Charge Transfer in Physics, Chemistry and Biology: Physical Mechanisms of Elementary Processes and an Introduction to the Theory
... The following expression was used to describe the distance dependent rate constant k (in atomic units) of a non-adiabatic electron transfer across a carbon nanotube/electrolyte solution interface (work terms are neglected for simplicity) [18], ...
... One can readily obtain for k conventional units cm s −1 , using the frequency factor (~10 12 s −1 for aqueous solutions [18] and reaction volume (ca 1 Å) or integrating the rate constant over r from the solution bulk to the closest approach distance). The LZ factor can be written as follows [18], ...
... One can readily obtain for k conventional units cm s −1 , using the frequency factor (~10 12 s −1 for aqueous solutions [18] and reaction volume (ca 1 Å) or integrating the rate constant over r from the solution bulk to the closest approach distance). The LZ factor can be written as follows [18], ...
Non-adiabatic electron transfer from a single wall carbon nanotube to a model H-like reactant is explored in the framework of quantum mechanical theory. As examples, we consider a pristine semiconducting (8, 0) nanotube, tubes intercalated with Li⁺ or Cl⁻ ions, and tubes doped with nitrogen. The electronic structure of the nanotubes has been calculated by DFT with periodic boundary conditions. The electronic coupling between the tube and the reactant, which is proportional to the reactant–electrode orbital overlap, is calculated as a function of distance, and used to estimate the effective Landau-Zener factor. Both doping and intercalation shift the nanotube Fermi level either to the valence band or to conduction band, which results in a rise of the rate constant of non-adiabatic electron transfer by up to four orders of magnitude.
... Modern electrochemistry reveals two main trends: first, the active use of experimental technique adopted from surface science; second, the modeling of reactions at molecular level. The quantum-mechanical theory of charge transfer in solutions has achieved a high level [1][2][3][4][5][6][7][8][9] ; the physical background of the elementary act of such processes is good understood. So far electrochemists collected many reliable experimental data: rate constants of various redox processes in different media and interfaces, current-voltage dependencies (see, for example, reviews [10,11] ). ...
... An important quantity in the electron transfer (ET) theory (assuming the two-state approximation) is the half of resonance splitting in the vicinity of the intersection of two free energy surfaces along the reaction coordinate describing initial (i) and final (f ) states (i.e., in the reaction transition state), DE e / 2 [6,7] : ...
... where x eff is classic polarization frequency (%10 13 s 21 for water [7] ); W i is the reactant work term; expð2rÞ is nuclear tunnelling factor; dx is a reaction volume which results from the integration over the reactant-electrode surface separation; DE a is the Franck-Condon (activation) energy barrier. If in adiabatic limit the activation barrier is not high, solvent dynamics effects can take place. ...
State-of-the-art in the area of quantum-chemical modeling of electron transfer (ET) processes at metal electrode/electrolyte solution interfaces is reviewed. Emphasis is put on key quantities which control the ET rate (activation energy, transmission coefficient, and work terms). Orbital overlap effect in electrocatalysis is thoroughly discussed. The advantages and drawbacks of cluster and periodical slab models for a metal electrode when describing redox processes are analyzed as well. It is stressed that reliable quantitative estimations of the rate constants of interfacial charge transfer reactions are hardly possible, while predictions of qualitatively interesting effects are more valuable.
... It is needless to repeat that charge transfer processes represent an important and ubiquitous phenomenon in physics, chemistry and biology [1][2][3][4][5]. A characteristic feature of charge transfer in polar media is a strong interaction of the charge with molecular environment. ...
... The saddle point lies near the initial equilibrium configuration only for the reactions that proceed in the so-called activationless region or near the maximum of the absorption band for light-induced electron transfer [2,14,15]. For the processes in the normal reaction region q k* is located approximately in the middle between the initial and final equilibrium values, i.e. the perturbation V is large. ...
... (a) Symmetric system, (b) nonsymmetric system. describes the fluctuations of the medium polarization in the model of the effective oscillators with dimensionless coordinates q k and the frequencies x k [2,20]. It is assumed that the transferable ''proton'' interacts dynamically with a single local mode along with the interaction with the medium polarization the letter being described in a standard way [2,20,21]. ...
Effects of deviation from the Born–Oppenheimer approximation (BOA) on the non-adiabatic transition probability for the transfer of a quantum particle in condensed media are studied within an exactly solvable model. The particle and the medium are modeled by a set of harmonic oscillators. The dynamic interaction of the particle with a single local mode is treated explicitly without the use of BOA. Two particular situations (symmetric and non-symmetric systems) are considered. It is shown that the difference between the exact solution and the true BOA is negligibly small at realistic parameters of the model. However, the exact results differ considerably from those of the crude Condon approximation (CCA) which is usually considered in the literature as a reference point for BOA (Marcus–Hush–Dogonadze formula). It is shown that the exact rate constant can be smaller (symmetric system) or larger (non-symmetric one) than that obtained in CCA. The non-Condon effects are also studied.
... As this effect is not included explicitly in the force field, it was addressed implicitly by the image charge. Here e 0 is electron charge, C is the Pekar factor (C ¼ 1 e opt À 1 e rad , e opt is the optical (high frequency) dielectric constant (1.8 for water [40] ), e rad is the ...
... [14] The reciprocal value of t L can be considered as a crude estimate of the effective frequency factor (n eff � 1=t L ), see eq. 1. A key factor of the heterogeneous ET rate constant in some kinetic regions is the electronic transmission coefficient, which is proportional in the weak coupling limit to the density of electronic states near the Fermi level of an electrode, 1ðe F Þ. [40] It is known [44] that for SWCNTs with metallic conductivity 1ðe F Þ � 1=d CNT . Thus both n eff and 1ðe F Þ might be regarded as additional factors which favor a higher ET rate in ultra-narrow carbon nanotubes and somewhat reinforces our main conclusion. ...
A Fe3+/2+ redox couple in conducting single wall carbon nanotubes filled with water molecules is investigated in the framework of the electron transfer theory and with classical molecular dynamics simulations. The diameter of the nanotubes ranges from 0.8 nm to 3.5 nm. It can be concluded qualitatively that the electron transfer rate significantly increases with decreasing nanotube diameter. This effect is explained basically in terms of the solvent reorganization energy. Other factors which can affect the reaction rate are discussed as well.
... Semi-classical Marcus theory has been extended to the quantum mechanical regime, see for example Refs [142,152]. A quantum mechanical approach that incorporates nuclear tunnelling processes can account for the low temperature speed-up of charge transfer, as observed for example in purple bacterial charge separation [66]. ...
... Semi-classical Marcus theory can be extended to a fully quantum-mechanical treatment based on the theory of non-radiative transitions that includes nu-clear tunnelling, and which gives a good prediction for the increasing rate of charge separation with decreasing temperature, see for example Ref. [152] for an overview. If it is assumed that charge separation is strongly coupled to some harmonic vibrational mode, then the rate is given by the Jortner rate given in Eq. (2.86). ...
Quantum biology is the application of the theory of open quantum systems to aspects of biology where classical physics fails to give an accurate de- scription. The most well-established area in quantum biology is the study of photosynthesis.
There exists a body of evidence that the primary photosynthetic processes of energy and charge transfer exhibit quantum mechanical properties essential for function and that cannot be described by classical physics. Open quantum systems models of excitation energy transfer and electron transfer in primary photosynthesis have elucidated many aspects of the relation between struc- ture and function in the photosynthetic complex, as well as contributed to an understanding of the role of the environment in quantum transport processes.
After an introduction to quantum biology, an overview of open quantum systems approaches to transfer processes in primary photosynthesis is given. Our results on decoherence-assisted transport in the context of photosyn- thetic excitation energy transfer, as well as our proposal of the direct role of spin in a protection mechanism during photosynthetic charge transfer, are presented. Finally, thoughts around an outlook for quantum biology are given.
... By adopting the idea that the rate determining step in the ORR is the addition of the first electron to the O 2 molecule [4,32], the rate constant (k) of this ET step in the adiabatic limit (strong electronic coupling) can be expressed as follows [33]: ...
... The ET rate constant in Eq. (2) depends also on nuclear tunneling factor[33,35]. The latter was assumed to be the same both for the ordered and the disordered PtCo alloy. ...
PtCo alloys have long been known to possess higher activity in the electrochemical oxygen reduction reaction compared to pure Pt. This work addresses the influence of the chemical order on the electrocatalytic activity of PtCo alloy, the phenomenon, which has been already documented in the literature but yet poorly understood. Single crystalline PtCo(0 01) alloy films were epitaxially grown on Mg(0 0 1) substrates and used as model electrodes to study the influence of the chemical order on the electrochemical oxygen reduction reaction. Electrodes with close Pt:Co atomic ratios in the bulk and near surface region show grossly different electrocatalytic activities against the chemical order, the ordered L1(0) phase being ca. 7 times more active than the disordered fcc A1 structure. Electronic structure calculations and quantum mechanical theory of electron transfer are utilized to provide a rational for this remarkable phenomenon.
... Most of works used spin-less model where the degeneracy of the electron energy levels in the metal due to the electron spin was ignored. A similar approximation was used in later works on adiabatic electrochemical electron transfer reactions [3][4][5][6][7][8] and in the works on electron tunneling in electrochemical contacts [9][10][11][12][13][14][15][16][17][18]. This model at first sight seems to be reasonable for single-electron outer-sphere electron transfer reactions in the absence of magnetic field when only one electron occupies the valence orbital of the redox group. ...
... General expressions for the transition probability for arbitrary frequency spectrum of vibrational subsystem can be found elsewhere (see e.g. Ref. [4]). It follows from Eqs. (1) and (2) that the transition probabilities for forward and backward reactions are related to each other by the detailed balance principle relationship ...
An effect of spin degeneracy of electron energy levels in the metal electrode on the observable characteristics of electrochemical systems in the absence of magnetic field is discussed. Single-electrode outer-sphere electron transfer reactions are considered as well as redox-mediated electron tunneling in electrochemical contacts. Particular attention is paid to the difference between the spin-less model and the limit of infinitely large Coulomb repulsion of the electrons occupying the same valence orbital in the redox group. Adiabatic and non-adiabatic regimes of the transitions are studied and the expressions for the tunnel current are obtained.
... The pairs of photoexcited hot carriers are separated by the electric field in the scl and electrons (or holes, depending on potential) are driven by the electric field of the scl to the electrode electrolyte interface. The transfer of hot carriers to species in an electrolyte is treated using conventional charge transfer theory [28,29]. The high energy of hot carriers results in higher charge transfer rates compared to equilibrium charge carriers, which accounts for the photoinduced currents. ...
... Thus, a steady concentration of hot electrons (n s J) with energy iU (iB 1) is maintained at the interface. According to the theory of charge transfer at semiconductor electrodes [28,29], the rate of electron transfer from the electrode to species in electrolyte is equal to ...
Photoelectrochemical reduction of oxygen and other observations of sustained electrochemically generated photocurrents with graphitic (HOPG) electrodes in aqueous solutions are reported. The photocurrents were observed over a wide pH range: 0 (0.5 M H2SO4)–14 (1 M NaOH). Photocurrent-potential, capacitance-potential and photocurrent light action spectral measurements were performed with basal plane and edge plane HOPG electrodes in order to elucidate the origin of the observed photocurrent. The photocurrent is attributed to hot electron–hot hole pairs photogenerated by direct transition between π-electronic states of the valence and conduction bands of graphite. Photogenerated carriers, holes or electrons, are driven by the electric field in the space charge layer (scl) to the electrode∣electrolyte interface, where they react directly with species in the electrolyte inducing anodic or cathodic photocurrent. The potential corresponding to the change of the photocurrent sign from cathodic to anodic was attributed to the flat band potential (EFB). The EFB of the basal plane electrode was 0 V versus SHE regardless of pH, while for the edge plane electrode EFB changed at a rate of 54 mV per unit of pH. The shift of EFB of the edge plane electrode with pH is ascribed to a change of the pH dependent surface dipole formed by oxygen containing surface redox groups.
... Solvated electrons and the products of solvated electron reactions with water and other species are known to be important in biology, where radicals such as O 2 •− ("superoxide" radical) and •OH (neutral hydroxyl radical) are highly reactive toward DNA and other biological molecules. 94 We have previously demonstrated the important role of solvated electrons in photochemical reduction of CO 2 to CO, where they are able initiate the one-electron reduction of CO 2 to its radical anion CO 2 ...
Solvated electrons in water have long been of interest to chemists. While readily produced using intense multiphoton excitation of water and/or irradiation with high-energy particles, the possible role of solvated electrons in electrochemical and photoelectrochemical reactions at electrodes has been controversial. Recent studies showed that excitation of electrons to the conduction band of diamond leads to barrier-free emission of electrons into water. While these electrons can be inferred from the reactions they induce, direct detection by transient absorption measurements provides more direct evidence. Here, we present studies demonstrating direct detection of solvated electrons produced at diamond electrode surfaces and the influence of electrochemical potential and solution-phase electron scavengers. We further present a more detailed analysis of experimental conditions needed to detect solvated electrons emitted from diamond and other solid materials through transient optical absorption measurements.
... 6 It is surprising to note that these chains develop almost independently; cross-links between different chains are sparse. Notable exceptions include Schmickler's work 7 bridging Anderson-Newns theory of chemisorption 8,9 and Marcus theory of solvent reorganization 10,11 to model electrocatalytic reactions, and the works of Kornyshev et al. 12,13 combining the jellium model, which is used to describe the metal side of electrochemical double layers, [14][15][16][17] and Levich-Dogonadze-Kuznetsov theory of nonadiabatic electron transfer [18][19][20][21] to study the charge-and distance dependence of the electronic overlap factor. ...
Electron transfer in electrocatalysis involves strong short-range electronic interactions and occurs in an electrochemical double layer. Describing the two elements on an equal footing is an essential but challenging task for theoretical electrocatalysis. This work addresses this challenge using a mixed quantum–classical treatment. This treatment features the combination of chemisorption theory, electron transfer theory, and double layer theory in a unifying framework. Electrostatic free energy terms and solvent reorganization energy, key parameters modulating the electron transfer process, are calculated from a three-dimensional continuum double layer model that considers the reactant structure, steric effect, and solvent orientational polarization. The presented model is reduced back to the Marcus theory by neglecting electronic interactions and to the Schmickler theory of electrocatalysis by neglecting double layer effects. Emphasis is placed on understanding the multifaceted double layer effects in electrocatalysis. Apart from modifying the driving force and reactant concentration that are considered in the Frumkin corrections, double layer effects also modulate the interfacial solvent reorganization energy, thus adding a new term to the transfer coefficient. An additional level of intricacy comes into play if the reactant zone needs to replace solvent molecules originally adsorbed on the metal surface when it approaches the metal surface. The resulting free energy penalty shifts the transition state away from the metal surface and thus increases the activation barrier. Understanding how the metal surface charging condition modulates the interfacial stiffness opens an additional channel of deciphering electrolyte effects in electrocatalysis.
... In accordance with Refs. [54,55], the following equation can be written in adiabatic (strong coupling) ET limit for the distance (x) dependent rate constant k of both steps, (2) and (3), as Eq. (18): ...
Nickel is a promising electrocatalyst of hydrogen electrode reaction in alkaline media. Its electrocatalytic activity for hydrogen oxidation and evolution reactions can be enhanced when its surface is partially covered by Ni (hydr)oxides or by associating Ni with Cu. In this work, the influence of the NaOH concentration on the hydrogen electrode kinetics on various Ni electrodes is investigated. On metallic Ni, the electrocatalytic activity is almost constant between pH 12 and 14, while it decreases by a factor two on partially oxidized Ni and NiCu electrode constant when the pH decreases from 14 to 12. Analyzing the current potential curves with the help of microkinetic modeling reveals that the Had and OHad binding energies on Ni do not depend on the pH while the rate constant of the Volmer and Heyrovsky reactions decrease with the pH. The pH effect on the electron transfer elementary act is briefly discussed in the framework of a quantum mechanical theory.
... Electron transfer processes are the heart of oxidation-reduction reactions which play important role in chemistry and biology. [1][2][3] Theoretical description of electron transfer rates at the level of the Marcus theory 4-10 is widely utilized for the description of variety of phenomena from photovoltaics, batteries design and catalysis in chemistry [11][12][13][14][15][16] to photosynthesis, vision and sense of smell in biology. [17][18][19][20][21][22][23] Interfacial electron transfer is behind many vital biological processes. ...
Utilization of electron transfer methods for description of quantum transport is popular due to simplicity of the formulation and its ability to account for basic physics of electron exchange between system and baths. At the same time, necessity to go beyond simple golden rule-type expressions for was indicated in the literature and {\em ad hoc} formulations were proposed. Similarly, kinetic schemes for quantum transport beyond usual second order Lindblad/Redfield considerations were discussed. Here we utilize recently introduced by us nonequilibrium Hubbard Green's functions diagrammatic technique to analyze construction of rates in open systems. We show that previous considerations for rates of second and fourth order can be obtained as a particular case of zero and second order Green's function diagrammatic series with bare diagrams. We discuss limitations of previous considerations, stress advantages of the Hubbard Green's function approach in constructing the rates and indicate that standard dressing of the diagrams is a natural way to account for additional baths/degrees of freedom when formulating generalized expressions for the rates.
... Electron transfer processes are the heart of oxidation-reduction reactions which play important role in chemistry and biology. [1][2][3] Theoretical description of electron transfer rates at the level of the Marcus theory 4-10 is widely utilized for the description of variety of phenomena from photovoltaics, batteries design and catalysis in chemistry [11][12][13][14][15][16] to photosynthesis, vision and sense of smell in biology. [17][18][19][20][21][22][23] Interfacial electron transfer is behind many vital biological processes. ...
Utilization of electron transfer methods for description of quantum transport is popular due to simplicity of the formulation and its ability to account for basic physics of electron exchange between system and baths. At the same time, necessity to go beyond simple golden rule-type expressions for rates was indicated in the literature and ad hoc formulations were proposed. Similarly, kinetic schemes for quantum transport beyond usual second order Lindblad/Redfield considerations were discussed. Here we utilize recently introduced by us nonequilibrium Hubbard Green's functions diagrammatic technique to analyze construction of rates in open systems. We show that previous considerations for rates of second and fourth order can be obtained as a particular case of zero and second order Green's function diagrammatic series with bare diagrams. We discuss limitations of previous considerations, stress advantages of the Hubbard Green's function approach in constructing the rates and indicate that standard dressing of the diagrams is a natural way to account for additional baths/degrees of freedom when formulating generalized expressions for the rates.
... Nevertheless, such distinction is minimized by normalization, which we used when comparing these two cases. To calculate the rate constant, k, of electron transfer, we used the Marcus 53 and quantum mechanical theories of electron transfer 54 ω ...
Since the discovery of graphene, this material is in the focus of intensive research in the field of electrochemical energy storage devices. Numerous experimental works demonstrate, however, contradictory results on the electron transfer kinetics at a graphene electrode, in particular on electrocatalytic properties of graphene edges. In this work, we explore the spatially resolved electrochemistry of basal plane and edge sites in the outer-sphere electron transfer step at the oxygen reduction reaction in acetonitrile solutions using theory and computations. We employ a quantum mechanical theory and explore the graphene – O2 orbital overlap resting on the results of Density Functional Theory calculations and some data obtained earlier from Molecular Dynamics simulations. The zigzag graphene edges provide ca. 4 times faster electron transfer (in comparison with the basal plane) at long separations, > 4 Å, i.e., in the diabatic regime. At the same time, effective rate constants obtained by integrating over the reaction layer do not reveal any noticeable difference for the ba-sal plane and edge sites. It is argued that some experimental data can be explained only assuming a barrier layer separating the graphene surface and O2 molecule.
... The background model takes into account both slow thermal and ultrafast nonequilibrium (hot) electronic transitions between the reacting particles, coherent and incoherent spin conversion in the ionic donor-acceptor pairs, temporal and spectral characteristics of the excitation pulse, internal conversion of photoexcited molecules and their vibrational relaxation. Transfer of the electron between the distant reactants as well as the effect of solvent relaxation on the ET rate (dynamic solvent effect) is treated within the Zusman model [5] and the perturbation theory [6] in the electronic coupling energy. ...
A numerical code for simulations of ultrafast charge separation/charge recombination (CS/CR) kinetics in photoreactions assisted by diffusion of the reactant molecules in viscous media is presented. The FLUT software package is based on the unified theory of CS/CR and employs the well-known concepts of the distance-dependent rate of electron transfer, the probability of hot electronic transition in the course of solvent and intramolecular relaxation, the diffusive escape of reactants from the solvent cage, spin evolution of radical-ion pairs, etc. The package is constructed as a hierarchy of C++ classes representing basic entities of the model. Several specific types of electronic couplings are implemented as the derived classes. The numerical method utilizes the Chebyshev time propagation algorithm with a spatial finite-differencing operator obeying the detailed balance condition. Validation of the code was performed using several benchmark tests; an agreement with exact analytic solutions in the limiting cases is obtained. The parallel version of the code is implemented with the OpenMP interface. The results of the CS/CR simulations in typical donor-acceptor systems are discussed.
... An electron can escape from one trap to another easily. Although the electron can tunnel in either direction, it is much easier for the electron to tunnel from the shallower trap to the deeper one than the other way round (It depends also on the shape of the trap, not just its depth) (Kuznetsov, 1995). The abnormal residual TL in the irradiated phosphor of Li 2 B 4 O 7 :Cu, B after varied periods of storage at room temperature can be explained by tunneling effect. ...
To determine the effects of various concentrations of the activators copper (Cu) and boron (B) on the thermoluminescence (TL) properties of lithium tetraborate, the phosphor was first synthesized and doped with five different concentrations of copper (0.1–0.005 wt%) using solution combustion method. 0.01 wt% Cu was the concentration which showed the most significant increase in the sensitivity of the phosphor. The second sort of Li2B4O7:Cu material was prepared by adding B (0.001–0.03 wt%) to it. The newly developed copper-boron activated lithium tetraborate (Li2B4O7:Cu, B) material with 0.01 wt% Cu and 0.001 wt% B impurity concentrations was shown to have promise as a TL phosphor. The material formation was examined using powder x-Ray Diffraction (XRD) analysis and Scanning Electron Microscope (SEM) imaging. Fourier Transform Infrared (FT-IR) spectrum of the synthesized polycrystalline powder sample was also recorded. The TL glow curves were analyzed to determine various dosimetric characteristics of the synthesized luminophosphors. The dose response increased in a “linear” way with the beta-ray exposure between 0.1–20 Gy, a dose range being interested in medical dosimetry. The response with changing photon and electron energy was studied. The rate of decay of the TL signal was investigated both for dark storage and under direct sunlight. Li2B4O7:Cu, B showed no individual variation of response in 9 recycling measurements. The fluorescence spectrum was determined. The kinetic parameters were estimated by different methods and the results discussed. The studied properties of synthesized Li2B4O7:Cu, B were found all favorable for dosimetric purposes.
... The charge transfer across nanowire (nanotube)/aqueous solution interface was addressed in Refs. 3, 6-9 on the basis of quantum mechanical theory 10,11 . Some qualitatively interesting features were predicted for the [Fe(CN) 6 ] 3-/4redox couple 3 (electrostatic catalysis, inverted Arrhenius plot) and the reduction of peroxodisulphate anion 9 (disappearance of a "pit" in current -voltage curves observed in diluted supporting electrolyte solutions). ...
In this paper, we report on calculations of the orbital overlap between Fe(III) and Cr(III) aquacomplexes and different electrode surfaces: Cu(111), Ag (111), Au(111), Pt(111), and corresponding monatomic wires. The electronic structure of the monocrystalline surfaces and nanowires are described in terms of the electronic spillover and density of electronic states at the Fermi level obtained from periodic density functional theory (DFT) calculations. The transmission coefficients (κ) characterizing the first stage of outer-sphere electron transfer for the reduction of aquacomplexes are calculated on the basis of Landau–Zener theory as a function of electrode–reactant separation; the electronic transmission coefficients for the [Cr(H2O)6]3+/2+ redox couple were found to be smaller than those for [Fe(H2O)6]3+/2+. Two different intervals can be clearly distinguished for Cu, Au and Pt: “a catalytic region”, where κ(wire) > κ(Me slab) and “an inhibition region”, where κ(wire) < κ(Me slab). A similar behavior exhibits the coupling constant estimated for a hydrogen atom adsorbed at the Au(111) surface and the Au monatomic wire. These effects originate from some specific features of electronic density profile for metal nanowires: at short distances the electronic density of nanowires is higher compared with the (111) metal surfaces, while at larger separations it decreases more sharply.
... Their colour reflects the relative energy balance resulting from the transfer of electronic charge from an electron donor to an electron acceptor and varies with changes in solvent permittivity. The CT complexes play an essential function in many branches of physics, chemistry and biology [118]. A perfect example of strong CT complex is tetracyano-p-quinodimethane (TCNQ) as a strong acceptor and tetrathiafulvalene (TTF) as a good donor; this complex shows one-dimensional electrical conductivity in a stacked crystalline solid [119]. ...
... Such qualitatively interesting features of this electrochemical redox process make it attractive to employ modern quantum mechanical theory of charge transfer in condensed media. 3,4 This has been done in works 1,2 with the help of quantum chemical modeling. The reduction of S 2 O 8 2− on a mercury electrode from solutions with variable viscosity (water−sugar and water−ethylene glycol (EG) mixtures) was investigated experimentally as well. ...
We explore solvent dynamics effects in interfacial bond breaking electron transfer in terms of a multi-mode approach and make an attempt to interpret challenging recent experimental results (the non-monotonous behavior of the rate constant of electroreduction of S2O82- from mixed water-EG solutions when increasing the EG fraction, see Zagrebin, P.A. et al, J. Phys. Chem.B, 2010, 114, 311). The exact expansion of the solvent correlation function (calculated using experimental dielectric spectra) in a series predicts the splitting of solvent coordinate in three independent modes characterized by different relaxation times. This makes it possible to construct a five-dimensional free energy surface along three solvent coordinates and one intra-molecular degree of freedom describing first electron transfer at the reduction of a peroxodisulphate anion. Classical molecular dynamics simulations were performed to study the solvation of a peroxodisulphate anion (S2O82-) in oxidized and reduced states in pure water and ethylene glycol (EG), as well as mixed H2O-EG solutions. The solvent reorganization energy of the first electron transfer step at the reduction of S2O82- was calculated for several compositions of the mixed solution. This quantity was found to be significantly asymmetric (the reorganization energies of reduction and oxidation differ from each other). The averaged reorganization energy slightly increases with increasing the EG content in solution. This finding clearly indicates that for the reaction under study the static solvent effect no longer competes with solvent dynamics. Brownian dynamics simulations were performed to calculate the electron transfer rate constants as a function of the solvent composition. The results of the simulations explain the experimental data, at least qualitatively.
... In electrocatalytic electrode reactions, one has in addition to these factors the influence of the electric field near the electrode(s) on the rate of elementary reaction steps. Treatments of the latter effect are customarily based on the assumption that the electric field is independent of the adsorbate coverage (for the review of various aspects of the theory of electrochemical reactions, see the monograph by Kuznetsov [2] and the textbooks by Schmickler [3] and Bard and Faulkner [4]). This approximation has widely been used in mean field and Monte Carlo simulations of the kinetics of electrochemical reactions (see, e.g., Refs. ...
The rate of electrochemical reactions including electron transfer is influenced by the electric field near the electrode(s). Scrutinizing this effect, we show that the dependence of the electric field near the flat single-crystal electrode on adsorbate coverage may result in increase (up to one order of magnitude) of a rate constant of electrochemical reaction with increasing coverage. In addition, we demonstrate that due to specific distribution of the electric field the rate of electrochemical reaction occurring on supported nm-sized metal particles may be higher (by a factor of 5) than on the flat surface. Finally, we briefly discuss the implications of these findings for interpretation of experimental data.
Рассматривается пульсирующий рэтчет с пространственно периодическим двухъямным потенциальным профилем, флуктуирующим на полпериода. Направление движения в таком рэтчете определяется тем, преодоление какого из барьеров, окружающих мелкую потенциальную яму, имеет большую вероятность. При относительно высоких температурах, в соответствии с законом Аррениуса, вероятности преодоления барьеров определяются их высотами, а при температурах, близких к абсолютному нулю, когда движение рэтчета происходит по туннельному механизму, важна также и форма барьера. Поэтому для узкого высокого и низкого широкого барьеров механизм преодоления может оказаться различным и, кроме того, зависящим от температуры. В результате возможно температурно-индуцированное изменение направление движения рэтчета. Представлена простая интерполяционная теория, иллюстрирующая этот эффект. Сформулированы простые критерии к форме потенциального рельефа, используя которые можно экспериментально наблюдать обращение движения.
The unique potential of fullerene C60 for various biological applications has ignited significant interest. However, its inherent non-polarity poses a critical challenge for its effective integration within biological systems. This study delves into the intricate physicochemical characteristics of the innovative [C60 + NO] complex using density functional theory and time-dependent density functional theory. The computational analyses encompass molecular charge, surface electrostatic potential, and dipole moment evaluations. Impressively, the dipole moment of the [C60 + NO] complex significantly increases to 12.92 D. Meticulous surface analysis reveals a subtle interplay between molecular structures, indicating weak interactions. The analysis of the absorption spectrum unveils a noteworthy red-shift of 200 nm subsequent to complex formation. To elucidate the electron transfer mechanisms, we explore photo-induced electron transfer through CAM-B3LYP. This exploration elucidates intricate pathways governing electron transfer, with complementary insights gleaned from Marcus theory's outputs, especially the Gibbs free energy of electron transfer. Changes in the physicochemical properties of approaching C60 and NO molecules reveal interesting results compared to separate molecules. These findings resonate profoundly in the context of potential biological and pharmaceutical utilization. With implications for the biomedical area, the outcomes linked to the [C60 + NO] complex kindle optimism for pioneering biomedical applications.
We study ultrafast charge rearrangement in dissociating 2-iodopropane (2−C3H7I) using site-selective core ionization at the iodine atom. Clear signatures of electron transfer between the neutral propyl fragment and multiply charged iodine ions are observed in the recorded delay-dependent ion momentum distributions. The detected charge-transfer pathway is only favorable within a small (few angstroms), charge-state-dependent spatial window located at C-I distances longer than that of the neutral ground-state molecule. These results offer insights into the physics underpinning charge transfer in isolated molecules and pave the way for a different class of time-resolved studies.
This chapter presents results on enhancing robustness of quantum systems with uncertainties using the sampling-based learning control (SLC) method. In Sect. 4.2, the SLC method is applied to state control of superconducting qubits, robust control of photoassociation of O+H and synchronizing a laser with molecules for charge transfer,,. Robust control based on SLC and b-GRAPE is presented for generating quantum gates in Sect. 4.3 and SLC of open quantum systems is investigated in Sect. 4.4.
In a previous work [J. Chem. Phys. 140, 174105 (2014)], we have shown that a mixed quantum classical (MQC) rate theory can be derived to investigate the quantum tunneling effects in the proton transfer reactions. However, the method is based on the high temperature approximation of the hierarchical equation of motion (HEOM) with the Debye-Drude spectral density, and results in a multistate Zusman type of equation. We now extend this theory to include quantum effects of the bath degrees of freedom. By writing the full HEOM into a multidimensional partial differential equation in phase space, we can define a new reaction coordinate, and the previous method can be generalized to the full quantum regime. The validity of the new method is demonstrated by using numerical examples, including the spin-Boson model, and the double well model for proton transfer reaction. The new method is found to resolve some key problems of the previous theory based on high temperature approximation, including possible numerical instability in long time simulation and wrong rate constant at low temperatures.
Oxides are often perceived as relatively simple materials that can provide exceptional electrocatalytic activities. However, the assumed simplicity of oxides bears little relation to reality, impeding the progress towards fundamental understanding of electrocatalytic reactions on such materials. Here, we discuss the complexity of oxide surfaces and oxide-electrolyte interfaces and emphasize the challenges and pitfalls that receive little attention but are vital for the future of the field. We also make critical remarks on the existing experimental and theoretical approaches in fundamental electrocatalysis.
Electron transfer reactions are the most important processes at electrochemical interfaces. They are determined by the interplay between the interaction of the reactant with the solvent and the electronic levels of the electrode surface. Theoretical treatments only based on Density Functional Theory calculations are not sufficient. This review emphasizes mainly the effect of the electronic structure of the electrode material on electron transfer under different kinetic regimes. Our goal is to understand experimental results in the framework of a theory valid for arbitrary strengths of electronic coupling.
Developing efficient algorithms is an important task in the learning control of quantum systems, since most learning control problems for quantum systems involve a heavy requirement for computational resources. In this paper, we employ a learning control algorithm with an adaptive target state developed in the chemical physics community for several classes of quantum control problems. For these problems applied to some new quantum control tasks, we further demonstrate that the algorithm using an adaptive target state can be more efficient than traditional learning control algorithms using a fixed target state. In the algorithm, the target state is updated according to the renormalized fragmentary yield in the desired region (or subspace) throughout the learning iterations. The adaptive target scheme is applied to three significant quantum control tasks, including the slow collision of a sodium cation and an iodine anion, the orientation of a LiH molecule, and population transfer between subspaces. Shaped laser pulses are obtained using the learning control algorithm, and numerical results are presented to demonstrate the advantages of the adaptive target scheme over the algorithm using a fixed target state. The adaptive target scheme is especially useful for learning control problems of quantum systems where the target state is not unique or known.
A model for simulating the transient electronic absorption spectra of donor-acceptor dyads undergoing ultrafast intramolecular charge transfer in solution has been developed. It is based on the stochastic multichannel point-transition approach and includes the reorganization of high-frequency intramolecular modes (treated quantum mechanically) and of low frequency intramolecular and solvent modes (described classically). The relaxation of the slow modes is assumed to be exponential with time constants taken from experiments. The excited-state dynamics is obtained by simulating the population distribution of each quantum state after optical excitation and upon electronic and vibrational transitions. This model was used to simulate the transient electronic absorption spectra measured previously with a pyrylium phenolate in acetonitrile. A very good agreement between the simulated and measured spectra was obtained assuming a three-level model including the ground state, the optically excited state and a dark state with large charge-transfer character and a substantially different geometry relative to that of the optically excited state. The merit of this approach to disentangle the contributions of both population changes and relaxation processes to the ultrafast spectral dynamics will be discussed.
Theoretical treatments of electrochemical reactions at semiconductors are usually based on theories which presume a weak interaction between the re- actant and the electrode. Here we develop a theory that is valid for arbitrary interaction strengths, and explore its consequences within a simple coupling scheme. Our model can be used as a framework for the investigation of spe- cific catalytic reactions including photoelectrocatalysis.
In laser-assisted collisions, a control field may fail if we cannot precisely synchronize the colliding particles and the laser pulse. In this paper, we show that laser pulses that are robust in this situation can be obtained by a sampling-based method to achieve a desired charge transfer probability with limited sensitivity to the arrival time of the laser pulses. The time-dependent wave packet method and an adaptive target scheme are used in optimal control calculations based on an adiabatic two-state model of a H + D⁺ collision system. Several samples with different pulse arrival times are selected to construct robust fields at two different collision energies and the validity of these fields are examined by tests with additional samples. Excellent performance was obtained with the robust fields in both cases.
Electron transfer and proton coupled electron transfer (PCET) reactions at electrochemical interfaces play an essential role in a broad range of energy conversion processes. The reorganization energy, which is a measure of the free energy change associated with solute and solvent rearrangements, is a key quantity for calculating rate constants for these reactions. We present a computational method for including the effects of the double layer and ionic environment of the diffuse layer in calculations of electrochemical solvent reorganization energies. This approach incorporates an accurate electronic charge distribution of the solute within a molecular-shaped cavity in conjunction with a dielectric continuum treatment of the solvent, ions, and electrode using the integral equations formalism polarizable continuum model. The molecule-solvent boundary is treated explicitly, but the effects of the electrode-double layer and double layer-diffuse layer boundaries, as well as the effects of the ionic strength of the solvent, are included through an external Green's function. The calculated total reorganization energies agree well with experimentally measured values for a series of electrochemical systems, and the effects of including both the double layer and ionic environment are found to be very small. This general approach was also extended to electrochemical PCET and produced total reorganization energies in close agreement with experimental values for two experimentally studied PCET systems.
IntroductionAdiabatic Proton Transfer General PictureAdiabatic Proton Transfer Free Energy Relationship (FER)Adiabatic Proton Transfer Kinetic Isotope Effects KIE Arrhenius BehaviorKIE Magnitude and Variation with Reaction AsymmetrySwain–Schaad RelationshipFurther Discussion of Nontunneling Kinetic Isotope EffectsTransition State Geometric Structure in the Adiabatic PT PictureTemperature Solvent Polarity EffectsNonadiabatic ‘Tunneling’ Proton Transfer General Nonadiabatic Proton Transfer Perspective and Rate ConstantNonadiabatic Proton Transfer Kinetic Isotope Effects Kinetic Isotope Effect Magnitude and Variation with Reaction AsymmetryTemperature BehaviorSwain–Schaad RelationshipConcluding RemarksAcknowledgmentsReferences General PictureAdiabatic Proton Transfer Free Energy Relationship (FER)Adiabatic Proton Transfer Kinetic Isotope Effects KIE Arrhenius BehaviorKIE Magnitude and Variation with Reaction AsymmetrySwain–Schaad RelationshipFurther Discussion of Nontunneling Kinetic Isotope EffectsTransition State Geometric Structure in the Adiabatic PT PictureTemperature Solvent Polarity Effects KIE Arrhenius BehaviorKIE Magnitude and Variation with Reaction AsymmetrySwain–Schaad RelationshipFurther Discussion of Nontunneling Kinetic Isotope EffectsTransition State Geometric Structure in the Adiabatic PT Picture General Nonadiabatic Proton Transfer Perspective and Rate ConstantNonadiabatic Proton Transfer Kinetic Isotope Effects Kinetic Isotope Effect Magnitude and Variation with Reaction AsymmetryTemperature BehaviorSwain–Schaad Relationship Kinetic Isotope Effect Magnitude and Variation with Reaction AsymmetryTemperature BehaviorSwain–Schaad Relationship
This chapter discusses the computational tools that were developed to specifically address liquid interfacial systems. It summarizes the microscopic insight gained about the structure and dynamics of neat liquid interfaces and the behavior of solute molecules adsorbed at these interfaces. The chapter focuses on molecular-level information that in recent years has been compared directly with experiments. This provides the necessary background for discussing the methodology and general insights that computer simulations provided for solute adsorption, transport, relaxation, and reactivity at liquid interfaces. Most simulation techniques applied to date to liquid interfaces are based on classical molecular dynamics and Monte Carlo methods. The development of nonlinear optical spectroscopic techniques has enabled the study of electronic transitions in solute molecules adsorbed at liquid interfaces, and the chapter focuses on some computational aspects of solute electronic spectroscopy at liquid/vapor and liquid/liquid interfaces.
We speculate about the existence of a “square-root Tafel dependence” for simple one stage anodic/cathodic electron transfer reactions in ionic liquids. In this dependence, the logarithm of the current depends linearly on the square-root of electrode potential. The modified law is a consequence of ion crowding in the electrical double layer at high charges of the electrode. It may be expected that this effect may be observed for slow reactions at large electrode polarisations, yet not triggering electrochemical decomposition of ionic liquids, and only if diffusion limitations on the transport of reactants are absent.
The possibilities of hydrogen atom tunneling transfer in biological liquids are discussed. Basic mechanisms of temperature and pressure effects on the tunneling rate constant are considered: the reorganization of reagents and the medium due to the transfer of H atoms and changes in the value and shape of the chemical reaction potential barrier upon intermolecular and soft intramolecular vibrations. Expressions are derived for the tunneling transition rate constant and kinetic isotopic effect as functions of temperature and pressure. It is found that the temperature dependence of the isotope effect is mainly affected by the second mechanism only. The theory is compared with the literature’s experimental data on the temperature dependence of the isotope effect. It is shown that experiments are described well by the theory at sensible values of the fitting parameters.
The approach to the calculation of the activation energy of electrochemical charge transfer is considered for systems characterized by asymmetry of inner-sphere reorganization (i.e. by the difference in inner-sphere reorganization energies for direct and reverse reactions). Model calculations are carried out and the region of parameters is evaluated for which the effect is pronounced. A quantum-chemical analysis of inner-sphere reorganization is presented for four cobalt and chromium chelate aminocarboxylate complexes and the corresponding contribution to activation energy is discussed in connection with the difference in rate constants.
We present a model for a tunnelling conductance of a chain molecule controlled by fluctuations of relative conformations between nearest neighbour units of the chain molecule. The model leads to a simple formula which in one regime reproduces recently reported Arrhenius dependences for alkanedithiols [W. Haiss, H. van Zalinge, D. Bethel, J. Ulstrup, D.J. Schiffrin, R.J. Nichols, Faraday Discuss. 131 (19) (2005)] with an activation energy proportional to the length of the molecule, whereas in other regimes it shows activationless behaviour corresponding to the tunnelling across more rigid molecules. The general formula covers the transition between the two limits, predicting some new dependences that could be interesting to verify experimentally.
A method of calculation of a free-energy surface (FES) of the proton transfer (PT) reaction in a polar aprotic solvent is developed. This is based on the two-state (valence bond) VB description of the solute combined with recent continuum medium models. Its essential new feature is an explicit quantum-chemical treatment of VB wave functions, including internal electronic structure of a chemical subsystem. The FES includes a pair of intrasolute coordinates, R, the distance between hydrogen-bonded atoms and s, the proton coordinate, together with the collective medium polarization mode. Two hydrogen-bonded systems immersed in a polar solvent (Freon) were considered. The first one is the H5O2+ ion, a model system which was used as a benchmark testifying the validity of our semiempirical calculations. The second system is the neutral (CN)(CH3)N–H⋯N(CH3)3 complex in Freon. PT for this system has been studied experimentally. The dependencies of basic parameters controlling FES properties (the overlap integral, the coupling matrix element and the reorganization energy Er) on intrasolute coordinates R and s are evaluated and discussed. In particular, for the neutral complex, Er depends on s linearly, and its dependence on R is weak. The FES, for the neutral system, has two potential wells separated by the energy barrier of ∼7 kcal/mol. Quantum-mechanical averaging over the proton coordinate, s, reduces the barrier from 7.0 to 1.2 kcal/mol. The value of the nonadiabatic parameter on the averaged FES is equal to 0.13. This implies that the PT in the second system corresponds to an intermediate dynamic regime and that proton tunneling effects are hardly significant for this reaction.
Intramolecular charge separation from the second singlet excited state of directly linked Zn--porphyrin--imide dyads and following charge recombination into the first singlet excited and the ground states has been investigated in the framework of a model incorporating four electronic states (the first and the second singlet excited, the charge separated and the ground states) as well as their vibrational sublevels. Kinetics of the transitions between these states are described in terms of the stochastic point-transition approach involving reorganization of a number of high frequency vibrational modes. The influence of the model parameters (the number of high frequency vibrational modes, the magnitude of the reorganization energies of the medium and the high frequency intramolecular vibrations, the solvent polarity) on the kinetics of population of the second and the first singlet excited states as well as the charge separated state has been investigated. Simulation of the kinetics of the charge separated state population allows quantitative reproducing the distinctive features of the two-humped kinetic curve observed in the experiment.
This paper considers the behavior of discrete and continuous mathematical
models for gene expression in the presence of transcriptional/translational
bursting. We treat this problem in generality with respect to the distribution
of the burst size as well as the frequency of bursting, and our results are
applicable to both inducible and repressible expression patterns in prokaryotes
and eukaryotes. We have given numerous examples of the applicability of our
results, especially in the experimentally observed situation that burst size is
geometrically or exponentially distributed.
Picosecond pump-probe spectroscopy is used to examine the dynamics of singlet excited state proton transfer in 2-(N-methylanilino)acetophenone in a series of solvent mixtures of benzene and acetonitrile. A factors and energies of activation are derived from temperature-dependent studies. The rate constants show a moderate solvent dependence, but Arrhenius parameters differ considerably between polar and unpolar solvents. It is concluded that proton transfer occurs via nonadiabatic tunneling into different vibrational channels of the product state.
Electron transmission through molecules and molecular interfaces has been a subject of intensive research owing to recent interest in electron transfer phenomena underlying the operation of the scanning tunneling microscope (STM) on one hand, and in the transmission properties of molecular bridges between conducting leads, on the other. In these processes, the traditional molecular view of electron transfer between donor and acceptor species give rise to a novel view of the molecule as a current carrying conductor, and observables such as electron transfer rates and yields are replaced by the conductivities, or more generally by current–voltage relationships, in molecular junctions. Such investigations of electrical junctions in which single molecules or small molecular assemblies operate as conductors, constitute a major part of what has become the active field of molecular electronics.
In this chapter, the author reviews the current knowledge and understanding of this field with particular emphasis on theoretical issues, and on the relationship between this new look at electron transfer phenomena and the traditional study of molecular electron transfer in solution. Different approaches to computing the conduction properties of molecules and molecular assemblies are reviewed, and the relationships between them is discussed.
The transfer of an electron between a transition metal donor and a transition metal acceptor can be promoted by the absorption of either thermal or light energy. The thermal processes have been experimentally determined to span a range of about 10²⁰ in their ambient rates of electron transfer. The same molecular properties that determine the electron-transfer reactivity also determine the correlated electron-transfer absorption or emission band energy, band shape, and absorptivity. The relationships between the thermal-kinetic and the spectroscopic observations are relatively simple when there is very little electron delocalization between the donor and acceptor, the weak-coupling limit. This limit applies to most bimolecular electron-transfer reactions of transition metals and to the spectroscopy of transition metal ion pairs; this is the limit treated well by Marcus theory. When the donor and acceptor are covalently linked, interactions of the donor and acceptor with the linker often increase the coupling in such a way as to enhance the reactivity and to alter the spectroscopy. Most of these linker-induced changes of reactivity and spectroscopy can be treated in terms of perturbation theory-based alterations of the behavior in the weak-coupling limit. Reactivity in some very strongly coupled complexes seems to fall outside the standard theoretical framework; many such systems seem to implicate nuclear motions of the linker in facilitating the donor–acceptor electronic coupling. Extensions to biological, multicentered, and heterogeneous systems containing transition metal complexes are briefly considered.
IntroductionMultilevel Electronic Processes in Chemistry and BiologyElectron Transfer Routes and Molecular Group Separation SchemesModulation of the Tunnel Factor and Self-consistent Electronic-Vibrational Interactions in Long-Range Electron TransferCoherent Two-step Electron TransferSome Observations on Relations between Thermal and Optical MultileLel Electronic TransitionsLong-Range Electron Transfer in Nontraditional Electrochemical Electron Transfer Systems: Tunnel Charge Lability. in Situ STM of Large Adsorbates, and Continuous Phase Transitions in Two-Dimensional Electrochemical Surface LayersSome Outlooks and Concluding Remarks
A method for introducing an effective coordinate for the solvation shell in a hydrogen-bonded cluster with the A–H…B reaction complex is presented. Due to the formation of additional hydrogen bonds between the A–H…B complex and the molecules of the solvation shell proton transfer (PT) is able to occur. An effective coordinate y is introduced which is approximately Jacobian. It is called upon to imitate the action of additional H-bonds which allow for the thermodynamical possibility of PT. To illustrate this method a model three-dimensional potential energy surface (PES), included explicitly the O–H and O…N stretching modes and y, was extracted from an ab initio surface. The PES was calculated for a cluster containing an phenol-ammonia complex surrounded by an ammonia shell (NH3)4. The two lowest proton adiabatic terms were computed. Two possible approaches for treating PT dynamics in the two-level approximation are discussed.
This is a review of researches on the calculation of the electronic characteristics of a free or perturbed metal surface; these include the electron density, the potential, the spectrum of surface states, the work function, the surface energy, response of the surface to an electric field, the chemisorption of atoms, and the adhesion of two metals. Preference is given to papers that employ the theory of the ground state of an inhomogeneous electron gas, the “density-functional method.” The various approximations to the density-functional used in the calculations, and the various models of the lattice (“jelly”, pseudopotential), are analyzed as to their applicability. Various sorts of experimental data are discussed on the basis of the theoretical results.
We consider a reversible dissociation–recombination reaction in solution which is described by a distribution of waiting times rather than a single dissociation rate constant. This is a non‐Markovian generalization of the backreaction boundary condition. We formulate the new boundary condition in terms of the residence time in the bound state and illustrate the theory by assuming a stable‐law density for the residence time. Explicit expressions are found for the Laplace transform of the survival probability in one and three dimensions, which can be inverted analytically for special values of the stable‐law parameter α and numerically for other values of α. We derive the long‐time behavior of the survival probability for arbitrary α, and note that the survival probability undergoes a first‐order phase transition in one dimension, in which its asymptotic value changes abruptly at α=1/2. In three dimensions it undergoes a second‐order phase transition at α=1, in which only the asymptotic slope of the survival probability changes discontinuously.
Scanning tunnelling microscopy (STM) involving electron tunnelling through large adsorbate molecules with discrete electronic levels accessible at low bias voltage, exhibits conceptual and physical analogies to other thermal and optical multi-level electronic processes. The analogies are most conspicuous if the adsorbate levels are strongly coupled to the environmental molecular, conformational or solvent nuclear motion but interact weakly with the conducting substrate and tip. These conditions would be appropriate for example for adsorbed large transition metal complexes or redox metalloproteins. In these limits electronic-vibrational coupling induces resonance between the local adsorbate level and either the substrate or the tip levels, and the STM current-voltage relations can be approached by methods known from the theory of related electronic transitions such as long-range molecular electron transfer and multi-photon optical processes. We provide a new theoretical frame for STM processes in this limit. The formalism rests on perturbative coupling of the adsorbate levels to the substrate and tip. Specific models incorporate strong coupling to the adsorbate and environmental nuclear motion, vibrational relaxation features, and the continuous electronic spectra of the substrate and tip. All these features are directly and transparently reflected in the current-voltage relations of the STM process.
Tunnelling is a useful concept in nonadiabatic solid-electrolyte processes in cases where the barrier is sufficiently unambiguously characterized, such as electron transfer at film-covered metal electrodes and through the space charge layer of semiconductors. Direct tunnel processes exhibt several features different from electrochemical processes at both pure metal and semiconductor electrodes. Their applicability as a tool for investigation of the interface properties is, however, limited because of the wide energy distribution of contributing levels which smears out electronic and vibrational structure in the current-voltage relationship. On the other hand, tunnelling via discrete barrier or surface levels, coupled or uncoupled from nuclear modes, displays rapid changes at voltages which depend in a characteristic way on the localization of the levels. the electronic-nuclear coupling etc. Current-voltage relationships of these “assisted” processes therefore contain much more “spectroscopic” information about the barrier region than for direct processes. Recent experimental investigations of tunnel processes which exhibit several of the expected features are discussed.
Long-range electron transfer (ET) in molecular and solid state systems always involves intermediate “environmental” electronic states. These participate in the superexchange mode if their energies are high, or sequentially when the energy is low and the states temporarily populated. The intermediate state nuclear motion can, however, be partially unrelaxed prior to the second ET step and the latter mechanism therefore differs from that of consecutive independent ET steps. We have analyzed the effect of intermediate state vibrational damping in a three-level process by a one-dimensional model and simple trajectory calculations. Damping is most reflected in the diabatic limit. Complete absence of damping gives a quadratic dependence on the electron exchange matrix elements. This differs from the fourth-order dependence obtained by second-order perturbation theory and a single reactive attempt at the intermediate-final state crossing. Vibrational damping drastically modifies this and the dependence on the electronic factor can now be of either second or fourth order, depending on the energies of the two crossing regions. This can have profound effects on the reaction free energy profile, external field dependence, etc. We finally discuss the first two ET steps in the bacterial photosynthetic reaction centre in terms of these views.
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Two-dimensional quasiclassical trajectories q1 (q2) are found for an illustrative model of two strongly shifted paraboloids (eigenfrequ ω1, ω2). It is shown that the motion originates along the caustic corresponding to the lower-frequency vibration (q2 = Q2 at ω1 < ω2) and proceeds along the classical upside-down barrier trajectory from the characteristic point (Q′1, Q2). Increase of ω1 or the vibrational quantum number n1 leads to diminishing of Q′1-Q1 so that the trajectory tends to the Miller periodic orbit. At ω1 ⪡ ω2Q′1 approaches the saddle-point value of q1. The statistically averaged trajectory corresponds to an instanton of period i/kBT. In the case of central symmetry of the paraboloids, the action along the trajectory obtained corresponds to the product of the Franck-Condon factors. For axially-symmetric positions of the paraboloids the increase in ω1/ω2 results in “corner cutting”, i.e. deviation of the trajectory from the minimum-energy path.
It is pointed out that a model which agrees well with the observed properties of semiconducting glasses is an attractive Hubbard model of localized states. Such a model has no gap for two-electron excitations but an energy gap for one-electron ones. The suggested physical model for a two-electron excitation is a new covalent bond in the structure, which is severely localized. It is also proposed that the one-electron excitation spectrum is wholly, or almost wholly, extended, and all observed gaps are identical with the mobility gap.
The rate of electron transfer reactions at titanium electrodes covered with thin platinum-doped passive films was investigated. The doping enhances the current considerably. It is suggested that the platinum atoms in the film serve as intermediate states in an electronic resonance-tunnelling mechanism. The corresponding model calculations show a satisfactory agreement with the experimental data. From these calculations the energy levels of the platinum states in the film are estimated.
1 Introduction.- 1.1 Nature of Elementary Chemical Processes.- 1.2 Development of Theories for Elementary Chemical Processes.- 1.3 Chemical Reactions as a Class of Radiationless Processes.- 2 Multiphonon Representation of Continuous Media.- 2.1 Nature of Solvent Configuration Fluctuations.- 2.2 Interaction with Ionic Charges.- 2.3 Relation to Macroscopic Parameters.- 3 Quantum Mechanical Formulation of Rate Theory.- 3.1 Elements of Scattering Theory.- 3.2 Channel States and Nature of the Perturbation.- 3.3 Evaluation of Transition Matrix Elements.- 3.3.1 Harmonic Oscillator Representation.- 3.4 The Role of a Continuous Vibration Spectrum.- 3.5 Relation to Experimental Data.- 3.5.1 The Electronic Factor.- 3.5.2 Intramolecular and Medium-induced Electronic Relaxation.- 3.6 Lineshape of Optical Transitions.- 4 The Effect of Intramolecular Modes.- 4.1 Special Features of Electron Transfer Processes.- 4.2 Quantum Modes in Electron Transfer Reactions.- 4.2.1 Displaced Potential Surfaces..- 4.2.2 Effects of Frequency Changes.- 4.2.3 Effects of Anharmonicity.- 4.3 Relation to Experimental Data.- 5 Semiclassical Approximations.- 5.1 One-Dimensional Nuclear Motion.- 5.1.1 Classical Nuclear Motion.- 5.1.2 Nuclear Quantum Effects.- 5.2 Many-Dimensional Nuclear Motion.- 5.3 Relation to Experimental Data.- 5.3.1 Outer Sphere Electron Transfer Processes.- 5.3.2 Nucleophilic Substitution Reactions.- 6 Atom Group Transfer Processes.- 6.1 General Features of Nuclear Motion.- 6.2 Semiclassical Approaches to Atom Group Transfer.- 6.3 Quantum Mechanical Formulation of Atom Group Transfer.- 6.3.1 Nuclear Tunnelling between Bound States.- 6.3.2 Adiabatic and Nonadiabatic AT.- 6.3.3 Relation to the Gamov Tunnelling Factor.- 6.4 Relation to Experimental Data.- 7 Higher Order Processes.- 7.1 Higher Order Processes in Chemical ET Reactions.- 7.2 Theoretical Formulation of Higher Order Rate Probability.- 7.2.1 Semiclassical Methods..- 7.2.2 The Effect of High-Frequency Modes..- 7.2.3 Adiabatic Second Order Processes.- 7.2.4 Quantum Mechanical Formulation.- 7.3 Relation to Experimental Data.- 8 Electrochemical Processes.- 8.1 Fundamental Properties of Electrochemical Reactions.- 8.1.1 The nonuniform dielectric medium.- 8.1.2 The continuous electronic spectrum.- 8.1.3 Adiabaticity effects in many-potential surface systems.- 8.2 Quantum Mechanical Formulation of Electrode Kinetics.- 8.2.1 Metal electrodes.- 8.2.2 Semiconductor electrode.- 8.3 Relation to Experimental Data.- 8.3.1 The current-voltage relationship.- 8.3.2 The nature of the substrate electrode.- 8.3.3 The electrochemical hydrogen evolution reaction (her).- 8.4 Electrode Processes at Film Covered Electrodes.- 8.4.1 Tunnelling mechanisms.- 8.4.2 Mobility mechanisms.- 9 Application of the Rate Theory to Biological Systems.- 9.1 General.- 9.2 Specific Biological Electron Transfer Systems.- 9.2.1 Primary Photosynthetic Events.- 9.2.2 Bioinorganic ET Reactions.- 9.3 Electronic Conduction in Biological Systems.- 9.4 Conformational Dynamics.- A1.- A1.1 Derivation of the Sum Rules(eq.(2.49)).- A1.2 Derivation of Eq.(2.56).- References.
A model for electron transfer involving a cavity in the solvent is considered. The separation of inertial and inertialess components of the polarization and of other dynamic variables of the medium is performed. An algorithm of the quantum chemical calculation of the electron wavefunctions of the reaction complex with due account of its interaction with the polarization of the medium is suggested.
A simple model is considered which allows to find an exact expression for the tunnelling factor of a reaction of light particle transfer in condensed medium without a priori assumption of the relation between the characteristic times of the motion along different degrees of freedom. Two opposite Born-Oppenheimer limits emerge from the exact result as particular cases. Characteristic parameters gouverning the inertial and inertialess behaviour of vibrational degrees of freedom are found.
Recent results on the theory of electron transfer reactions at superconducting electrodes are discussed. High temperature and low temperature approximations in the limits of weak and strong coupling with medium modes are considered. A mechanism explaining the hump of the current near the temperature of the superconducting transition is suggested.
Recent results in quantum mechanical theory of the elementary act of charge-transfer processes are reviewed. They involve nonadiabatic and adiabatic reactions of electron, proton and atom group transfer. Effects of stochastic motion along the reaction coordinate and fluctuational preparation of the potential barrier are discussed.
Electron transfer reactions at metal electrodes covered by insulating or semiconducting films may proceed by several mechanisms. At thick films, corresponding to bulk crystalline or amorphous semiconductors, they include electron transfer via the conduction or valence band of the film, electron transfer via localized bulk or surface states, or electron transfer between trapped polaron states. At thin films the dominating electron transfer mechanism is coherent or incoherent elastic tunnelling, or phonon-induced inelastic tunnelling.A survey of the conditions under which the various mechanisms are likely to prevail is given, and it is shown that in many cases the approach towards a theory of electron transfer reactions at film covered electrodes can be reduced to an analysis of elementary steps for which a theoretical formalism is wholly or partially available.
The Levich and Dogonadze theory of redox reactions in solutions is regarded as a suitable model to discuss the nature of quantum effects in electron transfer. In Section I the quantum mechanical framework—first order perturbation theory and the Born—Oppenheimer approximation—is reviewed; an inaccuracy in the corresponding treatment by Levich is corrected. In Section II the electron transfer rate is calculated. The result differs from the formula given by Levich; the difference becomes especially important at intermediate and at low temperatures. As a consequence quantum effects should already be observable at ordinary temperatures. It is shown that the high temperature approximation can also be obtained by a semiclassical calculation. These results are compared with the activated complex theory.
The general kinetics of the formation of polyatomic passive layers is discussed with regard to the processes at the phase boundaries metal/passive-layer and passive-layer/electrolyte-solution, and the potential gradient within the layer. The distribution of the electric potential ϕ(x) and the chemical composition n(x) of the oxide MeOn as a function of the distance x is studied. It is assumed that en electronic equilibrium exists from the metal throughout the whole layer. From this assumption there results a unique n(ϕ) relation independent of the structure of the layer, the layer thickness, and the cd.
By using an adiabatic dynamical simulation method, this report attempts to study a mixed classical-quantum model of strongly H-bonded complexes in polar solvents. The influence of the solvent on the proton-transfer rate constant and adiabatic proton dynamics is discussed. 21 refs., 3 figs.
A new Hamiltonian for the interaction of magnetic impurity spin with the conduction electrons is proposed. It is found that the conduction electrons may be condensed into the spin levels. For single impurity, the exact eigenstates are found. In the case of many impurities, virtual electron exchange is predicted for the first time. A single fermion and a single phonon operator interaction leads to hybrid interaction between bands of electrons along with some interesting effects.
Kubo's formula for the electrical conductivity is used to evaluate the Hall conductivity in the Random Phase Model of Amorphous Solids appropriate to the diffusive, short mean free path regime. The Hall mobility is found to be small in magnitude, temperature insensitive, and to exhibit the same anomalies in sign as found in the localized regime: namely, negative for a non-degenerate distribution of holes on electrons due to diffusive motion about a closed path of three sites.
Tunnelling term is included in the Hamiltonian of an anisotropic Ising model which is used to explain the thermodynamic properties of quasi-one-dimensional hydrogen-bonded ferroelectric crystals. A variational approach is used to treat the strong intrachain interactions accurately, while the weak interchain interactions are accounted for in a mean-field approximation. Also, fittings are presented of some experimental data pertaining to crystals of PbHPO4.Ein Tunnelterm wird in den Hamilton Operator eines anisotropen Ising-Modells eingefügt, der zur Erklärung der thermodynamischen Eigenschaften eines quasi-eindimensionalen ferroelektrischen Kristalls mit Wasserstoffbindungen benutzt wird. Zur exakten Behandlung der starken Intra-Ketten-Wechselwirkungen wird eine Variationsmethode eingeführt, während die schwachen Inter-Ketten-Wechselwirkungen in einer Mittel-Feld-Näherung berücksichtigt werden. Einige Anpassungen an experimentelle Werte, die sich auf PbHPO4-Kristalle beziehen, werden angegeben.
The electronic polaron approximation (EPA) by Toyozawa does not deal correctly with the short range part of the interaction between a quasi‐particle and the polarization field of a non‐metal, owing to the use of a continuum approximation for the polarization field. A modified EPA is derived, without this continuum approximation, from a weak coupling zero order tight‐binding model. It includes short range and local field effects and leads to an electron self‐energy ∑ without the explicit first Brillouin zone cut‐off found when Toyozawa's EPA is used. A simple practical approximation is derived in order to take into account the non‐dipolar contributions which cannot be neglected when large wave vectors are involved. A physical cut‐off, scaled by the atomic or ionic radius is thus incorporated in the theory. The numerical significance of the continuum approximation for the polarization field is tested by comparison, with the elaborate calculations of Kunz. It is found that the short wave length contributions to ∑ may in no way be neglected if one want to improve significantly over the much simpler classical approximation (due to Mott and Littleton), as used by Fowler.
The stochastic equations of motion describing liquid-state reaction dynamics, which were developed in the previous paper, are used herein to calculate the reaction rate constant. These equations involve the consideration of both the intrinsic chemical conversion and the solvent adjustment accompanying the conversion on the same basis. Special attention is paid to the situation where the solvent adjusts very slowly, such that it is this adjustment which controls the reaction rate. For this limiting case the effective solvation coordinate becomes the reaction coordinate.
Expressions for the probability of a nonadiabatic electron transfer between a donor-trap and an acceptor-trap in a polar medium are obtained. The transition probability is expressed in terms of the zeroth-order electron densities near the traps. The interaction of the electron with the phonon field is analyzed. A way of definition and calculation of the zeroth-order electron states with due account for their modulation by the phonon field is suggested. The effects of deviation from the Condon approximation due to the modulation of the electron densities and of the interaction with the phonon field causing the transition are estimated. The effects of this modulation on the kinetic parameters of the transition in the Condon approximation and in the improved Condon approximation are analyzed. It is shown that these effects are rather large for the processes involving solvated, trapped, or weakly bound electrons. The deviations from the Condon approximation should be taken into account for long-range electron transfer.
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The electron transfer capability of systems of the type H2H-(CH2)n-NH2 is studied using ab initio and semi-empirical methods. The coupling energy Δ between the nitrogen lone-pairs may, in the symmetric case, be defined as the energy difference between the symmetric and antisymmetric many-particle states of H2N-(CH2)n-NH2+ where ionization has occurred in the lone-pair orbitals. Δ may be related to the energy splitting between the corresponding orbital energies in the neutral molecule according to Koopmans' theorem. A “broken-symmetry” correction (BSC) both in space and spin is introduced. The two localized unrestricted Hartree-Fock solutions are subject to configuration interaction using transformation to “corresponding orbitals”. In saturated chains H2N-(CH2)n-NH2+, the lowering of |Δ| amounts to about 25%. Ab initio and semi-empirical results show the same trends in the contributions to Δ. In cases when different contributions cancel, the relative difference may be large. The conformation dependence is studied. For the case of sp3 lone-pairs, Δ is reduced for the cis case to a small fraction of the trans value and in some cases the sign is changed. The largest Δ is obtained for the all-trans conformation.
The line-shape of the exciton absorption band, in the case that k = 0 is at the bottom (or the top) of the exciton energy band and that there are no states with the same energy in order exciton
bands, is investigated in the limit of weak exciton-phonon coupling, with the use of the damping theory. The equations for
the energy dependent shift and broadening are solved with the aid of graphic calculation.
Except for the low temperature region, one can assume the exciton-phonon scattering to be elastic, and the line-shape in the
main part of the absorption band is determined essentially by the properties of long wavelength excitons and phonons. The
half-value width is rather small, and is proportional to (gT)2 where g is the exciton-phonon coupling constant and T is the absolute temperature.
The line-shape is strongly asymmetric, with a tail which is due to the indirect transition, on the high or low energy side
according as k = 0 is the bottom or the top of the exciton band.
Kramers's modeling of a chemical reaction by barrier-mediated diffusion is generalized to include a possible spatial variation of viscosity along the reaction coordinate. Expressions are derived for the rate coefficient k,k is influenced by the viscosity at the location of the barrier top, which is partially affected by the solvent viscosity eta. A predicted law delta(lnk)=εdelta(lneta) for 0
A phenomenological theory of polar solvation dynamics in electron transfer that accounts for the spatial‐ and frequency‐dependent dielectric function of the solvent is developed and described in a format appropriate to time‐dependent fluorescence Stokes shifts. The basic features of the relaxation dynamics are explored by using various analytical expressions for the dielectric function. The presence of spatial correlations persisting to frequencies higher than those corresponding to longitudinal solvent relaxation, τ−1L, yields significant or even substantial decay components with relaxation times shorter than τL. These are associated with motions of individual molecules within the solvent structural network. The implications of these predictions for solvation dynamics in activated charge‐transfer processes are noted.
Tunneling corrections to the classical transition state theory rate are considered for multidimensional anharmonic potentials. A theory is proposed for systematic accounting for the reaction barrier anharmonicity and associated curvature of the reaction path in terms of thermodynamic Green’s functions. A corresponding diagrammatic technique is developed for calculation of the rate constant. The theory allows summation of an infinite number of perturbation terms in a self‐consistent manner. It is shown that as a result of self‐consistent treatment, a renormalized potential function of the barrier can be introduced in place of the original potential which is the average of the latter over the thermal de Broglie wavelength of the tunneling particle. Even simple approximations for the rate calculated for such averaged potential barrier automatically takes into account a large body of anharmonic and quantum effects. The harmonic approximation of the renormalized potential coincides with the ‘‘best fit’’ quadratic potential surface introduced in the paper of G. Voth, D. Chandler, and W. H. Miller [J. Chem. Phys. 91, 7749 (1989)]. The theory also provides a method for calculation of further anharmonic corrections to this renormalized harmonic approximation.
The electronic structure of a solute in a polar and polarizable solvent depends on the nonequilibrium (or equilibrium) state of the solvent. Here we present a theory for this phenomenon, at the level of a dielectric continuum description of the solvent, characterized by an orientational polarization Por and an electronic polarization Pel. The entire range of electronic coupling between solute electronic states is considered. The present theory supersedes, in important respects, our earlier work [H. J. Kim and J. T. Hynes, J. Phys. Chem. 94, 2737 (1990); J. Chem. Phys. 93, 5194 (1990); 93, 5211 (1990)] by including a full quantization of Pel; this is a feature recently shown in a model study for weak electronic coupling [J. N. Gehlen, D. Chandler, H. J. Kim, and J. T. Hynes, J. Phys. Chem. (to be published)] to be necessary for, e.g., a correct description of electron transfer activation free energies and transition states. The quantization of Pel is effected via a coherent state formulation, coupled with a multiconfiguration self‐consistent representation of the solute‐Pel wave function. Nonequilibrium free energies and solute electronic structure are found and depend explicitly on the comparative time scales of a transferring electron in the solute and of Pel. The corresponding equilibrium relations are also found.
It is shown how a dynamical theory for proton transfer rates in solution can be implemented in a molecular???dynamics simulation for a model reaction system. The reaction is in the nonadiabatic limit, in which the transfer occurs via quantum tunneling of the proton. The importance of the coupling of the proton to the solvent and to an intramolecular vibration is illustrated, and the simulation results are successfully compared with analytic rate???constant expressions in several limiting regimes.
Condensed phase nonadiabatic transitions are studied in a semiclassical two state linear model. The effect of condensed phase is modeled by stochastic motion along the classical coordinate and by stochastic fluctuations of parameters of semiclassical Hamiltonian: slopes of diabatic terms, coupling of terms, and coordinate of avoided crossing. In the weak coupling limit simple formulas for nonadiabatic transition rate are derived without any assumptions about the mechanisms of stochastic classical motion and fluctuations of parameters. The rate appears to be independent of the mechanism of motion. The dependence on fluctuating parameters is expressed only through simple averages over parameter distributions, independent of correlation times of fluctuations.
The model for electron-transfer kinetics in solution is considered. In one model the appropriate energetic condition for charge transfer is met by a small number of vibration-rotation states in thermal equilibrium with the solution. Collisional activation (CA) between ion in the solution and the solvent is the origin of such states. In another model CA is neglected and the appropriate energy states are regarded as being reached by the fluctuations in the energy of the ion, as a result of its interaction with many surrounding solvent molecules; this is the energy fluctuation (EF) model. The dependence of the charge-transfer rate upon the interfacial potential difference for the two models is outlined, and the differences between the models is discussed. Comparison with spectroscopic data for H3O+ in solution suggests that the energy distribution in the vibration-rotation levels in this ion is continuous and the classical modes of vibration exist in water. A supposed discontinuity, which would have annulled the deduction of Tafel's law, was an origin of the EF model. In the EF model, the applicability of the Born-Landau equation, DeltaF0 = e2/2r(1/εopt - 1/εstat) is assumed. However, we show that this depends on a sufficiently large difference of the energy of an electron, trapped in the medium and bound to atoms in it. This difference is great if the medium is a solid, but not if it is a liquid. States suitable for acceptance or donation of electrons from ions to metals arise (at the equilibrium potential) much more frequently as a result of the equilibrium of the H3O+ ion with the solvent heat sink than those by electro-static fluctuation.
A unified theory of optical and thermal outer-sphere electron-transfer processes is outlined, in which the equations are obtained as special cases of general expressions for radiative and radiationless transition probabilities. For a particular donor—acceptor ion system, this leads to correlations between the rate of homogeneous thermal electron exchange and the frequency and bandwidth of the corresponding optical intervalence transfer absorption. These in turn can be correlated with thermal and optical transfer at a metal/solution interface. The relationship between the transmission coefficient in adiabatic thermal exchange and the transition probability in the corresponding optical transfer is discussed, and a number of predictions are made for the optical transfer of electrons between a molecule or ion and an electrode, which has so far not been observed.
The role of low- and high-frequency polarization of the medium in solution thermodynamics and electron transfer kinetics is discussed. For adiabatic reactions using classical polarization, the solvation of the transition state is weaker than that of the initial state due to delocalization of the electron density over both reactants in the former case. The influence of high-frequency polarization on initial- and transition-state solvations, the activation barrier, and retardation with respect to the electron are discussed. 24 refs., 2 figs.
The tunneling pathway model for electron transfer, which accounts for the unique covalent, hydrogen-bonded, and van der Waals contact linking donor and acceptor in a protein, gives a consistent description of electron-transfer rates in ruthenated proteins (cytochrome c, myogloblin, and cytochrome bâ), while simpler exponential decay models are not fully adequate. The authors report several new testable predictions of the pathway model relating electron-transfer rates to protein structure. The analysis predicts qualitative differences in the distance dependence of protein electron transfer at short (
We recently introduced a procedure for determining effective tunneling paths in nonthermal symmetric proton-transfer reactions. The procedure involves a "maximum probability path" (MPP) based on the ground-state vibrational wave function, which can be determined within the reaction surface Hamiltonian framework by ab initio electronic structure methods. In this paper we further define the MPP concept and explore this path using the malonaldehyde molecule and the formic acid dimer as examples, as well as calculations on simpler model systems. The results verify our qualitative expectations for tunneling in heavy-light-heavy systems. Furthermore, a comparison of the MPP to other more conventional paths, such as the IRC, helps elucidate certain physical aspects of multidimensional tunneling. Finally, simple extensions of the MPP procedure suggest possible directions for new methods for calculating tunneling splitting in complex molecules.
A theory is presented for the electronic structure and multidimensional free energy surfaces for hydrogen-bonded complexes AH⋯B capable of proton transfer to form an ion pair A-⋯HB+ in solution. Two diabatic states, neutral and ionic, are electronically coupled to each other and electrostatically coupled to the surrounding solvent and are treated via a nonlinear Schrödinger equation approach. The theory includes nonequilibrium solvation of the complex, a feature important for proton-transfer dynamics and spectroscopic phenomena. Representative calculations for a model OH⋯N complex are presented. The importance of the solvent polarization for the electronic structure of the complex is illustrated by comparison with the results obtained by solvation of the in vacuo electronic structure.
A Feynman path integral methodology is developed to study proton-transfer reactions in polar solvents. The present approach is specifically designed to efficiently probe the effects of intramolecular vibrations on the transfer rate as a general function of temperature and intramolecular couplings. A representative study of some nonlinear couplings which may affect the tunneling of the proton will be presented. A central feature of the method is that the actual intramolecular proton-transfer coordinate is taken as the reaction coordinate. Even so, the activated dynamics of the solvent polarization is shown to be accounted for.
A theoretical framework is discussed in which imaginary time Feynman path integration is adapted to define a quantum mechanical free energy for activated rate processes. Recently developed variational theories for the estimation of this factor, as well as the quantum dynamical corrections to the rate constant, are also discussed.
We consider the nonlinear Schrödinger equation arising when an explicit separation of the medium electron polarization is performed in the Hamiltonian describing the outer-sphere electron transfer (ET) reaction. The corresponding classical activation energy is determined by the point of intersection of two diabatic energy curves rather than by the stationary saddle point of the free energy functional, in agreement with the earlier treatment of Kim and Hynes. Analytical expressions for the free energy terms as functions of the ET coordinate are derived. The relative energy of this cross point is the same as the traditional activation energy of the Et process as given by the Marcus formula. In order to find the interaction matrix elements which promote the nonadiabatic transition and avoided crossing, an alternative quantum-mechanical description of the two-level ET model is given in terms of the wavefunction which allows independent reorganization of electrons belonging to the medium and to the chemical substrate. Explicit expression for matrix elements are presented and different kinetic regimes generated by this refined model are discussed.
The main results obtained during the last 5 yr in the quantum-mechanical theory of the reaction elementary act in solutions are summarized. The method of qualitative and quantitative analysis of experimental data is presented. The electron transfer reactions and ligand substitution reactions are considered as a example of the application of the general method. The perspectives of the theory development and the main problems of the theory are discussed.
Quantum simulation schemes based on the Feynman path integral molecular dynamics technique have been used to calculate the effective activation energy associated with nuclear reorganization in the self-exchange reaction of tuna cytochrome c. In addition, a quench technique is used to exhibit the instantons or most probable tunneling paths involved in the reorganization motion. At room temperature, the activation energy is calculated to be 8.8 kJ/mol, close to the estimate by Warshel et al., from purely classical considerations. The quantum contribution is small, 2.6 kJ/mol at room temperature. At lower temperature, the quantum tunneling becomes more significant and the free energy associated with the quantum correction factor begins to dominate, 4.2 kJ/mol at 150 K. The transient tunneling paths can deviate significantly from the transition direction, contrary to the picture one would expect for a purely harmonic system. Corrections to short time dynamics are discussed and shown to be small for tuna cytochrome c at room temperature, using an approximation based on the dispersed polaron method. In addition, the problem of conformational substates and their effect on the tunneling calculation is noted.
The field of chemisorption has recently seen great advances both in experimental spectroscopies for examining the geometry and electronic structures of adsorbed species and in numerical calculational techniques. The aim of this article is a critical analysis of these results, especially theoretical ones, with emphasis on the relationship to simple models of chemisorption. An important conclusion is the strong coupling nature of many chemisorption systems, i.e. that there exists something like a chemical bond between the adsorbate and the nearest-neighbour substrate atoms with their associated bonding and antibonding electronic levels.We start by outlining the phenomenological fundamentals, such as the structure of the adsorbed layer. Basic theoretical concepts, such as local density functional theory, on which much of the modern discussion of the electronic chemisorption problem depends, and the electronic structure of the clean surface, are then outlined. There follows a fuller discussion of the Anderson model, which is regarded as the underlying canonical model for considering chemisorption. A discussion, in detail, of the theoretical and some experimental aspects of a number of systems is then given. The principal systems considered, which are of contemporary interest, include hydrogen, oxygen and alkalis on free-electron and transition metals, chalcogenides on nickel, CO adsorption on various transition metals, and oxygen on silver.
The molecular structure of the photosynthetic reaction centre from Rhodopseudomonas viridis has been elucidated using X-ray crystallographic analysis. The central part of the complex consists of two subunits, L and M, each of which forms five membrane-spanning helices. We present the first description of the high-resolution structure of an integral membrane protein.