Reorganization Energy Research Articles

Theoretical prediction of holes and electrons mobility in coronene and perchlorocoronene crystals: Application of the Marcus–Jortner–Levich Theory

The study focused on theoretically predicting charge carrier mobilities in coronene and perchlorocoronene crystals using the Marcus–Levich–Jortner approach. By analyzing charge transfer integrals and considering hopping pathways, significant anisotropy in charge transfer was observed. Density functional theory (DFT) calculations at the B3LYP-D3/6-311++G(d,p) and ωB97X-D/6-311++G(d,p) levels were employed to assess the impact of normal modes on reorganization energies and charge transfer rates. Results indicated that perchlorination reduced mobility for both holes and electrons. Notably, maximum mobilities were observed at the melting points: 0.0120 cm2 V−1 s−1 for electrons and 0.0798 cm2 V−1 s−1 for holes in coronene at 455 K, and 0.0032 cm2 V−1 s−1 for electrons and 0.0426 cm2 V−1 s−1 for holes in perchlorocoronene at 560 K. This suggests that both crystals exhibit favorable charge transport characteristics near their respective melting points.

Thermodynamic Factors Controlling Electron Transfer among the Terminal Electron Acceptors of Photosystem I: Insights from Kinetic Modelling.

Photosystem I is a key component of primary energy conversion in oxygenic photosynthesis. Electron transfer reactions in Photosystem I take place across two parallel electron transfer chains that converge after a few electron transfer steps, sharing both the terminal electron acceptors, which are a series of three iron-sulphur (Fe-S) clusters known as FX, FA, and FB, and the terminal donor, P700. The two electron transfer chains show kinetic differences which are, due to their close geometrical symmetry, mainly attributable to the tuning of the physicochemical reactivity of the bound cofactors, exerted by the protein surroundings. The factors controlling the rate of electron transfer between the terminal Fe-S clusters are still not fully understood due to the difficulties of monitoring these events directly. Here we present a discussion concerning the driving forces associated with electron transfer between FX and FA as well as between FA and FB, employing a tunnelling-based description of the reaction rates coupled with the kinetic modelling of forward and recombination reactions. It is concluded that the reorganisation energy for FX- oxidation shall be lower than 1 eV. Moreover, it is suggested that the analysis of mutants with altered FA redox properties can also provide useful information concerning the upstream phylloquinone cofactor energetics.

DFT/TD-DFT study of novel triphenylamine-based dyes with azo moieties and π-spacer variations for enhanced dye-sensitized solar cell performance

This study involves a computational analysis of new D-π-A dyes obtained from triphenylamine (TPA), which contain various azo-dye components. The structural, molecular, electrical, and optical properties of these dyes were computed using Density Functional Theory (DFT) and Time-Dependent DFT, utilizing the B3LYP/6–31 G model. Our research specifically aimed to investigate the effects of incorporating different azo dye constituents in the para position of two phenyl groups of TPA. The results indicate that these alterations lead to notably broadened and red-shifted absorption spectra, as well as improved optoelectronic properties that are subject to additional tuning through the manipulation of the π-spacer. The excitation energies and HOMO-LUMO energy levels that have been estimated indicate the presence of effective electron injection and dye regeneration mechanisms. The results concerning the nonlinear optical (NLO) properties suggest that these dyes are likely to demonstrate superior performance in NLO applications. The factors encompassed in this study consist of light-harvesting efficiency (LHE), open-circuit photovoltage (VOC), electron injection driving force (ΔGinj), dye regeneration driving force (ΔGreg), excited state lifetime (τ) and reorganization energy (λtotal), which has a strong correlation with the electrical current density in a short-circuit (JSC) and DSSC's overall effectiveness. This scientific attempt contributes to the systematic advancement of efficient dyes, demonstrating the possibility for enhanced efficiency in DSSCs. Further validation of computational forecasts and advancement of renewable energy technology necessitate future experimental synthesis and testing.

Advancements in optical and optoelectronic characteristics of benzotriazole-based asymmetric non-fullerene acceptors for organic photovoltaics

The development of energy-efficient non-fullerene acceptors (NFAs) has attracted much attention in producing efficient organic solar cells (OSCs). These innovative materials are considered a promising alternative to traditional fullerene acceptors because they absorb a broader light range and potentially enhance the OSC performance. Herein, we engineered eleven new asymmetric caterpillar-shaped materials with an A1-D-A2-A1 core as NFAs for OSCs. These efficient materials offer a high absorption efficiency, wide energy level range, and efficient charge transport capabilities. To engineer these caterpillar-shaped NFAs (FT1-FT11), we performed various synergistic modulations of end-capped electron-withdrawing moieties and side-chain engineering on the synthetic reference molecule (FT). The electron and hole reorganization energies, binding energy, transition energy, transition density analysis, electrostatic potential, and photovoltaic characterizations have been performed for these designed caterpillar-shaped (FT1–FT11) series. The designed materials (FT1–FT11) showed a lower binding energy of 0.45 eV, a narrower bandgap (2.11 eV), higher absorption (ranges from 700.44 nm to 781.05), and low electron and hole mobilities (0.0029 and 0.0031) compared to reference molecule FT. To investigate the charge transport process, we employed polymer donor PTB7-Th and efficient NFA acceptor FT11 to establish a donor:acceptor complex (FT11:PTB7-Th). The complex study demonstrates a significant charge transformation at the donor-acceptor interface. As a result, the developed compounds (FT1–FT11), with their superior optoelectronic capabilities, could be viewed as a viable and sustainable choice to fabricate next-generation efficient OSCs.

From Coherence to Function: Exploring the Connection in Chemical Systems.

ConspectusThe role of quantum mechanical coherences or coherent superposition states in excited state processes has received considerable attention in the last two decades largely due to advancements in ultrafast laser spectroscopy. These coherence effects hold promise for enhancing the efficiency and robustness of functionally relevant processes, even when confronted with energy disorder and environmental fluctuations. Understanding coherence deeply drives us to unravel mechanisms and dynamics controlled by order and synchronization at a quantum mechanical level, envisioning optical control of coherence to enhance functions or create new ones in molecular and material systems. In this frontier, the interplay between electronic and vibrational dynamics, specifically the influence of vibrations in directing electronic dynamics, has emerged as the leading principle. Here, two energetically disparate quantum degrees of freedom work in-sync to dictate the trajectory of an excited state reaction. Moreover, with the vibrational degree being directly related to the structural composition of molecular or material systems, new molecular designs could be inspired by tailoring certain structural elements.In the realm of chemical kinetics, our understanding of the dynamics of chemical transformations is underpinned by fundamental theories, such as transition state theory, activated rate theory, and Marcus theory. These theories elucidate reaction rates by considering the energy barriers that must be overcome for reactants to transform into products. Those barriers are surmounted by the stochastic nature of energy gap fluctuations within reacting systems, emphasizing that the reaction coordinate, the pathway from reactants to products, is not rigidly defined by a specific vibrational motion but encompasses a diverse array of molecular motions. While less is known about the involvement of specific intramolecular vibrational modes, their significance in certain cases cannot be overlooked.In this Account, we summarize key experimental findings that offer deeper insights into the complex electronic-vibrational trajectories encompassing excited states afforded from state-of-the-art ultrafast laser spectroscopy in three exemplary processes: photoinduced electron transfer, singlet-triplet intersystem crossing, and intramolecular vibrational energy flow in molecular systems. We delve into the rapid decoherence, or loss of phase and amplitude correlations, of vibrational coherences along promoter vibrations during subpicosecond intersystem crossing dynamics in a series of binuclear platinum complexes. This rapid decoherence illustrates the vibration-driven reactive pathways from the Franck-Condon state to the curve crossing region. We also explore the generation of new vibrational coherences induced by impulsive reaction dynamics rather than by the laser pulse in these systems, which sheds light on specific energy dissipation pathways and thereby on the progression of the reaction trajectory in the vicinity of the curve crossing on the product side. Another property of vibrational coherences, amplitude, reveals how energy can flow from one vibration to another in the electronic excited state of a terpyridine-molybdenum complex hosting a nonreactive dinitrogen substrate. A slight change in vibrational energy triggers a quasi-resonant interaction, leading to constructive wavepacket interference and ultimately intramolecular vibrational redistribution from a Franck-Condon active terpyridine vibration to a dinitrogen stretching vibration, energizing the dinitrogen bond.

Alkyl-Doping Enables Significant Suppression of Conformational Relaxation and Intermolecular Nonradiative Decay for Improved Near-Infrared Fluorescence Imaging.

Despite various efforts to optimize the near-infrared (NIR) performance of perylene diimide (PDI) derivatives for bio-imaging, convenient and efficient strategies to amplify the fluorescence of PDI derivatives in biological environment and the intrinsic mechanism studies are still lacking. Herein, we propose an alkyl-doping strategy to amplify the fluorescence of PDI derivative-based nanoparticles for improved NIR fluorescence imaging. The developed PDI derivative, OPE-PDI, shows much brighter in n-Hexane (HE) compared with that in other organic media, and the excited state dynamics investigation experimentally elucidates the solvent effect-induced suppression of intermolecular energy transfer and intramolecular nonradiative decay as the underlying mechanism for the fluorescence improvement. Theoretical calculations reveal the lowest reorganization energies of OPE-PDI in HE among various solvents, indicating the effectively suppressed conformational relaxation to support the strongest radiative decay. Inspired by this, an alkyl atmosphere mimicking HE is constructed by incorporating the octadecane into OPE-PDI-based nanoparticles, permitting up to 3-fold fluorescence improvement compared with the counterpart nanoparticles. Owing to the merits of high brightness, anti-photobleaching, and low biotoxicity for the optimal nanoparticles, they have been employed for probing and long-term monitoring of tumor. This work highlights a facile strategy for the fluorescence enhancement of PDI derivative-based nanoparticles.

Regulation of organic solar cells performance through external electric field: From charge transfer mechanisms to photovoltaic properties

In organic solar cells (OSCs), comprehending the charge transfer mechanism at D/A interfaces is crucial for photoinduced charge generation and enhancing power conversion efficiency (PCE). The charge transfer mechanism and photovoltaic performance of the parallel stacking interface configuration of the PTQ10 polymer donor and T2EH non-fullerene acceptor (NFA) are systematically studied at the microscopic scale. The analysis of the electron-hole distribution of the PTQ10/T2EH excited states revealed the presence of multiple charge excitation modes and charge transfer pathways. Using Marcus theory, we examine the charge separation rate (KCS) of PTQ10/T2EH under external electric field (Fext) modulation, and it is clarified that reorganization energy (λ) is the main factor that affects the KCS. Our results show that Fext has a positive impact on the photovoltaic properties of PTQ10/T2EH thin films, as evidenced by the modulation of the open circuit voltage (VOC), voltage loss (VLOSS) and fill factor (FF). Overall, this study provides valuable theoretical insights for Fext to accelerate the charge separation process and enhance photovoltaic efficiency.

Theoretical justification for obtaining new nanomaterials by taking into account the influence of three main factors

By generalizing and transferring the results of the Marcus theory for homogeneous reactions to heterogeneous, Dogonadze and Kuznetsov created a quantum-mechanical theory of electrode reactions for ionic melts, the methodological principles of which allow more adequately evaluating the results of certain experimental methods of obtaining new substances with given properties. Such methods include high-temperature electrochemical synthesis (HES), the implementation of which is complicated by the lack of precise understanding of the mechanism of multi-electron processes of reduction of various ionic forms of refractory metals and nonmetals in ionic melts, including those containing molybdate and tungstate ions, both constantly and simultaneously.

Study of photosensitizer dyes for high-performance dye-sensitized solar cells application: A computational investigation

Dye-sensitized solar cells (DSSCs) offer a promising, cost-effective alternative to conventional photovoltaic systems. Organic sensitizers, capable of capturing a broad spectrum of sunlight, are key components in DSSCs, but their development and testing are often time-consuming and expensive. Quantum chemical calculations, specifically Density Functional Theory (DFT), have emerged as valuable tools to evaluate potential dye candidates, streamlining the design process and reducing costs. This study investigated the molecular structures and photophysical properties of three common dye classes used in high-performance DSSCs: natural pigments, anthocyanidin pigments, and squaraine dyes. Employing DFT and time-dependent DFT (TD-DFT) at the B3LYP/6–31G level, key parameters such as the HOMO-LUMO energy gap, free energy differences for electron injection and dye regeneration, short-circuit current density, total reorganization energy, and open-circuit voltage were analyzed. Additionally, maximum absorption wavelengths and oscillator strength values were calculated. Our findings provide valuable insights into the optical and electrical properties of these natural dyes, aiding DSSC manufacturers in selecting optimal sensitizers. This research highlights the potential of computational methods in accelerating dye development and improving the overall efficiency of DSSC technology.

Non-negligible Outer-Shell Reorganization Energy for Charge Transfer in Nonpolar Systems.

Many charge-transporting molecular systems function as ordered or disordered arrays of solid state materials composed of nonpolar (or weakly polar) molecules. Due to low dielectric constants for nonpolar systems, it is common to ignore the effects of outer-shell reorganization energy (λout). However, ignoring λout has not been properly supported and it can severely impact predictions and insights derived. Here, we estimate λout by two means: from experimental ultraviolet photoelectron spectra, in which vibronic progression in these spectra can be fitted with the widths of peaks determining the low-frequency component in reorganization energy, regarded to be closely associated with λout, and from molecular dynamic (MD) simulation of nonpolar molecules, in which disorder or fluctuation statistics for energies of charged molecules are calculated. An upper bound for λout was obtained as 505 and 549 meV for crystalline anthracene (140 K) and pentacene (50 K), respectively, by fitting of experimental data, and 212 and 170 meV, respectively, from MD simulations. These values are comparable to the inner-sphere reorganization energy (λin) arising from intramolecular vibration. With corresponding spectral density functions calculated, we found that λout is influenced both by low- and high-frequency dynamics, in which the former arises from constrained translational and rotational motions of surrounding molecules. In an amorphous state, about half of the λout's obtained are from high-frequency components, which is quite different from the conventional polar solvation. Moreover, crystalline systems exhibit super-Ohmic spectral density, whereas amorphous systems are sub-Ohmic.

PyCTRAMER: A Python package for charge transfer rate constant of condensed-phase systems from Marcus theory to Fermi's golden rule.

In this work, we introduce PyCTRAMER, a comprehensive Python package designed for calculating charge transfer (CT) rate constants in disordered condensed-phase systems at finite temperatures, such as organic photovoltaic (OPV) materials. PyCTRAMER is a restructured and enriched version of the CTRAMER (Charge-Transfer RAtes from Molecular dynamics, Electronic structure, and Rate theory) package [Tinnin et al. J. Chem. Phys. 154, 214108 (2021)], enabling the computation of the Marcus CT rate constant and the six levels of the linearized semiclassical approximations of Fermi's golden rule (FGR) rate constant. It supports various types of intramolecular and intermolecular CT transitions from the excitonic states to CT state. Integrating quantum chemistry calculations, all-atom molecular dynamics (MD) simulations, spin-boson model construction, and rate constant calculations, PyCTRAMER offers an automatic workflow for handling photoinduced CT processes in explicit solvent environments and interfacial CT in amorphous donor/acceptor blends. The package also provides versatile tools for individual workflow steps, including electronic state analysis, state-specific force field construction, MD simulations, and spin-boson model construction from energy trajectories. We demonstrate the software's capabilities through two examples, highlighting both intramolecular and intermolecular CT processes in prototypical OPV systems.

Isomer engineering of triptycene-fused multi-resonance TADF emitters: Implications for controlling blue electroluminescence performance

Concurrently achieving high efficiency, narrowband emission, and low efficiency roll-off remains a challenging task for realizing stable blue organic light-emitting diodes (OLEDs) with high color purity. To tackle this issue, herein, two isomeric pairs of B,N-doped multi-resonance thermally activated delayed fluorescence (MR-TADF) emitters, namely, 2,3-Tp-BuDABNA-R and 3,4-Tp-BuDABNA-R (R=H, 3,5-Xyl), featuring triptycene (Tp) moieties fused into the 2,3- and 3,4-positions of the B,N core, respectively, are presented. Both isomeric forms exhibit blue emissions with near-unity photoluminescence (PL) quantum yields in their doped host films. Notably, the 3,4-Tp-fused emitter shows 6-fold faster reverse intersystem crossing (RISC) compared to the 2,3-Tp-fused emitter, albeit with PL spectral broadening. Along with a smaller activation energy of RISC, the stronger spin–orbit coupling between the singlet (S1) and triplet (T2) excited states of the 3,4-Tp-fused emitter is suggested to facilitate the RISC process. Furthermore, a high-frequency vibration in the Tp-fused benzene ring, accompanied by a large reorganization energy, is found to mainly contribute to the spectral broadening of the 3,4-Tp-fused emitter. Blue TADF-OLEDs based on each isomeric emitter similarly achieve a high maximum external quantum efficiency of ∼ 26%. Remarkably, the devices with the 3,4-isomer exhibit significantly reduced efficiency roll-offs compared to the 2,3-isomer, demonstrating that isomer engineering of MR emitters can have a distinct impact on controlling device performance.

Electron Transfer Enhanced by a Minimal Energetic Driving Force at the Organic‐Semiconductor Interface

AbstractThe energetic driving force for electron transfer must be minimized to realize efficient optoelectronic devices including organic light‐emitting diodes (OLEDs) and organic photovoltaics (OPVs). Exploring the dynamics of a charge‐transfer (CT) state at an interface leads to a comprehension of the relationship between energetics, electron‐transfer efficiency, and device performance. Here, we investigate the electron transfer from the CT state to the triplet excited state (T1) in upconversion OLEDs with 45 material combinations. By analyzing the CT emission and the singlet excited‐state emission from triplet–triplet annihilation via the dark T1, their energetics and electron‐transfer efficiencies are extracted. We demonstrate that the CT→T1 electron transfer is enhanced by the stronger CT interaction and a minimal energetic driving force (<0.1 eV), which is explained using the Marcus theory with a small reorganization energy of <0.1 eV. Through our analysis, a novel donor–acceptor combination for the OLED is developed and shows an efficient blue emission with an extremely low turn‐on voltage of 1.57 V. This work provides a solution to control interfacial CT states for efficient optoelectronic devices without energy loss.

Electron Transfer Enhanced by a Minimal Energetic Driving Force at the Organic-Semiconductor Interface.

The energetic driving force for electron transfer must be minimized to realize efficient optoelectronic devices including organic light-emitting diodes (OLEDs) and organic photovoltaics (OPVs). Exploring the dynamics of a charge-transfer (CT) state at an interface leads to a comprehension of the relationship between energetics, electron-transfer efficiency, and device performance. Here, we investigate the electron transfer from the CT state to the triplet excited state (T1) in upconversion OLEDs with 45 material combinations. By analyzing the CT emission and the singlet excited-state emission from triplet-triplet annihilation via the dark T1, their energetics and electron-transfer efficiencies are extracted. We demonstrate that the CT→T1 electron transfer is enhanced by the stronger CT interaction and a minimal energetic driving force (<0.1 eV), which is explained using the Marcus theory with a small reorganization energy of <0.1 eV. Through our analysis, a novel donor-acceptor combination for the OLED is developed and shows an efficient blue emission with an extremely low turn-on voltage of 1.57 V. This work provides a solution to control interfacial CT states for efficient optoelectronic devices without energy loss.

Rapid Estimation of fwhm for OLED Emission Spectra Using Bond-Length and Bond-Order Alterations.

Color purity is crucial for achieving a high-quality organic light-emitting diode. An essential property for evaluating color purity is the full width at half-maximum (fwhm) of emission spectra. Theoretically estimating fwhm requires the detailed knowledge of vibronic coupling constants for all vibrational modes, which is time-consuming to obtain via quantum mechanical calculations, particularly for optimizing the geometry of the lowest singlet excited state. These calculations pose a bottleneck for high-throughput screening of narrowband emitters. To address this challenge, we propose a simple model to assess the reorganization energy based on the relationship between bond-length alteration (ΔBL) and vertical bond-order alteration (ΔBO) in optimized ground-state structures. Additionally, a one-mode model is employed to rapidly and effectively estimate fwhm. Our new model yields results comparable to experimental data and more accurate theoretical approaches based on vibronic coupling constants of all vibrational modes but with significantly lower computational costs. This model holds promise for the virtual screening of narrowband emitter candidates.

Periodically Twisted Molecular Wires Based on a Fused Unit for Efficient Intramolecular Hopping Transport.

Realizing efficient long-distance intramolecular charge transport based on a hopping mechanism is a key challenge in molecular electronics. In hopping transport, a smaller reorganization energy (λ) and energy difference between hopping sites (ΔEhs) should lead to a smaller activation energy and faster charge transfer. However, the development of π-extended molecules that meet these requirements is challenging. In this study, we successfully synthesized several nanometer-scale π-extended molecules composed of a fused π-conjugated unit as a hopping site for reducing λ. Conformational twists between fused units effectively localize π-conjugation in each unit, contributing to reducing ΔEhs. The expected electronic structures of the oligomers were confirmed using spectroscopic and electrochemical measurements. Single-molecule conductance measurements exhibited higher conductance and lower activation energy than those of nonfused oligothiophenes. First-principles calculations indicated that smaller λ and ΔEhs values explain the high conductance. These results highlight the efficiency of the proposed molecular design for effective intramolecular hopping transport.

(Invited) Computational Models of Proton-Coupled Electron Transfer in Energy Conversion and Storage

Proton-coupled electron transfer (PCET) is a fundamental class of chemical reactions that is central to many electrochemical energy conversion and storage processes. In this talk, I will discuss recent theoretical model development for interfacial and intermolecular PCET reactions. First, I will describe a rate constant model for the reaction between reduced anatase TiO2 surfaces and a 4-MeO-TEMPO nitroxyl radical. The rate constants are modeled using vibronically nonadiabatic PCET theory, where the transferring proton and electron are treated quantum mechanically. Hybrid functional periodic density functional theory (DFT) calculations are used to derive model parameters such as driving forces, reorganization energies, and proton vibrational wavefunctions. This modeling strategy highlights the role of inner-sphere bond reorganization and excited vibronic states on the PCET reactivity of metal oxides. Next, I will show how similar computational approaches can be used to predict whether electrochemical PCET is likely to proceed through concerted electron–proton transfer or sequential electron transfer and proton transfer steps. Using PCET reactions between redox-active quinones and functionalized imidazole bases as an example, mechanistic predictions are informed by DFT-calculated potential energy surfaces. These PCET modeling strategies have broad applicability to kinetic studies of electrochemical energy conversion and storage processes.

(Invited) Atomic-Scale Modeling of Charge-Transfer Kinetics at LixCoO2 Cathode-Electrolyte Interfaces

In the search for Li-ion batteries with improved performance, interface engineering is a comparably underexplored area with significant opportunities for innovation. Ion transport across interfaces is, for instance, a significant bottleneck for fast charging/discharging. At electrode interfaces, it is often essential to capture the coupled motion of ions and electrons in order to provide accurate computational input for interfacial design. In this work we employ the recently proposed theory of coupled ion-electron transfer kinetics[1,2] in conjugation with constrained and constant-potential density functional theory to examine Li+ transfer across battery electrolyte-cathode interfaces at the atomic scale. Here, we focus on the common Li x CoO2 intercalation-type cathode material and liquid-state organic electrolytes. We establish trends in transport kinetics for different states-of-charge (SOC, i.e., varied degree of lithiation, x), types of electrolytes, and counterions that closely reproduces experimentally observed trends. From our modeling, we can identify a few key descriptors to predict performance metrics in Li-ion transport of a given electrolyte-cathode combination. These include the Li-ion free energy of adsorption, the free energies of reorganization of the cathode material and electrolyte upon the addition/removal of electrons, as well as the electronic coupling between electron acceptor and donor states. As these quantities can be relatively straightforwardly estimated for any given electrode-electrolyte pair, we envisage broad applications of our modeling approach in the interfacial design for batteries and related fields.[1] Fraggedakis et al., Electrochimica Acta, 367, 137432 (2021)[2] Bazant, Faraday Discussion, 246, 60 (2023)

(Keynote) Unified Quantum Theory of Electrochemical Kinetics Based on Coupled Ion-Electron Transfer

A general theory of coupled ion–electron transfer (CIET) is presented, which unifies quantum kinetics of electron transfer (ET) with classical kinetics of bond-breaking ion transfer (IT) in a general framework of non-equilibrium thermodynamics [1]. In the limit of large reorganization energy, the theory predicts Marcus kinetics of “electron-coupled ion transfer” (ECIT). In the limit of large ion-transfer energies, the theory predicts generalized Butler–Volmer kinetics of “ion-coupled electron transfer” (ICET), where the charge transfer coefficient and exchange current are connected to microscopic properties of the electrode/electrolyte interface. In the ICET regime, the reductive and oxidative branches of Tafel’s law are predicted to hold over a wide but finite range of overpotentials, bounded by the ion-transfer energies for oxidation and reduction, respectively. The probability distribution of transferring electron energies in CIET smoothly interpolates between a shifted Gaussian distribution for ECIT (as in the Gerischer–Marcus theory of ET) to an asymmetric, fat-tailed Meixner distribution centered at the Fermi level for ICET. The latter may help interpret asymmetric line shapes in x-ray photo-electron spectroscopy (XPS) and Auger electron spectroscopy (AES) for oxidized metal surfaces in terms of shake-up relaxation of the ionized atom and its image polaron by ICET. In the limit of large overpotentials, the theory predicts a transition to inverted Marcus ECIT, leading to a universal reaction-limited current for metal electrodes, dominated by barrierless quantum transitions. Uniformly valid, closed-form asymptotic approximations are derived that smoothly transition between the limiting rate expressions for ICET and ECIT for metal electrodes, using simple but accurate mathematical functions. The theory is applied to lithium intercalation in lithium iron phosphate (LFP) and found to provide a consistent description of the observed current dependence on overpotential, temperature and concentration, learned from x-ray images [2]. CIET theory thus provides a critical bridge between quantum electrochemistry and electrochemical engineering.[1] M. Z. Bazant, Unified quantum theory of electrochemical kinetics by coupled ion-electron transfer, Faraday Discussions 246, 60-124 (2023).[2] H. Zhao, H. D. Deng, A. E. Cohen, J. Lim, Y. Li, D. Fraggedakis, B. Jiang, B. D. Storey, W. C. Chueh, R. D. Braatz, and M. Z. Bazant, Learning heterogeneous reaction kinetics from X-ray movies pixel-by-pixel, Nature 621, 289-294 (2023). Figure 1

Electronic Properties of [Ru(NH3)6][M(CN)6] Double-Complex Crystals and Their Thermal Decomposition to Prussian Blue Analogs

The complex ions, [Ru(NH3)6]2+/3+ and [Fe(CN)6]3-/4-, are well known redox couples exhibiting fast interfacial and homogeneous electron transfer rates in aqueous solution. Here we show that when mixed together, they form insoluble double-complex Ax[Ru(NH3)6][Fe(CN)6] crystals with rhombohedral symmetry, where A is a cationic species and 0≤x≤2. The analogous interactions of [Ru(NH3)6]2+/3+ with [Ru(CN)6]3-/4- produces Ax[Ru(NH3)6][Fe(CN)6] crystals. Electrical measurements of these single crystals exhibit ohmic current-voltage responses, with corresponding conductivities on the order of 0.1 mS/m. We propose a solid-state charge transport mechanism based on Marcus theory for homogeneous outer-sphere electron transfer.Thermal decomposition at mild temperatures removes ammine ligands and transforms these crystals into nanopolycrystalline Prussian blue analogs (PBAs) that retain the macroscopic geometry of the double-complex crystals but adopt the familiar highly defected porous crystal structure and intervalence charge transfer characteristics of PBAs.