Machine Learning Force Fields in Electrochemistry: From Fundamentals to Applications

Abstract

This article reviews the foundations and applications of machine learning force fields (MLFFs) in electrochemistry, highlighting their role as a transformative tool in materials science. We first provide an overview of MLFFs, then discuss their applications in ionics and electrochemical reactions, and finally outline future directions. Most MLFF approaches use invariant or equivariant descriptors derived from body-order expansions to represent many-body atomic interactions. These descriptors feed into linear regression models, kernel methods, or neural networks to construct potential energy surfaces for gases, liquids, solids, and interfaces involving inorganic and organic materials. MLFFs have enabled a wide range of advances, including all-atom molecular dynamics (MD), data extraction from MD, and accelerated materials discovery. In MD simulations, MLFFs allow accurate evaluation of ionic conductivity via the fluctuation–dissipation theorem and nonequilibrium MD under electric fields, applied to both solid and polymer electrolytes. For electrochemical reactions, MLFFs and Δ-ML models have been used to predict redox potentials in homogeneous and interfacial systems through thermodynamic integration. MLFFs also enable the extraction of key thermodynamic and kinetic information–such as free energy landscapes and local transport coefficients–from atomic trajectories, facilitating coarse-grained modeling of mass transport and reactions in complex electrolytes. In materials discovery, MLFFs have allowed high-throughput screening of 10⁷ to 10⁸ crystal structures, leading to the identification of promising Li-ion and Na-ion solid electrolytes. MLFFs are expected to continue evolving as a core technology in computational materials science, spanning a wide range from high-precision calculations to large-scale materials exploration.

Introduction

The global energy consumption has been increasing at an annual rate of 1.7% from 1990 to 2020. , The majority of this demand is met by fossil fuels, raising concerns about global warming due to the large amounts of CO₂ emissions. To address this issue, the development of a sustainable energy system based on renewable energy sources is desirable. Renewable energy sources such as solar and wind power provide an almost inexhaustible supply of resources for human activities. However, they are inherently variable and geographically dispersed. To effectively utilize these energy sources, technologies for storing electricity generated from renewable energy and converting stored energy back into electricity are essential.

Secondary batteries, fuel cells, and electrolytic cells have attracted significant attention as promising systems for converting electrical and chemical energy. Extensive fundamental and applied research has been conducted to develop cost-effective and durable systems while improving energy conversion efficiency and power density. − Energy conversion in electrochemical systems is realized through ion-electron transfer reactions at the electrodes and ion transport within the electrolyte. The efficiency and power density of these systems are strongly influenced by the reaction kinetics at the electrodes and the ionic conductivity of the electrolyte. Additionally, the durability of the system depends on the mechanical, thermal, and chemical stability of the materials. Furthermore, system performance and durability are not only determined by the intrinsic properties of individual materials but also by the methods used to assemble these materials into cells. The properties of the materials themselves are significantly affected by the applied potential at the electrodes and the environmental conditions to which they are exposed. Lastly, it goes without saying that cost–largely determined by the choice of materials and manufacturing processes–is a critical factor in the widespread adoption of these systems. Given this background, the primary focus of research and development in electrochemical systems includes the discovery of materials that enable superior performance and durability, the optimization of battery and cell assembly, and the development of control strategies to maximize the performance of materials and systems. Crucially, addressing all these challenges requires a deep understanding of atomic- and molecular-scale structures of materials, as well as mechanisms of charge carrier transport and reactions.

Electrochemistry is a discipline that investigates the structures of electrolytes and electrode surfaces, as well as their transformations. Shortly after its inception, microscopic structural models of electrolytes and electrodes were proposed. With the subsequent advancement of classical mechanics, electromagnetism, statistical mechanics, and quantum mechanics, mathematical models describing ionic conduction and electrode reactions were developed. − Today, electrochemistry has become closely integrated with first-principles (FP) calculations, which explicitly treat atomic nuclei and electrons and solve wave equations to predict material properties. This integration has enabled quantitative predictions of the thermodynamics and kinetics of ionics and reactions at electronic and atomic scales, establishing FP methods as a fundamental tool for mechanistic studies and the discovery of advanced electrochemical materials. − Furthermore, multiscale computational methods − have been developed to bridge the gap between atomic-scale FP calculations and larger-scale simulations.

However, FP calculations and the earlier-mentioned physics-based bottom-up multiscale simulation methods built upon them have inherent limitations. Because of the high computational cost of FP calculations, the typical length- and time-scales that can be handled are restricted to a few nanometers and a few hundred picoseconds. As a result, accurately predicting long-time scale ion transport and reaction dynamics in heterogeneous structures beyond this scale remains a significant challenge. Multiscale simulation techniques have been developed to address these limitations. For example, one approach involves parametrizing the potential energy surface (PES) obtained from FP calculations to construct an interatomic interaction model, which is used in molecular dynamics (MD) simulations. This classical MD approach , can be regarded as a type of multiscale simulation. Another typical method is coarse-grained MD, in which groups of atoms are mapped onto a single particle, and the potential of mean force acting on these particles are modeled based on MD-derived data. ,− Additionally, to extend the length- and time-scales of simulations, local transport coefficients of molecules or ions, obtained from classical MD simulations of heterogeneous structures, can be incorporated into kinetic Monte Carlo (kMC) simulations. ,, Another approach involves constructing energy diagrams for elementary reactions on solid surfaces and determining reaction rate constants based on transition state theory (TST). These rate constants can be incorporated into mean-field models or kMC simulations to comprehensively analyze the complex reactions at heterogeneous surface sites. ,,,, However, applying these methods to complex materials presents significant challenges. For instance, accurately parametrizing the PES of multielement materials using physics-based interaction models is challenging. Moreover, constructing such interaction models often requires extensive trial and error by human experts. This issue becomes particularly severe for phenomena involving formations and scissions of chemical bonds, limiting the range of materials that classical MD can handle and introducing significant errors in the predicted material properties. Challenges also arise when formulating reaction rate constants for elementary processes. In complex reaction networks involving a vast number of elementary steps, essential reaction pathways can be overlooked in modeling. In other words, a fundamental challenge in multiscale simulations is that errors can arise when transferring information from microscopic scales to larger scales.

In recent years, artificial intelligence (AI) and its subfield, machine learning (ML), have gained attention as promising approaches to addressing these issues. The utilization of AI and ML is ubiquitous in everyday life, playing a crucial role in applications such as chatbots, recommendation systems, autonomous driving, voice assistants, image recognition, language translation, and medical diagnosis support. Their impact on natural science is also profound, as evidenced by the 2024 Nobel Prize in Physics being awarded for the foundational discoveries and inventions related to artificial neural networks, and the 2024 Nobel Prize in Chemistry recognizing AI-driven protein structure prediction and design. The application of AI and ML continues to expand in the fields of electrochemistry and energy storage, where they are being utilized for a wide range of tasks, including materials discovery, synthesis, manufacturing process optimization, analysis, diagnostics and degradation prediction, and autonomous experimentation. A few years ago, AI and ML were predicted to drive a paradigm shift toward data-driven science, emerging as the “fourth paradigm” alongside experimental, theoretical, and computational science. Today, we are witnessing this innovation dynamically progress. However, this shift does not imply that data-driven science operates independently from the existing paradigms. Rather, it builds upon the foundations of experimental, theoretical, and computational science, integrating AI and ML into these established domains. Numerous excellent review articles − have already been published on the applications of AI and ML. Therefore, this work will not repeat those discussions.

One of the objectives of this article is to highlight the nature of the aforementioned hybrid paradigm shift by providing an overview of recent advancements in machine learning force fields (MLFFs), a class of supervised ML regression models for interatomic potentials, within bottom-up simulations of electrochemical phenomena. MLFFs can accurately reproduce PESs derived from FP calculations while accelerating finite-temperature MD simulations by orders of magnitude through bypassing explicit FP computations. In addition, variant models of MLFFs can be extended to predict free energy landscapes and transfer coefficients for use in coarse-grained simulations. As in other fields of materials science, AI and ML technologies that predict target properties from material descriptors while treating the underlying relationships as a black box are widely employed in electrochemistry. Compared to such approaches, the construction of bottom-up methodologies requires a significant amount of time, and their applicability may initially appear limited. However, once a bottom-up approach is established, it offers a key advantage: it enables the presentation of extensive information spanning microscopic (atomic and electronic) to mesoscopic and macroscopic scales in a form that is interpretable by humans. The rapid progress of MLFFs in recent years has led to an increasingly comprehensive understanding of diverse electrochemical phenomena.

The second objective of this article is to provide a comprehensive overview of the various forms of MLFF and explain their essential concepts. The development of MLFF has progressed rapidly, resulting in a seemingly overwhelming number of models, creating a somewhat complex landscape. However, the major forms of MLFF follow the physical framework of body-order expansion, as discussed later, and are designed to compactly and accurately represent many-body interactions between atoms. In other words, the development of MLFF formulation itself has been guided by theoretical science.

The third objective of this article is to describe methods for predicting key properties of electrochemical systems, as introduced at the beginning, with the aid of MLFFs and relevant ML technologies. Figure provides an overview of these approaches. The key properties of interest in this article include the transport coefficients of ions and molecules in electrolytes, as well as the thermodynamic and kinetic properties of electrochemical reactions at electrolyte–electrode interfaces. Over the years, research based on theoretical and computational sciences has established bottom-up approaches for predicting these properties. However, because of fundamental limitations in time- and length-scales, applying these methods to complex materials and diverse chemical systems remains challenging. To overcome these limitations and expand the applicability of computational approaches, MLFFs have rapidly gained traction. MLFFs enable the prediction of transport coefficients and reaction free energies at finite temperatures via molecular dynamics simulations. − MLFFs also enable the analysis of complex reaction networks that emerge during MD simulations, facilitating the automatic extraction of key reaction pathways. Additionally, they are utilized for coarse-graining models to extend simulations to larger scales beyond conventional MD calculations. ,,,,,,,, The application of MLFFs requires training data for constructing ML models. In recent years, the development of extensive databases − has accelerated the adoption of MLFFs. As in other fields, several excellent studies have been published on the application of MLFFs in atomic-scale property prediction and multiscale simulations. However, there are still relatively few comprehensive reviews that specifically address key electrochemical properties through the integration of fundamental theoretical and computational sciences with MLFF-based methodologies. Therefore, this article focuses on electrochemical properties and discusses the current state and future prospects of MLFFs and related technologies in this domain.

1.

Schematic of AI- and ML-aided evaluations, predictions and explorations of electrolyte and electrode materials.

The structure of this article is as follows. First, we provide a brief overview of the Hamiltonian and density functional theory (DFT), which serve as the foundation for simulating electrochemical phenomena. While there are many excellent textbooks on these topics, ,,, we revisit them concisely in this section to clarify what MLFFs aim to represent. Additionally, we touch upon the grand canonical DFT, which is essential for describing electrochemical reactions using FP methods. Second, we introduce MLFFs. The field of MLFFs is rapidly evolving, with a vast number of methods continuously being proposed, making it challenging to assess their relationships and relative advantages. This section explains representative approaches, focusing on how energy and materials are represented, and discusses the connections between different methodologies. Third, we outline methods to predict the properties of ionics and electrochemical reactions and introduce practical applications. We describe approaches for predicting ionic conductivity and the thermodynamics and kinetics of reactions by integrating FP calculations, MD, and MLFFs. Furthermore, we provide an overview of statistical analysis methods using MLFFs to extract information from atomic-scale simulations, which is then used in mesoscale and macroscale simulations of heterogeneous systems. Additionally, we introduce recent studies where MLFFs facilitate large-scale materials exploration beyond conventional approaches. Finally, we summarize the current state of the field and future perspectives, concluding this article.

Hamiltonian and Density Functional Theory

Hamiltonian

In this article, we focus on ionics and electrochemical reactions, whose theoretical description relies fundamentally on quantum mechanics. The computational approach that provides predictions for material properties is FP calculations. The starting point for these calculations is the relativistic Dirac equation for Fermions. , However, in many modern FP calculations, relativistic effects are incorporated into effective core potentials, such as pseudopotentials − and the Projector Augmented Wave (PAW) method. , By solving the nonrelativistic Schrödinger equation under these effective external potentials, material properties that account for relativistic effects can be predicted within this framework. − Therefore, in this work, we begin our discussion with the Schrödinger equation as the starting point.

Nonrelativistic quantum mechanics describes the stationary state of a system consisting of N _n nuclei and their surrounding electrons by the following many-body wave equation ,

$$Ĥ\mathrm{\Phi }({\mathbf{r}}^{N_{\mathrm{e}}},{\mathbf{R}}^{N_n})=E_{\mathrm{total}}\mathrm{\Phi }({\mathbf{r}}^{N_{\mathrm{e}}},{\mathbf{R}}^{N_n})$$ 1

$$Ĥ={\mathrm{K̂}}_n+{Ĥ}_{\mathrm{e}}$$ 2

where Φ represents the many-body wave function of the nuclei and electrons, K̂ _n denotes the kinetic energy operator of the nuclei, and Ĥ _e is the electronic Hamiltonian. The position coordinates of the N _e electrons are given by r ^{N _e} , while R ^{N _n} represents the position coordinates of the N _n nuclei. Even for the lightest hydrogen nucleus, its mass is 1836 times that of an electron, and electrons move approximately 100 times faster than the nuclei. Under such conditions, the kinetic energy of the nuclei is significantly smaller than other terms, and the electrostatic interactions between nuclei can be approximated as constants. In the majority of FP calculations, the Born–Oppenheimer approximation, which is based on this assumption, is applied. As a result, the wave equations for the nuclei and electrons can be separated as

$${Ĥ}_{\mathrm{e}}{\mathrm{\Psi }}_n({\mathbf{r}}^{N_{\mathrm{e}}},{\mathbf{R}}^{N_n})=E_n({\mathbf{R}}^{N_n}){\mathrm{\Psi }}_n({\mathbf{r}}^{N_{\mathrm{e}}},{\mathbf{R}}^{N_n})$$ 3

$$[{\mathrm{K̂}}_n+E_m({\mathbf{R}}^{N_n})]{\chi }_m({\mathbf{R}}^{N_n})=E_{\mathrm{total}}{\chi }_m({\mathbf{R}}^{N_n})$$ 4

where Ψ _n represents the antisymmetric many electron wave function corresponding to the n-th eigenstate of the N _e-electron system, and E _n is its eigenenergy. The function χ _m denotes the nuclear wave function associated with the m-th eigenenergy E _m . Using these definitions, the many-body wave function of the nuclei and electrons can be expressed as Φ­(r ^{N _e} , R ^{N _n} ) = χ _m (R ^{N _n} ) Ψ _m (r ^{N _e} , R ^{N _n} ). Condequently, the system can be approximated as an ensemble of nuclei moving on the energy surface defined by the eigenenergy E _m , which is obtained by solving the many electron wave equation, eq , for electrons surrounding the nuclei fixed at a given set of positions R ^{N _n} . In most FP calculations, this energy surface is typically taken as the electronic ground-state energy E ₀, and a classical approximation is often applied to eq . Under this approximation, the motion of the nuclei is described by the following classical Hamiltonian

$$H=\sum _{I=1}^{N_n}\frac{{\left|{\mathbf{P}}_I\right|}^2}{2M_I}+E_0({\mathbf{R}}^{N_n})$$ 5

where P _I is the momentum of the I-th nucleus with a mass of M _I . Alternatively, eq can be solved by incorporating quantum effects in nuclear motion based on Feynman’s path integral method. − As described by the following Hamiltonian, the system can be regarded as a canonical ensemble of PN _n classical particles moving on an effective potential

$$H=\sum _{s=1}^P\sum _{I=1}^{N_n}\frac{{\left|{\mathbf{P}}_I^{\prime s}\right|}^2}{2M_I^{\prime }}+\sum _{s=1}^P\left[\sum _{I=1}^{N_n}{M}_I{\omega }_P^2{({\mathbf{R}}_I^{s+1}-{\mathbf{R}}_I^s)}^2+\frac{1}{P}{E}_0({\mathbf{R}}^{N_n})\right]$$ 6

where P _I represents the momentum of a particle with a fictitious mass M _I , and ω _P = P/βℏ, where β is the inverse temperature. Equation is exact in the limit P → ∞; however, in practical numerical calculations, P is typically set to a finite value based on the empirical criterion P ≥ 4 ℏβω_max. Solving eq corresponds to the first-principles molecular dynamics (FPMD) method, while solving eq corresponds to the first-principles path integral molecular dynamics (FP-PIMD) method.

Once the PES E ₀ is obtained, it becomes possible to formulate equations of motion and partition functions for describing nuclear motion according to eq or eq . By numerically integrating the equations of motion via MD simulations or analyzing the minima and saddle points of E ₀, one can derive information on ionic conductivity, reaction rates, and a wide range of other material properties based on statistical mechanics–either exactly or approximately. ,,,,,,− In any case, predicting nuclear motion requires the ground-state PES E ₀, which is obtained by solving eq . Since exactly solving the many electron wave equation demands an enormous computational cost, approximations are applied in most calculations. Among these, the most widely used approach is DFT. ,,

Density Functional Theory

Ground State

Solving eq is equivalent to minimizing the energy of the system with respect to an antisymmetric many electron wave function as

$$E_0=E[{\mathrm{\Psi }}_0]=\underset{\mathrm{\Psi }}{\min }E[\mathrm{\Psi }]=\underset{\mathrm{\Psi }}{\min }\frac{⟨\mathrm{\Psi }|Ĥ|\mathrm{\Psi }⟩}{⟨\mathrm{\Psi }|\mathrm{\Psi }⟩}$$ 7

where Ψ₀ is the ground-state many electron wave function that minimizes E. However, both solving eq and minimizing with respect to this many electron wave function require an enormous computational cost, scaling as a power of the number of electrons N _e. The foundation of DFT was established by the Hohenberg–Kohn theorem and Levy’s constrained search. , These works demonstrated that the energy E can be expressed as a functional of the electron density ρ derived from an N-representable and antisymmetric many electron wave function Ψ. By minimizing this functional with respect to the electron density ρ, instead of the wave function Ψ, one can determine the ground-state energy E ₀ and obtain all relevant information about the electronic state

$$E_0=E[{\rho }_0]=\underset{\rho }{\min }E[\rho ]$$ 8

where ρ₀ is the ground-state electron density.

To determine the ground state, a specific form of the functional is required. In DFT, it is a natural approach to describe the energy as a functional of the electron density. − However, this formulation did not accurately represent material properties. To address this, Kohn and Sham introduced a reference system of noninteracting N _e electrons and proposed representing the energy functional as

$$E[\rho ]=T_{\mathrm{s}}[\rho ]+E_{\mathrm{h}}[\rho ]+E_{\mathrm{xc}}[\rho ]+E_{\mathrm{nn}}$$ 9

$$T_{\mathrm{s}}[\rho ]=\sum _{i=1}^{N_{\mathrm{e}}}\int {\psi }_i(\mathbf{r})(-\frac{{\nabla }^2}{2}){\psi }_i(\mathbf{r})\mathrm{d}\mathbf{r}$$ 10

$$E_{\mathrm{h}}[\rho ]=\frac{1}{2}\int \int \frac{\rho (\mathbf{r})\rho ({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}\mathrm{d}\mathbf{r}\mathrm{d}{\mathbf{r}}^{\prime }$$ 11

$$E_{\mathrm{nn}}=\frac{1}{2}\sum _{I=1}^{N_n}\sum _{J=1}^{N_n}\frac{Z_I{Z}_J}{|{\mathbf{R}}_I-{\mathbf{R}}_J|}$$ 12

$$\rho (\mathbf{r})=\sum _{i=1}^{N_{\mathrm{e}}}{\left|{\psi }_i(\mathbf{r})\right|}^2$$ 13

where T _s is the kinetic energy of the reference system, E _h is the electrostatic interaction energy between electrons, E _xc is the exchange-correlation energy, and E _nn is the electrostatic interaction energy between nuclei. The function ψ _i represents the i-th one-electron wave function of the reference system, while Z _I and R _I denote the nuclear charge and position vector, respectively, of the I-th nucleus. For simplicity, spin and Brillouin zone quantum numbers are not explicitly represented in the notation. Additionally, ψ _i satisfies the following orthonormality condition

$$\int {\psi }_i(\mathbf{r}){\psi }_j(\mathbf{r})\mathrm{d}\mathbf{r}={\delta }_{\mathit{ij}}$$ 14

Equation can be interpreted as introducing a reference system in which the kinetic energy functional can be exactly defined as eq , and defining E _xc as the difference between this energy and the energy of the interacting system. The energy minimization is performed with respect to ψ _i instead of ρ. The variational principle yields the following equation for ψ _i

$$ĥ{\psi }_i(\mathbf{r})={ϵ}_i{\psi }_i(\mathbf{r})$$ 15

$$ĥ=-\frac{{\nabla }^2}{2}+v_{\mathrm{h}}(\mathbf{r})+v_{\mathrm{xc}}(\mathbf{r})+v_{\mathrm{ext}}(\mathbf{r})$$ 16

$$v_{\mathrm{h}}(\mathbf{r})=\int \frac{\rho ({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}\mathrm{d}{\mathbf{r}}^{\prime }$$ 17

$$v_{\mathrm{xc}}(\mathbf{r})=\frac{\delta E_{\mathrm{xc}}}{\delta \rho }$$ 18

where ϵ _i represents the one-electron eigenenergy, and v _ext(r) denotes the external potential arising from nuclei and other sources. The term v _xc is referred to as the exchange-correlation potential. The eigen equation for the one-electron wave function ψ _i , given by eq , is known as the Kohn–Sham equation. To obtain the ground-state energy and electronic structure, the Kohn–Sham equation is solved to determine ψ _i . Then, using eq , the electron density ρ is computed from the N _e lowest-energy orbitals. The one-electron Hamiltonian in eq is then updated, and this procedure is iterated until convergence is achieved. The Kohn–Sham method reduces the computational cost of solving eq , which originally scales factorially with N _e (N _e!), to a problem that scales as N _e . Initially, in the Kohn–Sham method, the exchange-correlation energy E _xc was modeled using the exchange-correlation energy of a uniform electron gas. − Despite its simplicity, this model proved to be surprisingly accurate for describing the electronic structure of metals, attracting significant attention to the method. Later, issues were identified in describing the electronic states of semiconductors and insulators , as well as dispersion interactions. , To address the issues, improvements were made by incorporating nonlocal electronic interactions into the exchange-correlation energy, leading to enhanced accuracy. −

Canonical Ensemble

In practical FP calculations, instead of the ground-state DFT described in the previous section, DFT for electrons in thermal equilibrium under a canonical ensemble with a fixed electron number, temperature, and volume is often used. Here, the canonical ensemble refers to an ensemble defined for a subsystem of electrons moving around fixed nuclei. This finite-temperature DFT was developed shortly after the establishment of the ground-state theory. ,, In finite-temperature DFT, the Helmholtz free energy A of N _e electrons at temperature T _e can be expressed as a functional of the electron density ρ. By minimizing this functional with respect to ρ, the thermal equilibrium state in the canonical ensemble can be determined

$$A_0=A[{\rho }_0]=\underset{\rho }{\min }A[\rho ]$$ 19

The following form, incorporating the entropy of noninteracting electrons, was proposed as a functional for A

$$A[\rho ]=E[\rho ]-T_{\mathrm{e}}S[\rho ]$$ 20

$$S[\rho ]=-k_{\mathrm{B}}\sum _{i=1}^{\infty }[f_i\ln f_i+(1-f_i)\ln (1-f_i)]$$ 21

$$\rho (\mathbf{r})=\sum _{i=1}^{\infty }{f}_i{\left|{\psi }_i(\mathbf{r})\right|}^2$$ 22

where T _e is the electronic temperature, k _B is the Boltzmann constant, and f _i represents the electron occupancy of the i-th orbital. Similar to the previous section, applying the variational principle with respect to ψ _i yields equations of the same form as eqs to (). The key difference lies in the following equation for f _i

$$f_i={[1+{\mathrm{e}}^{({ϵ}_i-{\mu }_{\mathrm{e}})/k_{\mathrm{B}}{T}_{\mathrm{e}}}]}^{-1}$$ 23

where μ_e is the electronic chemical potential (Fermi energy), determined such that the occupancy f _i satisfies the following condition

$$\sum _i^{\infty }{f}_i=N_{\mathrm{e}}$$ 24

For numerical convenience, instead of using the Fermi–Dirac distribution given in eqs and (), alternative forms of the electronic entropy S _e–where the occupancy is expressed using an integral representation of the Gaussian or Hermite function–have been employed. In most FPMD simulations, DFT under a canonical ensemble with a fixed electron number N _e is applied. The nuclear motion is simulated by replacing E ₀ in eqs and () with A ₀. , This method is used in the FPMD simulations of ionic conduction in electrolytes, introduced later.

Grand Canonical Ensemble

In electrochemical reactions, the reaction center is modeled as an open system that allows electron exchange with a reservoir. Consequently, the reaction proceeds under conditions where the electrode potential, which corresponds to the electronic chemical potential, is held constant rather than the electron number. Under such conditions, the equilibrium state of electrons around the nuclei within the subsystem at a given moment follows the grand canonical ensemble. The nuclei are considered to move on the grand potential energy surface in equilibrium. In other words, E ₀ in eq is expressed in terms of the grand potential.

The formulation of DFT within the grand canonical ensemble was also developed shortly after the inception of DFT. ,, In grand canonical ensemble DFT, the grand potential is expressed as a functional of the electron density, and the equilibrium state of the system is determined by minimizing this functional with respect to the electron density

$${\mathrm{\Omega }}_0=\mathrm{\Omega }[{\rho }_0]=\underset{\rho }{\min }\mathrm{\Omega }[\rho ]$$ 25

Similarly to the formulations for the ground state and the canonical ensemble, the functional Ω is defined as

$$\mathrm{\Omega }[\rho ]=A[\rho ]-{\mu }_{\mathrm{e}}{N}_{\mathrm{e}}[\rho ]$$ 26

The Kohn–Sham equation is obtained using the same variational principle as in the canonical ensemble. The key difference from the canonical ensemble is that the occupancy f _i is determined by eq using the fixed μ_e, without the constraint condition in eq .

As indicated by the above equations, in the grand canonical ensemble, the electron number N _e within the subsystem varies to maintain a fixed μ_e. However, adjusting only the electron number is insufficient for FP calculations. The FP method, originally developed in solid-state physics to describe the electronic states of crystals, employs periodic systems to represent solids, liquids, and interfaces. In such a periodic system, if the total charge within the unit cell is not neutral, the energy diverges. Consequently, if only the electron number in the subsystem changes, the subsystem becomes charged, making energy calculations impossible. To resolve this issue, an additional charge must be introduced into the unit cell to compensate for changes in the number of electrons. In real electrochemical reactions, this role is fulfilled by the ion distribution in the electrolyte, which forms the electrical double layer. Within this layer, ions arrange themselves to establish electrochemical potentials in equilibrium with the bulk electrolyte in contact with the subsystem. Therefore, the grand canonical ensemble must also be applied to ions. Consequently, applying grand canonical ensemble DFT to electrochemical reactions requires explicit modeling of the electrical double layer.

As discussed in the previous section, FP calculations have traditionally targeted systems with a fixed N _e under the canonical ensemble. Standard FP calculation programs have also been designed to solve the Kohn–Sham equation within the canonical ensemble. However, electrochemical reactions require a different theoretical and numerical framework based on the grand canonical ensemble. Before the year 2000, such numerical methods and programs did not exist, and it was not initially clear how to treat electrochemical reactions using FP methods. However, as the importance of electrochemical systems, such as fuel cells and rechargeable batteries, grew, there emerged a demand to clarify the microscopic mechanisms of reactions, leading to the development of various computational methods. ,,,− Although the developed methods vary widely, they all share a common foundation: the theory of the grand canonical ensemble.

Machine Learning Force Field

Overview of Machine Learning Force Field

DFT has reduced the computational cost required to calculate the ground-state energy surface E ₀ from a power-law dependence on the number of electrons N _e to a cubic scaling. However, the computational cost remains substantial, limiting the typical time- and length-scales that FPMD can handle to only a few nm and a few hundred ps. To overcome this limitation, MLFFs − have been developed. MLFFs enable the calculation of the ground-state PES E ₀(R ^{N _n} ) without solving the wave equation. Similarly, attempts to represent PES as a function have also been made in classical force fields. − ,,, The key difference between classical force fields and MLFFs lies in their functional forms. Classical force fields rely on functions derived from physical laws, such as Coulomb interactions and Lennard-Jones potentials, whereas MLFFs utilize functions developed in the field of ML, such as neural networks and kernel functions. Classical force fields, based on physical models, can represent reasonable energy surfaces with a small number of parameters. This also means that relatively little training data is required to determine these parameters. However, due to their simple functional forms, classical force fields struggle to accurately describe complex energy surfaces, particularly those involving bond rearrangements. Additionally, determining the functional form and parameters often requires significant human intervention. Since the optimal functional form varies depending on the phenomenon being modeled, developing a universal parametrization algorithm is challenging. On the other hand, MLFFs can accurately describe complex energy surfaces for a wide range of phenomena using a unified functional form, and they allow the generation of accurate interatomic interaction models–without modifying programs or algorithms–as long as sufficient training data is available. As shown in eqs and (), the energy surface is an essential physical property that governs not only ionics and electrochemical reactions but also various physical processes. Consequently, establishing a universal representation of energy surfaces has a profound impact. Furthermore, although FP calculations are computationally demanding, they can generate significantly more training data than experimental measurements. Given their significance and advantages, MLFFs were among the earliest applications of AI and ML in materials science and are now widely used across various areas of the field.

MLFFs can be classified into four representative approaches, which will be discussed in the next section: linear regression, kernel regression, neural networks proposed by Behler and Parrinello − and their variants, and message-passing neural networks (MPNN) − [or convolutional neural networks (CNN) , ]. Figure presents key MLFF performance metrics as a function of publication year, including the errors in energy and atomic forces (σ_E and σ_F), the number of elemental species (N _el) in the materials for which properties were evaluated using MLFFs, and the number of structures (N _st) used to generate the FP training data. Since MLFF performance strongly depends on the choice of parameters, training data, and target materials, quantitatively comparing these data is challenging. However, a time-series overview of the performance of MLFFs applied to practical property evaluations provides valuable insight into their development. The application of ML to energy surfaces began with the introduction of neural networks by Blank et al. However, these early models were limited to representing the energy surfaces of small molecules using low-dimensional variables. A major breakthrough in this field came with the development of the neural network potential (NNP) by Behler and Parrinello, which enabled the representation of energy surfaces for condensed-matter systems composed of multiple atoms. Since then, various MLFFs have been developed, incorporating methods such as kernel regression, linear regression, MPNN, ,, and CNN. Figure (a),(b) show that, even at the time of the first NNP, the prediction error was already an order of magnitude lower compared to conventional classical force fields. , Although the plotted data show some scatter due to dependencies on MLFF parameters, training data, materials, and evaluation conditions, there is a clear trend of improving best accuracy over time. Figure (c) further illustrates the gradual increase in the number of elemental components (N _el) in materials studied using MLFFs, indicating the growing capability of these models to handle complex multielement systems. Another noteworthy trend is shown in Figure (d), which tracks the transition in the number of structures required for training (N _st). Over time, a bifurcation has emerged, with MLFF development progressing in two directions: one aiming to reduce N _st and the other increasing it. The reduction in N _st has been driven by the introduction of relatively simple regression methods, such as linear regression ,− and kernel regression, − along with advances in active learning, an efficient training strategy. ,,,, These developments have demonstrated that, when the target properties are well-defined, MLFFs can achieve sufficient accuracy for property evaluations even with a limited number of training structures. With these methods, even when accounting for the time required to train MLFFs, they can accelerate the overall simulation by 2 to 3 orders of magnitude compared to FP calculations. On the other hand, the trend toward increasing N _st has been driven by advancements in computing power, the expansion of FP databases, − ,, and the introduction of MPNN and CNN, which facilitate training on large data sets of multielement systems. Recently, training data sets on the order of N _st = 10⁶ to 10⁸ have been utilized, accelerating efforts to develop general-purpose MLFFs that cover the entire periodic table. ,,,,, While these general-purpose MLFFs still require fine-tuning for high-precision calculations, they are already being employed for the discovery of advanced materials across extensive search spaces. ,−

2.

Trends over time in the energy error (a) and force error (b) of MLFFs used in condensed matter simulations, the number of constituent elements in the studied materials (c), and the amount of structural data used for training (d). Yellow diamonds, red triangles, blue circles and green squares indicate linear regressions (SNAP, , MTP and ACE , ), kernel regressions (GAP, VASP-MLFF and FLAIR), Behler–Parrinello type NNPs, − and MPNNs (NequIP, Allegro and MACE) and graph-CNNs (M3GNet and CHGNet). Dashed black and gray lines are values reported for EAM and ReaxFF , potentials. Data were collected from refs ,− ,,− .

Table lists several MLFF software packages. MLFFs are rapidly evolving, with their applications continuing to expand and diversify. Dozens to hundreds of new methods have been emerging every six months, and many packages are distributed through repository hosting platforms like GitHub. The aim of this review article is not to overview all of these emerging simulation software packages. However, for readers who are mainly interested in applications to electrochemistry rather than fundamental studies of MLFF methods, the current complex circumstance makes it difficult to judge which software packages are suited for applications. Therefore, in this table, we focus on software that, based on our experience and the information available to us, is relatively easy to use and regularly maintained. In the early stages of MLFF development, packages were primarily written in traditional programming languages, such as C++ and Fortran, and simulations were mainly executed on CPU architectures. However, with the emergence and widespread availability of modern ML libraries such as PyTorch and TensorFlow, many recent implementations have increasingly adopted these frameworks and are commonly optimized for execution on GPU architectures. Several methods are integrated into well-established simulation engines such as LAMMPS, VASP, , CASTEP, and CP2K. Several are also compatible with the Atomic Simulation Environment (ASE). For model training and fine-tuning, FP electronic structure software is required. VASP and CASTEP, which incorporate efficient on-the-fly active-learning algorithms, provide robust platforms for seamlessly conducting data collection, model generation, and production simulations. There are also commercially available platforms featuring user-friendly graphical user interfaces (GUIs), such as MedeA and Materials Studio.

1. Software Packages That May Be Suitable for Users Focusing on Applications in Electrochemistry.

software	regression method	descriptor	release year and version
RuNNer	Behler–Parrinello type NN	invariant symmetry functions	Since 2007, available upon request from the developers.
GAP	Kernel-based method	invariant power spectrum and bispectrum	Since 2010, available via the QUIP package and implemented in both LAMMPS and CASTEP.
SNAP	linear regression	invariant bispectrum	Since 2015, available in LAMMPS and MedeA; latest version released in 2025.
MTP	linear regression	invariant moment tensor descriptors	Since 2016, available on GitHub and integrated into LAMMPS; the latest version, MLIP-3, was released in 2023.
N2P2	Behler–Parrinello type NN	invariant symmetry functions	Since 2017, available in LAMMPS and MedeA.
SchNet	convolutional NN	invariant interatomic distances	Since 2017, compatible with ASE; latest version updated to 2.1.1 in 2024.
deep potential	deep NN	invariant local geometory in local frame	Since 2018, available as DeePMD-kit on GitHub; version 3.0 released in 2023.
VASP	Kernel-based method	invariant power spectrum	Since 2019, available in VASP (v6.3.0 to 6.5.1); MLFF-generated models also usable in LAMMPS.
ACE	linear regression	invariant body-ordered symmetric basis	Since 2019, available via ACE1.jl (Julia) and PACE (C++) on GitHub; integrated into LAMMPS, CP2K, and MedeA.
M3GNet	graph CNN	invariant two and three body descriptors	Since 2022, available on GitHub.
NequIP	MPNN	equivariant learned atomic representations	Since 2021, available on GitHub and in LAMMPS; latest version 0.7.1.
MACE	MPNN	equivariant body-ordered symmetric basis	Since 2022, available on GitHub; integrated into LAMMPS.

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In this way, MLFFs are evolving into a core technology in computational science, supporting both high-accuracy property calculations and high-throughput materials exploration. In the following sections, we will describe the representative MLFF frameworks.

Energy Functional

MLFF does not utilize functions based on physical laws but employs functions developed within the field of ML. However, similar to classical force fields, MLFF also emphasizes physical laws to achieve a compact representation of the PES. A fundamental concept underlying many MLFFs is the body-order expansion of energy, as illustrated in Figure . This concept is not exclusive to MLFFs; many classical force fields are also constructed based on it. In the body-order expansion, energy is decomposed into one-body, two-body, three-body, and higher-order interactions, as described below

$$E({\mathbf{R}}^{N_n})=E^{(0)}+\sum _{I_1}{E}^{(1)}({\mathbf{R}}_{I_1})+\sum _{I_1,I_2}{E}^{(2)}({\mathbf{R}}_{I_1},{\mathbf{R}}_{I_2})+\sum _{I_1,I_2,I_3}{E}^{(3)}({\mathbf{R}}_{I_1},{\mathbf{R}}_{I_2},{\mathbf{R}}_{I_3})+···$$ 27

For simplicity, we denote the ground-state energy as E in this equation. Following the definition by Glielmo et al., we also define the (ν + 1)-th interaction term, E ^(ν+1), as the term that vanishes upon the (ν + 2)-th differentiation with respect to atomic coordinates

$$\frac{{\partial }^{\nu +2}{E}^{(\nu +1)}({\mathbf{R}}_{I_1},···,{\mathbf{R}}_{I_{\nu +1}})}{\partial {\mathbf{R}}_{I_1}···\partial {\mathbf{R}}_{I_{\nu +2}}}=0$$ 28

Equation is also referred to as the cluster expansion. Mayer demonstrated that cluster expansion is a low-density perturbative series in which higher-order cluster integrals contribute progressively less as the order ν + 1 increases. Under appropriate conditions, these contributions decay rapidly with increasing ν. This behavior is consistent with the empirical observation that electronic screening diminishes the significance of long-range interactions, resulting in smaller contributions from higher-order terms. The concept of body-order expansion plays a central role in efficiently computing many-body interactions. The computational cost of evaluating (ν + 1)-body interactions scales as N _n , making it necessary to truncate the expansion at lower orders for practical simulations. The (ν + 1)-body term can, in principle, be defined recursively as the excess energy of the full (ν + 1)-body system beyond the contributions from all constituent ν-body interactions. Alternatively, and more commonly in the context of MLFFs, the body-order expansion is treated as a global functional form, where the (ν + 1)-body contribution is determined by fitting the total energy expression to a database containing systems of varying sizes and configurations.

3.

Body-order expansion of the PES and related descriptors.

Many MLFFs represent the energy E of a material as the sum of the local energies E _I associated with its constituent atoms I ,,,− ,,,−

$$E({\mathbf{R}}^{N_n})=\sum _I^{N_n}{E}_I({\mathbf{R}}^{N_n})=\sum _I^{N_n}f({\mathbf{x}}_I)$$ 29

E _I can also be expressed in the form of a body-order expansion, as in eq . The local energy E _I is determined by the intrinsic properties of the atom I, which depend on its elemental species, as well as the positions of the surrounding atoms. It is commonly represented as a function f of a descriptor vector x _I , which consists of N _D-dimensional variables that encode the local structural environment as a function of the atomic positions R ^{N _n} . The introduction of x _I in the representation of E _I ensures strict invariance of the energy under a permutation of identical atoms, translation, and rotation–fundamental symmetries that must be preserved. While R ^{N _n} itself is not invariant under these operations, invariant descriptors can be constructed from R ^{N _n} . By expressing E _I as a function of these descriptors, the required symmetry of the energy can be strictly enforced. For example, in the first MLFF for condensed systems, the Behler–Parrinello NNP, a cutoff radius R _cut is introduced. The descriptor x _I is then constructed as a function of the atomic positions within this cutoff radius, incorporating two-body and three-body terms. These descriptors are transformed via nonlinear activation functions, thereby enabling the representation of higher-order interactions. The generation of higher-order interaction terms from lower-order interaction terms through a nonlinear function can be understood from the following Taylor expansion

$$E_I(\mathbf{x})=E_I({\mathbf{x}}_0)+{(\mathbf{x}-{\mathbf{x}}_0)}^{\mathrm{T}}\nabla E_I({\mathbf{x}}_0)+\frac{1}{2}{(\mathbf{x}-{\mathbf{x}}_0)}^{\mathrm{T}}\{\nabla \otimes \nabla E_I({\mathbf{x}}_0)\}(\mathbf{x}-{\mathbf{x}}_0)+···$$ 30

Many MLFFs developed subsequently have adopted a similar framework, albeit with different choices of descriptors and functional forms. This approach emphasizes the physical property that atomic interactions are generally stronger at shorter distances. However, interactions also include long-range effects, such as dispersion interactions and Coulomb interactions. Equation allows for an approximate representation of long-range interactions by distributing their influence into short-range interactions. For instance, an interaction originating from the atom I can influence the position of the atom J if it lies within the cutoff radius R _cut of I. In turn, changes in the position of the atom J can affect the position of the atom K if K is within R _cut of J, and so on. However, this indirect representation does not physically capture long-range interactions correctly. This issue becomes particularly evident in the failure to capture phenomena such as the LO-TO splitting of optical modes in polar materials and the subtle correlations associated with ionic motion.

To better capture long-range interactions, an approach has been proposed in which the total energy is decomposed into short-range and long-range interactions. The short-range interactions are represented using eq , while the long-range interactions are treated explicitly in the form of electrostatic interactions or dispersion forces ,,−

$$E({\mathbf{R}}^{N_n})=E_{\mathrm{short}}({\mathbf{R}}^{N_n})+E_{\mathrm{long}}({\mathbf{R}}^{N_n})$$ 31

An accurate and flexible representation of the electrostatic long-range energy functional E _long across a wide range of materials remains a challenging problem and is an active area of research in the development of MLFFs. Methods have also been proposed to incorporate not only the local environment of the atom I but also the surrounding environments of other atoms in the representation of E _I −

$$E_I({\mathbf{R}}^{N_n})=f({\mathbf{x}}_I,···,{\mathbf{x}}_J)$$ 32

Alternatively, a different approach that uses a descriptor x ^{N _D} representing the entire molecular or crystalline structure to directly predict the total energy without decomposition has been proposed ,,,,−

$$E({\mathbf{R}}^{N_n})=f({\mathbf{x}}^{N_{\mathrm{D}}})$$ 33

In both eqs and (), neural network is predominantly used. It should be noted, however, that the neural network formulations do not explicitly capture the physics of long-range electrostatic interactions. While they enhance the representational capability of the energy landscape by incorporating longer-range information compared to MLFFs that rely solely on the short-range term E _short, they may still fail to fully capture the subtle correlations arising from electrostatic interactions.

In the following subsections, we explain three commonly used representations of E _I : linear regression, kernel regression, and neural networks.

Linear Regression

In linear regression models, the energy E _I is expressed as a linear combination of the descriptor vector x _I ,,

$$E_I({\mathbf{R}}^{N_n})=f({\mathbf{x}}_I)=\mathbf{c}·{\mathbf{x}}_I$$ 34

where the vector c consists of N _D regression coefficients, which are determined to reproduce the training data. In materials composed of multiple elements, c depends on the elemental species. However, for simplicity, this elemental dependence is not explicitly indicated here. The representational capability of a function in linear regression depends on the completeness of the descriptors used as basis functions. Assuming that the descriptor components are linearly independent, increasing the number of basis functions enhances the representational capability, thereby improving the accuracy of the resulting ML model. However, as the number of basis functions increases, the degrees of freedom of the model increase, requiring a larger amount of training data to achieve high accuracy while avoiding overfitting. The feasibility of achieving high accuracy with a limited amount of training data depends on whether the descriptors can compactly represent atomic interactions. Systematic formulations of descriptors for many-body interactions have been proposed, , and it is common to use descriptors that can represent interactions up to four- or five-body terms as basis functions.

A key advantage of linear regression is that the regression coefficients can be determined by solving simple linear equations. Additionally, as shown in Figure (a), Bayesian inference enables the prediction of not only the mean of the regression coefficients but also their variance. Moreover, it is possible to statistically assess whether a given descriptor x _I falls within the interpolation region of the training data. , This error estimation capability enables the construction of active learning algorithms that select the structures from which training data should be acquired. ,,,, Furthermore, it allows the generated ML model to determine whether its predictions lie within the interpolation region of the training data. ,

4.

Illustrations of (a) Gaussian process regression, (b) the weight-space view of the kernel regression, (c) the Behler–Parrinello neural network, and (d) the crystal graph convolutional neural network. The inset in (a) shows the predicted and actual force errors (adapted with permission from ref Copyright 2019 American Physical Society). The schematics in (c, d) have been reconstructed based on refs and for high-resolution presentation.

Kernel Regression

In kernel regression, E _I in (eq ) is expressed as a linear combination of kernel functions ,− ,,,

$$E_I({\mathbf{R}}^{N_n})=f({\mathbf{x}}_I)=\mathbf{w}·\mathbf{k}({\mathbf{x}}_I)$$ 35

where the vector k is an N _K-dimensional vector whose elements are positive definite kernel functions k(x _I , x _{I K} ). The kernel function k(x _I , x _{I K} ) can be interpreted as a measure of similarity between the local environment of the atom I, described by x _I , and the local reference environment represented by the reference point x _{I K} in the descriptor space. The reference points are often selected from the local environments of atoms in structures within the training data.

As shown in Figure (b), from the perspective of weight space, eq can be interpreted as defining N _K reference points in the descriptor space, placing kernel functions at each of these points, and adjusting their weights to represent the PES. By selecting reference points from the training data of the target system, the energy surface of that system can be efficiently represented.

Equation can also be interpreted from the perspective of function space. Consider expanding the function f in terms of H basis functions ϕ _h (x _I ), which are nonlinear with respect to the descriptor x _I

$$f({\mathbf{x}}_I)=\mathbf{c}·\mathit{\varphi }({\mathbf{x}}_I)$$ 36

where ϕ is a feature vector whose elements are ϕ _h (x _I ). The optimization problem of the coefficients c in ridge regression with Tikhonov regularization is equivalent to the problem of optimizing w in eq , with the kernel function defined as k (x, x′) = ϕ^T(x)­ϕ­(x′). This transformation is known as the dual representation. Compared to linear regression, kernel regression offers several advantages. In linear regression, eq assumes a linear dependence of E _I on x _I ; however, in reality, there is no guarantee that E _I can be accurately expressed as a linear function of x _I . Therefore, in linear regression, it is necessary to predefine a descriptor x _I that allows for an accurate expansion of E _I . In kernel regression, a nonlinear mapping is applied to transform the N _D-dimensional descriptor space x _I into an H-dimensional feature space ϕ, converting a nonlinear problem in x _I into a linear problem in ϕ. However, this transformation alone does not fundamentally resolve the issue, as it merely replaces the task of selecting an appropriate representation of x _I with the selection of ϕ. The kernel regression addresses this issue by defining kernel functions as inner products of feature vectors rather than explicitly constructing ϕ. Similar to the basis functions used in linear regression, the form of the kernel function must be predefined. However, even if the form of the kernel function is fixed, the kernel functions k(x _I , x _{I K} ) used to expand E _I vary depending on the N _K reference points. As a result, the feature vector implicitly handled in the kernel regression changes according to the reference points as ,

$${\varphi }_h(\mathbf{x})=\sum _{I_{\mathrm{K}}=1}^{N_{\mathrm{K}}}k(\mathbf{x},{\mathbf{x}}_{I_{\mathrm{K}}})U_{I_{\mathrm{K}}h}{\mathrm{\Lambda }}_h^{-1/2}$$ 37

where Λ _h and U _{I K h} correspond to the eigenvalues and components of the eigenvectors of the N _K × N _K matrix K, whose elements are given by the kernel function k(x _{I K} , x _{J K} ). In kernel regression, reference points can be selected based on the training data, allowing for flexible construction of the feature vector. This method represents functions by setting inner products based on the data, without explicitly handling the high-dimensional feature space. This is known as the kernel trick.

From the perspective of interactions, as shown in the Taylor expansion in eq , kernel regression can be interpreted as generating higher-order interaction terms from descriptors corresponding to lower-order interactions through the nonlinear mapping from the descriptor space to the feature space.

While kernel regression provides flexibility in handling nonlinear problems, the coefficients w can still be determined by solving a system of linear equations. Consequently, regression coefficients can be efficiently determined in kernel regression as well. Furthermore, as illustrated in Figure (a), the uncertainty of the prediction can be readily evaluated using Bayesian inference and other statistical methods.

Equation was originally proposed to represent short-range interactions using the descriptor x _I , which characterizes the local environment within a cutoff radius from the atom I. Subsequently, methods have been developed to separate short-range and long-range interactions, introduce appropriate descriptors for each, and represent the PES in the form of eq . , This approach enables a flexible representation of long-range interactions by employing descriptors that distinguish structural differences over extended distances. However, eq does not explicitly incorporate the physical laws governing electrostatic interactions. As noted by Faller and co-workers, a major challenge in developing any MLFF–including those based on kernel regression–is the incorporation of physics-based long-range multipole interactions.

Neural Network

Among the various representation methods for MLFFs, neural networks are currently the most widely used, and numerous variations continue to be proposed. As mentioned earlier, the first method proposed for condensed systems, which remains widely used today, is the neural network potential (NNP) developed by Behler and Parrinello. This approach also adopts the form of eq . For instance, when using a network structure with two hidden layers, as illustrated in Figure (c), E _I can be expressed as

$$E_I({\mathbf{R}}^{N_n})=f(G_{I,1},···,G_{I,N_{\mathrm{D}}})=\phi (b_1^3+\sum _{l=1}^{N_{\mathrm{ND}}^2}{a}_{l1}^{23}\phi (b_l^2+\sum _{k=1}^{N_{\mathrm{ND}}^1}{a}_{\mathit{kl}}^{12}\phi (b_k^1+\sum _{\mu =1}^{N_{\mathrm{D}}}{G}_{I,\mu }{a}_{\mu k}^{01})))$$ 38

where G _I,μ represents descriptors known as symmetry functions, φ is an activation function, the parameter N _ND denotes the number of nodes in the m-th hidden layer, and a _kl and b _l are learnable parameters optimized to reproduce the training data. The descriptor G _I,μ consists of two-body and three-body variables that describe the local environment within a cutoff radius R _cut around the atom I. These descriptors are invariant under atomic permutation, translation, and rotation operations. However, as mentioned earlier, this formulation does not explicitly account for long-range interactions beyond the cutoff radius R _cut. To explicitly incorporate long-range interactions, methods have been proposed to decompose the total energy into short-range and long-range components, as shown in eq . The short-range interactions are represented by eq , while the long-range interactions are described using Coulomb interactions between point charges or Gaussian charges. According to their treatment of interatomic interactions, Behler and co-workers classified NNPs into different generations. The first generation consists of early models developed for small systems by Blank et al. The second generation includes models that use eq to represent short-range interactions among many particles in condensed systems. The third and fourth generations explicitly incorporate Coulomb interactions. In third-generation NNPs, the point charge of the atom I is expressed as a function of local environment descriptors using a neural network of the form given in eq . However, since charge distributions inherently depend on the long-range structure of the system, this approach suffers from accuracy limitations. Fourth-generation NNPs address this issue by defining the long-range interaction energy E _long in eq as the electrostatic interaction energy E _elect between Gaussian charges centered at atomic positions, with each Gaussian charge having a total atomic charge Q _I

$$E_{\mathrm{elect}}=\sum _{I=1}^{N_n}\sum _{I<J}^{N_n}\frac{Q_I{Q}_J\mathrm{erf}(R_{\mathit{IJ}}/\sqrt{2({\sigma }_I^2+{\sigma }_J^2)})}{R_{\mathit{IJ}}}+\sum _I\frac{Q_I^2}{2{\sigma }_I\sqrt{\pi }}$$ 39

The charge Q _I is determined through a process known as charge equilibration, , in which the energy associated with the charges is minimized with respect to Q _I

$$E_{Q\mathrm{eq}}=E_{\mathrm{elect}}+\sum _{I=1}^{N_n}({\chi }_I{Q}_I+\frac{1}{2}{\gamma }_I{Q}_I^2)$$ 40

where χ _I and γ _I represent the electronegativity and hardness of the atom I, respectively, and γ _I is a constant specific to the elemental species. The electronegativity χ _I is expressed using a neural network, as in eq . The parameters of the neural network are optimized to reproduce the atomic charges obtained from FP calculations, such as Hirshfeld charges. Consequently, the charge distribution is determined in a manner that reflects the influence of the global structure while ensuring the equilibration of chemical potential. This charge determination mechanism is physically well-founded and has been shown to provide a more accurate representation of long-range interactions between organic molecules compared to third-generation NNPs. Charge equilibration was originally proposed to predict charge distributions in molecules and solids within the framework of classical force fields. A conceptually similar model has been employed to describe charge distributions at electrolyte–electrode interfaces, and the method has since been further developed , and integrated into MD simulation packages such as MetalWalls. Therefore, this approach can be a promising direction for representing long-range interactions in MLFFs based on physical laws. However, challenges remain in the representation of electrostatic interactions. FP calculations do not uniquely define point charges, and constructing an ML model for electronegativity requires training data based on arbitrarily assigned charges. Various charge definitions exist, ,− but all involve some degree of arbitrariness, and there is no guarantee that the computed charges accurately capture electrostatic interactions. In addition, long-range interactions in materials involve not only monopole but also higher-order multipole contributions. Therefore, it should be recognized that the Gaussian charge model alone may not fully or accurately capture these interactions. Moreover, it remains uncertain whether the simple functional form of eq can universally capture the partial screening behavior inherent to dielectric materials. In addressing this issue, a promising approach would be to incorporate a more physically realistic charge distribution, uniquely determined from FP, along with its interactions with MLFFs. Further research on such approaches is necessary.

In recent years, methods have been proposed to incorporate long-range interactions within the framework of neural networks, utilizing convolutional neural networks (CNNs) ,,,, and message-passing neural networks (MPNNs). − ,, For instance, in the Crystal Graph Convolutional Neural Network (CGCNN) proposed by Xie and Grossman, which was the first application of CNNs to condensed systems, the crystal structure is represented as graph data, where atoms serve as nodes and atomic bonds act as edges, as shown in Figure (d). This graph representation is used as the network input. The information of each node and edge is represented by feature vectors h _I and e _IJ , respectively. The feature vector h _I consists of discretized numerical values that encode elemental properties such as position in the periodic table (group and period), electronegativity, and covalent radius. The feature vector e _IJ consists of discretized values of bond distances. The feature vector of the atom I is updated iteratively in a process known as convolution, which occurs in layers referred to as convolutional layers. During this process, the information from neighboring atomic nodes and edges is incorporated into the updated feature vector. For example, in a simple convolutional operation, the update at the t-th iteration is performed as

$${\mathbf{h}}_I^{t+1}=\phi [(\sum _{J\in N(I)}{\mathbf{h}}_J^t\oplus {\mathbf{e}}_{\mathit{IJ}}){\mathbf{W}}_{\mathrm{c}}^t+{\mathbf{h}}_I^t{\mathbf{W}}_{\mathrm{s}}^t+{\mathbf{b}}^t]$$ 41

where φ is a nonlinear activation function, N(I) represents the set of atoms bonded to the atom I, and ⊕ denotes a concatenation operation. The parameters W _c , W _s , and b ^t characterize the convolution process. This formulation shows how the properties of neighboring atoms J and their bonding information contribute to updating the feature vector of the atom I. By iteratively applying this process, information from distant atoms is incorporated into the feature representation of the atom I, resulting in a feature vector that reflects the long-range structural environment. Although more complex functions are used in practice, the essential mechanism remains the same: neighboring atoms exchange information through convolutional layers. The generated feature vectors are then further transformed by pooling layers and subsequently used for the regression of the total energy E. Following CGCNN, various methods utilizing crystal graph representations and CNNs have been proposed and applied to the regression of E and E _I . ,,,

MPNNs incorporate information from neighboring atoms in a manner similar to CNNs. ,,,,, In MPNNs, a message function M _t is used to generate a message m _I from the states of the nodes representing the neighboring atoms of the atom I at the t-th layer, denoted as σ _{J 1} , σ _{J 2} , ···, σ _{J ν}

$${\mathbf{m}}_I^t=M_t({\mathit{\sigma }}_{J_1}^t,{\mathit{\sigma }}_{J_2}^t,···,{\mathit{\sigma }}_{J_{\nu }}^t)$$ 42

Next, the node state of the atom I, σ _I , is updated using the update function U _t , based on the message m _I and the current node state σ _I

$${\mathit{\sigma }}_I^{t+1}=U_t({\mathit{\sigma }}_I^t,{\mathbf{m}}_I^t)$$ 43

If M _t and U _t are set as the concatenation and activation function, respectively, the formulation takes a form similar to eq . Various forms of MPNNs have also been proposed and applied to the regression of E and E _I . − ,

In general, neural networks provide more flexible functional representations compared to linear regression and kernel regression, but they also tend to require larger amounts of training data. However, this characteristic strongly depends on the complexity of the network and the type of descriptors, which will be discussed in the next subsection. To evaluate the uncertainty of predictions made by MLFFs based on neural networks, the Query by Committee (QBC) method is often employed. ,− In this approach, multiple ML models are generated from the same training data by initializing the network parameters differently, and the variance among their predictions is used as a measure of uncertainty. Compared to Bayesian and statistical methods commonly used in linear regression and kernel regression, QBC lacks a rigorous mathematical foundation. While Bayesian approaches for uncertainty estimation also exist for neural networks, QBC is often preferred in practice due to its lower computational cost.

Descriptor

Descriptors should have the ability to systematically and compactly represent many-body interactions, as their selection determines both the amount of training data required for constructing an ML model and the achievable accuracy. Following the proposal of the Behler–Parrinello NNP, various research groups introduced invariant descriptors for two- to four-body interactions that are rotationally invariant. , More recently, efforts have been made to unify these descriptors into a systematic many-body descriptor. ,,, Furthermore, recent advancements have extended invariant descriptors to equivariant descriptors, which transform covariantly under rotations. ,,,,− This extension was initially motivated by the need to predict physical properties such as polarizability and electron density, which exhibit equivariance under rotational transformations. , Shortly thereafter, a neural network architecture composed of nodes described by equivariant descriptors and edges that transform variables preserving equivariance was proposed, known as the equivariant neural network (E3NN). , In this approach, the invariant components of the final network output are used for energy regression. It was later discovered that combining equivariant descriptors, E3NN, and MPNN improves the accuracy of PESs for condensed systems, even with limited training data. This led to MLFFs that incorporate many-body equivariant descriptors through the frameworks E3NN and MPNN. ,,, The following sections provide an explanation of invariant and equivariant descriptors.

Invariant Descriptor

To provide a bird’s-eye view of various descriptors, we first introduce a framework that unifies major descriptors and explain how each individual descriptor corresponds to specific components within this framework. The formulation of descriptors representing the local environment of the atom I begins with the definition of the following local elemental distribution function

$${\rho }_I(\mathbf{r})=\sum _{J\ne I}{f}_{\mathrm{cut}}(R_{\mathit{IJ}})g(\mathbf{r}-{\mathbf{R}}_J+{\mathbf{R}}_I)$$ 44

where R _IJ denotes the distance between atoms I and J, and f _cut(R) is a cutoff function that smoothly approaches zero at a radius R _cut. The function g is a Dirac delta function or a broadened Gaussian function. For simplicity, the elemental dependence is not explicitly indicated in eq , but the distribution function is constructed separately for each elemental species based on the methodology described below. This approach treats different elemental species as entirely distinct particles without any similarity. While this method accurately distinguishes different elemental species, it has the drawback that the number of descriptors scales exponentially with the number of elemental species, leading to a significant computational cost in multielement systems. To address this issue, several strategies have been proposed, including tensor reduction (or element embedding) methods, in which learnable weights are assigned to elemental species as well as radial and angular channels. These approaches enable the collective treatment of multiple elemental distribution functions and descriptors. In addition, compression techniques have been developed to extract the most relevant descriptors from elemental distributions through data-driven selection. ,,,,,,,−

From the two-body distribution function in eq , the following (ν + 1)-body correlation function is constructed

$${\rho }_I^{\otimes \nu }({\mathbf{r}}_1,···,{\mathbf{r}}_{\nu })={\rho }_I({\mathbf{r}}_1)···{\rho }_I({\mathbf{r}}_{\nu })$$ 45

where the function ρ _i represents the probability of finding ν + 1 atoms, including the atom I, in a specific relative configuration, where the first atom is located at a relative position r ₁, the second atom at r ₂, and so on. This function is invariant under an atomic permutation and a translation but is not invariant under a rotation. A rotationally invariant distribution function is obtained by averaging over the rotational operation

$${\mathrm{\rho ̅}}_I^{\otimes \nu }({\mathbf{r}}_1,···,{\mathbf{r}}_{\nu })={\int }_{\mathrm{SO}(3)}\mathrm{d}\mathrm{R̂}{\rho }_I^{\otimes \nu }(\mathrm{R̂}{\mathbf{r}}_1,···,\mathrm{R̂}{\mathbf{r}}_{\nu })$$ 46

The function ρ̅ _I is a descriptor that represents the (ν + 1)-body relative configuration while preserving the required invariance. In practical MLFF constructions, discretized variables of the relative coordinates (r ₁, ···, r _ν) are used as descriptors. Owing to its analytical convenience, relative coordinates are often expressed in spherical coordinates, and ρ _I (r) is expanded in terms of orthogonal basis functions for the radial and angular components, χ _nl and Y _lm , respectively

$${\rho }_I(\mathbf{r})=\sum _n\sum _l\sum _m{A}_{I,\mathit{nlm}}{\chi }_{\mathit{nl}}(|\mathbf{r}|)Y_{\mathit{lm}}(\mathbf{r̂})=\sum _v{A}_{I,v}{\varphi }_v(\mathbf{r})$$ 47

where Y _lm is the spherical harmonics, χ _nl is, for example, chosen as a polynomial or a spherical Bessel function, the index v represents the set of indices (nlm), and ϕ _v (r) is defined as ϕ _v (r) = χ _nl (|r|)Y _lm (r̂). The expansion coefficients A _I,nlm are used to construct descriptors. ,, The expansion coefficients are computed using the radial functions and spherical harmonics as

$$A_{I,v}=\sum _{J\ne I}{f}_{\mathrm{cut}}(|{\mathbf{R}}_{\mathit{IJ}}|){\eta }_v({\mathbf{R}}_{\mathit{IJ}})$$ 48

$${\eta }_v(\mathbf{r})=h_{\mathit{nl}}(|\mathbf{r}|)Y_{\mathit{lm}}(\mathrm{r̂})$$ 49

In the Smooth Overlap of Atomic Positions (SOAP), which employs a Gaussian function for the function g in eq , the radial function h _nl is given by the following integral of the radial basis function χ _nl over the radial direction

$$h_{\mathit{nl}}(r)=\frac{4\pi }{{(\sqrt{2{\sigma }_{\mathrm{atom}}^2\pi })}^3}{\int }_0^{\infty }{\chi }_{\mathit{nl}}(r^{\prime })\exp (-\frac{r^{\prime 2}+r^2}{2{\sigma }_{\mathrm{atom}}^2}){\iota }_l\left(\frac{r{r}^{\prime }}{{\sigma }_{\mathrm{atom}}^2}\right)r^{\prime 2}\mathrm{d}{r}^{\prime }$$ 50

In the Atomic Cluster Expansion (ACE), which employs a Dirac delta function for the function g, the radial function h _nl is identical to the basis function χ _nl . Hence, η _v = ϕ _v . Consequently, the difference in the expansion coefficients between SOAP and ACE arises solely from the choice of the function g.

Using eq , the components of the invariant descriptor are obtained through the following integral

$$B_{I,\mathbf{v}}=\int \mathrm{d}\mathrm{R̂}{\mathrm{\Pi }}_{t=1}^{\nu }[\int \mathrm{d}{\mathbf{r}}_t{\eta }_{v_t}({\mathbf{r}}_t){\rho }_I(\mathrm{R̂}{\mathbf{r}}_t)]$$ 51

where v is a notation that represents all possible combinations of (v ₁, ···, v _ν).

For example, the descriptor for ν = 1 is given as

$$B_{I,v_1}={\delta }_{l_{1}0}{\delta }_{m_{1}0}\frac{1}{2l_1+1}{A}_{I,v_1}$$ 52

The component A _{I, v 1} with v ₁ = (n ₁00) corresponds to the expansion coefficient that appears when the radial distribution function is expanded using the basis function χ _{n 10}. Hence, n ₁ serves as an index for the interatomic distance between a pair of atoms depicted in Figure . ,, The descriptor using the two-body symmetry functions proposed by Behler and Parrinello can be interpreted as a descriptor that represents g using a Dirac delta function and utilizes the two-body symmetry functions as the basis functions. Similarly, the two-body component of the FCHL descriptor proposed by Faber et al. can be interpreted as a two-body descriptor that discretizes the radial direction in real space.

The descriptor for ν = 2 is given as

$$B_{I,v_1{v}_2}=p_{I,n_1{n}_2{l}_1}={\delta }_{l_1{l}_2}{\delta }_{m_1{m}_2}\frac{1}{2l_1+1}\sum _m{A}_{I,n_1{l}_{1}m}{A}_{I,n_2{l}_{2}m}$$ 53

The coefficient p _{I, n 1 n 2 l 1} corresponds to the expansion coefficient of the angular distribution function, which represents the relative configuration of three-body. , It is expanded using the two radial basis functions χ _{n 1 l 1} and χ _{n 2 l 1} , along with the Legendre polynomial P _{l 1} of order l ₁, which serves as the angular basis function. The two indices, n ₁ and n ₂, specify the lengths of the two sides of the triangle formed by three atoms depicted in Figure , while the other index, l ₁, represents the angle between these two sides. ,, This descriptor is known as the power spectrum and is used as the descriptor in the SOAP kernel. Similar to the two-body case, the three-body symmetry functions of Behler and Parrinello and the three-body descriptors proposed by Faber et al. are equivalent to this descriptor.

The descriptor for ν = 3 is given as

$$B_{I,v_1{v}_2{v}_3}=\underset{l_1{l}_2{l}_3}{b_{I,n_1{n}_2{n}_3}}=\frac{1}{2l_3+1}\sum _{m_1,m_2,m_3}{C}_{l_1{m}_1{l}_2{m}_2}^{l_3{m}_3}{A}_{I,v_1}{A}_{I,v_2}{A}_{I,v_3}$$ 54

In the tetrahedral structure formed by four atoms shown in Figure , the three edge lengths are specified by n ₁, n ₂, and n ₃, while the three angles are specified by l ₁, l ₂, and l ₃. The coefficient $b_{\begin{array}{ll}I,n_1{n}_2{n}_3& l_1{l}_2{l}_3\end{array}}$ is known as the bispectrum and has been used as a descriptor in GAP. The four-body descriptor proposed by Faber et al. can be considered a variant of this descriptor.

As demonstrated by Pozdnyakov and co-workers, both the power spectrum p _{I,n 1 n 2 l 1} and the bispectrum b _{I,n 1 n 2 n 3 l 1 l 2 l 3} possess inherent limitations in completely capturing the local structural environment. To address this issue, descriptors for ν = 4 and higher have been formulated as follows ,,

$$B_{I,\mathbf{v}}=\sum _{{\mathbf{v}}^{\prime }}{C}_{\mathbf{vv}\prime }{\mathcal{A}}_{I,\mathbf{v}\prime }$$ 55

$${\mathcal{A}}_{I,\mathbf{v}}={\mathrm{\Pi }}_{\xi =1}^{\nu }{A}_{I,v_{\xi }}$$ 56

where the coefficient C _vv′ is a generalized Clebsch–Gordan coefficient that integrates the coefficients appearing in eqs to (). In the Atomic Cluster Expansion (ACE), the elements of C _vv′ are formulated using Wigner symbols and are employed as basis functions in linear regression. This formulation in ACE explicitly represents the energy in the form of a body-order expansion, as in eq . Spherical harmonics can be expressed in Cartesian coordinates, providing a representation in which descriptors are given as products of moment tensors. The Moment Tensor Potential (MTP), proposed by Shapeev, utilizes this form as basis functions for linear regression.

The above constitutes an overview of invariant descriptors. Four additional points regarding invariant descriptors are discussed below. The first point concerns the equivalence of descriptors. As mentioned earlier, lower-order invariant descriptors, such as two-body and three-body terms, have been proposed in various forms. , Higher-order interaction terms can be derived from these lower-order descriptors through nonlinear transformations, such as neural network activation functions or kernel functions, as shown in eq . Although the proposed descriptors may appear to have different mathematical forms, the information they encode is fundamentally equivalent. In fact, a study comparing the Behler–Parrinello NNP, which has a completely different formalism, with the power spectrum-based SOAP kernel demonstrated that when trained on the same data set, both methods yielded nearly identical results in predicting the density isobar of liquid water. The equivalence of four-body and higher-order invariant descriptors has not yet been fully established. However, with careful evaluation, it is expected that MLFFs utilizing equivalent descriptors should achieve comparable performance.

The second point concerns the completeness of descriptors. As demonstrated by Glielmo et al., nonlinear transformations of lower-order terms alone cannot fully generate interaction terms of all orders. This leads to an inherent limitation in the accuracy of models due to the incomplete representation of many-body interactions. To overcome this issue, descriptors of arbitrary order must be systematically incorporated in the form of the body-order expansion. This formalization has been provided by the ACE and the MTP. However, since the number of descriptors grows exponentially with ν, it is necessary to truncate the expansion at a lower order in practical computations. For this reason, two-body and three-body nonlinearized methods, such as NNP and SOAP, are still widely used. Similarly, in ACE, the expansion is typically truncated at four- or five-body interactions in actual calculations.

The third point concerns the role of eqs and () in constructing many-body descriptors. If (ν + 1)-body descriptors were computed individually, the computational cost would scale proportionally to N _n to the power of ν + 1, making direct computation impractical. The approach described above circumvents this issue by first computing the distribution function in eq through the enumeration of atoms surrounding the central atom I and then constructing higher-order terms by taking their product in eq . This procedure is known as the density trick. ,

The fourth point concerns tensor reduction. While the density trick mitigates the steep scaling of descriptor calculations with respect to the body order ν + 1, the number of many-body descriptors still grows rapidly with the number of element types, scaling as N _el . As briefly mentioned at the beginning of this subsection, this scaling has hindered the direct application of higher-order descriptors in MLFFs. The challenge has been addressed through tensor reduction, which embeds chemical element information into a fixed-size latent space using a learnable transformation ${\mathbb{R}}^{N_{\mathrm{el}}}\to {\mathbb{R}}^K$ . This strategy has enabled the practical application of higher-order terms to multielemental materials across various MLFF frameworks. ,,,,,

Equivariant Descriptor

Around 2018, a method was proposed in which descriptors equivariant to a three-dimensional rotational operation were used as inputs to E3NN, while the invariant components of the output layer were employed to represent the PES. ,

A descriptor is defined as equivariant under rotational operations if it satisfies the following properties

$${\mathbf{x}}^L(\mathrm{R̂}{\mathbf{R}}^{N_n})={\mathbf{D}}_L(\mathrm{R̂}){\mathbf{x}}^L({\mathbf{R}}^{N_n})$$ 57

where D _L (R̂) is a matrix representing a rotational operation, known as the Wigner D-matrix. This descriptor takes different forms depending on L: for L = 0, it is a scalar with a single degree of freedom that remains invariant under the rotation; for L = 1 and L = 2, they are a three-dimensional vector with three degrees of freedom and a second-order symmetric tensor with five degrees of freedom, respectively, both of which transform equivariantly under the rotation. The parameter L is referred to as the rank. One of the orthogonal basis functions of the SO(3) group, the spherical harmonics, satisfies eq . Consequently, A _{I, v 1} , which is expressed as a linear combination of spherical harmonics, is equivariant. However, the product A _{I, v 1} A _{I, v 2} is not equivariant. A three-body equivariant descriptor can be constructed by linearly combining products of spherical harmonics using Clebsch–Gordan coefficients as expansion coefficients

$$B_{I,v_1{v}_2}^{\mathit{LM}}=\sum _{l_1{m}_1,l_2{m}_2}{C}_{l_1{m}_1{l}_2{m}_2}^{\mathit{LM}}{A}_{I,v_1}{A}_{I,v_2}$$ 58

Here, the fact that eq is an equivariant tensor of rank L is proven using the following equation, which the Clebsch–Gordan coefficients satisfy under the rotational operation

$$\sum _{l_1{m}_1,l_2{m}_2}{C}_{l_1{m}_1{l}_2{m}_2}^{\mathit{LM}}{D}_{l_1}^{m_1{m}_1\prime }(\mathrm{R̂})D_{l_2}^{m_2{m}_2\prime }(\mathrm{R̂})=\sum _{m^{\prime }}{D}_l^{\mathit{mm}\prime }(\mathrm{R̂})C_{l_1{m}_1\prime l_2{m}_2\prime }^{\mathit{lm}\prime }$$ 59

where D _l (R̂) represents a component of the Wigner D-matrix. A four-body equivariant descriptor can be obtained similarly to eq by taking the previously constructed three-body equivariant descriptor B _{I, v 1 v 2} , multiplying it by A _{I, v 3} , and performing a linear combination using the Clebsch–Gordan coefficients. In this manner, an equivariant descriptor for an arbitrary (ν + 1)-body system can be constructed. These equations can be summarized in a similar form to eq as ,,,,

$$B_{I,\mathbf{v}}^{\mathit{LM}}=\sum _{{\mathbf{v}}^{\prime }}{C}_{\mathbf{vv}\prime }^{\mathit{LM}}{\mathcal{A}}_{I,\mathbf{v}\prime }$$ 60

In E3NN, the state of each node is represented using equivariant descriptors, and information is transferred to the next layer while preserving equivariance, as expressed in eq . , Since E3NN is well-suited to the MPNN framework, MPNNs in the E3NN formalism were soon adopted for energy surface regression. ,,,,

At the end of this section, we explain MACE ,,, as an example of a state-of-the-art MLFF that utilizes many-body equivariant descriptors as inputs to E3NN and MPNN. Similar to other MPNNs, MACE performs message-passing through a T-layer neural network, where each node corresponds to one of the N _n atoms. The state of a node at layer t is represented by the atomic number z _I , position R _I , and equivariant descriptor h _I, kLM for the atom I. Here, L represents the rank of the equivariant descriptor, as previously explained. In addition to L and M, the descriptor also has an additional index k, referred to as the channel index, which takes values from 1 to N _ch. The number of channels, N _ch, is typically set between 64 and 256. As described later, in MACE, discretized information about elemental species and their surrounding elemental distributions is mapped to N _ch variables using learnable weights through the tensor reduction. Thus, k can be regarded as an index that encodes both elemental species and structural features. Since the state of a node is described by the three indices k, L, and M, the total number of variables per node is given by N _ch × (2L+1). The input for the I-th node is computed as

$$h_{I,k00}^0=\sum _z{W}_{\mathit{kz}}{\delta }_{z{z}_I}$$ 61

The weight W _kz is a learnable parameter determined through training, and the summation over z is taken over all atomic numbers. This technique, commonly referred to as element embedding (or tensor reduction), assigns element-specific weights and compresses the information into channel k, enabling the handling of multielement systems without exponentially increasing the number of descriptors. Messages are generated from the equivariant descriptors h _I, kLM . The message construction process begins by linearly mixing the equivariant descriptors of each node with respect to the channel variables

$${\mathrm{h̅}}_{I,\mathit{kLM}}^t=\sum _{\mathrm{k̃}}{W}_{k\mathrm{k̃}L}^t{\mathrm{h̅}}_{I,\mathrm{k̃}\mathit{LM}}^t$$ 62

where the weight W _k k̃L is also a learnable parameter determined through training. Next, an atomic basis function corresponding to the previously defined A _I, v is constructed as

$$A_{I,{\mathit{kl}}_3{m}_3}^t=\sum _{\mathrm{k̃},{\eta }_1}{W}_{k\mathrm{k̃}{\eta }_1{l}_3}^t\sum _{J\ne I}{\varphi }_{\mathit{IJ}\mathrm{k̃}{\eta }_1{l}_3{m}_3}^t$$ 63

where the function ϕ _{IJkη1 l 3 m 3} corresponds to the previously defined ϕ _v and serves as a one-particle basis function, forming an equivariant tensor of rank l ₃, and the index η₁ represents one of the radial basis function indices, which will be discussed later. A key characteristic of MPNNs is that they incorporate information from the neighboring atom J into the one-particle basis function, as described below

$${\varphi }_{\mathit{IJk}{\eta }_1{l}_3{m}_3}^t=\sum _{l_1{m}_1,l_2{m}_2}{C}_{l_1{m}_1{l}_2{m}_2}^{l_3{m}_3}{R}_{k{\eta }_1{l}_1{l}_2{l}_3}^t(|{\mathbf{R}}_{\mathit{IJ}}|)Y_{l_1{m}_1}({\mathrm{R̂}}_{\mathit{IJ}}){\mathrm{h̅}}_{J,{\mathit{kl}}_2{m}_2}^t$$ 64

The function R _{kη1 l 1 l 2 l 3} (r) corresponds to the previously defined radial basis function χ _nl . However, in MACE, the radial basis function has a more flexible form, depending not only on the channel index k but also on the angular indices l ₁, l ₂ and l ₃. This function is represented by transforming spherical Bessel functions through a three-layer multilayer perceptron (MLP). Unlike eq , in eq , the information associated with k̃ and η₁, which correspond to elemental species, radial components, and angular components, is compressed into channel k. Another important point to note is that at t = 0, the l ₃ = 0 component of eq provides an atomic basis function equivalent to that of ACE. Similar to eq , the product of ν atomic basis functions is taken to construct the product basis

$${\mathcal{A}}_{I,\mathit{k}\mathrm{lm}}^{\nu ,t}={\mathrm{\Pi }}_{\xi =1}^{\nu }{A}_{I,{\mathit{kl}}_{\xi }{m}_{\xi }}^t$$ 65

Following the same procedure as in eq , an equivariant basis function of rank L is constructed as

$$B_{I,{\eta }_{\nu }\mathit{kLM}}^{\nu ,t}=\sum _{\mathit{lm}}{C}_{{\eta }_{\nu }\mathit{lm}}^{\mathit{LM}}{\mathcal{A}}_{I,\mathit{klm}}^{\nu ,t}$$ 66

where the summation over l and m denotes a summation over all possible combinations of l _ξ and m _ξ, and the subscript η_ν corresponds to the index v′ in eq and serves as a collective index for the nonzero basis functions. The message is constructed by taking a linear combination of the basis functions as

$$m_{I,\mathit{kLM}}^t=\sum _{\nu }\sum _{{\eta }_{\nu }}{W}_{z_i{\eta }_{\nu }\mathit{kL}}^{\nu ,t}{B}_{I,{\eta }_{\nu }\mathit{kLM}}^{\nu ,t}$$ 67

the message m _I, kLM also forms an equivariant tensor of rank L similar to the basis functions. The descriptor for the (t + 1)-th layer is obtained by linearly mixing the message with the descriptor h _I,kLM from the t-th layer as

$$h_{I,\mathit{kLM}}^{t+1}=\sum _{\mathrm{k̃}}{W}_{\mathit{kL}\mathrm{k̃}}^t{m}_{I,\mathrm{k̃}\mathit{LM}}^t+\sum _{\mathrm{k̃}}{W}_{k{z}_{I}L\mathrm{k̃}}^t{h}_{I,\mathrm{k̃}\mathit{LM}}^t$$ 68

The above process can be repeated T times to obtain the invariant descriptor h _I,kLM at the T-th layer. In MACE, T is effectively set to 2. As a result, the equivariant descriptors h _I,kLM and h _I,kLM are generated. The invariant components of these equivariant descriptors are used to compute the energy E _I of the atom I as

$$E_I=\sum _{t=1}^2{\mathcal{R}}^t({\mathbf{h}}_I^t)$$ 69

$${\mathcal{R}}^t({\mathbf{h}}_I^t)=\{\begin{array}{ll}\sum _k{W}_k^t{h}_{I,k00}^t& \mathrm{if}t=1\\ \mathcal{F}(\{h_{I,k00}^t\})& \mathrm{if}t=2\end{array}$$ 70

The first term is a linear combination of the invariant components of the equivariant descriptors output from the first layer. As mentioned above, this is equivalent to the ACE formulation. In other words, the first term reflects only the information within a radius of R _cut. In the second term, the invariant components of the equivariant descriptors from the second layer–obtained after a single message-passing step–are nonlinearly transformed using the MLP function $\mathcal{F}$ . As a result, the second term incorporates information within a radius of 2R _cut. In practical calculations, four-body equivariant descriptors (ν = 3) are often used.

Neural networks utilizing equivariant descriptors, such as MACE ,,, and NequIP, have been shown to achieve high accuracy with a comparable or smaller training data compared to conventional MLFFs that rely on invariant descriptors. As a result, they have become one of the mainstream approaches in MLFF. However, the underlying reason for this improved efficiency remains unclear. The original motivation for using equivariant descriptors was based on the idea that interatomic interactions can be expressed as a sum of terms involving the products of multipole moments of the central atom and its neighboring atoms. The MACE framework follows this concept, where the network maintains an equivariant structure internally, takes products of equivariant multipoles (ϕ _{IJkη1 l 3 m 3} ), and ultimately utilizes the invariant components of the resulting descriptors to represent energy. However, the same applies to ACE and MTP, which also form invariant descriptors by taking products of equivariant multipoles (ϕ _I,v ). One possible explanation is that message-passing enables the incorporation of longer-range interactions. Another possible reason for this accuracy is that, in ACE and MTP, invariant descriptors are constructed from equivariant descriptors (spherical harmonics) using a predefined functional form, as in eq . In contrast, MACE and NequIP adopt a more flexible framework, where invariant descriptors are formed through multiple layers including the MPNN with learnable parameters, optimizing the composition of tensor products during training. The ability to flexibly represent diverse many-body interactions may certainly contribute to accuracy. However, MACE and NequIP achieve high-accuracy regression even with small training data, despite their increased model complexity. This suggests that the invariant descriptors obtained in the final network layer provide a compact representation of many-body interactions. However, the underlying physical reason remains unclear.

Applications to Electrochemistry

Once the energy surface E(R ^{N _n} ) is computed, properties related to ionics and electrochemical reactions can be predicted using analytical statistical mechanics models or simulations based on MD and MC methods, following the Hamiltonian described in eqs and (). Furthermore, the atomic-scale properties and mechanisms obtained from these simulations can be utilized in the modeling of larger-scale inhomogeneous systems. The use of MLFFs is expanding in both microscopic simulations and in bridging the gap from the microscopic to mesoscopic and macroscopic scales. The accessible time and length scales of MLFF-aided microscopic simulations range from 10^–10 to 10^–6 seconds and from 10² to 10¹¹ atoms, depending on factors such as system complexity, target properties, required accuracy, and available computational resources. ,− ,,− , MLFFs also offer an accurate and flexible framework for coarse-graining methods. Additionally, the acceleration of simulations for various materials through MLFFs has enabled the exploration of a much broader range of materials than was previously possible. This chapter discusses the application of MLFFs to ionics and electrochemical reactions, as well as the associated computational methods.

Ionics

The evaluation of properties related to ionic conduction in electrolytes and active materials based on FP calculations has been conducted using two approaches: dynamic calculations and static calculations. As discussed later, ionic conductivity is rigorously formulated based on the fluctuation–dissipation theorem, , and in principle, it can be computed using MD simulations. However, FPMD is computationally expensive, making it difficult to accurately obtain ionic conductivity or diffusion coefficients at the operating temperature of electrolytes within feasible simulation times. To address this limitation, alternative methods such as extrapolating high-temperature results to estimate properties at operating temperatures have been employed. ,− Even with such extrapolations, applying FPMD to a large number of materials remains challenging. For this reason, static calculations have also been widely used. In this approach, the minimum energy migration path between quasi-minimum sites in an optimized electrolyte structure at absolute zero temperature is determined. , The diffusion coefficient is estimated from the calculated activation energies using dilute diffusion theory or kMC simulations. ,− However, static calculations require significant human effort and computational resources, and there is no guarantee that they fully capture the overall ionic conduction process. Recently, advances in MLFFs have helped overcome the limitations of MD simulations. This section introduces the MD-based approaches.

Based on the Green–Kubo equation, , ionic conductivity is formulated as

$$\begin{array}{l}\sigma =\frac{e^2}{3V{k}_{\mathrm{B}}T}{\int }_0^{\infty }\sum _I\sum _J{Q}_I{Q}_J⟨{\mathbf{v}}_I(t)·{\mathbf{v}}_J(0)⟩\mathrm{d}t\\ =\frac{e^2}{3V{k}_{\mathrm{B}}T}{\int }_0^{\infty }\sum _I{Q}_I^2⟨{\mathbf{v}}_I(t)·{\mathbf{v}}_I(0)⟩\mathrm{d}t+\frac{e^2}{3V{k}_{\mathrm{B}}T}{\int }_0^{\infty }\sum _I\sum _{J\ne I}{Q}_I{Q}_J⟨{\mathbf{v}}_I(t)·{\mathbf{v}}_J(0)⟩\mathrm{d}t\\ =\underset{t\to \infty }{\lim }\frac{e^2}{6V{k}_{\mathrm{B}}\mathit{Tt}}[\sum _I{Q}_I^2<{|{\mathbf{R}}_I(t)-{\mathbf{R}}_I(0)|}^2>+\sum _I\sum _{J\ne I}{Q}_I{Q}_J<[{\mathbf{R}}_I(t)-{\mathbf{R}}_I(0)]·[{\mathbf{R}}_J(t)-{\mathbf{R}}_J(0)]>]\end{array}$$ 71

where e is the elementary charge, and V represents the system volume. The notation ⟨.⟩ denotes an ensemble average. In principle, ionic conductivity can be obtained by performing equilibrium MD under an appropriate ensemble and averaging the values inside the brackets. However, in practice, this method often fails to achieve convergence within the feasible length- and time-scales of equilibrium MD simulations.

To calculate ionic conductivity in electrolytes, an alternative approach is often adopted. In this approach, only the first term is evaluated while the second term is ignored. ,− ,, The first term corresponds to the contribution from self-diffusions, and the equation that uses only this term is known as the Nernst–Einstein equation. , In experimental studies, the Nernst–Einstein equation has also been frequently employed to estimate diffusion coefficients from conductivities measured via impedance spectroscopy, particularly in cases where direct measurements of the diffusion coefficients of charge carriers are challenging. A typical example is proton transport in aqueous electrolytes. − Proton conduction in aqueous environments is a key ion transport phenomenon in physical chemistry and biochemistry and also plays a critical role in proton exchange membrane (PEM) fuel cells and water electrolysis, where Nafion is used as a representative PEM as shown in Figure (a). Nafion absorbs water and forms clusters consisting of water molecules and sulfonic acid groups, which serve as ion-exchange sites. These clusters, typically a few nanometers in size, are connected to form continuous channels that enable proton transport. The environment within these channels resembles that of aqueous solutions, and the proton transport mechanism in Nafion is considered similar to that in aqueous media. In aqueous solutions, protons chemically bind with water molecules to form hydronium ions (H₃O⁺) and hydrated complexes such as Zundel ions (H₅O₂ ), where hydronium ions form hydrogen bonds with surrounding water molecules. ,,, However, proton transport does not necessarily occur via the simple diffusion of these complex ions. As shown in Figure (a), protons can transfer from one water molecule to another via hydrogen bonding with H₃O⁺. Furthermore, once an H₃O⁺ ion is formed by accepting a proton, it can subsequently donate a proton to another water molecule through hydrogen bonding. Importantly, the donated proton in this second step is not necessarily the same proton that was initially accepted. This continuous proton transfers between water molecules constitutes the Grotthuss mechanism. ,, In contrast, when protons remain bound to H₃O⁺ or hydrated complexes and move as part of these species, the transport process is referred to as the vehicle mechanism. A key property of the Grotthuss mechanism is that long-range charge transport is achieved through successive proton transfers, even without significant displacements of hydrogen and oxygen nuclei. In reality, however, proton conduction does not occur via an idealized Grotthuss mechanism alone; rather, it results from a complex interplay of both the Grotthuss and vehicle mechanisms. Nevertheless, the fact that the ionic conductivity of protons in aqueous solutions is more than four times higher than that of water molecules or other monovalent cations highlights the significant contribution of the Grotthuss mechanism. The presence of the Grotthuss mechanism complicates the identification of the diffusion coefficient of charge carriers. This is because tracking the trajectories of hydrogen or oxygen nuclei alone does not yield the trajectories of the actual charge carriers. For these reasons, the Nernst–Einstein equation has been used to estimate the diffusion coefficient of charge carriers from experimentally measured conductivity. In aqueous solutions, NMR measurements of the time scale of proton transfers as a function of pH and temperature have been conducted. It has been confirmed that the diffusion coefficient estimated using the measured time scale and an assumed displacement by a proton transfer closely agrees with the diffusion coefficient estimated from conductivity via the Nernst–Einstein equation. However, in complex electrolyte membranes such as Nafion, performing such a precise validation is challenging. ,,

5.

Applications of MD simulations to ionics: (a) diffusion coefficients of protons and water in hydrated Nafion, calculated from mean square displacements (open dots), compared with experimental values (closed dots) (adapted with permission from ref Copyright 2023 American Chemical Society), (b) Li-ion conductivity in LGPS, calculated using the Nernst–Einstein equation (σ_dilute), the Green–Kubo equation (σ_EMD), and the color diffusion MD method (σ_NEMD) (reproduced from ref under CC-BY 4.0), and (c) Li-ion conduction under an electric field, simulated using Behler–Parrinello-type neural network models for the Born effective charge tensor and interatomic potential (reproduced from ref under CC-BY 4.0).

The presence of the Grotthuss mechanism also makes MD simulations of proton transport challenging. Classical MD has been widely used to study ion transport phenomena in aqueous solutions. However, describing the formation and dissociation of chemical bonds in the Grotthuss mechanism requires specialized models. For systems with a limited number of elemental species, such as protons in pristine water, the empirical valence bond (EVB) method has been established and has contributed to elucidating proton transport mechanisms. However, for multielement systems like Nafion, constructing such models is time-consuming. FPMD can accurately describe these phenomena. However, evaluating proton conduction within the heterogeneous structure of Nafion requires an expensive computational cost. ,

Recently, the kernel method described in the previous section was applied to proton transport in Nafion. In this study, a total of 2.4 ns of MD simulations were performed with a 0.5 fs time step on hydrated Nafion, modeled using unit cells containing 832 to 1264 atoms, employing VASP 6.3.0. Prior to the production runs, an MLFF was trained on-the-fly via MD simulations of smaller hydrated Nafion fragments. A total of 278 structures were sampled over 10 days using 32 cores of an Intel Xeon Platinum 8358 processor (2.6 GHz). With the trained MLFF, each MD step took 8.8 s on 32 cores–already a significant speedup over the 1032 s required for full FP calculations. The latest version, VASP 6.5.1, further reduced this to just 0.5 s per step on 16 cores, thanks to algorithmic improvements. Overall, the MLFF approach enabled a three-orders-of-magnitude acceleration in total simulation time compared to the FP method, including the training runs. Figure (a) presents the calculated and experimental diffusion coefficients of charge carriers and water molecules as a function of the water uptake λ (the number of water molecules per sulfonic acid group) in Nafion. The diffusion coefficient of charge carriers was determined by identifying H₃O⁺ at each MD step and tracing its origin from the H₃O⁺ in the preceding step, enabling the construction of charge carrier trajectories. The mean squared displacement (MSD) of these trajectories was computed to determine the diffusion coefficient. For water molecules, the diffusion coefficient was calculated from the MSD of oxygen atoms. For both water molecules and charge carriers, the calculated diffusion coefficients agree well with experimental values. Both simulations and experiments show that as λ increases, the diffusion coefficient of charge carriers eventually exceeds that of water molecules. In the experiments, the diffusion coefficient of water was measured using NMR, while the diffusion coefficient of protons was estimated by converting the conductivity using the Nernst–Einstein equation. Strictly speaking, it was unclear whether these two values could be directly compared. On the other hand, the diffusion coefficients obtained from simulations were derived consistently from MD trajectories. This result suggests that while the Grotthuss mechanism is not as dominant as in aqueous solutions, it still contributes to proton transport in hydrated Nafion.

The method of estimating conductivity using only the self-diffusion contributions in the first term of eq ensures good convergence in simulations. However, the second term can sometimes have a significant impact on ionic conduction. A clear example is when cations and anions move together without dissociating. In this case, the first term contributes additively through the self-diffusions of both cations and anions, yielding a finite conductivity. However, in reality, the first and second terms cancel each other out, resulting in zero ionic conductivity. Thus, when ions exhibit correlated motion with other ions, the Nernst–Einstein equation can lead to large errors. For this reason, nonequilibrium MD simulations are used to directly simulate the behavior of charge carriers under an external electric field. One such method is the color diffusion method. , This approach assigns a color charge C _I to each ion I. An external electric field ${\mathcal{E}}_1$ is applied, and the force acting on the atom I is computed as

$${\mathbf{F}}_{I,1}=e{C}_I{\mathcal{E}}_1$$ 72

The current density due to the color charge is calculated as

$${\mathbf{J}}_1=\frac{e}{V}\sum _I{C}_I{\mathbf{v}}_{I,1}$$ 73

The same calculation is performed using two additional linearly independent electric fields, ${\mathcal{E}}_2$ and ${\mathcal{E}}_3$ , to obtain the corresponding current densities, J ₂ and J ₃. From these calculations, the conductivity tensor σ is obtained as

$$\mathit{\sigma }=\mathbf{J}{\mathcal{E}}^{-1}$$ 74

where J is a matrix collecting the current density vectors. The color diffusion method provides a well-converged result within a feasible simulation time. Figure (b) shows the simulation results obtained using the color diffusion method for Li₁₀GeP₂S₁₂ (LGPS), a representative solid electrolyte for Li-ion batteries. In this simulation, as shown in the figure, color charges of +1, −3, and −4 were assigned to Li, PS₄ tetrahedra, and GeS₄ tetrahedra, respectively. This study also compared the conductivity calculated using the Nernst–Einstein equation and the Green–Kubo formula from equilibrium MD simulations. FPMD was used in this study. As shown in the figure, the conductivity obtained from the Nernst–Einstein equation (σ_dilute) is significantly lower than those obtained using the Green–Kubo formula (σ_EMD) and the color diffusion method (σ_NEMD). Additionally, while σ_EMD and σ_NEMD are in good agreement, σ_NEMD exhibits faster convergence.

However, the assignment of color charges involves inherent arbitrariness. If color charges are assigned unrealistically, the resulting ionic conduction mechanism may also become unrealistic. At the same time, as noted in the previous section, there is no completely arbitrary-free method to assign point charges to atoms. As a more rigorous approach to simulating the dynamics of atoms under an electric field within the framework of linear response theory, a method utilizing the Born effective charge tensor has been proposed. The Born effective charge tensor is defined as the second derivative of the total energy of the system with respect to the electric field and atomic positions:

$$Z_{I,\mathit{ij}}=\frac{V}{e}\frac{\partial P_i}{\partial u_{I,j}}=\frac{1}{e}\frac{\partial F_{I,j}}{\partial {\mathcal{E}}_i}=\frac{1}{e}\frac{{\partial }^{2}E}{\partial {\mathcal{E}}_i\partial u_{I,j}}$$ 75

where i and j represent one of the three Cartesian directions: x, y, or z. The term P _i denotes the polarization in the i-direction, u _I,j represents the displacement of the atom I in the j-direction, and F _I,j is the force acting on the atom I in the j-direction. The Born effective charge offers a well-defined framework within linear response theory to compute the force on the atom I under an electric field ${\mathcal{E}}_1$ as

$${\mathbf{F}}_{I,1}=e{\mathbf{Z}}_I{\mathcal{E}}_1$$ 76

where Z _I is a tensor whose elements are given by Z _I,ij . The current density J ₁ is obtained from Maxwell’s equations under a static electric field as

$${\mathbf{J}}_1=\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}=\frac{e}{V}\sum _I{\mathbf{Z}}_I{\mathbf{v}}_{I,1}$$ 77

By performing the same procedure for two additional linearly independent electric fields, ${\mathcal{E}}_2$ and ${\mathcal{E}}_3$ , the conductivity tensor can be obtained. Recently, an ML method has been proposed that predict the Born effective charge tensor using the invariant and equivariant descriptors described in previous section. ,, This approach enables NEMD simulations under an electric field with high accuracy and without arbitrariness. Figure (c) shows the result of the 300 ps NEMD simulation of Li₃PO₄, modeled with a unit cell containing 127 atoms, conducted by Shimizu et al. as a demonstration. In this simulation, an ML model was constructed to predict the polarizability using a Behler–Parrinello-type neural network. The Born effective charge tensor was computed by taking the partial derivative of the ML model with respect to atomic positions. The Behler–Parrinello NNP was also used to describe interatomic interactions in the absence of an applied electric field. Although conductivity was not explicitly computed in this study, the results clearly show the Li-ion conduction along the direction of the applied electric field. A comparison of computation times between (b) FPMD-based simulations and (c) NNP-based simulations shows that the NNP-based approach in (c) allows for longer simulation times.

Electrochemical Reactions

Key properties of electrochemical reactions are thermodynamics and kinetics of elementary reactions, which are represented by free energy landscapes along reaction coordinates of elementary processes. Thermodynamic properties of an elementary process can be determined from a free energy difference between a reactant and a product along the reaction coordinate. Similarly, a free energy difference between a reactant and a transition state allows for the calculation of the rate constant based on TST. However, as briefly mentioned previously, the treatment of electrochemical reactions requires the theory of the grand canonical ensemble for electrons and ions, and modeling the electric double layer is essential for describing electrochemical reactions within this theoretical framework. However, the motion of ions and solvent molecules is slow, making it challenging for FP calculations to handle. This issue has limited the application of FP methods to electrochemical reactions. Since the early 2000s, research on FP methods for electrochemistry has become increasingly active, leading to the proposal of various approaches. ,,,− , More recently, MLFFs have been utilized in FP-based free energy calculations that incorporate the statistical averaging of ion and solvent motion at finite temperatures, enabling accurate predictions of electrochemical properties. ,,,,− This section discusses electrochemical reactions in homogeneous solutions and at electrode–electrolyte interfaces, introducing both conventional FP methods and finite-temperature MD calculations using MLFFs. The discussion highlights how theoretical, computational, and data-driven sciences are increasingly integrating and advancing in the field of electrochemical reactions. Besides the homogeneous solution and electrolyte–electrode interfaces, redox reactions within bulk solid electrodes are also practically important electrochemical processes. A typical example is redox reactions in Li battery cathode materials, which represent one of major applications of FP calculations. However, the formulas used for bulk electrode reactions are included within those used for interfacial electrochemical reactions, which will be discussed below. Accordingly, they are not discussed in this article.

Electrochemical Reactions in Homogeneous Solutions

The most fundamental thermodynamic property of an electrochemical reaction is the redox potential. Once the redox potential is determined, it becomes possible to predict whether a reaction will proceed in the oxidative or reductive direction at a given electrode potential. Additionally, in a homogeneous solution, the free energy difference (reaction free energy) associated with the reaction at any specified potential can be calculated.

Sprik and co-workers developed a statistically rigorous method for determining the reaction free energy based on thermodynamic integration (TI) , within FPMD. ,,, To illustrate this approach, consider an n-electron transfer reaction, Ox+ne^–→ Red. As discussed in the previous section, Red and Ox species in the electrolyte can be regarded as subsystems in contact with an electron reservoir characterized by the chemical potential μ_e. Nuclei in each state move according to the Hamiltonian defined by eq or (), with the PES corresponding to the grand potential of the respective state. To compute the free energy difference between the two states, Sprik et al. introduced the following grand potential, which linearly combines the grand potentials of Ox and Red using a coupling parameter λ

$$\mathrm{\Omega }({\mathbf{R}}^{N_n},\lambda )=\lambda A_{\mathrm{Red}}({\mathbf{R}}^{N_n})+(1-\lambda )A_{\mathrm{Ox}}({\mathbf{R}}^{N_n})-{\mu }_{\mathrm{e}}{N}_{\mathrm{e}}(\lambda )$$ 78

where N _e (λ) represents the number of electrons in the subsystem at λ. The motion of the subsystem containing the redox species is assumed to be described by the Hamiltonian in eq or (), with the grand potential surface Ω acting as the PES. The free energy change associated with the electron transfer reaction is derived using TI as

$$\mathrm{\Delta }\mathcal{A}={\int }_0^1{⟨\frac{\partial H}{\partial \lambda }⟩}_{\lambda }\mathrm{d}\lambda =\mathrm{\Delta }{\mathcal{A}}_0-{\mu }_{\mathrm{e}}n$$ 79

$$\mathrm{\Delta }{\mathcal{A}}_0={\int }_0^1{⟨A_{\mathrm{Red}}({\mathbf{R}}^{N_n})-A_{\mathrm{Ox}}({\mathbf{R}}^{N_n})⟩}_{\lambda }\mathrm{d}\lambda$$ 80

where ⟨.⟩_λ represents the ensemble average at the coupling constant λ. Here, we replace the Gibbs free energy with the Helmholtz free energy, supposing that the changes in volume during the electron-transfer half reaction are negligible. By definition, n = N _e(1) – N _e(0). It should be noted that the Helmholtz free energy $\mathcal{A}$ discussed here differs from the Helmholtz free energy A for the electronic subsystem introduced in the previous section, which is ensemble-averaged only over electronic states. In contrast, $\mathcal{A}$ also includes the ensemble average over nuclear motion. To distinguish between the two, the notation $\mathcal{A}$ is introduced. At equilibrium, since $\mathrm{\Delta }\mathcal{A}=0$ , the electronic chemical potential μ_e that equilibrates the electron transfer reaction is given by $\mathrm{\Delta }{\mathcal{A}}_0/n$ . Defining the chemical potential of an electron in vacuum as zero, the redox potential U _redox relative to vacuum is obtained as

$$U_{\mathrm{redox}}=-\frac{\mathrm{\Delta }{\mathcal{A}}_0}{\mathit{en}}$$ 81

This is the Nernst equation.

Sprik et al. ,,, developed a method to predict U _redox using FPMD calculations, where the energy surfaces A _Ox(R ^{N _n} ) and A _Red(R ^{N _n} ) were obtained from canonical ensemble DFT, as described previously. In the FPMD calculations, a system was constructed by placing redox species in liquid water under a periodic boundary condition (PBC). A uniform background charge was introduced to neutralize the solute charge, representing a dilute-limit.

Here, we note the choice of a reference for electronic energy levels in FP calculations using PBC. In general, FP calculations for materials under PBC cannot set the electrostatic potential reference to the vacuum level. In FP calculation programs, the reference is set such that the average value of the local component of the effective nuclear potential in the computational system is zero. Ox is in a state where it has lost one electron compared to Red. The removed electron is assumed to reside in an electron reservoir, whose chemical potential is defined by this artificial reference. As a result, the energy difference obtained from FP calculations, A _Red(R ^{N _n} ) – A _Ox(R ^{N _n} ), is referenced to the energy level of this electron reservoir. This reference is specific to the computational system and depends on the local potential constructed in each program, making direct comparison with experimental values impossible. To obtain a value comparable to the experimental result, alignment is required. Figure (a) illustrates the conceptual framework of a recently proposed alignment method. In this approach, in addition to the calculations of the homogeneous solution, FP calculations are performed for a water slab model in a vacuum, as shown in Figure (a). These calculations enable the determination of the 1s energy levels of oxygen atoms in water molecules near the center of the slab, using the vacuum level within the computational system as the reference. Similarly, from the FP calculations of the homogeneous solution at a coupling constant λ, as illustrated in the same figure, the 1s energy levels of oxygen atoms in water molecules located far from the redox species are also obtained. As mentioned earlier, the 1s energy levels in the homogeneous solution are shifted from those referenced to the vacuum level by an electrostatic potential offset, ⟨Δϕ⟩_λ, which is specific to the system and the computational program. If this shift did not exist, the average 1s energy level in one system would match that of the other, as the water molecules in both systems would be in a similar chemical environment. Therefore, by aligning the energy levels so that the average 1s energy level of one system coincides with that of the other, as shown in Figure (a), the shift, ⟨Δϕ⟩_λ, can be determined as

$$e{⟨\mathrm{\Delta }\varphi ⟩}_{\lambda }={\mu }_{\mathrm{vac}}-{⟨{ϵ}_{1\mathrm{s},\mathrm{slab}}⟩}_{\lambda }+{⟨{ϵ}_{1\mathrm{s},\mathrm{buk}}⟩}_{\lambda }$$ 82

where μ_vac represents the vacuum level. Using ⟨Δϕ⟩_λ, the free energy difference referenced to vacuum, $\mathrm{\Delta }{\mathcal{A}}_0$ , is obtained as

$$\mathrm{\Delta }{\mathcal{A}}_0={\int }_0^1{⟨A_{\mathrm{Red}}({\mathbf{R}}^{N_n})-A_{\mathrm{Ox}}({\mathbf{R}}^{N_n})⟩}_{\lambda }\mathrm{d}\lambda -\mathit{ne}{\int }_0^1{⟨\mathrm{\Delta }\varphi ⟩}_{\lambda }\mathrm{d}\lambda$$ 83

6.

ML-aided FP calculations of redox potentials in homogeneous aqueous systems: (a) model systems and alignment method, (b) three-step procedure for computing redox potentials using thermodynamic integration and thermodynamic perturbation theory (reproduced from ref under CC-BY 4.0), (c) particle insertion and element substitution methods (adapted with permission from ref Copyright 2025 American Institute of Physics), and (d) computed redox potentials of seven redox species in aqueous systems using five exchange-correlation functionals (reproduced from ref under CC-BY 3.0).

There is another important point to note regarding the FP calculation of the redox potential. Due to its relatively low computational cost, the exchange-correlation functional based on the generalized gradient approximation (GGA) ,, is commonly used in FP calculations. However, for some redox species in water, GGA can introduce significant errors exceeding 0.5 V. ,,, This issue arises because GGA underestimates the band gap of water. For instance, GGA predicts the valence band maximum of water to be several eV too shallow. As a result, the redox level in the aqueous solution hybridizes with this incorrect band edge and becomes pinned to an erroneous energy level. To avoid this issue, a hybrid functional that provides a more accurate water band gap is necessary.

Sprik et al. calculated redox potentials using FPMD with Gaussian localized basis sets, following the principles of the aforementioned method, though the details of their alignment procedure differ from those described above. ,,, However, the FPMD-based approach requires substantial computational resources. In particular, when using a complete basis set such as plane-wave basis functions, the computational cost becomes extremely high. For this reason, many studies on redox potentials in homogeneous solutions do not employ FPMD. Instead, alternative semiempirical approaches, such as implicit solvation models, are predominantly used. − However, continuum solvation models involve empirical parameters and may not accurately capture solute–solvent interactions. To address this issue, hybrid models have been utilized, where only the first solvation shell around the solute is explicitly represented as atoms, while the remaining solvent is treated as a continuum. ,− However, the results obtained from such models depend on the choice of the explicit atom region and the continuum solvation region.

Recently, a method for accelerating a series of calculations using MLFF has been proposed. , A conceptual diagram is shown in Figure (b). The calculation consists of three stages. In the first stage, the energy surfaces A _Ox(R ^{N _n} ) and A _Red(R ^{N _n} ) in eq are represented using MLFFs, and TI is performed. Additionally, the MD simulations required for the alignment calculations, which handle the water slab model over several nanoseconds, are accelerated using an MLFF. By replacing these FP calculations with MLFF-based calculations, the overall computational cost can be significantly reduced, even when including the cost of MLFF training. However, while MLFFs can represent the energy surfaces obtained from FP calculations with higher accuracy than classical force fields, they may still introduce non-negligible errors in redox potential calculations. The second stage is designed to correct for this issue. This correction is performed through TI in a manner analogous to eqs to (), transitioning from the MLFF energy surface A ^ML to the FP energy surface A ^FP as

$$A({{\mathbf{R}}^{N_n}}^{},\eta )=\eta A^{\mathrm{FP}}({{\mathbf{R}}^{N_n}}^{})+(1-\eta )A^{\mathrm{ML}}({{\mathbf{R}}^{N_n}}^{})$$ 84

$$\mathrm{\Delta }{\mathcal{A}}^{\mathrm{FP}-\mathrm{ML}}={\int }_0^1{⟨\frac{\partial H}{\partial \eta }⟩}_{\eta }\mathrm{d}\eta$$ 85

This correction requires FPMD; however, since the majority of the free energy difference has already been computed through TI using MLFF, the integrand in eq becomes relatively small, allowing the TI to converge rapidly within a few picoseconds of simulation. Through this approach, the accurate redox potential from FP calculations can be obtained. However, applying this method to the hybrid functional requires substantial computational resources to generate the hundreds of training data points necessary for the generation of MLFFs. To address this issue, the first two stages of the calculation are performed using GGA, and in the third stage, the free energy difference between GGA and the hybrid functional is used to correct the GGA results. At this stage, the Δ-ML method is employed to represent the difference between the energy surfaces obtained from two different exchange-correlation functionals in FP calculations. A key feature of the Δ-ML method is that it requires an order of magnitude fewer training data points than conventional MLFF while achieving an order of magnitude smaller errors. This is because the difference between the energy surfaces of the two exchange-correlation functionals varies much more smoothly with atomic positions than the individual energy surfaces themselves. As a result, an ML model with significantly higher accuracy can be generated with much less training data compared to standard MLFF. The free energy difference between GGA and the hybrid functional can be obtained, for example, using thermodynamic perturbation thoery (TPT) calculations based on the second-order cumulant expansion as

$$\mathrm{\Delta }{\mathcal{A}}^{\mathrm{HYB}-\mathrm{GGA}}\simeq {⟨\mathrm{\Delta }{A}^{\mathrm{\Delta }\mathrm{ML}}⟩}_{\mathrm{GGA}}-\frac{1}{2k_{\mathrm{B}}T}{⟨{(\mathrm{\Delta }{A}^{\mathrm{\Delta }\mathrm{ML}}-{⟨\mathrm{\Delta }{A}^{\mathrm{\Delta }\mathrm{ML}}⟩}_{\mathrm{GGA}})}^2⟩}_{\mathrm{GGA}}$$ 86

where ΔA ^ΔML represents the difference between the energy surfaces of the hybrid functional and GGA, as predicted by the Δ-ML method, and HYB denotes the hybrid functional. This approach reduces the number of hybrid functional calculations required for generating the ML model and allows the redox potential of the hybrid functional to be obtained without additional corrections.

In the above method, MLFF and Δ-ML can be regarded as ML models that enable the necessary “electron insertion” for computing redox potentials in electron transfer reactions. A similar concept can also be applied to proton-coupled electron transfer (PCET) reactions, represented as Ox + mH⁺ + ne^– → Red. In principle, the same approach can be followed: TI is performed using MLFFs to represent the energy surfaces of the reactant and product, the difference between the MLFF energy surface and that obtained by the FP method is corrected via TI, and the free energy difference between GGA and the hybrid functional is obtained using the Δ-ML method. However, in the case of PCET, in addition to “electron insertion”, it is also necessary to insert a proton. This so-called “particle insertion” can, in principle, be carried out through TI from a noninteracting state, where the inserted particle does not interact with surrounding particles, to an interacting state, following the same formalism as eqs to (). Such particle insertion methods have been widely used in classical MD simulations. However, caution is needed when implementing this approach with MLFF. In the noninteracting limit, the inserted particle and surrounding particles may overlap. MLFF is often not trained on such unrealistic configurations, and learning energy surfaces where particles experience extremely strong repulsive forces can degrade its accuracy. As a result, performing particle insertion TI solely with MLFF may lead to significant errors. Recently, a TI method using MLFF was developed to overcome this issue. , In this method, a virtual intermediate state is introduced between the noninteracting and fully interacting states, with model interactions defined to prevent unphysical particle overlap. By performing TI in two stages–first from the noninteracting state to the model interaction, and then from the model interaction to the full MLFF-based interaction–a smooth TI pathway is achieved. In addition to this particle insertion method, a TI for “element substitution”, in which an inserted particle in solution is replaced with another element, has also been proposed. Element substitution is performed by TI from the MLFF of the initial element to that of the substituted element. This technique has also been widely used in classical MD simulations. − As shown in Figure (c), inserting an element X using the particle insertion method and subsequently replacing it with another element Y via the element substitution method achieves the insertion of the element Y. A previous study has used both the particle insertion method and a combination of particle insertion and element substitution to evaluate the solvation free energies of proton, alkali cations, and halide anions. The results obtained from both methods were confirmed to be statistically consistent within the margin of error. This consistency indicates that both approaches provide reliable TI pathways.

By combining ML models for the “electron insertion”, “particle insertion” and “element substitution”, it becomes possible to compute the redox potentials of a wide range of electrochemical reactions in homogeneous solution. , Figure (d) presents the computed redox potentials referenced to the vacuum level for seven electrochemical reactions in water, using five functionals: RPBE + D3, − PBE0, PBE0 + D3, − HSE06, , and B3LYP. In these calculations, the TI and TPT MD calculations of eqs , (), and () were performed on unit cells containing 64 water molecules, after confirming system size dependence using unit cells with 32, 64, and 96 molecules. The shift ⟨Δϕ⟩_λ in eq was calculated using a water slab with 128 water molecules per unit cell, following a size-dependence check with slab systems containing 96, 128, 192, and 1024 water molecules. For each system, a single MLFF was generated, and MD simulations using the MLFFs were performed for durations ranging from several hundred picoseconds to at most several tens of nanoseconds, to ensure sufficient statistical sampling (see details in refs , ). All calculations were performed using VASP 6.3.0 and 6.5.0. The MLFF for each redox species was generated in approximately 2 days using 16 cores of an Intel Xeon Platinum 8358 processor (2.6 GHz), with each MD step taking just 0.1 s. A highly accurate Δ-ML model was developed for each functional and redox species using only 40 structures. Overall, the ML-assisted approach reduced the computational cost for evaluating redox potentials from 20 million core hours to just 16,800 core hours. The results indicate that PBE0 + D3 accurately reproduces experimental results. This approach is also applicable to methods beyond DFT with even higher accuracy. It provides an FP methodology for obtaining highly accurate redox potentials in homogeneous solutions while determining the solvation structure around redox species at finite temperature via MD simulations without arbitrariness. An important question is whether corrections to the MLFF, which require FPMD calculations, are necessary. Previous studies suggest that these corrections are essential for accurate predictions and error verification when computing redox potentials in a system containing approximately 200 atoms per unit cell, if the MLFF error is on the order of 1 to 10 meV atom^–1 or 10 to 100 meV Å^–1. In contrast, with Δ-ML, which achieves an order of magnitude smaller error (0.1 to 1 meV atom^–1, 1 to 10 meV Å^–1), accurate results have been obtained without any corrections. , MLFFs that employ the previously described equivariant descriptors are approaching this level of accuracy for certain systems. ,,,, In the future, MLFF alone is expected to be sufficient for predicting redox potentials without the need for corrections.

Interfacial Electrochemical Reactions

Electrochemical reactions at electrolyte–electrode interfaces are essential phenomena in secondary batteries, fuel cells, and electrolytic cells. A key theoretical framework for describing interfacial electrochemical reactions is the grand canonical ensemble theory for electrons. As mentioned earlier, this theory allows the number of electrons in the subsystem to fluctuate while maintaining a constant electronic chemical potential. On the other hand, the subsystem, which is represented under PBC, must remain electrically neutral. Therefore, to explicitly account for changes in the number of electrons, some form of electric double-layer model is required to compensate for the charge variation. Several models have been proposed, including fully explicit models that represent the entire subsystem using nuclei and electrons, ,,, simplified charge distributions such as Gaussian charges or background charges, ,, continuum electrolyte models, ,,,,,,,, and molecular statistical mechanics models. In all these models, the atoms in the subsystem are assumed to move along the surface of the grand potential Ω.

There are several ways to represent Ω. ,,, To improve the clarity of approximations, the Helmholtz free energy A, which is related to Ω through (eq ), is expanded in a Taylor series around N _e,0 with respect to the electron number N _e ,,

$$A=A_0+\mathrm{\Delta }{N}_{\mathrm{e}}{\frac{\partial A}{\partial N_{\mathrm{e}}}|}_{N_{\mathrm{e}}=N_{\mathrm{e},0}}+\frac{1}{2}\mathrm{\Delta }{N}_{\mathrm{e}}^2{\frac{{\partial }^{2}A}{\partial N_{\mathrm{e}}^2}|}_{N_{\mathrm{e}}=N_{\mathrm{e},0}}+O(\mathrm{\Delta }{N}_{\mathrm{e}}^3)$$ 87

where A ₀ represents the Helmholtz free energy at N _e = N _e,0. According to Janak’s theorem, the first derivative of the Helmholtz free energy A with respect to the electron number N _e is equal to the electronic chemical potential μ_e, 0 at N _e = N _e,0

$${\frac{\partial A}{\partial N_{\mathrm{e}}}|}_{N_{\mathrm{e}}=N_{\mathrm{e},0}}={\mu }_{\mathrm{e},0}$$ 88

The second derivative of A with respect to N _e is defined as the inverse of C:

$${\frac{{\partial }^{2}A}{\partial N_{\mathrm{e}}^2}|}_{N_{\mathrm{e}}=N_{\mathrm{e},0}}=\frac{1}{C}$$ 89

Using these equations along with eq , Ω can be expressed as

$$\mathrm{\Omega }=A_0-{\mu }_{\mathrm{e}}{N}_{\mathrm{e},0}-\mathrm{\Delta }{N}_{\mathrm{e}}\mathrm{\Delta }{\mu }_{\mathrm{e}}+\frac{\mathrm{\Delta }{N}_{\mathrm{e}}^2}{2C}+O(\mathrm{\Delta }{N}_{\mathrm{e}}^3)$$ 90

where Δμ _e = μ_e – μ_e,0. Assuming C to be a constant independent of N _e and neglecting terms of O(ΔN _e ), the value of ΔN _e that minimizes Ω can be determined from the condition ∂Ω/∂N _e = 0 as

$$\mathrm{\Delta }{N}_{\mathrm{e}}\cong C\mathrm{\Delta }{\mu }_{\mathrm{e}}$$ 91

This equation implies that C corresponds to the capacitance. From eq , Ω can be approximated as

$$\mathrm{\Omega }\cong A_0-{\mu }_{\mathrm{e}}{N}_{\mathrm{e},0}-\frac{\mathrm{\Delta }{N}_{\mathrm{e}}^2}{2C}$$ 92

This approach is referred to as the capacitor model. Furthermore, if the terms of order O(ΔN _e ) are also neglected, Ω can be approximated as

$$\mathrm{\Omega }\cong A_0-{\mu }_{\mathrm{e}}{N}_{\mathrm{e},0}$$ 93

Here, this is referred to as the first-order approximation.

Equations equivalent to the first-order approximation have been widely used, for example, in calculations of redox potentials for bulk electrodes such as cathode materials in Li-ion batteries. , In bulk electrodes, which are conductors, electrons are delocalized, and it is natural to assume that the interior of the electrode remains electrically neutral. Consequently, setting ΔN _e = 0 leads to an equation equivalent to the first-order approximation, which is valid in such cases. However, near the electrolyte, the electrode can become charged. To compensate for this charge, ions in the electrolyte redistribute, forming an electric double layer. This polarized layer generates a potential gap at the interface, which, in turn, adjusts the electrode potential. Before the year 2000, it was not obvious how to incorporate this interfacial electrochemical phenomenon within the framework of FP calculations. However, as electrochemical systems gained increasing significance, the demand for FP calculations of interfacial electrochemical properties increased. In response to this, Nørskov and co-workers proposed this approximation for modeling electrode–electrolyte interfaces. As shown in eq , this approximation allows the determination of Ω at any electrode potential μ_e based solely on an FP calculation at the reference state with electron number N _e,0. As discussed later, this method has successfully explained a wide range of interfacial electrochemical properties and is now the most widely used approach for FP calculations of electrochemical reactions. However, this approximation does not account for how the electron number N _e changes when the potential deviates from the reference μ_e,0. Consequently, it cannot predict properties related to the number of transferred electrons. To address this limitation, methods utilizing electric double-layer models have been proposed. In these methods, FP calculations are performed at multiple potentials (or electron numbers) to determine the potential dependence of Ω and the charge transfer properties. − Applications of these methods indicated that the capacitance C can be well approximated as a constant. Based on this, the capacitor model was developed. In the capacitor model, FP calculation results for a small number of conditions with different electron numbers are extrapolated to predict the potential dependence of Ω. ,,, For further developments beyond the first-order approximation, refer to existing review articles. ,,,

Here, we introduce a method for calculating the redox potential of interfacial reactions using the first-order approximation. As an example of a reaction, we consider the reduction of adsorbed OH on a solid surface in an aqueous solution under standard conditions: OH* + H⁺ + e^– → H₂O + *, where * represents a surface site, and OH* denotes the adsorbed OH species. This reaction is not only the rate-limiting step of the oxygen reduction reaction (ORR) on Pt, a representative electrode catalyst in fuel cells, , but its reverse reaction also corresponds to the first elementary step of the oxygen evolution reaction (OER) in water electrolysis. , Furthermore, this reaction serves as a typical example of PCET, and the method described here can be applied to many other interfacial electrochemical reactions. The free energy change associated with this reaction is expressed as

$$\mathrm{\Delta }\mathcal{A}=\mathcal{A}[{\mathrm{H}}_2\mathrm{O}]+\mathcal{A}[{}^{*}]-\mathcal{A}[{\mathrm{OH}}^{*}]-\mathcal{A}[{\mathrm{H}}^{+}]-{\mu }_{\mathrm{e}}$$ 94

where $\mathcal{A}[.]$ represents the free energy, obtained by taking the ensemble average over the nuclei and electrons of the chemical species inside the brackets. Here, we consider the hydrogen redox reaction: H⁺ + e^–↔1/2 H₂. The electronic chemical potential μ_e at which this reaction is in equilibrium satisfies the following condition

$${\mu }_{\mathrm{e}}^{\mathrm{SHE}}=\frac{1}{2}\mathcal{A}[{\mathrm{H}}_2]-\mathcal{A}[{\mathrm{H}}^{+}]$$ 95

Using eq , eq can be rewritten as

$$\mathrm{\Delta }\mathcal{A}=\mathcal{A}[{\mathrm{H}}_2\mathrm{O}]+\mathcal{A}[{}^{*}]-\mathcal{A}[{\mathrm{OH}}^{*}]-\frac{1}{2}\mathcal{A}[{\mathrm{H}}_2]+\mathit{eU}$$ 96

where U represents the electrode potential relative to the standard hydrogen electrode (SHE), which is defined as

$$U=-\frac{{\mu }_{\mathrm{e}}-{\mu }_{\mathrm{e}}^{\mathrm{SHE}}}{e}$$ 97

This approach, in which electrode potentials and free energy changes are computed by referencing the computationally determined SHE, is known as the Computational Standard Hydrogen Electrode (CSHE).

In the first-order approximation, nuclei in the subsystem move along the surface of Ω given in eq . Therefore, the free energy change in eq is obtained as

$$\mathrm{\Delta }\mathcal{A}\cong {\mathcal{A}}_0[{\mathrm{H}}_2\mathrm{O}]+{\mathcal{A}}_0[{}^{*}]-{\mathcal{A}}_0[{\mathrm{OH}}^{*}]-\frac{1}{2}{\mathcal{A}}_0[{\mathrm{H}}_2]+\mathit{eU}$$ 98

where ${\mathcal{A}}_0$ is the free energy based on the Hamiltonian with Ω as the energy surface, as given in eq , and it does not depend on U. Therefore, once ${\mathcal{A}}_0$ is determined, the reaction free energy at any potential can be predicted as a linear function of U. Using this linear equation, the redox potential of the interfacial electrochemical reaction can be calculated from the condition $\mathrm{\Delta }\mathcal{A}=0$ as

$$U_{\mathrm{OH}}=\mathrm{\Delta }{\mathcal{A}}_0/e$$ 99

$$\mathrm{\Delta }{\mathcal{A}}_0={\mathcal{A}}_0[{\mathrm{OH}}^{*}]+\frac{1}{2}{\mathcal{A}}_0[{\mathrm{H}}_2]-{\mathcal{A}}_0[{}^{*}]-{\mathcal{A}}_0[{\mathrm{H}}_2\mathrm{O}]$$ 100

Nørskov and co-workers developed a method for predicting the activity of electrocatalysts by combining this approach with statistical mechanical models, such as the ideal gas model and the harmonic oscillator model, within an analytical framework. For example, in the OH* reduction reaction described above, the free energies of nonadsorbed molecules such as H₂O and H₂ were calculated using the ideal gas model, based on molecular structures optimized at absolute zero and vibrational frequencies through FP calculations. Similarly, the free energies of the interfacial species * and OH* before and after the reaction were computed via the harmonic oscillator model. In this approach, the interfacial electrolyte was represented by an ice-like model, as shown in Figure (a). This methodology has been applied to a wide range of electrochemical reactions, including the hydrogen evolution reaction (HER), ORR, OER, CO₂ reduction reaction (CO₂RR), nitrogen reduction reaction (N₂RR), and lithium deposition reactions. These studies have shown that a broad range of electrocatalytic reactions follow the Sabatier principle, with their reaction rates represented by volcano plots, as shown in Figure (b–f), where catalytic activities are correlated with the stabilities of reaction intermediates. Owing to its ability to efficiently evaluate both the thermodynamics and kinetics of electrochemical reactions, this method has also been widely used for exploring advanced catalytic materials. − ,,

7.

An example of the ice-like model on a close-packed (111) surface (a) and volcano plots predicted by FP calculations for the hydrogen evolution reaction (HER) (adapted with permission from ref Copyright 2005 Electrochemical Society) (b), oxygen reduction reaction (ORR) (adapted with permission from ref Copyright 2004 American Chemical Society) (c), oxygen evolution reaction (OER) (adapted with permission from ref Copyright 2011 WILEY) (d), CO₂ reduction reaction (CO₂RR) (reproduced from ref under the CC-BY 4.0) (e), and N₂ reduction reaction (N₂RR) (adapted with permission from ref Copyright 2014 Royal Society of Chemistry) (f). Experimental data in (f) were taken from refs − .

However, the structure and dynamics of interfacial atoms at finite temperatures cannot simply be described as harmonic vibrations around the structure at absolute zero. This issue becomes particularly evident when dealing with complex interfacial structures that are inconsistent with the structure of ice. A typical example is the Pt surface functionalized with organic molecules, as shown in Figure (a). Recently, it has been confirmed that functionalizing Pt catalysts with organic molecules such as melamine enhances the ORR rate by several times on both nanoparticle catalysts and single-crystal surfaces. Additionally, on nanoparticle catalysts, melamine functionalization has been found to suppress Pt dissolution, thereby improving catalyst durability. This approach has attracted attention as one of the few strategies capable of enhancing both the activity and durability of ORR catalysts. However, determining the structure of interfacial water in this complex system is not straightforward. First, determining the adsorption structure of organic molecules in a solvated environment through structural optimization is challenging. Furthermore, in interfacial structures where symmetry is reduced by organic molecules, numerous quasi-stable configurations with comparable energies can exist, making it challenging to determine the solvent structure that dominates the free energy landscape. To address such challenges, employing the TI calculation using finite-temperature MD, as described in the previous section, is necessary. However, applying FPMD requires an expensive computational cost. Although a few previous studies have performed TI using FPMD, , the range of reaction coordinates that can be explored to map the free energy landscape is limited to the order of a few Å. With such a narrow exploration range, the evaluated free energy differences may be referenced to a system-specific value, making direct comparison with experimental results difficult. Moreover, the time-scale of reorientations of interfacial water molecules has been reported to be on the order of 1 to 10 ns, , making it difficult to obtain sufficient statistical averaging within the typical 10 ps time scale of FPMD simulations.

8.

An ML-aided FP calculation of interfacial electrochemistry: (a) interfacial model of melamine and OH adsorbed on the Pt(111) surface in water, (b) experimental and simulated volcano plot of ORR at an electrode potential of 0.9 V vs SHE on Pt(111) (black), Pt(111) with melamine (red), Pt_1–x Cu _x (111) (orange), other Pt alloys (purple), and transition metals (gray), where open dots and the solid black line represent calculations, and closed dots represent experimental data, ,− and (c) vibrational density of states of deuterated hydrogen atoms within a water molecule hydrogen-bonded to OH* on Pt(111) (black) and Pt(111) with melamine (red), compared to that in bulk liquid water (gray). Here, the relative change in ORR current density j compared to Pt(111) is plotted as a function of the redox potential of the OH* reduction reaction relative to Pt(111), U _OH – U _OH (adapted with permission from ref Copyright 2025 American Chemical Society).

Recently, the particle insertion method utilizing the MLFF described in the previous section ,, has been applied to this problem, leading to the elucidation of the mechanism behind the ORR enhancement through the melamine functionalization. In this approach, the free energy difference $\mathrm{\Delta }{\mathcal{A}}_0$ in eq is expressed as the sum of the reaction free energy $\mathrm{\Delta }{\mathcal{A}}_1$ for the bond formation between the adsorbed OH* and H atom at the solid–liquid interface, OH* + H → H₂O, and the reaction free energy $\mathrm{\Delta }{\mathcal{A}}_2$ for the dissociation reaction of H₂ molecule, 1/2H₂ → H

$$\mathrm{\Delta }{\mathcal{A}}_0=\mathrm{\Delta }{\mathcal{A}}_1+\mathrm{\Delta }{\mathcal{A}}_2$$ 101

$\mathrm{\Delta }{\mathcal{A}}_2$ can be readily computed by applying the ideal gas model to an H₂ molecule and an H atom in the gas phase. On the other hand, calculating the free energy of the bond formation between the OH* species and H atom at the solid–liquid interface is challenging. To address this, the particle insertion method utilizing MLFF was applied. As in the method described in the previous section, MLFFs were generated for both the initial and final states of the reaction. Using these MLFFs, TI calculation was performed over a total time scale of tens of nanoseconds, following the procedure of eqs to (), to compute the free energy change from an ideal gas-phase H atom, which does not interact with its surroundings, to a state where it is captured by the interfacial OH*. Furthermore, as described in eqs and (), TI was performed from the MLFF energy surface to the FP energy surface, thereby correcting for MLFF errors. From the obtained free energy difference $\mathrm{\Delta }{\mathcal{A}}_0$ , the redox potential U _OH for the OH* reduction reaction was determined. Using this U _OH value, the ORR reaction current was predicted based on the ORR reaction model. For comparison, evaluations were conducted not only on Pt(111) surfaces with and without adsorbed melamine but also on Pt monolayers on PtCu alloys [Pt/Pt_1–x Cu _x /Pt­(111)]. The liquid–solid interfaces were modeled using unit cells containing 48 Pt atoms and between 38 and 48 water molecules. A single MLFF was generated for each surface system through on-the-fly training, which was performed over approximately 3 days using 64 cores of the same processor described in the previous section for the Nafion system. The resulting MLFF achieved a simulation speed of 0.1 s per MD step using 8 cores of the same processor. For each system, more than 15 ns of MD simulations were conducted for the TI calculations. Figure (b) presents the computed ORR activity, expressed as the logarithm of the ratio of the reaction current j to the reaction current on Pt(111), j _Pt, plotted against the thermodynamic stability of adsorbed OH*, represented by the difference between U _OH and the OH* reduction redox potential U _OH on Pt(111) without melamine. Previous studies on PtCu alloys , have shown that the introduction of Cu shortens the Pt–Pt bond distance, weakening the interaction between Pt and OH*, which leads to an increase in U _OH. This, in turn, accelerates the OH* reduction reaction, enhancing the ORR activity. The TI method incorporating MLFF successfully reproduces these results. Similar to alloying, the melamine adsorption increases U _OH and enhances the ORR activity. The impact of the surface functionalization closely matches that observed for the most active alloy, Pt/Pt₂Cu/Pt, and results in a substantial enhancement–approximately a 9-fold increase in per-site activity compared to Pt(111). The computational result that melamine reduces OH* production and improves ORR activity is consistent with previous experimental findings on Pt(111). However, further analyses of interfacial electronic states, interfacial hydrogen bonding structures, and vibrational frequencies revealed that melamine destabilizes OH* through a mechanism fundamentally different from alloying. Figure (c) illustrates the effect of surface functionalization on the vibrational density of states for the O–H stretching mode in water molecules hydrogen-bonded to OH*. On the Pt(111) surface without melamine, water molecules form strong hydrogen bonds with OH*, causing a significant redshift in the O–H stretching frequency compared to that in liquid water. In contrast, the O–H stretching frequency is blueshifted on the Pt(111) surface in the presence of melamine. This indicates that melamine disrupts the hydrogen bonds between adsorbed OH* and surrounding water molecules. This study concludes that such interfacial hydroben bond disruption leads to the destabilization of OH*. Capturing this disruption using FP calculations, which impose an ice-like solvent structure at absolute zero, is challenging. By incorporating MLFF, finite-temperature MD simulations exceeding the reorientation time scale of interfacial water (several nanoseconds) , become feasible, enabling free energy calculations that account for fluctuations of interfacial water molecules.

Not only for the first-order approximation but also for capacitor models and even higher-order approximations, MLFF is expected to be applicable. By constructing an MLFF that represents interatomic interactions in the subsystem at each electrochemical potential (μ_e) or each electron number (N _e) under the electric double layer, and performing TI calculations for electron or particle insertion as in the examples discussed above, it should be possible to evaluate reaction free energies while explicitly considering variations in the electrode potential. Moreover, in addition to improving the accuracy of MLFF, incorporating electrostatic interactions and electric field response theories ,,,,, could enhance the transferability of MLFF. This would allow efficient predictions for various potential states based on training with a limited number of potential conditions. Furthermore, leveraging the high computational efficiency of MLFF, explicitly incorporating a large ion reservoir in direct contact with the subsystem would enable highly accurate grand canonical ensemble simulations, not only for electrons but also for ions, using a fully atomistic model. Thus, in the future, attempts to explicitly incorporate electric double layers using MLFF will become a major focus of research.

Knowledge Extractions and Simulations on Inhomogeneous Systems

MLFF enables simulations on larger length- and time-scales than FP calculations by representing the PES, allowing for precise predictions of transport coefficients and reaction properties. However, this alone is insufficient to address complex phenomena involving multiple elementary processes within heterogeneous structures in electrochemical systems. To accomplish this, it is necessary to extract key information from MD simulations and transfer it to simulations at even larger length- and time-scales. As shown in Figure , MLFF has been utilized for this purpose as well. In this section, we present coarse-grained simulations of transport phenomena in polymer electrolytes and detailed analyses of complex surface reactions, explaining the methods used for information extraction and their applications.

Multiscale Molecular Simulations

In the electrolytes and electrodes of fuel cells, electrolytic cells, and secondary batteries, heterogeneous structures larger than a few nanometers are formed, and transport phenomena at submicrometer to larger scales can become the dominant factor influencing performance. − , Experimentally elucidating the locations of chemical species and their transport pathways within such heterogeneous media is extremely challenging. MD simulations can provide rigorous predictions of transport properties based on statistical mechanics. However, even classical MD simulations are typically limited to time-scales of 100 ns to 1 μs and length-scales of 10 to 100 nm, making it difficult to fully address the above issues. An alternative approach is to treat materials as continuous medium and describe transport phenomena by solving diffusion equations or fluid dynamics equations. While this approach can cover the relevant length- and time-scales, the solubility and effective diffusion coefficients of chemical species used in such calculations are often adjusted to reproduce experimental results, making it difficult to consider this a truly predictive simulation method.

Recently, with the advancement of MLFF, systematic methods have emerged that enable bridging the gap between the atomic scale and the mesoscale. One representative technique is coarse-graining, a method in which groups of atoms are combined into a single pseudoparticle, and interactions between these coarse-grained particles are represented as functions of their positions. This approach reduces the number of particles to be handled, increases the MD time step, and expands the accessible length- and time-scales. In general, coarse-graining enables simulations on time scales ranging from 1 μs to 1 ms and length scales from 100 nm to 1 μm. Similar to classical MD, coarse-graining has been extensively studied, and many physics-based interaction models have been proposed. − There are various coarse-graining techniques, and numerous excellent review articles have been published. For further details, we refer the reader to these refs ,− . Coarse-grained models have been developed to represent the free energy landscape $\mathcal{E}({\mathbf{R}}^{N_{\mathrm{CG}}})$ obtained from MD simulations

$$\mathcal{E}({\mathbf{R}}^{N_{\mathrm{CG}}})=-k_{\mathrm{B}}T\ln \int \mathrm{d}{\mathbf{R}}^{N_n}\exp (-\frac{E({\mathbf{R}}^{N_n})}{k_{\mathrm{B}}T})\delta ({\mathbf{M}}^{N_{\mathrm{CG}}}({\mathbf{R}}^{N_n})-{\mathbf{R}}^{N_{\mathrm{CG}}})+\mathrm{const}$$ 102

where R ^{N _CG} is a variable that collects the position vectors of N _CG coarse-grained particles. M ^{N _CG} is a mapping function that transforms the atomic coordinates R ^{N _n} of N _n atoms into the coordinates of the coarse-grained particles, R ^{N _CG} . In many applications, the linear mapping shown below is used

$${\mathbf{M}}_I^{N_{\mathrm{CG}}}({\mathbf{R}}^{N_n})=\sum _J{c}_{\mathit{IJ}}{\mathbf{R}}_J$$ 103

The relationship between the inputs and outputs of $\mathcal{E}({\mathbf{R}}^{N_{\mathrm{CG}}})$ is analogous to that of MLFF, allowing MLFF techniques to be utilized without significant modification. − Figure (a) presents simulation results for a variant of Chignolin, a 10-amino-acid protein, using SchNet, a type of MPNN. In this case, the mean forces obtained from the classical force field CHARMM22 were used as training data. Mean forces can also be obtained from all-atom MD simulations using MLFF, allowing MLFF to provide a seamless framework for hierarchical connections–linking FP calculations to all-atom models and all-atom models to coarse-grained models.

9.

Applications of ML models to multiscale simulations: (a) the structure of the miniprotein chignolin obtained using the coarse-grained SchNet model with MPNN (adapted with permission from ref Copyright 2020 American Institute of Physics), (b) the structure of Nafion at λ = 10, and (c) the global diffusion coefficient D _global in Nafion, simulated using the kMC method with local A(r) and D(r) predicted by the SOAP kernel method. Images (b, c) are adapted with permission from ref Copyright 2025 American Chemical Society.

Another notable application of MLFF-related technology is the development of methods that extract not only the free energy landscape but also local transport coefficients from all-atom MD simulations and construct ML models to represent these properties. In this application, the free energy landscape was calculated using the classical force field DREIDING based on the following equation

$$A(\mathbf{r})=-k_{\mathrm{B}}T\ln {⟨\exp (-\frac{U({\mathbf{R}}_{\mathrm{solt}},{\mathbf{R}}_{\mathrm{solv}})}{k_{\mathrm{B}}T})⟩}_{\mathbf{M}({\mathbf{R}}_{\mathrm{solt}})=\mathbf{r}}$$ 104

where ⟨.⟩ _M(Rsolt)=r denotes an ensemble average taken with the center of mass of the solute, M _solt, fixed at r. The local diffusion coefficient of the solute, D(r), was calculated using the flat-bottom potential method, which estimates the diffusion coefficient from the local mean square displacement of the solute. The method was applied to the diffusion of H₂ molecules in the Nafion polymer electrolyte. As shown in Figure (b), the electrolyte was modeled using 200 Nafion molecules, each composed of 10 monomers per polymer chain. All-atom MD simulations were conducted for hydrated Nafion with water uptakes of λ = 6, 10, and 14. A uniform grid r with a resolution of approximately 0.4 to 0.8 nm was set within the unit cell, and A(r) and D(r) of H₂ molecules were computed at each grid point. The computed values of A(r) and D(r) were used as training data to generate ML models based on the kernel regression explained previously. In this study, both A(r) and D(r) were assumed to be represented as functions of the elemental distribution function around r. For representing the elemental distribution functions, the three-body invariant descriptor in eq was used, and the SOAP kernel was employed for regression. Using the predicted values from the ML models, kMC simulations were performed based on the following transition probabilities

$$P_{\mathbf{i}\to \mathbf{i}+\mathrm{\Delta }\mathbf{i}}=\frac{1}{6}\frac{D(\mathbf{i})+D(\mathbf{i}+\mathrm{\Delta }\mathbf{i})}{2D_{\max }}\min [1,\exp (-\frac{A(\mathbf{i}+\mathrm{\Delta }\mathbf{i})-A(\mathbf{i})}{k_{\mathrm{B}}T})]$$ 105

where i represents the grid position, Δi denotes the displacement of movement, A(i) and D(i) correspond to the free energy and local diffusion coefficient at grid point i, respectively, and D _max represents the maximum diffusion coefficient. This kMC simulation is equivalent to solving the following diffusion equation in a heterogeneous field

$$\frac{\partial \rho (\mathbf{r},t)}{\partial t}=\nabla ·[D(\mathbf{r})\nabla \rho (\mathbf{r},t)+\rho (\mathbf{r},t)D(\mathbf{r})\nabla \frac{A(\mathbf{r})}{k_{\mathrm{B}}T}]$$ 106

Figure (c) presents the global diffusion coefficient of H₂ molecules obtained from the kMC simulation, plotted against the number of training data points, 3N, where N represents the number of data points for each λ. The ML model successfully reproduces the kMC results obtained using A(i) and D(i) from all-atom MD simulations (shaded region) with approximately 15,000 data points. As a result, the overall computational speed improved by an order of magnitude. The acceleration factor will increase as the system size grows. In this study, not only the global diffusion coefficient but also the spatial distributions of A(i) and D(i) within Nafion were presented. These distributions identified regions with high H₂ solubility and diffusivity. The results suggested that H₂ molecules primarily diffuse through the boundary regions between the Nafion main chain region and the cluster region. The kMC simulation is highly efficient and is expected to enable the simulation of transport phenomena on length scales beyond 1 μm and time scales of up to 1 s. This advancement indicates that, in the near future, entire electrolyte membranes and electrodes with thicknesses of several micrometers can be simulated.

Simulations of Complex Reactions

Many electrochemical reactions are complex multistep processes composed of a vast number of elementary steps. Even the simple hydrogen evolution reaction (HER), 2H⁺ + 2e^– → H₂, consists of three elementary steps: the Volmer reaction (H⁺ + e^– → H*), the Tafel reaction (2H *→ H₂), and the Heyrovsky reaction (H* + H⁺ + e^– → H₂). Even on highly symmetric single-crystal surfaces, identifying the reaction pathways is not straightforward, as the reaction intermediate H* can adopt various configurations due to interactions with the surrounding solvent molecules. Elucidating these pathways requires extensive and long-time FPMD simulations with significant computational cost. , In more complex reactions such as CO₂ reduction and N₂ reduction, which can generate numerous reaction intermediates and products, the number of possible elementary steps exceeds tens of thousands, even on single-crystal surfaces, making the selection of reaction pathways extremely challenging. ,, Furthermore, on electrode surfaces with defects or nanoparticle catalysts, surface symmetry is reduced, leading to a diverse range of surface sites. −

Traditionally, dominant reaction pathways have been identified based on experimental observations such as the dependence of the overall reaction rate on reactant concentrations and electrode potential, the types of reaction intermediates and products, as well as the chemical intuition of researchers. ,,, FP calculations have then been performed on a limited number of elementary steps to construct reaction models. ,, A typical procedure for FP calculations involves the following steps: constructing initial structures of reactants and products based on experimental information, identifying local minima and saddle points on the energy surface E(R ^{N _n} ) through structural optimizations, ,, determining the structures, energies, and vibrational frequencies of reactants, products, and transition states, and finally, calculating reaction rate constants k based on TST. Alternatively, in some cases, a reaction coordinate is predefined, and the activation free energy is determined using a TI method based on FPMD. However, in complex reactions involving tens of thousands of elementary steps, this approach may overlook important reaction pathways and lead to inaccurate predictions. Furthermore, the methods that utilize experimental data for model construction cannot predict the properties of unknown reactants.

To facilitate more systematic predictions, a method leveraging correlations identified in FP calculation data has been developed. An early approach in this direction is the descriptor-based method. Through detailed analyses of surface reactions, FP calculation data on surface reactions have been accumulated, , leading to the discovery of key linear relationships, such as scaling relations, which describe linear correlations between formation energies of reaction intermediates and the Brønsted-Evans–Polanyi (BEP) relationship, which expresses a linear correlation between the reaction energy and the activation energy, as illustrated in Figure (a). By leveraging these relationships, the high-dimensional variables in complex reactions can be effectively reduced to a smaller set of key parameters. Using these relations, reaction rate constants can be determined, allowing the prediction of complex reaction kinetics through chemical reaction models. This approach has made it possible to systematically summarize the activity of various electrocatalytic reactions in the volcano-type correlations shown in Figure . ,,,,,−

10.

ML-aided analyses of complex catalytic reactions: (a) the Brønsted-Evans–Polanyi (BEP) relationship between activation energy (E _a) and reaction energy (ΔE) observed in an FP data set of dimer dissociations from ref ; (b) identification of a key subset from the full reaction network of syngas (CO and H₂) conversion to CO₂ (reproduced from ref under CC-BY 4.0); (c) energy diagrams of the direct decomposition of NO on a cuboctahedral Rh₁₃₈Au₆₃ alloy nanoparticle and adsorption energies of reaction intermediates (N, O, and NO) on a cuboctahedral Rh₅₇₈Au₁₃₃ alloy nanoparticle (adapted with permission from ref Copyright 2017 American Chemical Society); and (d) a contour plot illustrating 14,958 reaction pairs (IS, TS, and FS) and low-energy pathways for methanol synthesis from a CO, CO₂, and H₂ mixture on Cu(211) (adapted with permission from ref Copyright 2022 American Chemical Society).

The outstanding success of this descriptor-based approach, together with advancements in AI and ML, has driven the development of more systematic methods for exploring reaction pathways. Figure (b) presents an application of this approach by Ulissi and co-workers to the gas-phase synthesis of C1 and C2 compounds from H₂ and CO. In this method, the key reaction pathways are extracted through the following process. First, the main reaction intermediates and elementary steps are identified, and all possible reaction pathways connecting these elementary steps are generated. In this study, 99 reaction intermediates and 190 elementary steps were used to generate thousands of possible reaction pathways, as shown in the figure. Next, an ML model is developed to predict the formation energies of reaction intermediates using an FP calculation database for surface reactions. In this study, Gaussian process regression based on the kernel method was employed for training. To construct input features, reaction intermediates were represented using SMILES, and their approximately 50-dimensional descriptors were reduced to around 10 dimensions through principal component analysis. From the predicted formation energies of reaction intermediates, the reaction energies of elementary steps were computed. Using these reaction energies, the activation energies of elementary steps were estimated based on the BEP relationship. From these activation energies, reaction rates for elementary steps were evaluated, allowing dominant reaction pathways to be identified. FP calculations were performed on these dominant reaction pathways, and the resulting data were added to the database. The ML model was subsequently updated using this new database, and the updated model was used again to extract dominant reaction pathways. By repeating this process, the accuracy of the predicted dominant reaction pathways improved, ultimately leading to the identification of the key reaction pathways shown in the figure.

With the advancement of MLFF, descriptors for representing three-dimensional atomic configurations have also evolved, enabling the prediction of the energies of reactants and products on three-dimensional surface sites of alloy nanoparticles. Figure (c) illustrates an application of this approach to the direct NO decomposition reaction, 2NO → N₂ + O₂, on Rh_1–x Au _x alloy nanoparticles. , In this study, the power spectrum, one of the three-body descriptors, as defined in eq , was employed to predict the formation energies of three reaction intermediates: N*, O*, and NO*. The nanoparticles were modeled as cuboctahedral structures with diameters ranging from 2 to 5 nm, consisting of 201 to 3355 atoms, as shown in the figure. The atomic positions in the nanoparticles were determined based on the lattice constants of an optimized bulk fcc crystal, following the so-called lattice gas model. From these atomic configurations, the power spectrum was computed and used as input for the ML model. For regression, Bayesian linear regression with a SOAP kernel was employed. The training data consisted of FP calculations for single-crystal surfaces with local structures similar to terraces and edges of the nanoparticles. As with the nanoparticle model, descriptors were computed from atomic configurations determined by the lattice constants of an unrelaxed bulk fcc crystal. However, the formation energies used for training were those obtained after structural relaxation. As a result, the model enabled the prediction of the formation energies for relaxed reaction intermediates based on unrelaxed atomic configurations. Similar to the method of Ulissi et al., the reaction energies of elementary steps were calculated from the predicted formation energies of reaction intermediates. The activation energies of the elementary steps were predicted from the formation energies using the BEP relationship. In addition to the ML model for predicting formation energies, another ML model was developed to evaluate the stability of nanoparticles. A lattice gas model was again used, and the training data consisted of the total energies of bare single-crystal surfaces after structural relaxations. The same SOAP-kernel-based regression method was applied, allowing the ML model to predict the total energies of alloy nanoparticles after relaxation based on their unrelaxed atomic configurations. Using this ML model, MC simulations were performed to determine the stable elemental distributions within the alloy nanoparticles. The NO direct decomposition activity was then evaluated on the identified stable surfaces. An example of the results is shown in Figure (c). The ML model predicted the formation energies of the three reaction intermediates at various sites on the alloy nanoparticle. Based on these predictions, site-specific free energy diagrams were generated, and the overall activity of the nanoparticle was assessed as the sum of reaction rates across all sites. The energy diagrams in the figure demonstrate that the ML model accurately reproduces FP calculation results. On the Au(211) single-crystal surface, the reaction intermediates are too unstable, resulting in large energy barriers for their formation. Conversely, on the Rh(211) single-crystal surface, the intermediates are overly stabilized, leading to large energy barriers for their removal. In contrast, on the Rh₁₃₈Au₆₃ alloy nanoparticle, the stability of reaction intermediates at corner sites is optimized, balancing the formation and removal processes. This indicates that the ML model has learned the scaling relationships from FP calculation data on single-crystal surfaces and effectively expresses the Sabatier principle. This study concluded that for a nanoparticle with a diameter of 2 nm and x = 0.31, the number of highly active corner sites increases, leading to maximum catalytic activity.

As MLFF has become capable of representing complex PESs of multielement systems, it has become possible to automatically explore reaction pathways and evaluate energy diagrams using the PES obtained from MLFF, without assuming the lattice gas model. Figure (d) illustrates an application of this approach to methanol (CH₃OH) synthesis from a mixed gas of CO, CO₂, and H₂ on a Cu/Zn alloy surface. The MLFF used in this study was the first-generation NNP developed by Behler and Parrinello. Training data were collected through an active learning approach using reaction pathway exploration simulations based on the stochastic surface walking (SSW) method, which enables unbiased PES searching based on the bias-potential-driven dynamics and Metropolis MC method. Initially, training data were gathered using FP calculations with the SSW method, and an MLFF was generated. Next, additional SSW simulations were performed using the MLFF, during which structures appearing in the simulations were randomly selected for FP calculations. These new data were added to the training data, and the MLFF was updated. By iterating this process, FP data for a total of 63,539 structures were collected. The final MLFF achieved root-mean-square errors (RMSEs) of 2.99 meV atom^–1 and 0.08 eV Å^–1 for the entire training data set and 2.1 meV atom^–1 for the energy of the test data set. Using MLFF-based SSW simulations, the stable elemental distribution of the alloy single-crystal surface was determined. Subsequently, reaction pathway exploration was performed on this surface using the SSW method. Figure (d) presents the results for the Cu(211) single-crystal surface. A total of 1,220,000 local minima on the energy surface were identified, along with 14,958 elementary steps and the corresponding structures of reactants, products, and transition states. The figure shows a frequency distribution visualized by compressing these data into two-dimensional collective variables (CV₁, CV₂). The red-colored regions indicate structures with high occurrence frequencies. Within this distribution, the lowest-energy reaction pathway from CO₂ to CH₃OH, as evaluated by MLFF, is highlighted. This pathway was also confirmed by FP calculations to be a low-energy pathway. Similar calculations were performed for Cu–Zn alloy single-crystal surfaces. It was concluded that step sites where Zn atoms avoid direct adjacency serve as the active sites for CH₃OH synthesis and that CO₂ is the primary source of CH₃OH. Additionally, it was revealed that during the reaction, –Zn–OH–Zn– chains appear at the edges, deactivating the edge sites. CO plays a crucial role in reducing these chains and restoring the active sites.

The three cases (b), (c), and (d) all focus on gas-phase catalytic reactions. Although, to the best of our knowledge, comprehensive exploration and analysis of reaction pathways such as those presented here have not yet been conducted for electrochemical reactions at liquid–solid interfaces, the applicability of MLFFs to interfacial electrochemistry has already been demonstrated. ,,,,,− In principle, the method proposed by Shi and co-workers is also applicable to liquid-phase reactions and holds promise as an effective approach for analyzing complex electrochemical reactions.

Materials Explorations

One of the ultimate goals of computational materials science is to explore unknown materials through computational methods and identify those with useful functionalities. Electrochemical systems can be considered one of the fields where FP calculations have been most effectively utilized. A pioneering example of this approach is the calculation of redox potentials for Li insertion reactions in cathode materials for Li-ion batteries using eq . Ceder and co-workers applied this method to propose Li _x Al _y Co_1–y O₂ and experimentally demonstrated that, in agreement with FP-based predictions, increasing the Al content y leads to an increase in the open-circuit voltage (OCV). Since then, FP calculations have become an established standard for predicting the OCV of cathode materials. The same equation has also played a crucial role in the search for electrochemical catalyst materials. As described in the previous section, the use of scaling relations and the BEP relationship has enabled dimensionality reduction of key factors, allowing the activities of a wide range of electrochemical catalytic reactions to be systematically represented as volcano plots based on a few key descriptors, as shown in Figure . ,− These descriptors were used as indicators to search for alloy catalysts, leading to the proposal of advanced materials. ,,

The exploration of materials using FP calculations has also been extended to solid electrolytes by incorporating simplified interatomic interaction models. For example, Kahle and co-workers proposed a Li-ion solid electrolyte through a multistep screening process, as illustrated in Figure (a). First, band gaps were evaluated through crystal structure optimization using FP calculations, narrowing down the initial set of 1400 candidate materials to 900 insulating materials. Next, for the filtered materials, the self-diffusion coefficient of Li ions was calculated using MD simulations with an interaction model known as the pinball model, reducing the candidates to 130 materials. In the pinball model, Li⁺ ions are treated as independent particles that move within the field generated by a fixed crystalline host. The interaction between the host and the ions is computed using the electronic density and wave functions of the host, obtained from FP calculations performed solely on the host. Since the energy surface governing the motion of Li ions as independent particles is determined only from the structural optimization of the host, the computational efficiency is significantly improved. Finally, further evaluation was conducted using FPMD simulations, leading to the selection of five candidate materials. Jang and co-workers also conducted a multistep screening process and proposed a novel Na-ion solid electrolyte. Figure (b) outlines this process. Initially, electrostatically stable crystal structures were selected from an initial pool of 523 million candidates based on energy estimates derived from the Ewald summation, narrowing the set to 850 structures. Next, FP-based structural optimization was performed to assess the stability, further reducing the candidates to 170 materials. These were subjected to FPMD simulations with a large time step of 10 fs, refining the selection to 44 materials. Finally, for the remaining candidates, FPMD simulations were conducted with a 1 fs time step at temperatures ranging from 500 to 900 K over 100 ps. Using the Nernst–Einstein equation, ionic conductivities were calculated, and electrolytes with conductivities on the order of 10 mS cm^–1 were identified. A key common feature of both studies is the use of physics-based models, such as the pinball model and the Ewald summation, to efficiently screen large sets of candidate materials.

11.

Four examples of FP-based materials exploration: (a) high-throughput computational screening of solid-state Li-ion conductors (adapted with permission from ref Copyright 2020 Royal Society of Chemistry), (b) high-throughput exploration of Na-ion sulfide solid electrolytes (reproduced from ref under the Creative Commons Attribution-NonCommercial License), (c) large-scale exploration of stable crystal structures (reproduced from ref under CC-BY 4.0), and (d) large-scale screening of stable and ion-conductive solid electrolytes (adapted with permission from ref Copyright 2024 American Chemical Society).

In recent years, with the advancements of MLFFs, general-purpose MLFFs have been developed to represent interactions in arbitrary materials composed of any element from the periodic table. These models are now being used for large-scale materials screening. One of the pioneering studies is the stability exploration conducted by Cheng and Ong. In this study, a general-purpose MLFF called M3GNet, based on CNN as described previously, was employed. M3GNet was trained on the FP calculation database from the Materials Project. The exploration began by substituting elements in crystal structures from the ICSD database. Through this process, 31 million virtual crystal structures were generated. Their energies were evaluated using the MLFF, and the structures with the lowest energy on the convex hull were selected. At this stage, the number of candidates was reduced to 2000 structures, which were further evaluated using FP calculations. Finally, 1578 materials were identified as stable. By employing the MLFF that better reproduces FP calculations in the screening process, this approach enables a more accurate assessment of material stability compared to previous methods.

Merchant et al. conducted a large-scale stability exploration using a multistep process incorporating ML models. Figure (c) presents a diagram outlining the overall scheme. The exploration was carried out through two routes: the structural pipeline and the compositional pipeline. As the first step, crystal structures from the Materials Project and the OQMD database were used as the source. By applying a method called Symmetry Aware Partial Substitution, which efficiently generates new crystal structures by substituting elements while considering symmetry, a total of 10⁹ candidate crystal structures were generated. In the structural pipeline, the screening was performed using the ML model GNoME. While the details of GNoME remain unclear, it is an MPNN that predicts energy from graph representations of crystal structures. The model was pretrained on FP data from 69,000 crystal structures in the Materials Project. Structures identified as stable by GNoME were clustered, and their stability was further evaluated using FP calculations. The newly obtained data were added to the database, and the GNoME model was updated. By iterating this process, the accuracy of GNoME improved to 11 meV atom^–1, leading to the proposal of 2.2 million crystal structures that were more stable than those reported in previous studies. The compositional pipeline employed a crystal structure prediction (CSP) method. An ML model predicting stability based on elemental composition was used to identify stable compositions, from which 100 candidate crystal structures were generated. For these structures, a CSP known as Ab initio Random Structure Searching (AIRSS) was applied to determine the most stable crystal structures. Among the structures obtained through these processes, 381,000 structures were found to be the lowest-energy structures on the convex hull, and 736 of them were reported to have been experimentally synthesized. Furthermore, the vast amount of FP data collected in this study was used to train a general-purpose MLFF based on NequIP, a type of MPNN employing equivariant descriptors and network, as described previously. This MLFF, referred to as the GNoME potential, was utilized in MD simulations, leading to the reported discovery of 623 new superionic conductors.

More recently, Chen et al. explored solid electrolytes using MLFF. Figure (d) outlines the overall process. In this study, approximately 30 million candidate crystal structures were generated by substituting elements in crystal structures from the ICSD database. Structural optimization was performed using M3GNet, and approximately 600,000 crystal structures within a 50 meV atom^–1 energy range from the convex hull were selected. These structures were sequentially evaluated using ML models for band gap and redox potential predictions, FP calculations for stability assessment, and M3GNet-based MD simulations for ion diffusion, ultimately narrowing down the candidates to 147 structures. Further screening incorporating factors such as cost and density reduced the number of final candidate materials to 18. Among these, Na _x Li_3–x YCl₆ was selected and synthesized. They reported that doping Li into the parent phase Na₃YCl₆ led to a two-order-of-magnitude increase in ionic conductivity, from 10^–8 to 10^–6 S cm^–1, demonstrating an interesting effect. The authors have developed and provided a cloud-based system that integrates the functionalities of ML models and FP calculations.

As evident from Figure , these two recent studies propose a systematic and large-scale materials exploration by utilizing ML models for high-speed screening of extensive candidate pools. Before breakthroughs in data-driven science, the initial stages of this screening process relied heavily on the intuition of experienced scientists–an art-like process that was often difficult for others to logically explain. Since the early days of materials informatics, AI and ML have been recognized as technologies capable of systematizing this process. Today, this transformation is gradually becoming a reality, with MLFF playing a crucial role as one of its driving forces.

Conclusions

We presented an overview of technological trends and future prospects for AI and ML applications in electrochemical systems. In particular, we focused on MLFFs which can represent interatomic interactions across diverse materials within a unified framework. MLFFs offer general applicability comparable to FP calculations while being several orders of magnitude faster. This advantage has driven the rapid growth of MLFF applications in ionics and electrochemical reactions. MLFFs enable finite-temperature MD simulations of ionic conductivity, as well as reaction thermodynamics and kinetics, with sufficient statistical sampling–capabilities that were previously out of reach for conventional classical force fields or FP calculations. In addition, systematic coarse-graining techniques have been developed to extract interparticle interactions and transport coefficients from MD data, allowing large-scale simulations over extended length and time scales while preserving statistical mechanical rigor. Furthermore, with advancements in algorithms, computing power, and FP databases, training on FP data for 10⁵ to 10⁶ crystal structures has become possible, leading to the development of general-purpose MLFFs, which are now being used for large-scale materials discovery. As discussed in Introduction section, the development and application of MLFF are built upon the foundations of classical mechanics, quantum mechanics, and statistical mechanics, which have been established through theoretical and computational sciences. This represents a prime example of how data-driven science is merging with traditional scientific disciplines to drive further advancements.

While MLFF is an extremely promising technology, many challenges remain for further applications. Finally, based on the overview of MLFF development and applications presented in this paper, we summarize below the key research challenges for MLFF. As the first topic, we outline the challenges related to the advancement of MLFF itself.

• Accuracy and transferability: For both inactive and active learning algorithms, the current use of MLFF consists of two main processes: (1) data preparation and training, and (2) simulations for property evaluations. One common issue in MLFF applications arises when a trained model is used for long MD simulations on large systems. During simulations, atomic configurations may emerge that lie outside the interpolation range of the training data, potentially causing the simulation to fail. Here, failure does not simply refer to minor errors, such as inaccuracies typical of classical force fields. Instead, it refers to catastrophic breakdowns, such as atoms overlapping unnaturally or the simulation cell expanding (or collapsing) uncontrollably. In such cases, the model must be retrained, and the simulation restarted from the beginning. This results in repeated trial-and-error cycles that substantially increase both human effort and computational cost. To mitigate this issue, it is essential to improve the transferrability of MLFF models, enabling them to achieve high accuracy with a minimal amount of training data.

• Error detections and model validations: Even if more training data become available and the training efficiency of MLFF improves, it remains difficult to guarantee a priori that MLFF will provide accurate results for all materials under all conditions. There are already countless MLFFs, and the number of MLFF models and their applications will continue to grow. In this situation, it is essential to continuously demonstrate the validity of the presented computational results. The validity of MLFF predictions should not simply be judged by whether the predicted properties match experimental results. This is because the FP calculations that MLFF aims to reproduce may themselves produce results that deviate from experimental findings. FP calculations are not necessarily expected to match experimental outcomes, and if MLFF yields results that coincide with experiments due to erroneous regression, it becomes difficult to reproduce the process, and the results lose scientific validity. The validity of MLFF predictions should first be evaluated by confirming that the regression results statistically remain within the interpolation range of the training data in the descriptor space. For linear regression or kernel regression, this can likely be assessed through statistical analyses, such as Bayesian error estimation and projection error evaluation. For neural networks, in practice, QBC may be the only feasible approach for such validation. Establishing uncertainty evaluation methods and carefully examining the relationship between estimated uncertainty and actual errors for various properties are crucial. It is necessary for the scientific community to systematically accumulate knowledge on the conditions under which MLFF provides reliable predictions, as well as on effective methods for assessing their validity.

• Physical interpretation of MLFF: MLFF has been advancing rapidly, continuously improving in both accuracy and applicability. At the same time, however, the representation of interatomic interactions has become increasingly complex. The development of MLFF began with the representation of many-body interactions using nonlinear functions of two-body and three-body invariant descriptors, following the body-order expansion concept illustrated in Figure . Subsequently, fully many-body invariant descriptors were introduced, leading to the emergence of linear regression models that directly realize the body-order expansion framework. However, state-of-the-art methods today, which are considered superior in terms of both data efficiency and accuracy, rely on complex MPNN models incorporating many-body equivariant descriptors. While it is undeniable that this technology has driven a major breakthrough, why is this formulation necessary? What is the optimal representation for efficiently capturing interatomic interactions? Answering these fundamental questions is crucial for the continued development of MLFF.

• Robust and user-friendly MLFF software packages: Dozens to hundreds of MLFF methods and training strategies are proposed every six months, and many software packages are publicly available. However, the differences between these methods are often difficult to interpret–even for specialists in MLFFs–making it challenging for users to identify the approach best suited to their application. This situation contrasts sharply with the well-established landscape of traditional MD and FP simulation packages. In addition to advancing state-of-the-art MLFF algorithms, the development of robust and user-friendly software packages is crucial for promoting the broader adoption of MLFFs.

The challenges related to the applications of MLFFs in electrochemistry are as follows.

• All-atom simulations of electric double layers: In FP calculations, it is challenging to represent the entire electric double layer using an explicit atom model. As a result, model charges and implicit solvation models are often introduced to evaluate electrochemical properties. However, as discussed in this article, implicit solvation models rely on empirical parameters. Similar to the challenges of error detection and model validation in MLFFs, agreement between the results of a model that combines empirical models with FP calculations and experimental data does not, by itself, demonstrate the soundness of the model, as FP calculations themselves can deviate from experimental results. To validate these models, the finite-temperature behavior of the electric double layer must be evaluated at the FP level. The MLFF-based approach, following eq , represents one attempt to address this issue. However, it still relies on a first-order approximation based on eq . A more advanced approach–ideally one without approximations–is needed to explicitly represent the polarization effects of the electric double layer and to explicitly evaluate the potential dependence of electrochemical properties with higher accuracy. Ultimately, the most rigorous method would be to conduct all-atom MD simulations in a sufficiently large system where electrochemical reactions do not alter the chemical potentials of electrons and ions in the subsystem. MLFF is expected to make this possible. Furthermore, in such efforts, incorporating long-range interactions and potential response behavior into MLFF is expected to enhance training efficiency and applicability, leading to improved computational accuracy and efficiency.

• Unified multiscale simulation framework and software: All materials follow quantum mechanics. Therefore, FP calculation methods and their software can be applied to a wide range of phenomena. This makes it easier to generalize methods and standardize computational programs. However, when the system size increases from the atomic scale to the mesoscale, complex heterogeneous structures emerge, and the number of involved elementary processes becomes enormous, necessitating approximations. As a result, different approximations are developed for different phenomena, leading to fragmentation of methods and a loss of universality. This is also why it is difficult to develop generalized coarse-grained mesoscale models, as well as universal kMC or mean-field models for complex reaction networks. The case studies introduced in this article suggest that MLFF may provide a unified framework for systematically handling diverse elementary processes. For example, MLFFs can be used to represent interactions between coarse-grained particles, free energy landscapes, and local diffusion coefficients, as well as to automatically extract reaction pathways–all within a single ML framework. The development of a unified mesoscale algorithm and software framework based on MLFF represents an important and promising research challenge.

• ML-aided materials explorations: The large-scale acceleration of materials discovery using MLFF has become a mainstream approach in materials exploration. A key challenge in this approach is to extract new physically interpretable principles for material design from the vast amount of generated data. MLFF is an information science-based approach, yet it has been developed in accordance with fundamental physical principles. One major advantage of simulations using MLFF is that they provide complete atomic coordinate information and, if necessary, insights into electronic structure. In materials development, it is not only important to identify promising candidate materials through high-throughput screening but also to physically elucidate the mechanisms underlying their high performance and establish new guiding principles for material design. For discovering such principles, materials informatics approaches that uncover hidden patterns within large data sets can be effective. Ultimately, it is essential to formulate mechanisms and guiding principles based on physical laws that can be understood by humans.

MLFF is a core technology that enables accurate property evaluation, multiscale simulations, and large-scale materials exploration. Its applications are not limited to ionics and electrochemical reactions, as discussed in this article, but will undoubtedly continue to expand across a wide range of condensed matter physics and materials science.

Vocabulary

Machine learning force field (MLFF): a supervised regression model that represents the potential energy surface of a material based on interatomic interactions.

Δ-machine learning (Δ-ML): a supervised regression model that represents the difference between potential energy surfaces calculated by two different first-principles methods.

Body-order expansion: A systematic approach to modeling interactions in a system by decomposing the total potential energy into contributions from one-body (individual atoms), two-body (pairs of atoms), three-body (triplets), and higher-order terms. This expansion represents increasingly complex many-body effects and is commonly used in the development of interatomic potentials.

Equilibrium Molecular Dynamics (EMD): A simulation technique where a molecular system is allowed to evolve at thermodynamic equilibrium, i.e., no net external forces or gradients. EMD is used to calculate properties like diffusion coefficients, radial distribution functions, and thermodynamic averages by sampling equilibrium trajectories.

Non-Equilibrium Molecular Dynamics (NEMD): A simulation method where external perturbations, such as temperature gradients, shear, or electric fields, are applied to drive the system out of equilibrium. NEMD is commonly used to study transport properties like thermal conductivity, viscosity, and response to external fields.

Thermodynamic integration (TI): A method for computing free energy differences by integrating the ensemble average of a derivative (usually of the potential energy) along a continuous path between two thermodynamic states.

Thermodynamic perturbation theory (TPT): An approach for estimating the free energy difference between two systems by treating one as a small perturbation of the other, using statistical averages obtained from simulations of the reference system.

The authors declare no competing financial interest.