposed one-dimensional approach are shown to be in good agreement with experimental results. The energy diffusion equation and the rate expression for electron transfer obtained from the present theory, therefore, open up the possibility of describing the dynamics of electron transfer in complex systems, through a simpler approach.     

Electrochemical gating of single osmium molecules tethered to Au surfaces

The electrochemical study of electron transport between Au electrodes and the redox molecule Os[(bpy)₂(PyCH₂ NH₂CO-]ClO₄ tethered to molecular linkers of different length (1.3 to 2.9 nm) to Au surfaces has shown an exponential decay of the rate constant k _ET ⁰ with a slope β = 0.53 consistent with through bond tunneling to the redox center. Electrochemical gating of single osmium molecules in an asymmetric tunneling nano-gap between a Au(111) substrate electrode modified with the redox molecules and a Pt-Ir tip of a scanning tunneling microscope was achieved by independent control of the reference electrode potential in the electrolyte, E _ref ? E _s, and the tip-substrate bias potential, E _bias. Enhanced tunneling current at the osmium complex redox potential was observed as compared to the off resonance set point tunneling current with a linear dependence of the overpotential at maximum current vs. the E _bias. This corresponds to a sequential two-step electron transfer with partial vibration relaxation from the substrate Au(111) to the redox molecule in the nano-gap and from this redox state to the Pt-Ir tip according to the model of Kuznetsov and Ulstrup (J Phys Chem A 104: 11531, 2000). Comparison of short and long linkers of the osmium complex has shown the same two-step ET (electron transfer) behavior due to the long time scale in the complete reduction-oxidation cycle in the electrochemical tunneling spectroscopy (EC-STS) experiment as compared to the time constants for electron transfer for all linker distances, k _ET ⁰.     

Quantum effect of intramolecular high-frequency vibrational modes on diffusion-controlled electron transfer rate: from the weak to the strong electronic coupling regions

The Sumi-Marcus theory is extended by introducing two approaches to investigate electron transfer reactions from weak-to-strong electronic coupling regime. One of these approaches is the quantum R-matrix theory, useful for dealing with the intramolecular vibrational motions in the whole electronic coupling domain. The other is the split operator approach that is employed to solve the reaction-diffusion equation. The approaches are then applied to electron transfer in the Marcus inverted regime to investigate the nuclear tunneling effect on the long time rate and the survival probabilities. The numerical results illustrate that the adiabatic suppression obtained from the R-matrix approach is much smaller than that from the Landau-Zener theory whereas it cannot be predicted by the perturbation theory. The jointed effects of the electronic coupling and solvent relaxation time on the rates are also explored.