Machine learning-aided first-principles calculations of redox potentials

Green energy and a circular economy require electrochemical technologies for energy conversion and storage. The redox potential of electron-transfer half-reactions (Ox + ne− → Red) in liquids determines level alignment with electrodes and stability windows of ions/molecules, informing the design of redox couples, solvents, additives, and catalysts. However, accurate first-principles predictions of redox potentials remain challenging, with typical errors around 0.5 V. Prior thermodynamic integration approaches (e.g., computational standard hydrogen electrode) found semi-local functionals yield large errors due to misplacement of water band edges and hybridization with redox levels. Hybrid functionals improved accuracy but showed significant variability depending on pseudopotentials and code implementations. Because hybrid functionals are computationally expensive, many studies rely on continuum solvation or QM/MM, introducing approximations and uncertainty in absolute predictions. There is a need for a rigorous, statistically converged first-principles framework. The present work aims to: (1) accurately calculate redox potentials of Fe, Cu, and Ag ions in water; (2) establish a computationally feasible pathway ensuring statistical accuracy; and (3) systematically explore density functionals to guide future studies.

Previous first-principles thermodynamic integration methods using CSHE aligned to electrostatic potentials highlighted large errors with semi-local functionals for aqueous transition-metal redox couples. Hybrid functionals reduced errors but results varied across pseudopotentials and codes (e.g., CP2K vs alternative implementations), with reported spreads for Cu2+/Cu+ from ~1.13 to 0.20 V and for Ag2+/Ag+ from 0.90 to 1.72 V. Approximate methods such as continuum solvation and QM/MM can reproduce experiments but rely on model assumptions, limiting rigor. For interfacial electrochemistry, many studies still approximate nuclear motion (e.g., harmonic models) or solvent response (RISM, continuum), motivating development of fully first-principles approaches. Prior work also proposed alignment of energy levels via average electrostatic potentials and separate slab calculations to connect to vacuum. The current study refines alignment by referencing to the O 1s level of water, mitigating finite-size effects and enabling absolute potential scales. Literature indicates semi-local functionals misplace water’s band edges; hybrid functionals and beyond-DFT treatments improve electronic structure but at high cost, suggesting ML-augmented pathways as promising.

Overview: The free energy difference ΔA between reduced and oxidized states defines the redox potential Uredox = ΔA/(en). The study computes ΔA via thermodynamic integration (TI) along a coupling parameter λ connecting oxidized (λ=0) and reduced (λ=1) states. The potential energy surface is described within grand-canonical DFT, with U(λ) = λU1 + (1 − λ)U0. Absolute potential referencing requires accounting for the electron chemical potential relative to vacuum; direct access to vacuum in periodic bulk is not available, so an alignment scheme is used. Absolute alignment via O 1s referencing: The energy difference term (U1 − U0) is measured relative to the average local potential in bulk simulations, and the O 1s core level of water far from the redox species (ε1s,bulk) is computed along the TI path. A separate water slab calculation yields the vacuum level relative to the O 1s of water in the slab (ε1s,slab). The absolute free energy difference is ΔA = ∫0→1 ⟨U1 − U0⟩λ dλ − ne Δφ, with Δφ = ∫0→1 ⟨ε1s,bulk⟩λ dλ − ε1s,slab. This aligns bulk and vacuum references while reducing finite-size errors by using an internal core-level marker. ML-aided integration strategy: Direct hybrid-functional TI is prohibitive. The authors introduce a multi-step scheme: (1) Perform TI from oxidized to reduced states using machine-learned force fields (MLFF): ML(Ox) → ML(Red). This captures non-linearities and enables long, statistically converged MD at low cost. (2) Correct MLFF to semi-local FP for both oxidation states via TI: ML(Ox) → FPℓ(Ox) and ML(Red) → FPℓ(Red), where FPℓ denotes semi-local functional (RPBE+D3). These integrands are small and nearly linear because MLFF reproduces FP structures, allowing short MD runs. (3) Bridge semi-local FPℓ to hybrid FPnl using Δ-machine learning (Δ-ML) that learns the energy difference ΔU = Unl − Uℓ with high accuracy from limited training (few tens of FPnl calculations). Thermodynamic perturbation theory (second-order cumulant) with Δ-ML predicts the free energy change FPℓ → FPnl with very few hybrid evaluations. The overall free energy is assembled from these steps. Sampling and efficiency: MLFFs enable million-step MD sampling for water slabs (1.5 ns total), yielding 3000 independent snapshots for O 1s–vacuum alignment with only ~2200 core hours including ML training, versus ~1 million core hours for brute-force semi-local FP. For the hybrid functional step, Δ-ML plus perturbation reduces cost from ~20 million core hours (direct TI) to ~16,800 core hours for generating semi-local FPMD configurations used in perturbation averaging. Computational details: VASP with PAW is used. Exchange-correlation: semi-local RPBE with Grimme D3 for MLFF training and baseline FPℓ; hybrid PBE0 with and without D3 for FPnl at exact-exchange fractions 0.25 and 0.50. System sizes: bulk cells with 32, 64, and 96 H2O to assess size effects (production at 64 H2O); separate pure-water slab (128 H2O; 12.5×12.5×50 Å cell) for vacuum alignment. K-points: 2×2×2 for 32-H2O bulk, Γ for larger systems; plane-wave cutoff 520 eV. MLFFs: kernel-based, local-energy sum with SOAP kernels and two-/three-body descriptors; active on-the-fly learning at elevated temperatures (400 K) to broaden phase-space coverage. A-ML (Δ-ML) learns Unl − Uℓ from 40 structures per system drawn from 20-ps semi-local FPMD at 298 K; RMSE for A-ML is 1–2 orders of magnitude smaller than MLFF RMSE. MD: Langevin thermostat, H mass 2 amu, 1 fs timestep. TI numerics: Simpson’s rule with 5 λ-points for oxidized→reduced path; trapezoidal with 3 points for ML→FPℓ corrections; ensemble averages over 10–80 ps per point after equilibration. For Δφ, use trapezoidal average of O 1s in fully reduced and oxidized states. Thermodynamic perturbation uses second-order cumulant, validated by near-Gaussian ΔU distributions. Absolute potentials reported versus vacuum; conversion to SHE uses 4.0 V for SHE absolute potential.

• Absolute redox potentials with PBE0 (25% exact exchange) agree closely with experiment: Fe3+/Fe2+ = 0.92 V (exp. ~0.77 V), Cu2+/Cu+ = 0.26 V (exp. ~0.15 V), Ag2+/Ag+ = 1.99 V (exp. ~1.98 V). Average RMSE ≈ 0.11 V; adding D3 does not materially change errors.
• Semi-local RPBE+D3 shows large, non-systematic errors: underestimates Ag, overestimates Cu versus experiment, underscoring necessity of hybrid functionals for accurate band-edge/redox alignment.
• Dependence on exact exchange: Increasing exact exchange lowers Cu2+/Cu+ potential and raises Ag2+/Ag+; Fe3+/Fe2+ shows a weaker, nonmonotonic trend. PBE0 with 50% exact exchange worsens agreement (e.g., Cu becomes too low, Ag too high).
• Machine-learning accuracy vs property accuracy: Standard MLFFs (few meV/atom energy RMSE, tens of meV/Å force RMSE) still induce 30–250 mV deviations in redox potentials relative to full FPℓ. Δ-ML models, with >10× smaller RMSE, reduce free-energy deviations to below 10 mV.
• Structural insights: Hydration coordination numbers match prior studies. Fe remains 6-fold coordinated across redox states; Cu changes from 5–6 (Cu2+) to 2–3 (Cu+); Ag changes from 5–6 (Ag2+) to 4–5 (Ag+). MLFF reproduces FP radial distribution functions and running coordination reasonably well.
• Water surface properties: RPBE+D3 MLFF predicts water surface tension 79±5 mN/m (128 H2O) and 84±5 mN/m (1024 H2O) at 298 K, near experimental ~72 mN/m and within previous simulation ranges; interfacial dipole orientations are bimodal, consistent with SFG analyses and classical SPC/E trends.
• Efficiency gains: O 1s–vacuum alignment with ML sampling reduces slab-alignment compute from ~1,000,000 to ~2,200 core hours. Δ-ML with thermodynamic perturbation reduces hybrid-functional TI from ~20,000,000 to ~16,800 core hours (limited by generating semi-local FPMD configurations).

The study addresses the longstanding challenge of quantitatively predicting absolute aqueous redox potentials from first principles by combining extensive statistical sampling with a rigorous alignment to the vacuum level. Referencing to water’s O 1s core level stabilizes the absolute scale and mitigates finite-size effects. The ML-assisted multi-step integration strategy captures anharmonic and non-linear solvent and coordination dynamics at low cost, while Δ-ML plus perturbation theory accurately transfers free energies from semi-local to hybrid functionals with minimal hybrid evaluations. The strong performance of PBE0 with 25% exact exchange indicates that accurate electronic structure, particularly water band edges and redox-level hybridization, is crucial for realistic potentials; dispersion corrections play a minor role for these couples. The observed mapping between ML model accuracy and redox free-energy accuracy clarifies requirements for reliable ML surrogates and validates Δ-ML corrections to achieve sub-10 mV agreement with target FP methods. Overall, the framework delivers near-experimental accuracy with orders-of-magnitude smaller computational cost and is generalizable to other electron-transfer reactions and, potentially, to more advanced electronic-structure levels beyond standard DFT.

The work introduces a practical, accurate, and scalable first-principles workflow for absolute redox potentials in solution by combining ML force fields for efficient sampling, TI-based corrections to semi-local DFT, and Δ-ML thermodynamic perturbation to hybrid functionals. Using PBE0 with 25% exact exchange, the method reproduces the Fe3+/Fe2+, Cu2+/Cu+, and Ag2+/Ag+ redox potentials with an average error of ~0.11 V and correct trends versus experiment. The O 1s-based alignment provides a robust bridge to the vacuum level. The approach dramatically reduces computational cost while retaining statistical rigor, and the integration pathway is broadly applicable to diverse ET reactions. Future directions include applying the final Δ-ML perturbation step to methods beyond DFT (e.g., many-body perturbation theory), extending to interfacial electrochemistry with explicit electrodes, exploring other redox couples and solvents, and investigating the origin of the optimal exact-exchange fraction for aqueous systems.

• Semi-local functionals yield non-systematic errors for redox potentials; accurate results rely on hybrid functionals and thus on the quality of Δ-ML corrections and limited hybrid benchmarks.
• The mechanism for why PBE0 with 25% exact exchange provides balanced accuracy remains unclear; PBE0 still underestimates the liquid water band gap.
• MLFFs, even with standard meV/atom accuracy, can induce tens to hundreds of mV errors in redox potentials; careful correction via TI and Δ-ML is necessary. Risks remain for ML extrapolation outside training data.
• Assumptions include negligible volume change for one-electron transfers, exclusion of side reactions, and Gaussian-like ΔU distributions for the second-order cumulant approximation in perturbation theory.
• Finite-size effects are mitigated via O 1s alignment and size checks up to 96 H2O, but residual size and sampling limitations may persist.
• Pseudopotential and PAW choices can impact results; while tested for Cu semi-core effects in PAW, broader transferability across codes/pseudopotentials is not exhaustively assessed.