Grand canonical electronic density-functional theory: Algorithms and applications to electrochemistry

Toward Realistic Models of the Electrocatalytic Oxygen Evolution Reaction

The electrocatalytic oxygen evolution reaction (OER) supplies the protons and electrons needed to transform renewable electricity into chemicals and fuels. However, the OER is kinetically sluggish; it operates at significant rates only when the applied potential far exceeds the reversible voltage. The origin of this overpotential is hidden in a complex mechanism involving multiple electron transfers and chemical bond making/breaking steps. Our desire to improve catalytic performance has then made mechanistic studies of the OER an area of major scientific inquiry, though the complexity of the reaction has made understanding difficult. While historically, mechanistic studies have relied solely on experiment and phenomenological models, over the past twenty years ab initio simulation has been playing an increasingly important role in developing our understanding of the electrocatalytic OER and its reaction mechanisms. In this Review we cover advances in our mechanistic understanding of the OER, organized by increasing complexity in the way through which the OER is modeled. We begin with phenomenological models built using experimental data before reviewing early efforts to incorporate ab initio methods into mechanistic studies. We go on to cover how the assumptions in these early ab initio simulations─no electric field, electrolyte, or explicit kinetics─have been relaxed. Through comparison with experimental literature, we explore the veracity of these different assumptions. We summarize by discussing the most critical open challenges in developing models to understand the mechanisms of the OER.

Identification of K+-determined reaction pathway for facilitated kinetics of CO2 electroreduction

Cations such as K+ play a key part in the CO2 electroreduction reaction, but their role in the reaction mechanism is still in debate. Here, we use a highly symmetric Ni-N4 structure to selectively probe the mechanistic influence of K+ and identify its interaction with chemisorbed CO2-. Our electrochemical kinetics study finds a shift in the rate-determining step in the presence of K+. Spectral evidence of chemisorbed CO2- from in-situ X-ray absorption spectroscopy and in-situ Raman spectroscopy pinpoints the origin of this rate-determining step shift. Grand canonical potential kinetics simulations - consistent with experimental results - further complement these findings. We thereby identify a long proposed non-covalent interaction between K+ and chemisorbed CO2-. This interaction stabilizes chemisorbed CO2- and thus switches the rate-determining step from concerted proton electron transfer to independent proton transfer. Consequently, this rate-determining step shift lowers the reaction barrier by eliminating the contribution of the electron transfer step. This K+-determined reaction pathway enables a lower energy barrier for CO2 electroreduction reaction than the competing hydrogen evolution reaction, leading to an exclusive selectivity for CO2 electroreduction reaction.

Revealing the Size and Potential Dependent D2O Microkinetics on Pt Nanoparticles Using Grand Canonical Ensemble Modeling

Revealing the potential and nanoparticle size effect is significant for understanding the electrochemical microkinetic behaviors under real reaction conditions. Herein, an efficient strategy of combining the robust fully converged constant potential (FCP) algorithm and the size dependent site distribution rule assumption was proposed. A simple reaction of isotopic D2O/H2O adsorption and dissociation on Pt nanoparticles was set as the model reaction. The results show that the cathodic negative potential and the anodic positive potential would result in the D2O orientation of the D-down/O-down physisorption configuration. Microkinetic simulations by this strategy obtained electrochemical widows for D2O/H2O dissociation, and the optimal Pt nanoparticle diameter was predicted to be 1.8 nm, which agrees well with the experimental observation of ∼2 nm threshold. The kinetic isotope effect (KIE) rate constant ratio at the optimal potential of -0.80 V vs SHE was calculated to be ∼1.83. This work provides a guideline in studying electrochemical electrode-electrolyte interactions on nanoparticles.

Modeling of Nanomaterials for Supercapacitors: Beyond Carbon Electrodes

Capacitive storage devices allow for fast charge and discharge cycles, making them the perfect complements to batteries for high power applications. Many materials display interesting capacitive properties when they are put in contact with ionic solutions despite their very different structures and (surface) reactivity. Among them, nanocarbons are the most important for practical applications, but many nanomaterials have recently emerged, such as conductive metal-organic frameworks, 2D materials, and a wide variety of metal oxides. These heterogeneous and complex electrode materials are difficult to model with conventional approaches. However, the development of computational methods, the incorporation of machine learning techniques, and the increasing power in high performance computing now allow us to tackle these types of systems. In this Review, we summarize the current efforts in this direction. We show that depending on the nature of the materials and of the charging mechanisms, different methods, or combinations of them, can provide desirable atomic-scale insight on the interactions at play. We mainly focus on two important aspects: (i) the study of ion adsorption in complex nanoporous materials, which require the extension of constant potential molecular dynamics to multicomponent systems, and (ii) the characterization of Faradaic processes in pseudocapacitors, that involves the use of electronic structure-based methods. We also discuss how recently developed simulation methods will allow bridges to be made between double-layer capacitors and pseudocapacitors for future high power electricity storage devices.

Theoretical Investigation of the Potential-Dependent CO Adsorption on Copper Electrodes

Despite the importance of CO adsorption in many electrocatalytic reaction mechanisms, there has been little investigation of the dependence of the free energy of CO adsorption on the applied potential. Herein, we report on the potential-dependent adsorption of CO on Cu electrodes using a grand-canonical density functional theory approach. We demonstrate that, within the working potential range of electrocatalytic CO2 reduction on Cu(111) and Cu(100), the CO adsorption strength can change by over 0.1 eV. Our analyses explain the potential dependence through an interfacial capacitance loss upon CO adsorption as well as orbital relaxation induced by the electrode potential. Via sensitivity analysis with respect to two electrolyte model parameters (solvent dielectric constant and Debye screening length), we find that the surface excess charge density is a useful descriptor of the CO adsorption free energy.

On the Thermodynamic Equivalence of Grand Canonical, Infinite-Size, and Capacitor-Based Models in First-Principle Electrochemistry

First principles-based computational and theoretical methods are constantly evolving trying to overcome the many obstacles towards a comprehensive understanding of electrochemical processes on an atomistic level. One of the major challenges has been the determination of reaction energetics under a constant potential. Here, a theoretical framework was proposed applying standard electronic structure methods and extrapolating to the infinite-cell size limit where reactions do not alter the potential. Today, electronically grand canonical modifications to electronic structure methods, holding the potential constant by varying the number of electrons in a finite simulation cell, become increasingly popular. In this perspective, we show that these two schemes are thermodynamically equivalent. Further, we link these methods to capacitive models of the interface, in the limit that the capacitance of the charging components (whether continuum or atomistic) are equal and invariant along the reaction pathway. We benchmark the three approaches with an example of alkali cation adsorption on Pt(111) showing that all three approaches converge in the cases of Li, Na and K. For Cs, however, strong deviation from the ideal conditions leads to a spread in the respective results. We discuss the latter by highlighting the cases of broken equivalence and assumptions among the approaches.

Revealing Interface Polarization Effects on the Electrical Double Layer with Efficient Open Boundary Simulations under Potential Control

A major challenge in modeling interfacial processes in electrochemical (EC) devices is performing simulations at constant potential. This requires an open-boundary description of the electrons, so that they can enter and leave the computational cell. To enable realistic modeling of EC processes under potential control we have interfaced density functional theory with the hairy probe method in the weak coupling limit (Zauchner et al. Phys. Rev. B 2018, 97, 045116). Our implementation was systematically tested using simple parallel-plate capacitor models with pristine surfaces and a single layer of adsorbed water molecules. Remarkably, our code's efficiency is comparable with a standard DFT calculation. We reveal that local field effects at the electrical double layer induced by the change of applied potential can significantly affect the energies of chemical steps in heterogeneous electrocatalysis. Our results demonstrate the importance of an explicit modeling of the applied potential in a simulation and provide an efficient tool to control this critical parameter.

Facet-Dependent Oxygen Evolution Reaction Activity of IrO2 from Quantum Mechanics and Experiments

The diversity of chemical environments present on unique crystallographic facets can drive dramatic differences in catalytic activity and the reaction mechanism. By coupling experimental investigations of five different IrO2 facets and theory, we characterize the detailed elemental steps of the surface redox processes and the rate-limiting processes for the oxygen evolution reaction (OER). The predicted complex evolution of surface adsorbates and the associated charge transfer as a function of applied potential matches well with the distinct redox features observed experimentally for the five facets. Our microkinetic model from grand canonical quantum mechanics (GC-QM) calculations demonstrates mechanistic differences between nucleophilic attack and O-O coupling across facets, providing the rates as a function of applied potential. These GC-QM calculations explain the higher OER activity observed on the (100), (001), and (110) facets and the lower activity observed for the (101) and (111) facets. This combined study with theory and experiment brings new insights into the structural features that either promote or hinder the OER activity of IrO2, which are expected to provide parallels in structural effects on other oxide surfaces.

Two-Dimensional Layered Heterojunctions for Photoelectrocatalysis

Two-dimensional (2D) layered nanomaterial heterostructures, arising from the combination of 2D materials with other low-dimensional species, feature a large surface area to volume ratio, which provides a high density of active sites for catalytic applications and for (photo)electrocatalysis (PEC). Meanwhile, their electronic band structure and high electrical conductivity enable efficient charge transfer (CT) between the active material and the substrate, which is essential for catalytic activity. In recent years, researchers have demonstrated the potential of a range of 2D material interfaces, such as graphene, graphitic carbon nitride (g-C3N4), metal chalcogenides (MCs), and MXenes, for (photo)electrocatalytic applications. For instance, MCs such as MoS2 and WS2 have shown excellent catalytic activity for hydrogen evolution, while graphene and MXenes have been used for the reduction of carbon dioxide to higher value chemicals. However, despite their great potential, there are still major challenges that need to be addressed to fully realize the potential of 2D materials for PEC. For example, their stability under harsh reaction conditions, as well as their scalability for large-scale production are important factors to be considered. Generating heterojunctions (HJs) by combining 2D layered structures with other nanomaterials is a promising method to improve the photoelectrocatalytic properties of the former. In this review, we inspect thoroughly the recent literature, to demonstrate the significant potential that arises from utilizing 2D layered heterostructures in PEC processes across a broad spectrum of applications, from energy conversion and storage to environmental remediation. With the ongoing research and development, it is likely that the potential of these materials will be fully expressed in the near future.

Computational chemistry for water-splitting electrocatalysis

Electrocatalytic water splitting driven by renewable electricity has attracted great interest in recent years for producing hydrogen with high-purity. However, the practical applications of this technology are limited by the development of electrocatalysts with high activity, low cost, and long durability. In the search for new electrocatalysts, computational chemistry has made outstanding contributions by providing fundamental laws that govern the electron behavior and enabling predictions of electrocatalyst performance. This review delves into theoretical studies on electrochemical water-splitting processes. Firstly, we introduce the fundamentals of electrochemical water electrolysis and subsequently discuss the current advancements in computational methods and models for electrocatalytic water splitting. Additionally, a comprehensive overview of benchmark descriptors is provided to aid in understanding intrinsic catalytic performance for water-splitting electrocatalysts. Finally, we critically evaluate the remaining challenges within this field.

Complementary probes for the electrochemical interface

The functions of electrochemical energy conversion and storage devices rely on the dynamic junction between a solid and a fluid: the electrochemical interface (EI). Many experimental techniques have been developed to probe the EI, but they provide only a partial picture. Building a full mechanistic understanding requires combining multiple probes, either successively or simultaneously. However, such combinations lead to important technical and theoretical challenges. In this Review, we focus on complementary optoelectronic probes and modelling to address the EI across different timescales and spatial scales - including mapping surface reconstruction, reactants and reaction modulators during operation. We discuss how combining these probes can facilitate a predictive design of the EI when closely integrated with theory.

Incorporating Electrolyte Correlation Effects into Variational Models of Electrochemical Interfaces

We propose a way for obtaining a classical free energy density functional for electrolytes based on a first-principle many-body partition function. Via a one-loop expansion, we include coulombic correlations beyond the conventional mean-field approximation. To examine electrochemical interfaces, we integrate the electrolyte free energy functional into a hybrid quantum-classical model. This scheme self-consistently couples electronic, ionic, and solvent degrees of freedom and incorporates electrolyte correlation effects. The derived free energy functional causes a correlation-induced enhancement in interfacial counterion density and leads to an overall increase in capacitance. This effect is partially compensated by a reduction of the dielectric permittivity of interfacial water. At larger surface charge densities, ion crowding at the interface stifles these correlation effects. While scientifically intriguing already at planar interfaces, we anticipate these correlation effects to play an essential role for electrolytes in nanoconfinement.

Electrochemical Nitrate Reduction Catalyzed by Two-Dimensional Transition Metal Borides

We investigated 2D transition metal borides (MBenes) for the efficient conversion of nitrate to ammonia. MBenes have been previously shown to bind oxygen with extraordinary strength, which should translate toward selective adsorption of nitrate in aqueous media. Using Density Functional Theory, we screened MBenes by computing their nitrate and water adsorption energies, seeking materials with strong nitrate binding and weak water binding. We identified MnB, CrB, and VB as the best materials for selective nitrate adsorption and proceeded by computing their free energies for generating ammonia. Of the three candidates, CrB requires the lowest overpotential, making it the best candidate. To further decrease the overpotential, we doped the CrB MBene with secondary transition metals and found the addition of Mn to the active site further reduced the overpotential. We then computed the reaction mechanism grand canonically to observe the effect of applied potential on the free energy landscape.

Constant Potential Thermodynamic Integration for Obtaining the Free Energy Profile of Electrochemical Reaction

In this work, we advanced an efficient free energy sampling method based on constrained ab initio molecular dynamics (cAIMD) with a fully explicit solvent layer to depict the electrochemical reaction process at constant surface charge density, named the "Constant-Potential Thermodynamic Integration (CPTI)" method. For automatically adjusting surface charge density at different states, we built an "on-the-fly" procedure which is capable of managing all the necessary steps during cAIMD simulations, including the system pre-equilibrium, surface charge density updating, and force sampling. We applied it to predict the potential-dependent free energy profiles of CO2 adsorption on a single-atom catalyst. The results show that our method can not only account for changes in electrostatic potential energy associated with potential but also consider the potential-induced solvation effects. Our approach enables the accurate simulation of electrochemical environment by presenting the complete solid-liquid interface and efficient computation of electrocatalytic reaction energetics based on a robust potential descriptor.

Efficient conversion of propane in a microchannel reactor at ambient conditions

The oxidative dehydrogenation of propane, primarily sourced from shale gas, holds promise in meeting the surging global demand for propylene. However, this process necessitates high operating temperatures, which amplifies safety concerns in its application due to the use of mixed propane and oxygen. Moreover, these elevated temperatures may heighten the risk of overoxidation, leading to carbon dioxide formation. Here we introduce a microchannel reaction system designed for the oxidative dehydrogenation of propane within an aqueous environment, enabling highly selective and active propylene production at room temperature and ambient pressure with mitigated safety risks. A propylene selectivity of over 92% and production rate of 19.57 mmol mCu-2 h-1 are simultaneously achieved. This exceptional performance stems from the in situ creation of a highly active, oxygen-containing Cu catalytic surface for propane activation, and the enhanced propane transfer via an enlarged gas-liquid interfacial area and a reduced diffusion path by establishing a gas-liquid Taylor flow using a custom-made T-junction microdevice. This microchannel reaction system offers an appealing approach to accelerate gas-liquid-solid reactions limited by the solubility of gaseous reactant.

Revision of the oxygen reduction reaction on N-doped graphenes by grand-canonical DFT

Nitrogen-doped graphenes were among the first promising metal-free carbon-based catalysts for the oxygen reduction reaction (ORR). However, data on the most efficient catalytic centers and their catalytic mechanisms are still under debate. In this work, we study the associative mechanism of the ORR in an alkaline medium on graphene containing various types of nitrogen doping. The free energy profile of the reaction is constructed using grand-canonical DFT at a constant electrode potential in combination with an implicit electrolyte model. It is shown that the reaction mechanism differs from the generally accepted one and depends on the surface potential and doping type. In particular, as the potential decreases, coupled electron-proton transfer changes to sequential electron and proton transfer, and the potential at which this occurs depends on the doping type. It has been shown that oxygen chemisorption is the limiting step. The electrocatalytic mechanism of the nitrogen dopants involves reducing the oxygen chemisorption energy. Calculations predict that, at different potentials, different types of nitrogen impurities most effectively catalyze the ORR.

An implicit electrolyte model for plane wave density functional theory exhibiting nonlinear response and a nonlocal cavity definition

We have developed and implemented an implicit electrolyte model in the Vienna Ab initio Simulation Package (VASP) that includes nonlinear dielectric and ionic responses as well as a nonlocal definition of the cavities defining the spatial regions where these responses can occur. The implementation into the existing VASPsol code is numerically efficient and exhibits robust convergence, requiring computational effort only slightly higher than the original linear polarizable continuum model. The nonlinear + nonlocal model is able to reproduce the characteristic "double hump" shape observed experimentally for the differential capacitance of an electrified metal interface while preventing "leakage" of the electrolyte into regions of space too small to contain a single water molecule or solvated ion. The model also gives a reasonable prediction of molecular solvation free energies as well as the self-ionization free energy of water and the absolute electron chemical potential of the standard hydrogen electrode. All of this, combined with the additional ability to run constant potential density functional theory calculations, should enable the routine computation of activation barriers for electrocatalytic processes.

Fundamental Drivers of Electrochemical Barriers

We find that ion creation and destruction dominate the behavior of electrochemical reaction barriers, through grand-canonical electronic structure calculations of proton deposition on transition metal surfaces. We show that barriers respond to potential in a nonlinear manner and trace this to the continuous degree of electron transfer as an ion is created or destroyed. This explains both Marcus-like curvature and Hammond-like shifts. Across materials, we find the barrier energy to be driven primarily by the charge presented on the surface, which, in turn, is dictated by the native work function, a fundamentally different driving force than in nonelectrochemical systems.

Enhanced CO2 Reactive Capture and Conversion Using Aminothiolate Ligand-Metal Interface

Metallic catalyst modification by organic ligands is an emerging catalyst design in enhancing the activity and selectivity of electrocatalytic carbon dioxide (CO2) reactive capture and reduction to value-added fuels. However, a lack of fundamental science on how these ligand-metal interfaces interact with CO2 and key intermediates under working conditions has resulted in a trial-and-error approach for experimental designs. With the aid of density functional theory calculations, we provided a comprehensive mechanism study of CO2 reduction to multicarbon products over aminothiolate-coated copper (Cu) catalysts. Our results indicate that the CO2 reduction performance was closely related to the alkyl chain length, ligand coverage, ligand configuration, and Cu facet. The aminothiolate ligand-Cu interface significantly promoted initial CO2 activation and lowered the activation barrier of carbon-carbon coupling through the organic (nitrogen (N)) and inorganic (Cu) interfacial active sites. Experimentally, the selectivity and partial current density of the multicarbon products over aminothiolate-coated Cu increased by 1.5-fold and 2-fold, respectively, as compared to the pristine Cu at -1.16 VRHE, consistent with our theoretical findings. This work highlights the promising strategy of designing the ligand-metal interface for CO2 reactive capture and conversion to multicarbon products.

Controlled Electrochemical Barrier Calculations without Potential Control

The knowledge of electrochemical activation energies under applied potential conditions is a prerequisite for understanding catalytic activity at electrochemical interfaces. Here, we present a new set of methods that can compute electrochemical barriers with accuracy comparable to that of constant potential grand canonical approaches, without the explicit need for a potentiostat. Instead, we Legendre transform a set of constant charge, canonical reaction paths. Additional straightforward approximations offer the possibility to compute electrochemical barriers at a fraction of computational cost and complexity, and the analytical inclusion of geometric response highlights the importance of incorporating electronic as well as the geometric degrees of freedom when evaluating electrochemical barriers.

Electrically gated molecular thermal switch

Controlling heat flow is a key challenge for applications ranging from thermal management in electronics to energy systems, industrial processing, and thermal therapy. However, progress has generally been limited by slow response times and low tunability in thermal conductance. In this work, we demonstrate an electronically gated solid-state thermal switch using self-assembled molecular junctions to achieve excellent performance at room temperature. In this three-terminal device, heat flow is continuously and reversibly modulated by an electric field through carefully controlled chemical bonding and charge distributions within the molecular interface. The devices have ultrahigh switching speeds above 1 megahertz, have on/off ratios in thermal conductance greater than 1300%, and can be switched more than 1 million times. We anticipate that these advances will generate opportunities in molecular engineering for thermal management systems and thermal circuit design.

Electrocatalytic Mechanisms for an Oxygen Evolution Reaction at a Rhombohedral Boron Monosulfide Electrode/Alkaline Medium Interface

Rhombohedral boron monosulfide (r-BS) with a layer stacking structure is a promising electrocatalyst for an oxygen evolution reaction (OER) within an alkaline solution. We investigated the catalytic mechanisms at the r-BS electrode/alkaline medium interface for an OER using hybrid solvation theory based on the first-principles method combined with classical solution theory. In this study, we elucidate the activities of the OER at the outermost r-BS sheet with and without various surface defects. The Gibbs free energies along the OER path indicate that the boron vacancies at the first and second layers of the r-BS surface (VB1 and VB2) can promote the OER. However, we found that the VB1 is easily occupied by the oxygen atom during the OER, degrading its electrocatalytic performance. In contrast, VB2 is suitable for the active site of the OER due to its structure stability. Next, we applied a bias voltage with the OER potential to the r-BS electrode. The bias voltage incorporates the positive excess surface charge into pristine r-BS and VB2, which can be understood by the relationship between the OER potential and potentials of zero charge at the r-BS electrode. Because the OH- ions are the starting point of the OER, the positively charged surface is kinetically favorable for the electrocatalyst owing to the attractive interaction with the OH- ions. Finally, we qualitatively discuss the flat-band potential at a semiconductor/alkaline solution interface. It suggests that p-type carrier doping could promote the catalytic performance of r-BS. These results explain the previous measurement of the OER performance with the r-BS-based electrode and provide valuable insights into developing a semiconductor electrode/water interface.

Substantial impact of surface charges on electrochemical reaction kinetics on S vacancies of MoS2 using grand-canonical iteration method

The surface charges of catalysts have intricate influences on the thermodynamics and kinetics of electrochemical reactions. Herein, we develop a grand-canonical iteration method based on density functional theory calculations to explore the effect of surface charges on reaction kinetics beyond the traditional Butler-Volmer picture. Using the hydrogen evolution reaction on S vacancies of MoS2 as an example, we show how to track the change of surface charge in a reaction and to analyze its influence on the kinetics. Protons adsorb on S vacancies in a tough and charge-insensitive water splitting manner, which explains the observed large Tafel slope. Grand-canonical calculations report an unanticipated surface charge-induced change of the desorption pathway from the Heyrovsky route to a Volmer-Tafel route. During an electrochemical reaction, a net electron inflow into the catalyst may bring two effects, i.e., stabilization of the canonical energy and destabilization of the charge-dependent grand-canonical part. On the contrary, a net outflow of electrons from the catalyst can reverse the two effects. This surface charge effect has substantial impacts on the overpotential and the Tafel slope. We suggest that the surface charge effect is universal for all electrochemical reactions and significant for those involving interfacial proton transfers.

A Combined Reaction Path Search and Hybrid Solvation Method for the Systematic Exploration of Elementary Reactions at the Solid-Liquid Interface

We present a combined simulation method of single-component artificial force induced reaction (SC-AFIR) and effective screening medium combined with the reference interaction site model (ESM-RISM), termed SC-AFIR+ESM-RISM. SC-AFIR automatically and systematically explores the chemical reaction pathway, and ESM-RISM directly simulates the precise electronic structure at the solid-liquid interface. Hence, SC-AFIR+ESM-RISM enables us to explore reliable reaction pathways at the solid-liquid interface. We applied it to explore the dissociation pathway of an H2O molecule at the Cu(111)/water interface. The reaction path networks of the whole reaction and the minimum energy paths from H2O to H2 + O depend on the interfacial environment. The qualitative difference in the energy diagrams and the resulting change in the kinematically favored dissociation pathway upon changing the solvation environments are discussed. We believe that SC-AFIR+ESM-RISM will be a powerful tool to reveal the details of chemical reactions in surface catalysis and electrochemistry.

CO2 Reduction to Methane and Ethylene on a Single-Atom Catalyst: A Grand Canonical Quantum Mechanics Study

In recent years, two-dimensional metal-organic frameworks (2D MOF) have attracted great interest for their ease of synthesis and for their catalytic activities and semiconducting properties. The appeal of these materials is that they are layered and easily exfoliated to obtain a monolayer (or few layer) material with interesting optoelectronic properties. Moreover, they have great potential for CO2 reduction to obtain solar fuels with more than one carbon atom, such as ethylene and ethanol, in addition to methane and methanol. In this paper, we explore how a particular class of 2D MOF based on a phthalocyanine core provides the reactive center for the production of ethylene and ethanol. We examine the reaction mechanism using the new grand canonical potential kinetics (GCP-K) or grand canonical quantum mechanics (GC-QM) computational methodology, which obtains reaction rates at constant applied potential to compare directly with experimental results (rather than at constant electrons as in standard QM). We explain the reaction mechanism underlying the formation of methane and ethylene. Here, the key reaction step is direct coupling of CO into CHO, without the usual rate-determining CO-CO dimerization step observed on Cu metal surfaces. Indeed, the 2D MOF behaves like a single-atom catalyst.

Thermodynamic and kinetic modeling of electrocatalytic reactions using a first-principles approach

The computational modeling of electrochemical interfaces and their applications in electrocatalysis has attracted great attention in recent years. While tremendous progress has been made in this area, however, the accurate atomistic descriptions at the electrode/electrolyte interfaces remain a great challenge. The Computational Hydrogen Electrode (CHE) method and continuum modeling of the solvent and electrolyte interactions form the basis for most of these methodological developments. Several posterior corrections have been added to the CHE method to improve its accuracy and widen its applications. The most recently developed grand canonical potential approaches with the embedded diffuse layer models have shown considerable improvement in defining interfacial interactions at electrode/electrolyte interfaces over the state-of-the-art computational models for electrocatalysis. In this Review, we present an overview of these different computational models developed over the years to quantitatively probe the thermodynamics and kinetics of electrochemical reactions in the presence of an electrified catalyst surface under various electrochemical environments. We begin our discussion by giving a brief picture of the different continuum solvation approaches, implemented within the ab initio method to effectively model the solvent and electrolyte interactions. Next, we present the thermodynamic and kinetic modeling approaches to determine the activity and stability of the electrocatalysts. A few applications to these approaches are also discussed. We conclude by giving an outlook on the different machine learning models that have been integrated with the thermodynamic approaches to improve their efficiency and widen their applicability.

Stable and oxidative charged Ru enhance the acidic oxygen evolution reaction activity in two-dimensional ruthenium-iridium oxide

The oxygen evolution reactions in acid play an important role in multiple energy storage devices. The practical promising Ru-Ir based catalysts need both the stable high oxidation state of the Ru centers and the high stability of these Ru species. Here, we report stable and oxidative charged Ru in two-dimensional ruthenium-iridium oxide enhances the activity. The Ru0.5Ir0.5O2 catalyst shows high activity in acid with a low overpotential of 151 mV at 10 mA cm-2, a high turnover frequency of 6.84 s-1 at 1.44 V versus reversible hydrogen electrode and good stability (618.3 h operation). Ru0.5Ir0.5O2 catalysts can form more Ru active sites with high oxidation states at lower applied voltages after Ir incorporation, which is confirmed by the pulse voltage induced current method. Also, The X-ray absorption spectroscopy data shows that the Ru-O-Ir local structure in two-dimensional Ru0.5Ir0.5O2 solid solution improved the stability of these Ru centers.

Theoretical Study on B-doped FeN4 Catalyst for Potential-Dependent Oxygen Reduction Reaction

Electrochemical reactions mostly take place at a constant potential, but traditional DFT calculations operate at a neutral charge state. In order to really model experimental conditions, we developed a fixed-potential simulation framework via the iterated optimization and self-consistence of the required Fermi level. The B-doped graphene-based FeN4 sites for oxygen reduction reaction were chosen as the model to evaluate the accuracy of the fixed-potential simulation. The results demonstrate that *OH hydrogenation gets facile while O2 adsorption or hydrogenation becomes thermodynamically unfavorable due to the lower d-band center of Fe atoms in the constant potential state than the neutral charge state. The onset potential of ORR over B-doped FeN4 by performing potential-dependent simulations agree well with experimental findings. This work indicates that the fixed-potential simulation can provide a reasonable and accurate description on electrochemical reactions.

Potential-dependent transition of reaction mechanisms for oxygen evolution on layered double hydroxides

Oxygen evolution reaction (OER) is of crucial importance to sustainable energy and environmental engineering, and layered double hydroxides (LDHs) are among the most active catalysts for OER in alkaline conditions, but the reaction mechanism for OER on LDHs remains controversial. Distinctive types of reaction mechanisms have been proposed for the O-O coupling in OER, yet they compose a coupled reaction network with competing kinetics dependent on applied potentials. Herein, we combine grand-canonical methods and micro-kinetic modeling to unravel that the nature of dominant mechanism for OER on LDHs transitions among distinctive types as a function of applied potential, and this arises from the interplay among applied potential and competing kinetics in the coupled reaction network. The theory-predicted overpotentials, Tafel slopes, and findings are in agreement with the observations of experiments including isotope labelling. Thus, we establish a computational methodology to identify and elucidate the potential-dependent mechanisms for electrochemical reactions.

Unraveling the reaction mechanisms for furfural electroreduction on copper

Electrochemical routes for the valorization of biomass-derived feedstock molecules offer sustainable pathways to produce chemicals and fuels. However, the underlying reaction mechanisms for their electrochemical conversion remain elusive. In particular, the exact role of proton-electron coupled transfer and electrocatalytic hydrogenation in the reaction mechanisms for biomass electroreduction are disputed. In this work, we study the reaction mechanism underlying the electroreduction of furfural, an important biomass-derived platform chemical, combining grand-canonical (constant-potential) density functional theory-based microkinetic simulations and pH dependent experiments on Cu under acidic conditions. Our simulations indicate the second PCET step in the reaction pathway to be the rate- and selectivity-determining step for the production of the two main products of furfural electroreduction on Cu, i.e., furfuryl alcohol and 2-methyl furan, at moderate overpotentials. We further identify the source of Cu's ability to produce both products with comparable activity in their nearly equal activation energies. Furthermore, our microkinetic simulations suggest that surface hydrogenation steps play a minor role in determining the overall activity of furfural electroreduction compared to PCET steps due to the low steady-state hydrogen coverage predicted under reaction conditions, the high activation barriers for surface hydrogenation and the observed pH dependence of the reaction. As a theoretical guideline, low pH (<1.5) and moderate potential (ca. -0.5 V vs. SHE) conditions are suggested for selective 2-MF production.

Grand Canonical Ensemble Modeling of Electrochemical Interfaces Made Simple

Grand canonical ensemble (GCE) modeling of electrochemical interfaces, in which the electrochemical potential is converged to a preset constant, is essential for understanding electrochemistry and electrocatalysis at the electrodes. However, it requires developing efficient and robust algorithms to perform practical and effective GCE modeling with density functional theory (DFT) calculations. Herein, we developed an efficient and robust fully converged constant-potential (FCP) algorithm based on Newton's method and a polynomial fitting to calculate the necessary derivative for DFT calculations. We demonstrated with the constant-potential geometry optimization and Born-Oppenheimer molecular dynamics (BOMD) calculations that our FCP algorithm is resistant to the numerical instability that plagues other algorithms, and it delivers efficient convergence to the preset electrochemical potential and renders accurate forces for updating the nuclear positions of an electronically open system, outperforming other algorithms. The implementation of our FCP algorithm enables flexibility in using various computational codes and versatility in performing advanced tasks including the constant-potential enhanced-sampling BOMD simulations that we showcased with the modeling of the electrochemical hydrogenation of CO, and it is thus expected to find a wide spectrum of applications in the modeling of chemistry at electrochemical interfaces.

Electrochemical Reduction of CO2 via Single-Atom Catalysts Supported on α-In2Se3

In this work, the electrochemical CO₂ reduction reaction (CO₂RR) over transition metal and α-In₂Se₃ monolayer catalysts was investigated by density functional theory (DFT) and an effective screening medium method-reference interaction site model (ESM-RISM). On the basis of the scaling relationship between the adsorption free energies of intermediates, we constructed the relationships between oxygen-bound intermediates with *O and carbon-bound intermediates with *CHO. The calculation results indicate that *OCHO intermediates are more favorable for the first hydrogenation of CO₂ on M@In₂Se₃ catalysts; thus, the adsorption energy of oxygen-bound species determines the catalytic performance of M@In₂Se₃. The Co@In₂Se₃↓-C was predicted to be the most promising catalyst with a low limiting potential of -0.385 V as determined by the computational hydrogen electrode method. Constant potential calculations also demonstrate that the M@In₂Se₃ catalysts hold great potential for highly efficient CO₂RR. This work provides a fundamental understanding for the rational design of ferroelectric single-atom catalysts for the purpose of highly efficient electrocatalytic CO₂ reduction.

Approximating constant potential DFT with canonical DFT and electrostatic corrections

The complexity of electrochemical interfaces has led to the development of several approximate density functional theory (DFT)-based schemes to study reaction thermodynamics and kinetics as a function of electrode potential. While fixed electrode potential conditions can be simulated with grand canonical ensemble DFT (GCE-DFT), various electrostatic corrections on canonical, constant charge DFT are often applied instead. In this work, we present a systematic derivation and analysis of the different electrostatic corrections on canonical DFT to understand their physical validity, implicit assumptions, and scope of applicability. Our work highlights the need to carefully address the suitability of a given model for the problem under study, especially if physical or chemical insight in addition to reaction energetics is sought. In particular, we analytically show that the different corrections cannot differentiate between electrostatic interactions and covalent or charge-transfer interactions. By numerically testing different models for CO₂ adsorption on a single-atom catalyst as a function of the electrode potential, we further show that computed capacitances, dipole moments, and the obtained physical insight depend sensitively on the chosen approximation. These features limit the scope, generality, and physical insight of these corrective schemes despite their proven practicality for specific systems and energetics. Finally, we suggest guidelines for choosing different electrostatic corrections and propose the use of conceptual DFT to develop more general approximations for electrochemical interfaces and reactions using canonical DFT.

Fractional Charge Density Functional Theory and Its Application to the Electro-inductive Effect

We employed the chemical potential equalization principle to demonstrate that fractional electrons are involved in the electro-inductive effect as well as the vibrational Stark effect. By the chemical potential model, we were able to deduce that the frontier molecular orbitals of immobilized molecules can provide valuable insight into these effects. To further understand and quantify these findings, we introduced fractional charge density functional theory (FC-DFT), a canonical ensemble approach for open systems. This method allows for the calculation of electronic energies, nuclear gradients, and the Hessian matrix of fractional electronic systems. To correct the spurious delocalization error commonly found in approximate density functionals for small systems, we imposed the Perdew-Parr-Levy-Balduz (PPLB) condition through linear interpolation of two adjacent integer points (LI-FC-DFT). Although this approach is relatively simple in terms of molecular modeling, the results obtained through LI-FC-DFT calculations predict the same trend seen in experimental reactivity and the frequency change of immobilized molecules.

Design of a Four-Atom Cluster Embedded in Carbon Nitride for Electrocatalytic Generation of Multi-Carbon Products

The development of efficient and stable catalysts for the electrocatalytic CO₂ and CO reduction reactions (CORR) is under active investigation, but the problems of poor selectivity and low efficiency for C₂ products still exist. We design a two-dimensional carbon nitride material (C₅N₂H₂) that contains an eight N-atom structure capable of coordinating four-metal atom clusters and supporting simultaneously two carbon oxide molecules needed for the C₂ coupling. The designed material has excellent electrical conductivity and stability. After high-throughput screening of catalytic performance of multiple four-metal clusters embedded into the framework, we systematically investigate the CORR process of 11 candidates. We find that Cu₄-C₅N₂H₂ has superior selectivity and low limiting potential for generating ethylene, while Cu₂Zn₂-C₅N₂H₂ is selective and efficient to synthesize ethanol. Further, we discover a novel type of descriptor related to 2D material flexibility to evaluate the potential-determining step for generating ethylene. Our report both broadens the possibilities for few-atom CO reduction and demonstrates a novel substrate flexibility-related descriptor to predict the catalytic performance of materials.

Comparing Ab Initio Molecular Dynamics and a Semiclassical Grand Canonical Scheme for the Electric Double Layer of the Pt(111)/Water Interface

The theoretical modeling of metal/water interfaces centers on an appropriate configuration of the electric double layer (EDL) under grand canonical conditions. In principle, ab initio molecular dynamics (AIMD) simulations would be the appropriate choice for treating the competing water-water and water-metal interactions and explicitly considering the atomic and electronic degrees of freedom. However, this approach only allows simulations of relatively small canonical ensembles over a limited period (shorter than 100 ps). On the other hand, computationally efficient semiclassical approaches can treat the EDL model based on a grand canonical scheme by averaging the microscopic details. Thus, an improved description of the EDL can be obtained by combining AIMD simulations and semiclassical methods based on a grand canonical scheme. By taking the Pt(111)/water interface as an example, we compare these approaches in terms of the electric field, water configuration, and double-layer capacitance. Furthermore, we discuss how the combined merits of the approaches can contribute to advances in EDL theory.

Ab initio study of water dissociation on a charged Pd(111) surface

The interactions between molecules and electrode surfaces play a key role in electrochemical processes and are a subject of extensive research, both experimental and theoretical. In this paper, we address the water dissociation reaction on a Pd(111) electrode surface, modeled as a slab embedded in an external electric field. We aim at unraveling the relationship between surface charge and zero-point energy in aiding or hindering this reaction. We calculate the energy barriers with dispersion-corrected density-functional theory and an efficient parallel implementation of the nudged-elastic-band method. We show that the lowest dissociation barrier and consequently the highest reaction rate take place when the field reaches a strength where two different geometries of the water molecule in the reactant state are equally stable. The zero-point energy contributions to this reaction, on the other hand, remain nearly constant across a wide range of electric field strengths, despite significant changes in the reactant state. Interestingly, we show that the application of electric fields that induce a negative charge on the surface can make nuclear tunneling more significant for these reactions.

Adaptive Electric Fields Embedded Electrochemical Barrier Calculations

Electrochemical interfaces are grand canonical ensembles of varying electrons. Simulating them by standard first-principles methods is a challenging task, since the number of electrons (or charge) is fixed in the calculation. Under the constant charge framework, we developed a constant potential simulation method realized by adding an adaptive electric field to a charge neutral system. Electric field is the controlling variable. In addition, we defined an internal reversible hydrogen electrode potential (ϕ_IRHE), which can ensure the model independence of our method. To validate our method, the reaction energies of some electrochemical reactions are calculated, the results are comparable with the computational hydrogen electrode model and experiments. At last, the evolution of transition state structures and charge transfer coefficients of some electrochemical reactions on Ag(111) surface were discussed by our method.

Density-Potential Functional Theory of Electrochemical Double Layers: Calibration on the Ag(111)-KPF6 System and Parametric Analysis

The density-potential functional theory (DPFT) of electrochemical double layer (EDL) is upgraded by adopting (generalized) gradient approximations for kinetic, exchange, and correlation functionals of metal electrons. A new numerical scheme that is more stable and converges faster is proposed to solve the DPFT model. The DPFT model is calibrated with existing differential double-layer capacitance (C_dl) data of the EDL at Ag(111)-KPF₆ aqueous interface at five concentrations at room temperature. Metal electronic effects are essential to explain why the two peaks of the camel-shaped C_dl curves are almost symmetric in spite of the size difference of the hydrated cations and anions. A systematic parametric analysis is then conducted in terms of key EDL properties, including the potential of zero charge and the differential capacitance. The parametric analysis, on the one hand, elucidates how quantum mechanical behaviors of metal electrons as well as interactions between metal electrons and the electrolyte solution impact the EDL properties and, on the other hand, identifies key parameters of the DPFT model, which should be calibrated using first-principles calculations and/or advanced experiments in the future.

Prediction of Three-Metal Cluster Catalysts on Two-Dimensional W2N3 Support with Integrated Descriptors for Electrocatalytic Nitrogen Reduction

In the electrocatalytic nitrogen reduction reaction (NRR), nitrogen (N2) is chemically inert, it is difficult to break the triple bond, and the subsequent protonation step is very challenging. Suitable catalysts with high selectivity and high activity are needed to promote the electrocatalytic NRR. We screen a large number of clusters composed of three metal atoms embedded into a two-dimensional metal nitride, W2N3, with a N vacancy, and calculate the reaction energetics. The VNiCu cluster has the best catalytic activity among all the catalysts proposed so far. The Fe3 and Fe2Co clusters are excellent catalysts as well. In all cases, spin polarization is needed to observe the catalytic effect. We establish the optimal NRR path and confirm that it remains unchanged in the presence of a solvent. We find three groups of descriptors that can well predict the materials' properties and exhibit linear relationships with the NRR limiting potential.

Exploring the Potential Energy Surface of Pt6 Sub-Nano Clusters Deposited over Graphene

Catalytic systems based on sub-nanoclusters deposited over different supports are promising for very relevant chemical transformations such as many electrocatalytic processes as the ORR. These systems have been demonstrated to be very fluxional, as they are able to change shape and interconvert between each other either alone or in the presence of adsorbates. In addition, an accurate representation of their catalytic activity requires the consideration of ensemble effects and not a single structure alone. In this sense, a reliable theoretical methodology should assure an accurate and extensive exploration of the potential energy surface to include all the relevant structures and with correct relative energies. In this context, we applied DFT in conjunction with global optimization techniques to obtain and analyze the characteristics of the many local minima of Pt₆ sub-nanoclusters over a carbon-based support (graphene)-a system with electrocatalytic relevance. We also analyzed the magnetism and the charge transfer between the clusters and the support and paid special attention to the dependence of dispersion effects on the ensemble characteristics. We found that the ensembles computed with and without dispersion corrections are qualitatively similar, especially for the lowest-in-energy clusters, which we attribute to a (mainly) covalent binding to the surface. However, there are some significant variations in the relative stability of some clusters, which would significantly affect their population in the ensemble composition.

Electrochemistry from the atomic scale, in the electronically grand-canonical ensemble

The ability to simulate electrochemical reactions from first-principles has advanced significantly in recent years. Here, we discuss the atomistic interpretation of electrochemistry at three scales: from the electronic structure to elementary processes to constant-potential reactions. At each scale, we highlight the importance of the grand-canonical nature of the process and show that the grand-canonical energy is the natural thermodynamic state variable, which has the additional benefit of simplifying calculations. We show that atomic forces are the derivative of the grand-potential energy when the potential is fixed. We further examine the meaning of potential at the atomic scale and its link to the chemical potential and discuss the link between charge transfer and potential in several situations.