Redox Potential-Based Modeling

• Redox potential-based models are computational frameworks that map free-energy changes to electrode potentials, providing a unified view of electron-transfer thermodynamics.
• They integrate structural, electronic, and transport variables across battery materials, biomolecules, and semiconductors to predict system performance.
• Applications span first-principles calculations, machine-learning techniques, and Monte Carlo simulations to accurately capture charge balance and environmental effects.

A redox potential-based model is a computational representation in which electron-transfer thermodynamics are expressed directly through electrode potentials, free-energy differences, chemical potentials, or equivalent affinity variables. In recent literature, the term spans sparse lattice Hamiltonians for intercalation cathodes, explicit-solvent first-principles free-energy calculations, fluctuation-relation schemes for protein redox shifts, constant-pH/constant-redox-potential molecular dynamics, semiconductor impurity models with solvent coupling, and bond-graph descriptions of chemoelectrical transduction (Zhong et al., 2023, Jinnouchi et al., 2024, Oliveira et al., 2023, Gu et al., 29 Apr 2025).

1. Thermodynamic basis

The common thermodynamic core is the mapping between a free-energy change and an electrochemical potential. In aqueous electrochemistry this is written as

$E^{\circ} = - \frac{\Delta G^{\circ}}{n F},$

with the corresponding conversion between absolute and standard-hydrogen-electrode scales

$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$

For non-standard conditions, the Nernst equation supplies the composition dependence,

$E = E^{\circ} - \frac{R T}{n F} \ln Q.$

These relations appear explicitly in first-principles frameworks for aqueous ions and molecules, and they also underlie protein redox-potential estimators that infer $\Delta G$ from simulation data and then convert it to $E$ (Jinnouchi et al., 2024, Oliveira et al., 2023).

In battery thermodynamics, the same idea is expressed through the lithium chemical potential. For intercalation in disordered rocksalt cathodes, the equilibrium voltage is written as

$$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$

and the average voltage between two compositions follows from the corresponding free-energy difference (Zhong et al., 2023). In bond-graph formulations of biomolecular energy transduction, the same mapping is recast as a Faraday-equivalent chemical potential,

$\phi_F = \mu/F,$

so that chemical potential can be treated as an electrical-like effort in volts and the power balance takes the same form in chemical and electrical domains (Gawthrop, 2016).

This suggests that the defining feature of a redox potential-based model is not a single numerical method but a particular choice of state variable: redox thermodynamics is made primary, and structural, electronic, or transport variables are organized around its prediction.

2. Principal computational formulations

Representative realizations include rank-sparse cluster-expansion Hamiltonians with semigrand-canonical Monte Carlo for disordered cathodes, machine-learning-assisted thermodynamic integration and perturbation for aqueous redox, fluctuation-relation estimators for proteins, multidimensional constant-$E$/constant-pH replica exchange, charge-aware machine-learning potentials, and semiconductor impurity models with self-consistent solvent coupling (Zhong et al., 2023, Jinnouchi et al., 2024, Oliveira et al., 2023, Cruzeiro et al., 2018, Kocer et al., 2024, Gu et al., 29 Apr 2025).

Formulation	Primary variables	Representative use
Sparse lattice Hamiltonian + SGCMC	Li content, oxidation-state decorations, $\mu_{\mathrm{Li}}$	Disordered rocksalt cathodes
TI/TPT with ML surrogates	$\Delta G$, local potential gap, absolute potential	Aqueous redox and SHE
MD + Crooks–Bayes	Non-equilibrium work distributions, $$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$0	Heme-protein redox shifts
Constant-$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$1/constant-pH REMD	Discrete protonation and redox states	Biomolecular titration/redox coupling
4G-HDNNP / charge-aware MLP	Global charges, oxidation-state fidelity	Fe$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$2/Fe$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$3 in water
Haldane–Anderson + Holstein	Defect occupancy, hybridization, solvent coordinate	Semiconductor-coupled redox

The lattice formulation is especially important where configurational disorder and multiredox chemistry are inseparable. For Li$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$4Mn$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$5Nb$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$6O$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$7F$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$8, the energy is represented by a cluster expansion with a screened Ewald term, and the electronic degrees of freedom are retained explicitly by treating Mn$$E_{\mathrm{vs\,SHE}} = E^{\mathrm{abs}} - E_{\mathrm{SHE}}^{\mathrm{abs}}.$$9/Mn$E = E^{\circ} - \frac{R T}{n F} \ln Q.$0/Mn$E = E^{\circ} - \frac{R T}{n F} \ln Q.$1 and O$E = E^{\circ} - \frac{R T}{n F} \ln Q.$2/O$E = E^{\circ} - \frac{R T}{n F} \ln Q.$3 as distinct species on fixed lattice sites (Zhong et al., 2023).

At the opposite end of the scale, explicit-solvent first-principles schemes compute redox free energies directly from finite-temperature sampling. The workflow built around RPBE+D3, PBE0 or PBE0+D3, machine-learned force fields, $E = E^{\circ} - \frac{R T}{n F} \ln Q.$4-machine learning, thermodynamic integration, and thermodynamic perturbation produces quantitative aqueous redox potentials and an absolute standard hydrogen electrode potential by referencing to vacuum through an O 1s-aligned local potential gap (Jinnouchi et al., 2024).

Protein-focused formulations replace direct free-energy integration with estimators based on fluctuation relations or constant-pH/constant-$E = E^{\circ} - \frac{R T}{n F} \ln Q.$5 state sampling. The MD+CB method uses oxidized and reduced equilibrium trajectories, instantaneous state switches, the Crooks relation, and Bayesian inference to obtain posterior estimates of $E = E^{\circ} - \frac{R T}{n F} \ln Q.$6 and hence redox-potential shifts (Oliveira et al., 2023). In AMBER, discrete protonation and oxidation states can be sampled under fixed pH and redox potential, and the same framework can be accelerated with replica exchange along the $E = E^{\circ} - \frac{R T}{n F} \ln Q.$7, pH, and $E = E^{\circ} - \frac{R T}{n F} \ln Q.$8 dimensions (Cruzeiro et al., 2018).

3. Charge balance, electrostatics, and environmental coupling

A central technical issue is that redox potential is rarely a purely local quantity. In ionic materials, explicit charge balance must be enforced. In LMNOF, the cluster expansion is trained only on charge-balanced structures, while semigrand-canonical sampling uses charge-neutral table-exchange moves such as Li$E = E^{\circ} - \frac{R T}{n F} \ln Q.$9 + Mn$\Delta G$0 $\Delta G$1 Mn$\Delta G$2 + vacancy and Li$\Delta G$3 + O$\Delta G$4 $\Delta G$5 O$\Delta G$6 + vacancy, so that the sampled manifold remains electroneutral at every Monte Carlo step (Zhong et al., 2023).

In explicit-solvent electrochemistry, electrostatics enters through both sampling and referencing. The aqueous first-principles framework defines the local potential gap by aligning to oxygen 1s core levels in bulk and slab water, and uses that correction in the free energy for proton insertion and in the decomposition of the absolute SHE potential. The same study reports a nuclear quantum correction of +0.30 eV for proton-related vibrational free energy in $\Delta G$7 (Jinnouchi et al., 2024).

Charge-aware machine-learning models make this nonlocal dependence explicit. Fourth-generation high-dimensional neural network potentials combine local short-range energy networks with global charge equilibration, so that the inferred charges depend on the entire system composition rather than only a local cutoff sphere. For FeCl$\Delta G$8 and FeCl$\Delta G$9 in water, this allows Fe ions to adopt charges consistent with Fe$E$0 and Fe$E$1 irrespective of the positions of the chloride counterions (Kocer et al., 2024).

In semiconductor redox theory, the same issue appears as hybridization between a redox level and band states. The extended Haldane–Anderson model with Holstein-type solvent coupling determines the actual charge on the redox species self-consistently through Green’s functions, while an instant-equilibrium solvent coordinate generates an effective negative-$E$2 term. The resulting self-consistency is reported as essential for reproducing ionization potentials, electron affinities, and redox potentials obtained from density functional theory (Gu et al., 29 Apr 2025).

At the continuum level, redox potential enters boundary kinetics rather than the Hamiltonian. In asymptotic multicomponent electrochemical-cell models derived from Poisson–Nernst–Planck theory, the Nernst relation defines the equilibrium potential inside Butler–Volmer boundary conditions, and the thin-double-layer reduction shows that redox reactions modify double-layer charging at short times and double-layer capacitance at long times (Jarvey et al., 2022).

4. Energy-storage materials and electrochemical devices

In disordered rocksalt cathodes, redox potential-based modeling is used to connect atomic disorder, multiredox chemistry, and observable voltage profiles. For LMNOF, a basis with pairs up to 7 Å and triplets and quadruplets up to 4 Å generated 858 candidate effective cluster interactions, which were reduced by $E$3-regularized mixed-integer regression to 169 non-zero terms from 463 charge-balanced DFT structures. The resulting model reproduces the charging curve at 300 K, including a sloped high-voltage Mn-dominated regime and a slope flattening around $E$4 associated with the onset of oxygen redox; an undecorated cluster expansion instead gives a featureless slope and misses the separation of Mn and O contributions (Zhong et al., 2023).

For solid redox films, cyclic voltammetry itself can be cast as a redox potential-based model. The ion-coupled electron-transfer treatment based on semi-infinite linear diffusion uses an interfacial Nernst boundary condition and yields two direct regressions: the CV mid-peak potential versus $E$5 ion activity or pH, and the capacity versus the inverse square root of scan rate. Applied to MnO$E$6, the alkaline mid-peak slopes of approximately $E$7 to $E$8 mV/pH indicate one H$E$9/e$$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$0, whereas the acidic slope of approximately $$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$1 mV/pH indicates two H$$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$2/e$$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$3 (Malaie, 18 Mar 2025).

At the device scale, the same concept appears in multiphysics flow-battery models. The Titanium–Manganese redox flow battery model in COMSOL couples Nernst equilibrium potentials, Butler–Volmer kinetics, porous-electrode charge conservation, species transport, and sulfuric-acid dissociation kinetics to compute dissociation rate, overpotential, current density, and electrode potential. In that study, compressing the electrode from 4.5 mm to 3 mm reduces overpotential and increases current density and electrode potential (Saha et al., 2022).

Physics-informed machine learning can be attached to the same electrochemical structure. The two-dimensional all-vanadium redox flow battery model used for PINN training contains 6 governing equations and 24 boundary conditions. The baseline PINN predicts cell voltage correctly but shows a constant-like shift in the potentials; adding constraints derived from the current collector boundary removes that shift, and a small amount of labeled data improves the enhanced model further (Chen et al., 2023).

A more empirical atomistic realization is redoxSQE, where atoms carry integer oxidation states and bond split charges, and the instantaneous battery voltage is defined as the difference in electrochemical potentials at the front atoms of the two electrodes (Dapp et al., 2013).

5. Biomolecular, solution, and semiconductor realizations

For proteins, redox potential-based models often seek relative shifts rather than absolute electrode potentials. The MD+CB method, applied to the de novo heme protein m4D2 and five mutants, estimates redox free energies from forward and backward instantaneous work samples using the Crooks detailed fluctuation relation and a Bayesian posterior. The reported correlation with experiment is approximately 0.85 across all variants and approximately 0.97 for the single mutants, and the method compares favorably with a continuum electrostatic PB+MC scheme (Oliveira et al., 2023).

Constant-redox-potential molecular dynamics in AMBER provides a different route. Discrete oxidation and protonation states are sampled under fixed $$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$4 and pH, and replica exchange can be performed along the redox-potential, pH, and temperature dimensions. For N-acetylmicroperoxidase-8 axially connected to a histidine peptide, 2D-REMD improves convergence relative to 1D-REMD, and 3D E,T,pH-REMD improves further; the simulations reproduce the expected increase in nearby pK$$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$5 upon reduction and the increase in $$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$6 upon protonation (Cruzeiro et al., 2018).

In aqueous solution, first-principles redox models target absolute scales. A PBE0+D3-based framework with machine-learning-assisted sampling reports $$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$7 V and an average error of 80 mV across seven redox couples, including transition-metal ions and molecular species (Jinnouchi et al., 2024). A related ML-aided TI/TPT framework for Fe$$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$8/Fe$$V(x) = -\frac{\mu_{\mathrm{Li}}^{\mathrm{cathode}}(x) - \mu_{\mathrm{Li,ref}}}{F},$$9, Cu$\phi_F = \mu/F,$0/Cu$\phi_F = \mu/F,$1, and Ag$\phi_F = \mu/F,$2/Ag$\phi_F = \mu/F,$3 yields PBE0 predictions of 0.92, 0.26, and 1.99 V versus SHE, with an RMSE of 0.11 V across the three ions (Jinnouchi et al., 2023).

Charge-aware machine learning extends these ideas to reactive solution dynamics. Fourth-generation machine-learning potentials reproduce the distinct Fe–O first-shell peaks at approximately 2.12 Å for Fe$\phi_F = \mu/F,$4 and approximately 2.03 Å for Fe$\phi_F = \mu/F,$5, and they qualitatively capture oxidation-state switching in mixed-valence Fe$\phi_F = \mu/F,$6Cl$\phi_F = \mu/F,$7 simulations (Kocer et al., 2024).

Semiconductor electrochemistry introduces band-structure control. The extended Haldane–Anderson/Holstein model shows how the valence-band maximum, conduction-band minimum, density of states, and hybridization affect ionization potentials, electron affinities, redox potentials, and the asymmetry of reorganization energies; the paper further argues that in the strong-coupling limit the shrinking separation between $\phi_F = \mu/F,$8 and $\phi_F = \mu/F,$9 lowers reorganization and clarifies a route to catalytic enhancement (Gu et al., 29 Apr 2025).

Strongly correlated redox systems require yet another extension. For actinide couples, density functional theory combined with a generalized Anderson impurity model and explicit 5f correlations improves redox-potential estimates and the description of disproportionation tendencies relative to mean-field DFT alone (Horowitz et al., 2010). In biomolecular energy transduction, bond graphs provide a systems-level counterpart: the Faraday-equivalent chemical potential allows redox half-reactions, proton pumping, and membrane potential to be represented within a unified energy-based network model of the mitochondrial electron transport chain (Gawthrop, 2016).

6. Validation, limitations, and directions of development

Despite their diversity, these models share several limitations. In disordered cathodes, finite-temperature vibrational contributions are neglected, electronic entropy is only partially captured by charge decoration, and kinetics and oxygen-gas evolution are outside the equilibrium model; the simulated LMNOF voltage is also systematically lower than experiment, consistent with exchange–correlation limitations even for $E$0SCAN without Hubbard $E$1 (Zhong et al., 2023). In aqueous first-principles models, explicit electrode–surface interactions are absent from the vacuum-aligned water-slab construction, and long trajectories are still required even after ML acceleration (Jinnouchi et al., 2024).

Machine-learning accelerators introduce distributional limits rather than removing them. Benchmarking of the MACE-OMol-0 foundation potential shows strong performance for proton-coupled electron transfer but diminished accuracy for electron-transfer reactions involving reactive ions and multi-electron transfers; in Test Set A, the FP-only MAE is 0.869 V overall, with approximately 0.146 V for 1e$E$2 ET and approximately 1.735 V for 2e$E$3 ET, whereas the recommended hybrid workflow reduces the corresponding 1e$E$4 ET MAE to 0.032 V (Chen et al., 28 Oct 2025). For protein disulfides, dummy-atom thermodynamic integration reproduces the large-protein 7Q53 redox potential within about 13 mV, but small proteins show systematic offsets that the authors attribute to the current inability of the dual-topology protocol to reclassify short-range electrostatic 1–4 terms during the alchemical transformation (Mejia-Rodriguez et al., 23 Sep 2025).

Continuum and asymptotic models have a different scope. The multicomponent electrochemical-cell reduction assumes the thin-double-layer limit, dilute solution, and equal diffusivities, and therefore does not directly address steric effects, dielectric decrement, strong ion correlations, or explicit porous-electrode microstructure (Jarvey et al., 2022). This suggests that redox potential-based models are best viewed as a family of thermodynamically anchored representations rather than a single closed formalism.

Current extensions in the surveyed literature are therefore method-specific but directionally consistent: add vibrational free energies and explicit electronic entropy in lattice models, improve sampling and referencing in explicit-solvent first-principles schemes, include multi-ion and mixed-valence environments in charge-aware machine-learning potentials, incorporate explicit electrodes and constant-potential methods in interfacial electrochemistry, and use sparse data or integral constraints to stabilize physics-informed surrogates (Zhong et al., 2023, Jinnouchi et al., 2024, Kocer et al., 2024, Chen et al., 2023). Across these developments, the unifying objective remains the same: to encode redox thermodynamics as a directly computable state function and then resolve how structure, disorder, solvation, charge transfer, and transport perturb that function.