This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Applications of redox processes range over a number of scientific fields. This review article summarizes the theory behind the calculation of redox potentials in solution for species such as organic compounds, inorganic complexes, actinides, battery materials, and mineral surface-bound-species. Different computational approaches to predict and determine redox potentials of electron transitions are discussed along with their respective pros and cons for the prediction of redox potentials. Subsequently, recommendations are made for certain necessary computational settings required for accurate calculation of redox potentials. This article reviews the importance of computational parameters, such as basis sets, density functional theory (DFT) functionals, and relativistic approaches and the role that physicochemical processes play on the shift of redox potentials, such as hydration or spin orbit coupling, and will aid in finding suitable combinations of approaches for different chemical and geochemical applications. Identifying cost-effective and credible computational approaches is essential to benchmark redox potential calculations against experiments. Once a good theoretical approach is found to model the chemistry and thermodynamics of the redox and electron transfer process, this knowledge can be incorporated into models of more complex reaction mechanisms that include diffusion in the solute, surface diffusion, and dehydration, to name a few. This knowledge is important to fully understand the nature of redox processes be it a geochemical process that dictates natural redox reactions or one that is being used for the optimization of a chemical process in industry. In addition, it will help identify materials that will be useful to design catalytic redox agents, to come up with materials to be used for batteries and photovoltaic processes, and to identify new and improved remediation strategies in environmental engineering, for example the reduction of actinides and their subsequent immobilization. Highly under-investigated is the role of redox-active semiconducting mineral surfaces as catalysts for promoting natural redox processes. Such knowledge is crucial to derive process-oriented mechanisms, kinetics, and rate laws for inorganic and organic redox processes in nature. In addition, molecular-level details still need to be explored and understood to plan for safer disposal of hazardous materials. In light of this, we include new research on the effect of iron-sulfide mineral surfaces, such as pyrite and mackinawite, on the redox chemistry of actinyl aqua complexes in aqueous solution.

One fundamental type of process that control energy fluxes in nature is redox processes, which involves electron transfer reactions that relate to a number of scientific fields, such as chemistry, biology, geochemistry, and mineralogy.

Reduction of hazardous toxic elements such as Cr(VI) and As(V) by redox active minerals where the role of redox chemistry is not well understood. The toxic Cr(VI) exhibits as CrO_{4}^{2−} ion and this is relatively more toxic to the environment and regarded as one of the dangerous carcinogenic agent [

An(VI) (An = U, Np, and Pu) is one of the most stable oxidation state for the actinides and exists as a linear oxo cation, in particular for the U and Pu. In contrast, for the Np the stable oxidation state is V for the actinyl species. These actinyl ions are soluble in aqueous environment and highly mobile; this makes these ions for their active transport process into the geosphere. These are radioactive materials, highly hazardous and lead to long term severe contaminations. Reduction process of these highly mobile and soluble actinyl ions are often catalyzed by the redox-active semiconducting minerals, for example iron sulfides (pyrite and mackinawite) [

After the reduction process, either by the redox-active mineral surface, or by any organic reductants, for example quinones, or radicals, for example OH radicals, the actinyl(V) reduced species is formed, which is unstable with respect to disproportionation in aqueous environment. This disproportionation process takes place via the formation of a cation-cation intermediate and followed by protonation steps as computationally proposed by Steele _{2}] or colloidal precipitated species. If there is a chelating ligand in solution, for instance carbonate ligands lead to form U(IV) carbonate complex rather than uraninite [

In addition, the redox-active mineral surface plays an important role in the reduction process and acts as a template and adsorbs the formed product on its surface and retains more often. However, there are different possibilities for the reduction to take place; either it could take place in solution, or on the surface adsorbed complexes. If the reduction takes place in solution, then there are two possibilities here, it could be adsorbed onto mineral surface and then either go for disproportionation or second reduction. Even disproportionation reaction for the actinyl(V) may possible in solution itself even before adsorption. Another interesting possibility is that the An(VI) species might be first adsorbed on the redox active mineral surface and then the reduction process, followed by either surface mediated disproportionation or the proton coupled second direct reduction. These various possibilities and complicated redox mechanistic pathways are challenging to density functional theory (DFT) methods, since treating a surface process using small cluster models has its limitations. However, this could give a deeper understanding to these fundamental redox processes that have been taking place for hundreds of years perhaps millions of years. Understanding these semiconductor redox active minerals surface mediated redox process of the environmentally toxic and hazardous actinide materials will help us to design effective remediation processes for immobilization and eventually even reused and recycled. This prevents long-standing and long term serious contamination to the environment.

Aqueous sediments contain a variety of dissimilatory metal-reducing bacteria (DMRB), which are often involved in reduction processes of heavy metals [

Another interesting aspect in understanding redox processes are E_{H}-pH or Pourbaix diagrams in which the electrode potentials of relevant species are plotted against the pH of the solution. Depending on the pH of the solution, different species often exists in equilibrium with each other. By combining, for example Fe with As or Fe with organic compounds will help us to understand the redox active behavior of hazardous pollutants in the environmental conditions.

One-electron redox process of a redox reaction can be simply defined in terms of a half-cell reaction as follows:
^{m} + e^{−} → Red^{m−1}

The accurate prediction of redox potentials using appropriate computational approaches can help us understand redox mechanisms of geochemical reactions and aid us in designing and optimizing redox-sensitive remediation techniques. In addition, the controlled modification of geochemical or industrial redox processes can optimize the design of redox-active catalytic agents, which can be utilized in large-scale industrial applications.

In this paper, we are going to give an overview and summarize the available computational methods for predictions of redox potentials in solution. Advantages and disadvantages of different methods reported in the literature are being examined and a suitable approach for the prediction of redox potentials of semiconducting mineral surfaces will be recommended. In addition, we will describe the prediction of reduction potentials for redox reactions involving organic materials, transition metal complexes, and actinides in solution. However, our main intention is, after having evaluated computational approaches for the prediction of reduction potentials from the literature, to develop and apply a reliable computational approach and use it to investigate the redox chemistry of semi-conducting minerals and redox-active surface-mediated chemical transformations.

Several computational settings have to be taken into account when tackling the computation of reduction potentials, which include the basis sets used, solvation models, free energy corrections, zero point energy (ZPE) corrections, standard state corrections, spin-orbit coupling interactions, and relativistic effects. Although, to a certain degree, relativistic effects can be included into pseudopotentials (PP) of actinide elements, the number core electrons replaced by the pseudopotential, which is 78 electrons for large-core PPs and 60 electrons for small-core ones. Using small-core PPs in combination with DFT methods was found to reproduce experimental reduction potentials in aqueous solution more closely, which will be described in more detail later.

In the introduction section, we have demonstrated the motivation for this article based on the wider applications of redox processes in various fields such as chemistry, biology, and mineralogy. The remaining sections of this article are organized as follows. We will introduce the essential background for calculating redox potentials followed by describing the importance of the reference electrode potential and the thermodynamic cycle in various redox potential prediction methods. This is followed by how these computational tools are applied to redox potentials predictions of organic compounds, transition metal complexes, actinides, and semiconducting materials in solution.

After describing the computational treatment of reference electrodes for aqueous and non-aqueous solutions, we will briefly explain the relation between the thermodynamic cycle and the calculation of redox potentials. Finally, we will illustrate various methods to predict the Gibbs free energy of redox processes in solution, which is the essential part of calculating redox potentials. In addition, we will outline the absolute potentials of reference electrodes and their values in aqueous and non-aqueous solutions.

Standard reduction potentials are typically referenced to the standard hydrogen electrode (SHE), whose reaction is given as:
_{3}O]^{+} + e^{−} → ½H_{2} + H_{2}O

When the SHE is used as a reference electrode in cyclic voltammetry experiment, its value is defined to be zero at pH = 0 and atmospheric pressure of H_{2} = 1 bar. Similar to the SHE, there are other reference electrodes such as the saturated calomel electrode (SCE) and Ag^{+}/AgCl electrode that are in practice easier to handle in aqueous solution than the SHE. Although the experimentalist would determine the redox potentials of complexes with respect to any suitable and practical reference electrode, the determined redox potentials can then be converted with respect to the SHE or any other reference electrode of interest. However, care should be taken when converting experimental redox values determined with respect to one reference electrode

Typically, experimental redox potentials are referenced or reported with respect to reference electrodes, for instance the SHE, in literature. Although during the experiment the value of the SHE is considered zero, its absolute value is not zero; a range of absolute reference values for the SHE have been reported from 4.24 to 4.73 eV [

In some cases, it may be necessary to use non-aqueous solutions, e.g., if the reactants are not stable or soluble in water. The ferrocene redox couple has been widely used as a reference redox couple for non-aqueous solutions. The ferrocene molecule is a metallocene type transition metal organo-metallic complex, in which the Fe(II) ion is sandwiched between two cyclopentadienyl (Cp) anionic ligands. Depending on the Cp ring conformations, two structural isomers with different symmetries are possible, eclipsed (d5h,

Ferrocene (eclipsed) structure in vertical (

The ferrocene/ferrocenium ion (Fc/Fc^{+}) redox system was found to show a solvent-independent redox behavior in a range of 22 non-aqueous solvents and was thus recommended as a reference electrode system by the IUPAC for electrochemical studies in non-aqueous solutions [^{+} reference electrode has been widely used and accepted as a reference electrode system for non-aqueous solutions. The half-cell reaction for Fc/Fc^{+} reference electrode system is given as follows:
^{+} + e^{−} → Fc

Interconversion of electrode potentials between different reference electrodes is a problematic issue unless similar experimental conditions, for example electrolyte concentration, ionic strength, and solvents, are used during the redox potential determinations [^{+} potentials, different electrolyte concentrations were investigated and reported. Depending on the electrolyte concentration, the Fc/Fc^{+} redox potential varies within a range of ~0.1–0.15 eV, and while this is significant as far as computational predictions are concerned, these uncertainties in experimental redox potentials could also introduce systematic errors in computational predictions [

Absolute potentials for the Fc/Fc^{+} reference electrode system in various solvents have recently been determined computationally using a high-level G3 method (G3 method involves several post self-consistent field (SCF) calculations where the energy expression is given as, E_{0}[G3(MP2)] = QCISD(T)/6-31G(d) + ∆E(MP2) + ∆E(SO) + E(HLC) + E(ZPE), where SO is the spin-orbit interaction, HLC is the “high level correction”, and ZPE is the zero point energy correction). Although the computational study had used the G3 method, the gas-phase ionization potential for Fc was predicted to be 0.20 eV smaller than the experimental ionization potential. Despite this underestimation, the absolute potential for the Fc/Fc^{+} reference electrode in dimethylsulfoxide (DMSO), acetonitrile (ACN) and dichloroethane (DCE) solvents were accurately predicted by encompassing the conductor-like polarizable-continuum (CPCM) and the charge-density based solvation model (SMD).

To get the reduction potentials from computations, the thermodynamic cycle (see

Thermodynamic cycle for the calculation of Gibbs free energies of a one-electron reduction process.

The thermodynamic cycle involves several terms that correspond to the gas or solution phase. The free energy of the electron is not taken into account since the addition of another half-cell reaction for the reference electrode automatically cancels the energy of the electron out. It could also be argued that the ionic convention of an electron leads to a zero value of its free energy in the gas and solution phase.

For the free electron, there are three different conventions, namely ionic, Fermi-Dirac, and Boltzmann statistics conventions (the ionic convention is based on the “gaseous ion energetics”, in which the heat capacity of a free electron is assigned a value of zero. The other two conventions are based on different statistical treatments, though the resulting differences in the energy are small (0.04 eV) between these two conventions). Based on the ionic convention, the free energy of a free electron is considered zero in the gas as well as in the solution phase. In contrast, the other two conventions give slightly different heat capacity values for the free electron. However, the resulting overall free energy value for the redox reaction is not significantly different between the ionic and Fermi-Dirac conventions of the electron, hence this is typically ignored [^{o}(s) = ∆G^{o}_{(gas)} + ∆∆G^{o}(s) + ∆ZPE + ∆FEc
^{o}_{(s)} = ∆G^{o}_{(s,Red)} − ∆G^{o}_{(s,Oxd)}
_{(s)} = ∆G^{o}_{(gas)} + [(∆G^{o}_{(s,Red)} + ∆G^{1atm→1M})] − [(∆G^{o}_{(s,Oxd)} + ∆G^{1atm→1M})] + ∆ZPE + ∆FEc
_{red} = ∆G*(s) − ∆G_{(ref.elec)}
^{o}_{(g)} is the reduction free energy in the gas-phase, ∆G^{o}_{(s)} is the standard state reduction free energy of the redox reaction in the solution-phase, ∆G^{o}_{(s,Oxd)} is the standard state solvation free energy of the oxidized species, ∆G^{o}_{(s,Red)} is the standard state solvation free energy of the reduced species, ZPE is the zero-point energy correction obtained from frequency calculations at stationary equilibrium geometry, FEc is the free-energy correction from thermal contributions ∆G*_{red} is the standard reduction free energy of the redox reaction referenced with respect to reference electrode, and ∆G_{(ref.elec)} is the free energy of the reference electrode. Here, the (^{o}) notation corresponds to the standard state at 1 atm and 298.15 K, whereas the (*) notation corresponds to 1 mol/L. The term ∆G^{1atm→1M} equals to transfer a reagent from its gas phase at 1 atm to its dissolved state at 1 mol/L, which numerically equals to 7.93 kJ/mol. In the above redox equation (see ^{1atm→1M}, in this case. Another way to look at the Gibbs free energy of mixing is using G = E_{elec} + ZPE + E_{trans} + E_{rot} + E_{vib} + RT − TS, where E_{elec} is the SCF energy, ZPE the zero point energy, ^{o} + RT ln([C]/[A][B]), in which * refers to 1 mol/L whereas the ^{o} refers to 1 atm standard states for all species. Based on the ideal-gas assumption, the concentrations of [A], [B], and [C] are defined as 1/25.4 mol/L (at 298.15 K). Inserting these values into the above equation lead to ∆G* = ∆G^{o} + RT ln(25.4) or ∆G* = ∆G^{o} + 7.93 kJ/mol.

According to the Nernst equation, the free energy of a reduction reaction is related to an experimentally determined reduction potential.
_{0} = −∆G*_{red}/^{−1} V^{−1} or 96485 C mol^{−1}) and E_{0} is the experimentally determined redox potential (in V).

In this section, different computational methods that were employed to predict the reduction potentials are briefly described. Two frequently used methods for reduction potential predictions are the direct and isodesmic reaction method.

The reference-electrode half-cell reaction, for instance the half-cell reaction of the SHE or Fc/Fc^{+} reference electrode, can be included into the redox reaction of interest and the reference electrode value would be calculated in a way similar to the half-cell reaction of the reactant of interest. Combining the redox half-cell reaction Equations (1) and (2) will, e.g., produce the following complete redox reaction.
^{m} + ½H_{2} + H_{2}O → Red^{m−1} + [H_{3}O]^{+}

The reduction free energy for the overall redox reaction (Equation (9)) can then be calculated according to the expression in Equation (6). The overall redox equation (Equation (9)) already contains the half-cell reaction of the reference electrode (SHE) in the overall redox reaction, hence the ∆G_{(ref-elec)} term vanishes. The free energy value of the reference electrode half-cell reaction is calculated using the relevant species involved in the redox reaction of the reference electrode system. There are experimentally determined absolute values available for these reference electrodes in different solvents; for example, the absolute values of the SHE in H_{2}O, methanol, ethanol, acetonitrile, and dimethylsulfoxide (DMSO) are different, these absolute values were derived from thermodynamic parameters [^{+} reference electrode with respect to the SHE is also known. By knowing the correct interconversion factors, the determined redox potential can be converted into the reference electrode of interest. This issue of interconversion has been recently addressed [

Absolute potentials of the standard hydrogen electrode (SHE) (V) in different solvents and from different sources.

Solvents | Trassatti | Fawcett | Kelly ^{a} |
Kelly ^{b} |
---|---|---|---|---|

Water | 4.44 | 4.42 | 4.24 | 4.28 |

Methanol | 4.19 | 4.17 | 4.34 | 4.38 |

Ethanol | 4.21 | 4.24 | - | - |

Acetonitrile | 4.60 | 4.59 | 4.48 | 4.52 |

Dimethylsulfoxide | - | 3.83 | 3.92 | 3.96 |

Notes: ^{a} the integrated heat capacity and entropy values for the free electron were based on the Fermi-Dirac statistics; ^{b} the integrated heat capacity and entropy values for the free electron were based on the Boltzmann statistics.

In the isodesmic model, rather than using a reference electrode reaction in the overall redox reaction, calculated redox potentials are referenced with respect to the redox half-cell reaction of a reference complex (Equation (10)). The fact that the inclusion of the redox potential of the reference complex automatically determines the reference potential with respect to the reference electrode as to the determined experimental redox potential. Error cancellations lead to minimize systematic errors in the redox potential prediction. Moreover calculating reference electrode potentials accurately is difficult, which in turn introduces systematic errors to redox potential predictions.
_{ref} + e^{−} → Red_{ref}
_{ref} → Oxd_{ref} + Red

Free energies of the species involved in the isodesmic redox reaction (Equation (3)) are calculated using the chosen DFT method. The reduction free energy (∆G*_{(s)}) is calculated for the isodesmic model redox reaction according to Equation (11) using the thermodynamic cycle scheme as explained earlier (see _{red} = ∆G*_{(s)} − ∆G_{(ref)}
_{(ref)} is the experimental redox free energy of the reference complex used as isodesmic model. This method had been successfully applied not only to organic compounds [

This section summarizes computational methods used to calculate redox potentials, with an overview on the suitability of different density functional theory (DFT) methods, basis sets, and solvation methods in the first part and an introduction to thermodynamic integration at the end.

DFT is based on the two theorems proposed by Kohn and Hohenberg [_{e}[ρ(_{ne}[ρ(_{ee}[ρ(_{XC}[ρ(

In Equation (13), T_{e}[ρ(_{ne}[ρ(_{ee}[ρ(_{XC}[ρ(

There are several ways to define the exchange-correlation functional term (E_{XC}[ρ(

Unless the choice of DFT functional produces accurate geometries as compared to high-quality experiments, other predicted properties are likely to be inaccurate, for instance the redox potential. It is worth to be cautious in choosing appropriate DFT functionals for redox potential predictions. Although the B3LYP hybrid (Hartree-Fock-DFT) functional has been found to accurately predict redox potentials for organic compounds, for transition metal complexes, the GGA DFT functional BP86 performed better than the B3LYP one [

In general, basis sets such as the double or triple zeta basis with additional diffuse and polarization functions are sufficient to capture the energetics of redox processes [

Solvation is a crucial factor in redox potential determination. Solvation can be modeled by static or dynamic simulations of explicit water molecules around the ion or complex, by replacing the water by a homogenous dielectric fluid, or by a combination of the above, e.g., by calculating the first or first two hydration spheres with explicit water and the surroundings by a dielectric fluid. Several solvation models for the dielectric fluid approach are available in the literature and incorporated in some quantum-mechanical programs, such as PCM (polarizable continuum model) [

In CPCM solvation calculations, first, a cavity, mimicking the water-free region around the aqueous complex, is built by placing interlocking spheres around each atom or group of atoms of solute. Then, the surface around the cavity is mapped by small regions, called tesserae. Inside the cavity the solute is placed in a vacuum whereas outside the cavity the value of the dielectric constant is equal to the solvent of interest, for instance, ε = 78.39 for water. However, for the solute-cavity description, different radii are available in some programs) (e.g., in

Implicit solvation models such as PCM, CPCM, and the Poisson-Boltzmann finite (PBF) element method were benchmarked for the predicted standard redox potentials of eighty Ru-based complexes in solution using the DFT-HF B3LYP hybrid method [

Explicit solvation by adding explicit water molecules around the solute can describe the hydrogen bonding environment more accurately [

In transition metal coordination chemistry, the field strength of the coordinating ligand on the metal center and the coordination number are the determining factors for the amount of crystal field splitting. The field exerted by the ligands splits up the degeneracy of the metal d-orbitals for TM coordination complexes. The higher this energy difference between the energy levels of orbital is (the former d orbitals now have so-called t_{2g} or e_{g} symmetry), the most likely is a low-spin (LS) configuration of the transition metal. In contrast, for degenerate orbitals or small amounts of crystal-field splitting, Hund’s rule calls for a high-spin (HS) configuration [

Conjugated double bonds between organic multi-dentate ligands that complex metal centers are often involved in redox reactions or change the reduction potential of the metal by modifying their immediate local electronic environment. This phenomenon is termed non-innocence. The fact that the incoming electrons tends to occupy low-lying delocalized ligand-based orbitals leads to either under- or over-estimation of redox potentials. While for experimentalists, the nature of the redox process of metal-bound ligands is often difficult to define, be it either metal-based or ligand-based, the source and sink of the electron density can be more easily and quantifiably tracked using calculations. Spin cross over and ligand non-innocence in redox potential predictions are discussed by Hughes

Spin-orbit coupling (SO) effect is usually not taken into account in most transition metal redox chemistry calculations, as this effect is generally insignificant (and only the spin-orbit coupling difference between the oxidized and reduced complex plays a role). Redox potentials of M(2+/3+) (M = Ru and Os) complexes were investigated using the CASSCF approach. The calculated SO coupling on the predicted redox potentials were found to be −0.07 and −0.30 eV for the Ru and Os complexes, respectively. This shows that even for the second row transition metal redox chemistry, the SO effect is not very significant since SO effect value found for the Ru(2+/3+) redox is −0.07 eV, which is negligible. However, for the fourth row transition metal redox chemistry, SO effects have to be taken into account since the SO value found for the Os(2+/3+) redox is −0.30 eV, which is significant [^{1} electron of the uranyl(V) ion is about −0.31 eV [

Using quantum-mechanical molecular-dynamics simulations, the free energy of a redox process can be calculated. In order to calculate the free energy of a redox process, the thermodynamic integration method [_{Oxd}(1 − η) + E_{Red}(η)
_{Oxd}
_{Red}
_{Oxd} is the energy of the oxidized species (where η = 0) and _{Red} is the energy of the reduced species (where η = 1). The derivative of the energy of the redox reaction with respect to the integration parameter, η can be written in terms of the vertical energy gap, where the vertical energy gap is defined as the difference between the E_{Oxd} and E_{Red} terms.
_{Red} − E_{Oxd} = ∆E

Using the canonical ensemble formalism, integration of the expectation value

More often, only the initial state (η = 0) corresponds to the oxidized state and the final state (η = 1) corresponds to the reduced state are studied according to linear response approximation. However, this has to be validated; on the other hand one could argue that the intermediate states are chemically meaningless entities. Thus, the free energy difference for the redox process can be thermally averaged over the energies of the oxidized and reduced species corresponding to the η values, 0 and 1, respectively.

In addition, there are studies that investigated the redox transformations in which the redox process involves protonation or deprotonation, where three states for the integration parameter (η) are investigated (η = 0, 0.5 and 1) [

According to Marcus theory of electron transfer, the oxidized and reduced species attain a certain configuration favorable for the electron transfer, after the electron transfer these species tend to relax themselves to their equilibrium state. Corresponding free energies for these relaxation processes are reorganization free energies (λ). These reorganization free energies (λ) for the oxidized and reduced species can also be deduced from the calculated reduction free energy of the redox process. The relevant expression to compute the reorganization free energies for the oxidized and reduced species are shown below (Equations (23) and (24)).

After getting the reduction free energy from Equation (22), this can be further modified into reduction potential according to the Nernst equation (see Equation (8)) as explained earlier. Then, the obtained reduction potential has to be referenced with respect to the reference electrode potential, for instance the SHE and this makes the calculated reduction potential to be directly comparable with the experimental reduction potential.

Neutral organic molecules occur in a wide variety of chemistries, such as aliphatics, aromatics, phenols, quinones, amines, and nitro compounds. The redox chemistry of these compounds is interesting in terms of the described approaches, since most organic transformations take place by electron transfer in solution. Electrochemical organic transformations are often used as an efficient way to perform complicated organic syntheses. Various aspects involved in the electrochemical synthesis of organic compounds, for example, mechanism of redox processes, kinetics of electrode reactions, homogeneous or heterogeneous electron transfers, coupled electron transfer processes, have been reviewed [

Molecular and mechanistic-level details of reduction of carbon dioxide (CO_{2}) by organic compounds, for instance by pyridine that reduces CO_{2}, is of fundamental interest. Once these redox processes are properly understood using computational approaches, this knowledge can be applied to reduce CO_{2} emission [

One approach to grasp the theory of one-electron transfer processes is Marcus theory. Based on this theory, the one-electron redox process happens adiabatically and as soon as the electron is transferred from the oxidized species to the reduced species, relaxation of the complex and surrounding solvent molecules are expected to happen. By applying this, several authors have predicted the adiabatic electron affinity of different species, which is equivalent to calculating the energy of oxidized species with an additional electron. However, subsequent structural relaxation was not taken into account. The calculated adiabatic ionization potentials (IP) were also correlated with respect to the available experimental redox potentials.

Quinones are often involved in biological electron transfer and redox reactions. These organic molecules undergo one and two-electron transformations. Redox potentials of quinones were calculated using both the direct and isodesmic method of redox potential predictions. One-electron redox potentials of eight quinones with different substituents were determined using the DFT/B3LYP/PCM approach. A correlation (E_{red} = −2.115 − 12.845E_{HOMO}) between the calculated HOMO orbital energy to the experimental redox values was obtained. Redox potentials were predicted to be within a MUE of 0.03 eV of the experimental values [_{ox} (in eV) = 0.949 + 0.134Σσ, ^{2} = 0.973, where σ is the Hammett constant of the substituent). This study claims that using these empirical correlations, the redox potentials of unknown arylimidazole derivatives and the effect of various substitutions on the redox behavior can be obtained with reasonable accuracy [

One-electron reduction potentials of 116 (_{ox} (in eV) =1.66Σσ + 0.54. Then using this empirical relation, a large number of substitution effects on the reduction and oxidation potentials of quinones can be obtained without much further computational effort. In addition, the calculated electron affinities were plotted against the Hammett constants of the substituents and excellent correlations were obtained [

By employing the B3LYP/PCM method, redox potentials of 270 different organic compounds in acetonitrile solvent were calculated [

Reduction potentials for 74 different organic (cyano aromatics, quinones, flexible pi molecules, polyaromatic hydrocarbons (PAH), heterocyclic amines, and ^{+} reference potential (5.22 eV) in acetonitrile solution. The estimated MUE for the calculated redox potentials was 0.03 eV. The calculated absolute redox potentials plotted against the experimental redox potentials produced an excellent straight line correlation fit and an accurate Fc/Fc^{+} reference potential (5.17 eV) was obtained [

Reduction potentials of 25 cyclic nitroxide (pyrrolidine, piperidine, isoindoline, and azaphenalene) organic compounds in acetonitrile and water solution were calculated using ab-initio methods (G3 and B3LYP) incorporating solvation effects using the PCM solvation model [

Implicit solvation models, such as SM8, SMD, CPCM, IEFPCM, and COSMO-RS combined with the CBS-QB3 method were examined for the redox potential predictions to obtain a suitable solvation model [

Another important field for the application of redox chemistry is DNA bases because the nucleotide bases are the fundamental constituents of DNA. These bases are of two categories, namely purine- and pyrimidine-type bases, in which adenine and guanine are purine bases, whereas cytosine, thymine, and uracil are pyrimidine bases. One-electron oxidation potentials of these DNA bases were estimated computationally using DFT B3LYP and complete basis set (CBS-QB3) methods incorporating solvation effects using the solvation model density (SMD). However, both of these DFT and CBS-QB3 methods were found to underestimate the one electron oxidation potentials of nucleotide bases in acetonitrile solvent by MUEs of 0.33 and 0.21 eV, respectively [

Recently, the oxidation mechanism of guanine and the redox potentials of intermediates along the proposed mechanistic pathways were determined using the DFT B3LYP and CBS-QB3 methods [

Flavonoids are another important class of organic compounds who play the role of antioxidants and biological one- and two-electron catalysts. Reduction potentials of 28 flavins in water solvent with respect to the SHE reference potential (4.44 eV) were calculated using the B3LYP method including solvation effects by employing the CPCM solvation model. The estimated MUE for the calculated redox potentials was 0.06 eV. In addition, substitution effects were systematically evaluated for the redox behavior of flavins. Linear correlations were obtained for the Hammett substitution constants when plotted against the computed and experimental redox potentials. In addition, the calculated HOMO orbital energies with respect the calculated redox potentials revealed an excellent linear correlation [

In this section, we summarize the previous computational investigations on redox potential calculations for inorganic compounds, for example, carboranes and oxo acids, such as chloro-, bromo-, and nitro-oxo acids in solution. Compared to redox calculations on organic compounds, published reports on computational redox calculations of inorganic compounds are few in number.

Carboranes are a type of inorganic cluster compounds which contain carbon, boron, and hydrogen atoms, and often H-atoms are substituted by different groups, for instance, chloride, methyl, ^{+} reference potential in acetonitrile solvent for few carboranes using the B3LYP/PCM method, for which the predicted oxidation potentials were in agreement with the experimental oxidation potentials.

Boron hydrides are structurally similar to carboranes. Recently, reduction potentials for a series of hypercloso boron hydrides, B_{n}_{n}_{12}X_{12} (X = F, Cl, OH, and CH_{3}) [

Employing the hybrid DFT B3LYP method in combination with the PCM method for solvation effects, reduction potentials for chloro-, bromo-, and nitro- oxo acids in acidic and basic environments were calculated with respect to the SHE reference potential [

In this section, we will discuss computational reduction potential predictions for transition metal complexes, from the 3d to 5d series, and actinyl complexes (5f series) in aqueous and non-aqueous solution.

First redox potential calculations of the 3d transition metal aqua complexes are discussed. Uudsemaa

Reduction potentials for organic compounds and TM complexes (including metallocenes and coordination complexes) were calculated with respect the SCE reference potential (4.188 eV) using the PCM solvation model for solvation effects combined with the hybrid DFT B3LYP method [

The hybrid DFT B3LYP method with the solvation effects included using the PB continuum solvation model predicted the reduction potentials of 95 octahedral 3d TM complexes coordinating with different ligands [

The hybrid DFT B3LYP functional combined with the IEF-PCM (integral equation formalism-polarizable continuum model) solvation model resulted in a good prediction of reduction potentials for a series of TM complexes (including metallozenes, metallozenedichlorides, bipyridyine complexes, carbonyl complexes, and maleonitriledithiolate complexes) in non-aqueous solution such as dichloromethane (DCM), acetonitrile, and dimethylformamide (DMF) only if corrected for an inaccurate reference potential [^{+} reference potential. In contrast, the GGA BP86 and the PBE functionals performed better than the hybrid DFT functional and the MUE of the predicted reduction potentials was ~0.30 eV smaller than the MUE calculated for the B3LYP predicted reduction potentials. All the predicted B3LYP reduction potentials in this study had to be shifted by a constant value of 0.43 eV to improve the linear correlation between the calculated and the experimental redox potentials [_{3}CN solvent were underestimated and systematic shifts of −0.82 and −0.53 eV had to be added to reproduce the experimental oxidation and reduction potentials, respectively. In contrast, the GGA BP86 method predicted the redox potentials with a MUE of 0.12 eV [

Recently, the reduction potentials of group eight (Fe, Ru, and Os) metal octahedral complexes were calculated using two different DFT functionals, PBE and M06L combined with the COSMO-RS and SMD solvation models, respectively [

A new approach for the redox potential calculation of TM complexes has been recently proposed. This approach is called as the pseudo counter-ion solvation, which is basically described by adding an oppositely charged sphere (

To elucidate the ligand additive effects on the redox properties using the B3LYP method for computations [^{−}, Cl^{−}, H_{2}O, CH_{3}CN, and N_{2} ligands. The linear regression fitting method is then applied to correlate the adiabatic oxidation energy, calculated for the oxidation reaction (Equation (25)), with the number of CO ligands and the straight line expression is shown below (Equation (26)).
_{n}_{6−n}]^{2+Q} → [M(CO)_{n}_{6−n}]^{3+Q} + e^{−}
^{−}, Cl^{−}, H_{2}O, CH_{3}CN and N_{2}).
_{adiabatic} = I + S∙[_{adiabatic} term refers to the adiabatic energy difference for the oxidation process, the term “I” refers to the intercept, the term “S” refers to the slope of the straight line and the term

Redox potentials for 30 octahedral tungsten-alkylidyne complexes with a variety of different coordinating ligands were calculated by employing the hybrid B3LYP DFT method [_{xy}

Actinides are 5f elements. Except the protactinium (Pa), uranium (U) and thorium (Th) elements, the remaining elements are manmade elements. The interests to study the redox chemistry of actinides have been growing over the years [

More importantly, interactions of these elements with the geologically most abundant minerals, transport, speciation, precipitation, and migration behavior have to be understood and indeed these processes are complicated processes. Because minerals present in geo-sphere can promote different reactions such as precipitation, adsorption, reduction, and surface mediated chemical reactions. In general, mineral surfaces are more chaotic and actively involved in bio mineralization processes and surface mediated reactions. Redox active minerals will help us to design more effective remediation strategies for these actinide elements, recycling process and reuse in future.

Reduction potentials of actinyl (U, Np, and Pu) aqua complexes were calculated using the DFT B3LYP method with respect to the calculated SHE reference. The An (An = U, Np, and Pu) atoms were described by large core (LC) PPs and basis sets, this lead to an inadequate description of valence states of these elements and resulted in an overestimation reduction potentials by ~2.5 eV relative to experimental values. Even though the calculated reduction potentials are overestimated, spin-orbit coupling interaction and multiplet effects were found to be significant to get the experimental trend [

High level _{2}]^{2+} + Fe(II) → [UO_{2}]^{+} + Fe(III) [

The actinyl(VI, V) aqua complexes are linear oxo-cations with water molecules coordinating to the metal center in the equatorial plane whereas the An(IV, III) aqua complexes do not have axial oxygen atoms and the coordination number can go up from 8 to 10. The following overall redox reactions including the SHE reference were employed to calculate the actinyl(V) to An(IV) and An(IV) to An(III) redox potentials [_{2}(H_{2}O)_{5}]^{+} + ½H_{2} + 3[H_{3}O]^{+} → [An(H_{2}O)_{8}]^{4+} + 2H_{2}O
_{2}O)_{8}]^{4+} + ½H_{2} + H_{2}O → [An(H_{2}O)_{8}]^{3+} + [H_{3}O]^{+}

Austin _{2}]^{2+/+} (An = U, Np, and Pu) and [AnO_{2}(L)_{n}^{m} (where L = H_{2}O, OH^{−}, Cl^{−}, AcO^{−}, and CO_{3}^{2−})) redox potentials in aqueous solution with respect to the calculated SHE reference electrode potential using DFT methods. Solvation effects were incorporated with the CPCM solvation model and the solute cavities were described using the universal force field (UFF) radii [

Moreover, a series of DFT functionals ranging from GGA to hybrid DFT functionals and recently developed M0x functionals from Truhlar

Reduction potentials for actinyl(VI/V) in aqueous solution and uranyl(VI/V) complexed with organic multi-dentate ligands in a range of non-aqueous solutions, such as DMSO, dimethylformamide (DMF), dichloromethane (DCM), acetonitrile (ACN), and pyridine, using the DFTB3LYP and M06 functionals combined with the CPCM/UAKS solvation model were investigated. The importance of reference electrode potential, solute cavity description and the effect of explicit solvation were elucidated. In addition, the effect ionic strength on the redox potentials was studied, however this effect was found to be negligible. The effect of counter ions on the calculated redox potentials were also studied, because of strong binding of counter ions with the metal-bound anionic ligands did not produce any noticeable trend. The reason for this could be attributed to the tight binding and the employed static model, whereas in solution the counter ions are dynamic and solvated all the time. Both the direct and isodesmic methods applied for the redox potentials predictions in solution. The results suggests that the reduction potentials of actinyl complexes in solution can be predictable with in ~0.1–0.2 eV of experimental values using the hybrid DFT B3LYP/CPCM/UAKS method. The isodesmic model of redox potential prediction gets upper hand over the direct method; this is mainly due to cancellation of errors. It should be noted that the reference electrodes considered were the SHE and Fc/Fc^{+} for aqueous and non-aqueous solutions, respectively. Moreover, the uranyl(VI/V) redox was found to be altered by introducing relevant substitutions into the periphery of the uranyl bound multi-dentate ligand. Electron-releasing substituents increase the electron density around the metal center which in turn increases the uranyl(VI/V) redox potentials, while electron-drawing substituents decrease the reduction potential. A direct correlation between the calculated redox potentials and the Hammett parameters of the peripheral ligand substituent groups were established, which could be used for ligand design, e.g., to control separation chemistry of actinides [

Electron affinities of a series of organometallic complexes, [Cp*_{2}UX_{2} (X = BH_{4}, Me, and NEt_{2}Cl), Cp3UX (X = Cl, BH_{4}, SPh, S^{i}Pr, and O^{i}Pr), L_{2}U(BH_{4}) (L = Cp_{2}, tmp_{2}, tBuCp_{2}, Cp*tmp, and Cp*_{2}), and L_{3}UCl (L = Cp, MeCp, TMSCp, tBuCp, and Cp*)] were calculated using the DFT BP86/ZORA method and the solvation effects were included with the COSMO solvation model. The calculated electron affinities in THF solvent were highly correlated with experimental U(IV/III) reduction potentials. Similarly, the calculated LUMO and HOMO orbital energies showed linear correlation with experimental reduction potentials. This implies that the electronic nature of the ligand plays a significant role in altering the redox properties of U(IV) organometallic complexes in THF solution [

In

Previous calculations have used the hybrid DFT B3LYP [

Actinyl ion aqua model complexes contain five water molecules in their equatorial plane. Although, four and six water molecules in the equatorial plane of actinyl ions were computationally proposed [_{2}(H_{2}O)_{5}]^{2+} + e^{−} → [AnO_{2}(H_{2}O)_{5}]^{+}

Pyrite is an iron-disulfide mineral with a bulk formula of FeS_{2}. A small stoichiometric cubic pyrite cluster of molecular formula Fe_{4}S_{8} is used to examine pyrite-actinyl interactions and effects of pyrite on the reduction potentials of actinyl aqua complexes are investigated. The Fe_{4}S_{8} pyrite cluster model has four Fe(II) ions, and one of the Fe(II) is attached to a [AnO_{2}(H_{2}O)_{5}]^{2+} aqua complex through one of the actinyl oxygen atom, a so-called cation-cation type interaction. Using these cluster models, reduction potentials for the pyrite attached actinyl aqua complexes are calculated. In our calculations, the spins of these cluster models are treated as high-spin (HS) configurations, because the Fe(II) ions present in the pyrite fragment are coordinatively unsaturated. The Fe_{4}S_{8} coordinates of the super-cluster (pyrite-actinyl) were kept frozen in all optimizations in order to minimize the computational cost, and the structural relaxation of the pyrite moiety is expected not to have significant impact on the calculated redox potentials of pyrite-actinyl complex cluster models. A half-cell reaction used to calculate the reduction potentials of pyrite-actinyl cluster is given in Equation (30) and the optimized structure of [Fe_{4}S_{8}-UO_{2}(H_{2}O)_{5}]^{2+} cluster model is shown in _{4}S_{8}-AnO_{2}(H_{2}O)_{5}]^{2+} + e^{−} → [Fe_{4}S_{8}-AnO_{2}(H_{2}O)_{5}]^{+}

DFT-optimized geometry of pyrite-uranyl cluster ([Fe_{4}S_{8}-UO_{2}(H_{2}O)_{5}]^{2+}) model.

Mackinawite is a layered iron sulfide, its general formula is FeS. In this study, a stoichiometric sheet-like cluster of molecular formula Fe_{8}S_{8} is used as a model for mackinawite semiconducting mineral. The Fe(II) atoms in the cluster are treated as low spins as in bulk mackinawite. Although this cluster model contains coordinatively undersaturated Fe(II) atoms at corners, we have not explored yet the relative thermodynamics of high-_{4}S_{8}-UO_{2}(H_{2}O)_{5}]^{2+} cluster model is shown in _{8}S_{8}-AnO_{2}(H_{2}O)_{5}]^{2+} + e^{−} → [Fe_{8}S_{8}-AnO_{2}(H_{2}O)_{5}]^{+}

By applying the thermodynamic cycle scheme, free energies for the above one-electron reduction half-cell reactions (Equations (29)–(31)) can be evaluated in gas and aqueous solution phases. From the reduction free energies, utilizing the Nernst relationship between the free energy and electrode potential, absolute reduction potentials were obtained. The calculated absolute reduction potentials are then referenced with respect to the standard hydrogen electrode (SHE) reference potential (4.44 eV). The calculated reduction free energies are not adjusted by zero-point energy (ZPE) and free-energy (FE) corrections, since these additions have only a minor influence on the potential. Computationally obtained one-electron reduction potentials in aqueous solution for different model complexes are collated in

DFT-optimized geometry of mackinawite-uranyl cluster ([Fe_{8}S_{8}-UO_{2}(H_{2}O)_{5}]^{2+}) model.

Experimental and calculated reduction potentials of actinyl aqua model complexes in aqueous solution (eV).

Models | Experimental | [AnO_{2}(H_{2}O)_{5}]^{2+/+} |
[Fe_{4}S_{8}-AnO_{2}(H_{2}O)_{5}]^{2+/+} |
[Fe_{8}S_{8}-AnO_{2}(H_{2}O)_{5}]^{2+/+} |
|||||
---|---|---|---|---|---|---|---|---|---|

Opt. ^{a} |
Adiabatic ^{b} |
Opt. | Adiabatic ^{b} |
Opt. | Adiabatic ^{b} |
||||

E_{0} |
E_{0}+SO^{c} |
E_{0} |
E_{0}+SO^{c} |
E_{0} |
E_{0} |
E_{0} |
E_{0} |
||

U | 0.088 | −0.173 | 0.137 | −0.504 | −0.194 | 0.017 | 0.145 | −0.256 | 0.234 |

Np | 1.159 | 0.820 | 1.210 | 0.471 | 0.861 | 0.036 | 0.154 | −0.267 | −0.699 |

Pu | 0.936 | 1.332 | 1.422 | 0.975 | 1.065 | 0.036 | 0.163 | 3.174 | −0.658 |

MUE | - | 0.33 | 0.20 | 0.44 | 0.24 |

Notes: ^{a} reduction potentials calculated based on optimized geometries; ^{b} reduction potentials calculated based on adiabatic approach; ^{c} spin-orbit(SO) interaction corrections are −0.31, −0.34, and −0.09 eV for uranyl, neptunyl, and plutonyl redox reactions, respectively taken from Hay

By applying a similar approach to Steele

Calculated redox potentials for pyrite-actinyl cluster models revealed a change in redox potentials for all the actinyl aqua complexes. Interestingly, the difference between the adiabatic and the full optimization redox calculation do not show significant variation, the values only differ by ~0.1 eV. Our results confirm that the redox-active semiconducting pyrite mineral surface plays a critical role in altering the redox potential of actinyl aqua complexes. Although there is no experimental proof for the redox potentials change in the presence of minerals for Np and Pu complexes, recent powder-micro electrode study by Renock

The semiconducting mackinawite surface has shown a significant influence on the redox behavior of actinyl aqua complexes. The adiabatic method results in a reduction potential of 0.234 eV for the mackinawite-uranyl cluster. In contrast, the full optimization method produced −0.256 eV, which is ~0.5 eV lower than the former one, which is due to the relaxation of the reduced species. For the mackinawite-neptunyl cluster, the scenario is different. The adiabatic method produces a ~0.5 eV smaller reduction potential compared to the full optimization method. However, the full optimization method results in similar redox potentials (−0.256 and −0.267 eV) for both the mackinawite adsorbed uranyl and neptunyl cluster models. The aqueous plutonyl complex on mackinawite behaves differently than the U and Np equivalents. The full optimization method produces a reduction potential of 3.17 eV for the mackinawite-plutonyl cluster model which is ~2 eV higher than the plutonyl(VI/V) reduction potential. However, the adiabatic method produced a reduction potential value of −0.658 eV which is close to the value obtained for the mackinawite-neptunyl reduction potential, −0.699 eV. The reason for these diverging results of aqueous actinyl complexes on mackinawite is unclear. When the role of pyrite and mackinawite in altering the redox behavior of actinyl aqua complexes are compared, mackinawite has significantly more impact (~0.2–0.3 eV) than the pyrite and this inference is based on the negative reduction potentials obtained for the mackinawite adsorbed actinyl aqua complexes.

We showed that the stoichiometric cluster model approach can be applied to study the interaction semiconducting mineral surfaces on the redox properties of surface-adsorbed actinyl aqua complexes. Our computational approach precisely reproduced the experimental uranyl(VI/V) reduction potential on pyrite mineral surface (experimental value is 0.003 eV and our calculated value is 0.017 eV). Computational investigations are ongoing to underpin the surface-mediated redox process of adsorbed actinyls on semiconducting mineral surfaces such as iron-sulfides and iron-oxides.

Li-based battery materials have been widely used in electronics applications. Myriad of Li-based materials have been synthesized and the important properties of these materials such as the electric conductivity, chemical composition, thermal stability, diffusion, efficiency, and service life have been characterized over the years [^{+} ion intercalation potentials for Li ion battery materials, such as layered Li_{x}_{2} (M = Co and Ni), Li_{x}_{2}, olivine-structured Li_{x}_{4} (M = Mn, Fe, Co, and Ni), and spinel-like Li_{x}_{2}O_{4}, Li_{x}_{2}O_{4} were determined using

Insertion of Li into a transition metal oxide can be expressed as:
_{x}_{x}_{y}_{x+}_{x}_{y}

MO is the transition-metal oxide material. The relation between the voltage of the cell and the Li chemical potential is given by the following equation.
_{x}_{1}MO_{y}_{x}_{2}MO_{y}_{x}_{1}MO_{y}_{x}_{2}MO_{y}

Conventional GGA DFT studies were found to underestimate the experimental Li intercalation potentials for lithium oxide materials, such as Li_{x}_{2} (M = Co and Ni), Li_{x}_{2}, and Li_{x}_{4} (M = Mn, Fe, Co, and Ni), and the calculated MUEs for the predicted Li^{+} ion intercalation potentials were 0.75 eV less than the experimental Li^{+} ion intercalation potentials. This underestimation was attributed to the lack of cancellation of self-interaction. In contrast, the use of the DFT+U method was found to predict the Li^{+} ion intercalation potentials for these materials to be within 0.15 eV of the experimental intercalation potential. Moreover, a recent study using the newly-developed hybrid DFT functional HSE06 has predicted the Li^{+} ion intercalation potentials for the above mentioned materials as accurate as the DFT+U methods and reproduced the experimental intercalation potentials within a MUE of 0.19 eV [_{4} and Li_{2}CoSiO_{4} materials were first calculated using the DFT+U approach and later synthesized and characterized by experimental studies. The theoretically predicted Li ion intercalation potentials have been found to be in excellent agreement with the experiments, which shows that the material design using prior

Similar to the Li based materials, Ce(4+/3+) reduction free energies of ceria were calculated using classical molecular-dynamics simulations [^{n}^{+} species forms M^{(n−1)+}, and the overall free energy difference for the formation of reduced species can be calculated. The authors have used a Buckingham potential and shell-models to describe the interactions in CeO_{2}. Using the following reaction scheme, the free energy for the Ce (4+/3+) reduction process was estimated.

According to the Kröger–Vink notation,^{4+} ion in its lattice position and ^{3+} and the next term designates an oxygen vacancy. Moreover, the Ce(4+/3+) reduction free energies in mixed oxides such as CeO_{2}–MO (M = Zr, Ca, Mn, Ni, and Zn) and CeO_{2}–M_{2}O_{3} (M = Sc, Mn, Y, Gd, and La) [_{2}–MO mixed oxides. Similarly, in CeO_{2}–M_{2}O_{3} mixed oxides, an increase in trivalent metal was found to predict a decrease in the Ce(4+/3+) reduction free energies. Although reduction free energies for the Ce (4+/3+) reduction process in cerium oxide materials were modeled using classical molecular dynamics approaches, these studies did not include reference electrode systems.

Oxidation and reduction potentials of photo catalytic semiconductors were recently determined using DFT methods [^{+} + 2e^{−} → H_{2} (_{2} + 4H^{+} + 4e^{−} → 2H_{2}O (

The thermodynamic parameters, for example the Gibbs free energies of the water redox reaction species such as H_{2}, H^{+}, O_{2}, and H_{2}O are available in literature [

The next task is to calculate the reduction and oxidation potentials for a semiconductor with respect to these well-known water redox reactions. As an example, naturally-occurring and semiconducting zinc sulfide (ZnS) [

Reduction:
_{2} → Zn + H_{2}S

Oxidation:
_{2}O → ZnO + S + H_{2}

The conceptual basis used in this study is relatively simple, “obtain the Gibbs free energies of formation from already-available databases and calculate the free-energy (∆G) for oxidation and reduction reactions with respect to the above water redox reactions (Equations (36) and (37))”. It should be noted that the oxidation and reduction reaction proposed with these well-known oxidation and reduction reaction of water should produce thermodynamically stable compounds. Otherwise obtaining thermodynamic parameters from experimental data bases is not possible, and the relevant energies have to be calculated from

Now, we discuss computational studies on the calculation of redox potentials of surface-grafted substituted ferrocenes (SFCs). In order to make the SFCs, ferrocene was functionalized with ethyl, vinyl or ethynyl groups and then attached to Si(100) surfaces. Using electrochemical methods, redox potentials of these adsorbed SFCs were determined. The DFT B3LYP method was used to calculate redox potentials of the surface-functionalized SFCs in CH_{3}CN solution, and the solvation effects were included using the PCM solvation method. The Si(100) surface-functionalized SFCs were modeled using mono and di hydrogenated Si surface cluster models. The calculated redox potentials are in good agreement with the experiments [

In this review article, we have tried to provide a perspective view on various computational redox potentials predictions methods and outlined the frequently used methods available in the literature and briefly discussed the methods and the applicability and transferability of the approaches to other complexes, for instances, to transition-metal complexes, actinides, semiconductors, mineral surfaces, and surface-bound species. In order to give some guidance on the suitability of computational settings for the calculation of reduction potentials, we summarize some of these computational parameters in the following.

^{−}) have been found to produce large systematic errors in the prediction of redox potentials of actinyl aqua complexes in solution.

We hope that the methods described in this review serve as a helpful tool to choose appropriate computational methods for redox-potential predictions in areas of chemistry, biology, and mineralogy. These techniques may also help explore wherever redox chemistry is playing an important role, for instance, in geochemistry and biochemistry, in which the transport of electrons is crucial for many life processes. In addition, there are areas of cross-fertilization for example in designing battery materials with more efficiency and cost effectiveness than currently used materials or in understanding the corrosion of materials.

Moreover, this study has profound implications, in particular to the environmental science where a deeper understanding of redox reactions taking place on semiconducting mineral surfaces would help us design new remediation methods for radioactive actinide materials. For example, reductive immobilization processes slow the transport of these elements into the geo- and hydrosphere down and prevent long-term damages.

Rather than applying conventional approaches for the materials synthesis without any prior knowledge about the materials is often a black-box approach, in which success rate is serendipitous. Utilizing the gained knowledge from credible

This study was funded by the U.S. Department of Energy, Office of Basic Energy Sciences, Heavy Element Chemistry program under Grant DE-FG02-06ER15783. In addition, we would like to express our sincere thanks to the two anonymous reviewers for their constructive comments and suggestions for the improvement of this manuscript.

Both Krishnamoorthy Arumugam and Udo Becker were involved in writing and revising all parts of the manuscript. While Krishnamoorthy Arumugam did the bulk of the calculations, the latter were inspired and revised by Udo Becker.

The authors declare no conflict of interest.

_{7}proteins proposed from DFT calculations

_{2}(L)

_{n}]

^{m}

_{2}O, Cl

^{−}, CO

_{3}

^{2−}, CH

_{3}CO

_{2}

^{−}, OH

^{−}) in aqueous solution, studied by density functional theory methods

_{2}in products of U(VI) reduction

_{2}]

^{2+}by cytochrome

_{7}proteins proposed from DFT calculations

_{2}]

^{2+}(An = U, Np, Pu) in aqueous solution, and by Fe(II) containing proteins and mineral surfaces, probed by DFT calculations

_{3}CN

_{2}oxidation and production pathways catalyzed by nickel molecular electrocatalysts

^{3+}|Ru

^{2+}reduction potential

_{2}

^{2+}and AnO

_{2}

^{+}aquo complexes for An = U , Np , and Pu

^{2+}/Ru

^{3+}redox reaction: The Marcus perspective

_{a}, redox reactions and solvation free energies

_{a}for benzoquinone from density functional theory based molecular dynamics

_{2}

_{11}Me

_{10}

^{−}anions

_{n}H

_{n}(

_{12}X

_{12}(X = F, Cl, OH, and CH

_{3})

_{a}values of hydrated transition metal cations by a combined density functional and continuum dielectric theory

_{2}

^{2+}/AnO

_{2}

^{+}(An = U, Np, Pu, Am) and Fe

^{3+}/Fe

^{2+}couples

_{3}UX

_{*2}UX

_{2}

_{2}(H

_{2}O)

_{n}

^{2+}, NpO

_{2}(H

_{2}O)

^{+}, and PuO

_{2}(H2

_{2}O)

^{2+}complexes (

_{4}: Surface energy, structure, Wulff shape, and surface redox potential

_{2}–ZrOO

_{2}catalysts

_{2}–ZrOO

_{2}solid solutions

_{2}–MO and CeO

_{2}–M

_{2}O

_{2}mixed oxides: A computer simulation study