Static theoretical investigations of organic redox active materials for redox flow batteries

New efficient redox flow batteries (RFBs) are currently of great interest for large-scale storage of renewable energy. Further development requires the improvement of the redox active materials. Quantum chemical calculations allow the screening of large numbers of redox active molecules for required static molecular properties. In particular, redox potentials are calculated in high-throughput studies. In addition, calculations of solubility and reactivity and in-depth electronic structure analysis are performed for smaller numbers of molecules. In this review, we provide an overview of the static theoretical investigations carried out on the known classes of molecules that are considered as redox active materials in RFBs. We will focus on electronic structure methods such as density functional theory and wave function-based methods. Furthermore, investigations using the increasingly important machine learning techniques are presented. For each class of redox active molecules considered, significant theoretical results are presented and discussed. In addition, the different quantum chemical approaches used are examined, in particular with regard to their advantages and limitations. Another focus of this review is the comparison of theoretically predicted results with available experimental studies. Finally, future challenges and trends in the theoretical studies of redox active materials are highlighted.


Introduction
The economic need for 'green' energy is driving investment in renewable and environmentally friendly energy sources such as solar and wind energy [1].Current international conflicts are accelerating this transition as many countries begin to strive for energy self-sufficiency [2].Local energy production and storage would also have a positive impact on the social and economic development [3].The social impact is particularly strong for small-scale local energy production through photovoltaics [4].
One of the main problems with renewable energy sources is their volatility (seasonal and daily changes in solar radiation, wind, etc).Therefore, the energy produced needs to be stored during periods of high output in order to be available during periods of low output [5].A switch to renewable energy will only be possible with suitable energy storage solutions [6].First, the energy storage solutions need to be cost effective.Second, they need to vary in size to cover everything from solar panels on a single-family home to industrial power generation.From this point of view, redox flow batteries (RFBs) have great potential because their power and capacity are independently scalable [7][8][9].
Historically, the first RFBs were based on reactions between transition metal ions in different oxidation states, such as vanadium or chromium.These batteries utilized V 3+ /V 2+ or Cr 3+ /Cr 2+ on the negative electrode (anode) side and VO 2 + /VO 2+ or Cr 3+ /CrO 2− 4 on the positive electrode (cathode) side in aqueous solution under acidic conditions [10].However, due to the cost and toxicity of these transition metals, there is increasing interest in organic redox active materials for RFBs.Aqueous RFBs with organic (redox) active materials are of particular interest because they are potentially more environmentally friendly than systems based on transition metal compounds [11].Furthermore, the organic components may be synthesized from renewable resources in the long term [12,13].
A number of different classes of organic materials are currently explored for use as redox active materials in the two liquid phases (anolyte and catholyte) of RFBs.Besides organic molecules, organometallic complexes have great potential.The properties of both can be altered by chemical modification of the organic molecules.An important part of developing competitive, organic based RFBs is finding active materials with optimal properties.Redox-active materials determine the capacity and stability of the battery and thus essentially determine the parameters of the RFB.Volumetric capacity, redox potentials, cell voltage, solubility and chemical stability are such fundamental parameters that depend on the redox active materials of the RFB.Some of these parameters can be systematically studied using static theoretical methods [14,15].
Quantum chemical investigations of organic redox molecules for RFBs have two main objectives.First, quantum chemical calculations are used to gain a deeper understanding of experimentally studied redox active materials.In this context, the geometric structure, electronic structure, and possible side reactions of the redox active materials are investigated [16].Second, many molecules are screened for desired properties.Computational screening is an advantageous first step compared to time-consuming and expensive laboratory synthesis [16,17].However, it is beneficial for the computational predictions if experimental benchmark data are available for some molecules of the compound class under investigation.
Density functional theory (DFT) is usually the method of choice in these studies.With DFT, properties such as the redox potential and solubility can be predicted in a fast and reliable manner.In addition to DFT, machine learning (ML) tools are also starting to be used today.It should be noted that a large number of different types of organic compounds also give rise to a large number of electronic structures with different electronic affinities, chemical bonding types and molecular energy levels.In some cases, these cannot be accurately described by a single theoretical approach [18,19].
This review focuses on theoretical calculations of important classes of organic redox active materials considered for potential use in RFBs [20].We present and discuss the following classes of active materials: (1) stable organic radicals, (2) quinones, (3) nitrogen-containing redox active materials, (4) heterocycles containing nitrogen and sulphur, (5) pyrylium based systems.Besides organic molecules, organometallic complexes (6) are also included into the review.In many cases, there is an overlap between the different groups of active materials (1)- (5).Here we classify the compounds according to their electrochemical and structural properties.The basic structures of the organic redox active materials, their redox reactions, and important modifications as well as their design are discussed from a theoretical point of view.The available results from quantum chemical calculations of the geometric and electronic structures as well as the redox potentials of catholytes and anolytes are evaluated.The advantages, disadvantages, and existing problems of the presented theoretical results are discussed.Finally, a general overview of the different theoretical methods is presented.The applicability, advantages, and disadvantages of the different theoretical methods for different classes of redox-active material are discussed.An overview of the classes of redox-active molecules and properties considered, as well as the methods used in theoretical studies, is shown in figure 1.
All redox potentials presented in this review are given against the standard hydrogen electrode (SHE).If the redox potential is given against a different electrode in the original publication, then the potential was recalculated.Due to possible inconsistencies with different reference electrodes, we present the SHE values.The conversion data in table 1 were used to recalculate to SHE values.

Basics of organic RFBs
RFBs consist of three major technical components.The two electrolyte tanks, in which the catholyte and anolyte is stored separately, and the electrochemical cell in the core of the RFB, which is used for electrochemical conversion.The electrochemical cell consists of the two electrodes, mostly carbon felts or plates, and a separator that divides the catholyte and anolyte compartments (figure 2).The electrolytes (catholyte and anolyte) circulate between the tanks and the electrochemical cell.Catholyte and anolyte are electrolytes containing the solvated redox active materials with positive or negative potential, respectively.A supporting electrolyte is also included to increase the ionic conductivity.The redox active materials are molecules that can be reversibly oxidized and reduced during the electrochemical conversions in the RFB.The separator is usually a membrane that is permeable to the supporting electrolyte but impermeable to the redox active material.RFBs represent systems between a fuel cell and a conventional battery, allowing independent scaling of the power and capacity of the RFB.If the anolyte and catholyte contain the same active material, the system is called symmetrical [7].804 [24] The voltametric energy density W V of a RFB is the product of the voltage U (difference between two half-cell redox potentials of the electrolytes) and the volumetric capacity C V of the electrolyte, see equation (1): (1) The volumetric capacity of a material depends on the solubility of the redox active molecules and the number of electrons transferred.The number of electrons transferred is specific to each electrochemical reaction.The energy density is calculated from the redox potentials and concentrations of the two active materials.The voltage is the difference between the redox potentials of the anolyte and catholyte.The redox potentials can be systematically predicted from quantum chemical calculations of the redox reactions of the redox active molecules [15].In addition to the determination of the redox potential, the calculation of the thermodynamically stable form of reactant and product in solution is very important.These can be used to take into account different effects such as solvation of the redox active molecule, and side reactions with the solvent or the conductive salt.For example, the degree of protonation in aqueous electrolytes strongly influences the redox potential and possibly also the reactivity or stability [25].
Furthermore, kinetic parameters such as reaction rate constants, diffusion coefficients and the liquid phase thermodynamic equilibrium are important characteristics for RFBs as they influence the resulting power.However, calculating them from first principles requires computationally demanding approaches such as ab initio molecular dynamics simulations, the Monte-Carlo method [26], or the calculation of potential energy surfaces (PESs) and transition states according to Marcus theory [27][28][29].The theoretically reliable and fast predictable properties of the model systems address important practical parameters of RFBs but are of course not a description of the real battery as a whole.
The three basic efficiency parameters of an RFB experimentally used are the Coulomb efficiency η C , the voltage efficiency η V , and the energy efficiency η E .The first two denote the ratio between charge and discharge capacities and voltages, respectively, while η E is their product [7].Typical η E values for RFBs range from 50% to 90% [30].These parameters are experimentally determined and usually show the deviation from the 'ideal' theoretically predicted voltages and capacities.Compared to Li ion batteries, RFBs typically have a lower voltametric energy density in the order of 20-50 Wh l −1 [31].However, the storage of the redox active material in the liquid phase in separate reservoirs leads to an advantage in scalability and the unique properties of these systems, which makes them particularly attractive as large-scale energy storage devices for renewable energy sources.The liquid state of the redox active materials also allows the use of various colloidal systems such as sols and emulsions in electrolytes, which provide technological flexibility [32,33].Quantum chemical modeling of colloidal systems is much more difficult and time-consuming due to the system size and environmental effects [17,34].Therefore, the present review focuses on single molecules in solutions as redox-active materials.

General overview of organic redox-active materials
An important advantage of organic molecules as redox active materials or as a part of a redox active system is that their properties can be tailored.Solubility, kinetic properties, and redox potentials can be adjusted by modifying the basic organic structure, for example by introducing certain functional groups [35,36].This allows the properties of organic RFBs to be controlled and tuned.
Aprotic or protic organic liquids as well as water are of interest as solvents for the redox active materials.Aprotic organic solvents have better electrochemical stability and a larger potential window than protic solvents [37].With these solvents, organic RFBs can be constructed with higher energy densities than in aqueous systems because redox couples with higher voltage differences can be utilized.However, the ionic conductivity in organic solvents is much lower than in aqueous solutions, which limits the applicable current densities [7].Aqueous electrolytes are potentially more environmentally friendly [11] than organic ones and are associated with lower costs, rendering them particular interesting in current research [30].The lower cell voltage and therefore energy density of aqueous systems can be overcome in stationary RFB applications by increasing the volume of the electrolyte [38].In addition, aqueous systems usually have higher conductivity than aprotic organic systems, and by this a higher power density [7].
A typical disadvantage of organic active materials compared to metal ion based RFBs is the limited long-term stability of the electrolytes.Side reactions of the organic molecules lead to the degradation of the active material and thus to the ageing of the electrolytes, which is one of the major problems of organic RFBs [30,39].
In the following chapters, we will first present, evaluate and classify the commonly used quantum chemical methods for theoretical investigations of organic redox-active materials.Then, the most promising classes of organic redox-active materials, for which theoretical calculations have also been carried out, are discussed.For each class considered, the best-known and most representative examples are presented.

Calculating properties of organic redox active materials 3.2.1. Theoretical calculation of the redox potential
In the first step of molecular engineering and design, candidates for organic redox-active materials can be systematically predicted in silico using methods from quantum chemistry.Quantum chemistry offers a variety of computational methods and approaches for ab initio and first principles studies of redox active materials.Artificial intelligence (AI) and ML can also be used to predict molecular properties.The training sets for AI and ML can in turn be calculated using quantum chemical methods.Ghule et al showed that these approaches can be used to calculate molecular properties with similar accuracy to quantum chemical calculations, but in much less time [40].Only recently have researchers begun to apply ML tools for the investigation of redox active materials.The corresponding publications are highlighted in this review, but so far, the majority of investigations utilize quantum chemical methods.
The computational methodology for studying redox active materials by quantum chemical methods typically involves three steps.First, a molecular model system is constructed with a given number of atoms arranged in a guess for the minimum energy structure.Second, the total electronic energy of the system is calculated using one of the methods discussed in the next chapter, optimizing the structure until the minimum energy is reached.Finally, the free energy is calculated, taking into account temperature, concentration, and vibrational and rotational degrees of freedom.
The choice of the appropriate model system is a crucial step in this process.Considerations such as the chemical state of the molecules under specific external conditions and effects such as protonation, and isomerization must be taken into account.The model system should reflect the most realistic structure of the system being studied under the given conditions.If multiple conformers of the molecules are present in solution, these may also need to be considered.Typically, the redox potential based on the Gibbs free energy, the solubility expressed by the Gibbs free energy of solvation, and the stability by considering side reactions or descriptors are calculated by quantum chemical calculations [41].From these data, general statements for the design of the redox active molecule are then identified.Furthermore, novel redox active materials can be predicted.Other possible reference parameters of the redox properties are energies of the highest occupied molecular orbitals (HOMOs) and lowest unoccupied molecular orbitals (LUMOs), or ionization energies and electron affinities [19,[42][43][44][45].A detailed review of calculations for electrochemical reactions has been published by Marenich et al [29].A general equation of the standard redox potential is given by: where ∆G reac 0 is the Gibbs free energy of the electrochemical reaction, F is the Faraday constant, n is the number of electrons transferred and E 0 SHE is the potential of the SHE.The SHE potential in equation ( 2) is known and accepted by IUPAC as 4.44 ± 0.02 V at 298.15 K [46].It should be noted that reference values of 4.05-4.42V have been reported from different approaches [29].For quantum chemical calculations, the value for E 0 SHE depends on which Gibbs energy of solvation of a proton was used to parameterize the implicit solvent model [47].E 0 SHE = 4.28 V is considered by many quantum chemical studies as the accurate value for the parameterization of implicit solvent models [29,47,48].This value does not include the contributions from the surface potential of water.In chemically balanced single-phase reactions these contributions cancel each other out.If the contribution of surface potential needs to be included, E 0 SHE = 4.42 V can be used [29].Newer solvent models such as SM8 and solvation models based on density (SMD) utilize E 0 SHE = 4.28 V. Older but still common implicit solvent models use other values such as E 0 SHE = 4.47 V for polarized continuum model (PCM) [49] and E 0 SHE = 4.34 V for SM6 [50].For the solvent models COSMO, COSMO-RS and DCOSMO-RS the Gibbs energy of solvation of a proton has not been used for parameterization [51,52].Therefore, it is currently unclear which E 0 SHE should be applied to them.In these cases, the authors of the review recommend using E 0 SHE = 4.28 V. Overall, pre-calculated parameters from literature such as SHE potentials, free energy of protons and electrons transferred in a reaction can be sources of error.Depending on the theoretical approach or the experimental conditions, the parameters from the literature have to be chosen carefully.The SHE potential does not affect the trends of theoretically calculated results, but it contributes to the need to obtain the absolute values of calculated parameters that can be compared to experimental results.
However, solvation Gibbs energies can also be a potential source of inaccuracy in theoretical calculations.Solvation effects are usually treated with implicit solvent models.Besides the inaccuracy of the implicit solvent model itself the correct solvation Gibbs energy of a proton is of particular importance.It not only contributes to the potentials of the SHE, which are needed to compare the results to experimental values.It is also often used indirectly to parameterize the implicit solvent model, since the experimental solvation free energies of ionic solutes are obtained with thermochemical cycles that include the solvation free energy of the proton [47].The most commonly used experimental proton solvation free energy values in literature, based on Boltzmann statistic, are −1112.5kJ mol −1 and −1098.9kJ mol −1 as it was discussed by Hammerich and Speiser [47].They correspond to redox potential of SHE with values of 4.28 V and 4.42 V, respectively.The first value is in good agreement with the previously discussed theoretical value for SM8 and SMD models and the second one includes an estimate of the surface potential contribution.
Sometimes an implicit solvent model is not sufficient and the specific inclusion of explicit solvent molecules is required.Nevertheless, it is recommended [53,54] as best practice in quantum chemistry to avoid explicit solvent molecules wherever possible.Properties tend to converge poorly with the number of explicit solvent molecules.In addition, the structure optimizations are usually long and tedious because the explicit solvent molecules lead to a flat PES with many local minima [53,54].The correct inclusion of the solvation effects will differ from case to case and requires careful consideration of possible sources of error.
The calculated redox potentials are the most practical parameters for theoretical research on RFBs.They can be calculated quantum chemically quickly and usually with sufficient accuracy [55].Furthermore, readily available laboratory measurements allow easy comparison with experiments.Due to the large number of the external model parameters involved in the calculation such as free energies or reference potentials, redox potentials are not the most reliable values for a high-precision comparison of the electronic structure of redox-active molecules in theoretical studies but are mainly relevant for practical research.
Finally, the calculations of organic redox active materials can be divided in two important directions: (1) prediction of and search for novel active materials by screening of potential redox active molecules, and (2) investigation of properties, reactivities and degradation of specific molecules of interest.Detailed theoretical investigations of organic molecules and their redox reactions in different environments with respect to their possible degradation mechanism usually require more theoretical consideration than the calculation of thermodynamic parameters such as redox potentials.The different degradation mechanisms need an assumed degradation reaction and system-specific quantum chemical modeling.The systematic theoretical research of the stability of redox-active molecules is therefore a specific and complex problem [30,56].In screening approaches, for example, the redox potentials of many redox active molecules are calculated, and degradation is usually not taken into account.

Theoretical quantum chemical methods for calculation
In this section, we briefly present some basic information about the quantum chemical approaches commonly used to investigate redox active materials.In the quantum chemical calculations, the molecules under investigation are usually considered as single molecules.Solvent effects are considered using solvent models based on the continuum approach.The quantum chemical method of choice represents the DFT, which is based on the Hohenberg-Kohn theorems and within the framework of the Kohn-Sham theory.The most problematic aspect of DFT is the unknown exchange-correlation density functional of energy, which is always approximated in the Kohn-Sham theory.The basic idea of DFT is the density functional dependence of the system energy E [ρ] on the electronic density ρ.The general equation of Kohn-Sham-DFT is thus: where T s is the kinetic energy of electrons, V ext is the interaction of electrons with cores, U is the Coulomb interaction between the electrons and V xc is the exchange and correlation interaction of the electrons.The exchange-correlation term is approximated in DFT calculations.
Typically, DFT calculations for redox active molecules are based on the gradient generalized approximated (GGA) exchange-correlation density functionals such as PBE or BP86, or hybrid density functionals such as B3LYP, using non-periodic quantum chemistry software packages.The DFT calculations are performed with double ζ basis sets, sometimes triple ζ basis sets or rarely with plane-wave basis sets.DFT calculations generally provide results with a relatively small amount of computational time for small and medium-sized molecules.
Due to their approximative nature, some of the density functionals contain systematic errors, such as delocalization errors, which cause excess charge delocalization and lead to incorrect electronic densities and molecular properties [57,58].In general, GGA functionals overestimate the delocalization and hybrid functionals with the exact Hartree-Fock exchange terms underestimate the delocalization [59].Different density functionals give specific deviations and errors for different classes of compounds and computational conditions in electrochemical calculations.For example, different hybrid functionals have different accuracies for the prediction of redox potentials, and their average error varies with the size of the basis set [60,61].Some newly developed density functionals, such as strongly constrained and appropriately normed meta-GGA functionals, show promising results for ionization energies of atomic model systems [62], but were not yet used for the molecular systems discussed in this review.
Moreover, standard DFT calculations do not take into account or strongly underestimate medium and long-range dispersion interactions.Dispersion corrections are therefore required.There are several approaches to include these dispersion interactions in the DFT calculations.The most common approaches are the DFT-D approach of Grimme, and the van der Waals corrected density functionals (vdW-DFT) [63].DFT-D approaches are based on distance dependent damped potentials introduced as an additive correction and are developed in several generations D2, D3 and D4.The D4 correction is the most up-to-date approach in the series and the only one that takes into account the charge of the molecule.The use of dispersion corrections or improved vdW functionals is particularly important for redox active molecules where interand intramolecular interactions are present [64,65].
For large systems (∼1000 atoms) [66] or when large numbers of molecules need to be investigated, semi-empirical methods and tight binding approaches are used [67].The tight binding approaches are based on DFT but have a much lower computational cost.The idea of these methods is to use a valence-only minimal basis self-consistent approach with the electronic density parameterized for several chemical elements.Recently developed extended tight binding (xTB) methods give promising results [67].These methods do not provide quantum chemical accuracy [68].However, they are more precise and physically correct than classical force fields and allow the calculation of much larger systems than with DFT.In addition, they can be efficiently used for preoptimization of structures and conformational analysis.
Beyond the accuracy of electronic structure calculations, environmental effects have a very significant influence on the thermodynamics and kinetics of chemical reactions.The most important environment of active material molecules in RFBs represents the solvent.Solvation effects can be explicitly described by quantum mechanics (QM)/molecular mechanics (MM) models or by implicit solvent models.Common implicit solvent models are the conductor-like screening model (COSMO), and the PCM [19].These models depend solely on the dielectric constant of the solvent and do not include effects depending on the solvent nature.Newer implicit solvent models such as SMD and direct COSMO-RS include terms to account for dispersion interactions and hydrogen bonding to the solvent.Nevertheless, implicit solvent models can show larger errors in the results, especially if the interactions with the first solvation shell are strong (e.g.hydrogen bonding to water molecules) [69].Explicit solvent models with solvent molecules in the model system provide more accurate and reliable results.However, such calculations require much more computational effort and the model systems become much more complicated, as shown by Sterling and Isegava [19,70].Therefore, the implicit solvent models are usually used for the calculation of redox active molecules.See also section 3.2.1 for a discussion of the solvation energy of a proton.
Wave function-based methods are based on the direct solution of the time-independent Schrödinger equation with an approximated wave function Ψ of a molecule (equation ( 4)).Equation ( 4) is the most fundamental ansatz of wave function-based quantum chemical calculations, linking the wave function Ψ with the system energy E as an eigenvalue of the Hamiltonian operator H.
Usually, the Hartree-Fock method is employed as s basic approximation, and the electron correlation is calculated in addition.Hartree-Fock theory-based electron correlation methods are called post-Hartree-Fock methods.The post-Hartee-Fock methods such as the coupled cluster (CC) approach, the complete active space self-consistent field (CASSCF) method or the Møller-Plesset perturbation theory, provide a more detailed description of the molecules before and after the electrochemical reactions than the DFT.The use of electron correlation wave function-based methods can provide very detailed information about electronic states and reaction mechanisms at the electronic transition level.The results obtained are more accurate but more constrained by the size of the model systems due to the high computational cost.In some studies considered here, redox properties calculations of selected organic and organometallic systems of limited size have been performed using the wave function based methods described [71][72][73][74][75].
At present, however, the post-Hartree-Fock methods are less widely applied than DFT-based methods.The higher computational complexity of the post-Hartree-Fock approach limits the size of the model system.This high computational cost also means that these methods are not suitable for large screening studies.The faster basis set convergence and the lower computational effort compared to the electron correlation methods are the well-known advantages for the preference of DFT in computational studies.In this context, the scaling factors of the computational methods (proportional to the computational effort) corresponding to the system size N are O(N 4 ) [O(N 3 ) with resolution of identity] for DFT, O(N 5 ) for MP2 and O(N 7 ) for CCSD(T).
A review with prediction of the limitations of electronic structure methods was published by Grimme and Schreiner and is illustrated in figure 3 [76].
The family of Quantum Monte-Carlo methods uses stochastic computational methods to evaluate multidimensional integrals (variational Monte-Carlo) and to find electronic states (diffusion Monte-Carlo) are used in [77].These types of methods are very promising for accurate calculations of various molecular properties such as redox potentials, but only a few studies involve calculations of redox related properties [78][79][80].
In addition to the chosen theoretical methods, there are other computational details that can influence the accuracy of the results.It is important to consider factors such as basis set sizes and the specifics of the problem being investigated.The basis set superposition errors (BSSE) can also play a role, for example in the description of the intermolecular interaction of redox-active molecules (dimerization of redox active molecules) [81].The BSSE can be significant for weakly interacting molecules and using a small basis set.Unrestricted calculations may suffer from spin contamination.Discussion of these fundamental methodological problems is beyond the scope of this review, but it is important to be aware of these methodological errors so that they can be corrected if necessary.

Summary of properties calculations
In summary, the standard theoretical method for ground state calculations of organic redox active molecules is DFT.It is the most universal and convincing method for screening but also degradation of different classes of active materials for RFBs [82].However, errors in DFT calculations have been demonstrated in the modeling of some classical organic reaction mechanisms of nucleophilic addition and substitution, which are important for the description of for example degradation mechanisms and side reactions of active materials [83].An accurate consideration of compounds such as radicals and reaction mechanisms with different spin multiplicity, symmetry breaking and spin contamination may require more expensive electron  multireference correlation wave function-based methods.These are currently only rarely used for molecular design of redox active molecular materials and are more demanding than DFT calculations [84][85][86].It can be concluded that there is currently no universal theoretical approach for investigating organic redox active materials in all practical cases.Each theoretical approach contains specific approximations and has specific advantages and disadvantages depending on the research case and system size.

Stable organic radicals
Radicals as organic redox active materials can be neutral molecules or ionic species.The solvent can be aqueous or non-aqueous.Typically, ionic radicals are products of electrochemical reactions following electron transfer during reduction or oxidation [87].The class of radicals encompasses a variety of redox active materials, and this chapter discusses several typical neutral radicals with corresponding reduction and oxidation reactions and their modifications.In general, these radicals are based on cyclic systems with heteroatoms.

TEMPO-based radicals
The class of 2,2,6,6-tetramethylpiperidine-1-oxyls (TEMPO) is a very well-studied cathode material.Research on TEMPO in the context of RFBs has focused on finding the most suitable modification to make it suitable as a catholyte.Experimental studies exhibit many possible modifications and a great synthetic potential of the TEMPO core structure for the development of active materials.For example, the soluble molecule N,N,N-2,2,6,6-heptamethyl-piperidinyloxy-4-ammonium chloride was found to be an efficient electrolyte in high capacity aqueous RFBs [88][89][90].TEMPO-based radicals are also used in combination with other active molecule fragments, such as phenazine, for symmetrical RFBs as a bipolar active material for both anolyte and catholyte [91].
The cathodic and anodic reactions of TEMPO-based molecules with a -R substituent and the corresponding one-electron transitions are presented in figure 4. The oxidation of the TEMPO molecule leads to the formation of an oxoammonium cation (TEMPO) + with oxidation potential E 0 = 0.73 V vs. SHE in water [92] (0.72 V [93]).The reduction results in the hydroxylamine anion (TEMPO) − .In these processes, the single occupied molecular orbital (SOMO) becomes either the LUMO or the HOMO (figure 5).The HOMO and LUMO show a localization of the unpaired electron at the N-O molecular fragment.There is also a stronger delocalization of the energetically lower lying electron pairs in the σ-bounded ring system.
The orbitals and spin states for reduced and oxidized TEMPO were calculated and analyzed, among others by Mao et al for different oxidation states at the B3LYP/6-3+1G(d,p) (CPCM) level of theory [95].The CPCM solvent model was used to simulate methanol and acetonitrile.Calculations were performed for systems with pyridinium cationic lipid substituent.The computational results show that the functional group plays a very important role in the electronic structures and related properties of TEMPO derivatives.The TEMPO molecule converts to a singlet state after the one-electron reduction, while the substituted pyridinium lipid becomes a triplet.Calculated UV-vis spectra at the TDDFT level of theory were also provided in good agreement with the experiment and serve to interpret the spectra.
Theoretical investigations of the electrochemical properties at DFT/B3LYP and HF/MP2 levels of theory with PCM in water, DMF and ACN of TEMPO-based nitroxide radicals show a particular positive redox potential shift of the oxidation reaction with electron-withdrawing substituents R in position 4 (figure 4) [16].The trend of these shifts is in good agreement with experimental results from the work of Mendkovich et al [96].Due to the nature of the TEMPO structure without π-electron conjugation in the piperidine ring, the inductive (I) effect of substituents (-R) primarily affects the redox potentials through changes in the σ-bond electron density.For example, -H as a substituent in the basic structure of TEMPO causes a neutral inductive effect and leads to lower redox potentials than the cyano-group, oxygen-containing groups, or halogens, which show a stronger inductive withdrawing effect.These shifts are particularly observed for systems in aqueous solution.In this context, it was shown at the MP2/aug-cc-pVTZ level of theory that the Gibbs free energy ∆G r 0 for the oxidation of the unsubstituted system in vacuum increases by more than 0.3 eV (0.3 V redox potential shift according to equation ( 2)) when a single -CN group is introduced.This potential shift effect was similarly observed by Hodgson et al for various TEMPO-based derivatives at DFT-B3LYP, MP2 and QCISD levels of theory in combination with their own n-layered integrated molecular orbital and MM' ONIOM method [97].These calculations have been carried out in the gas phase and with the PCM model in aqueous medium.The different theoretical methods give good accuracy of redox potentials with a difference of 0.01 V from each other.
An example of DFT failure for TEMPO based structures is the calculation of charge transfer in a TEMPO-thiophene-based derivative.The results are presented in the paper of Zens et al [98].The use of TDDFT to calculate PESs provides qualitatively incorrect results with incorrect energy level behavior of the PES in regions of state crossing.The use of the CASSCF method with perturbatively included dynamic correlation (NEVPT2) confirms both the strong static correlation and the importance of the dynamic correlation.Moreover, this method gives a correct result for the experimentally observed charge transfer according to the Marcus theory.The state average CASSCF method is used to correctly predict the structure reorganization.The influence of electronegative functional groups in the molecular chains between TEMPO and thiophene core on the charge transfer confirms an electron-withdrawing effect (figure 6).This effect favors the intramolecular oxidation of the TEMPO groups.
A detailed analysis of the conformers and magnetic properties of the stable TEMPO-based radical 4-amino-2,2,6,6-tetramethylpiperidine-1-oxyl-4-carboxylic acid (figure 7) in aqueous solution (PCM) and in vacuum was reported by D'Amore et al using the PBE0/6-31G(d) approach and the HF-based quadratic configuration model including single and double excitations.Based on the calculations, the most preferred conformational structures and Ramachandran maps were reported.Furthermore, the dependence between the conformational stability and the dielectric properties of the solvent for a given substitution pattern was demonstrated.The calculated magnetic properties agree well with experimental observations [99].

The 2-phenyl-4,4,5,5-tetramethylimidazo-line-1-oxyl oxide (PTIO) radical
The next important radical system for symmetric non-aqueous RFBs is PTIO [100][101][102].The radical can be reversibly oxidized and reduced to oxoammonium and aminoxyl ions at a theoretical cell voltage of 1.73 V in acetonitrile.The reaction of the nitroxide radical group is similar to the TEMPO one-electron reduction and oxidation reactions (figure 8) [101].Symmetric non-aqueous RFBs based on PTIO have promising efficiencies, but there is a continuous loss of capacity due to the decomposition of the active material.
Similar to TEMPO, different modifications of PTIO are possible; for example, a PTIO-based bipolar radical with various substitution patterns at the phenyl group [100].There are currently no published theoretical results for PTIO-based systems, and calculations are expected to be challenging due to the complicated charge delocalization in the radical and the possible multi-reference character.DFT calculations may lead to an incorrect description, similar to what was demonstrated for some symmetric radical-cationic systems such as butadiene's by Oxgaard and Wiest [103].Qualitatively different results have been reported that depend on the calculated electron localization and delocalization.Post-Hartree-Fock calculations can also be problematic for the description of PTIO-radicals due to the large size of the molecule.In addition, the instability of this radical with different ionic moieties requires a systematic consideration of side reactions and degradation processes.Therefore, the theoretical investigation of these systems represents a particularly important step, but also a challenge.

Blatter radicals
The electrochemistry of the 1,2,4-benzotriazin-4-yl (Blatter) radicals as a bipolar active material for the construction of RFBs was reported with symmetrical electrolyte composition with cell potentials ranging from 0.91 to 1.32 V in ACN (figure 9).The reduction and oxidation potentials for the reactions of the Blatter radical (R = Ph) are −0.60 V and 0.44 V, respectively [104].Blatter radicals exhibit good chemical stability in all charge states, making them particularly promising active materials.
Theoretical calculations of the electronic structure of the Blatter radical have revealed a high degree of electron delocalization within the SOMO over the two conjugated ring systems (figure 10), as well as a paramagnetic character of the molecule.During the one-electron redox reactions, the SOMO converts to the LUMO in the oxidized form, and to the HOMO in the reduced form.In contrast to TEMPO, the Blatter    −756 [111] 424 [111]

Dithiazolyle-and dithiadiazolyle-based radicals
Another group of potentially interesting systems for RFBs are radicals based on heterocyclic nitrogen-sulphur systems such as dithiazolyle-and dithiadiazolyle-based radicals (table 2) and their derivatives with different functional groups on the benzene ring and their isomers [107][108][109][110][111]. As with the previous radicals, these radicals can be either oxidized or reduced by a one-electron process (figure 11).The experimentally determined redox potentials of the benzo-dithiazolyle-(BDTA) and 4-phenyl-dithiadiazolyle-(PDTDA) radicals and their acronyms are given in tables 2 and 3. Theoretical investigations of these radicals at the DFT-PBE level of theory provide reliable predictions of the redox potential without qualitative errors for the various forms of the systems.The HOMOs, SOMOs and LUMOs of cations, radicals and anions are presented in figures 12 and 13.Interestingly, due to the structural deformation, the anion exhibits an overlap between the phenyl group and the heterocyclic fragment.This is expected to result in different chemical binding properties compared to the flat structures of the cation and anion.This is particularly evident for B132DTA in figure 12.
All BDTA HOMOs contain strongly conjugated and delocalized π-systems over both rings.This leads to a significant influence of substituents with mesomeric effects.The two ring systems in PDTDA are separated   by a σ-bond.Due to possible rotations and weaker π-overlap between the two ring systems, the inductive effect of the substituted phenyl group influences the redox potential.Theoretical calculations confirm the weak influence of substituents with mesomeric effect on PDTTDAs [97].DFT-based screening studies of the PDTDAs and BDTAs radicals also show a positive influence of electron-withdrawing and -donating effects on the redox potential shift due to substitution of the benzene ring.The potential shift is particularly strong for delocalized aromatic conjugated systems of dithiazolyl radicals with conjugation of π-electron systems as seen from the orbital structure.Potential shifts are greater than 200 mV for PDTDA and greater than 500 mV for BDTA [112].

Aromatic compounds with ether groups
Another group of aromatic systems that are not radicals in neutral form but form ionic radicals in charged RFBs are aromatic systems with ether groups.These materials are usually based on alkoxybenzenes (figure 14(a)).The transfer of one electron by reduction or oxidation reactions produces ionic radicals with strong delocalization.These molecules are used in solution or attached to polymers, but the low energy density of some of these materials makes their application problematic.The most common core structure is 1,4-di-tert-butyl-2,5-dimethoxybenzene with various substituents on the ether [7,113].The oxidation potential of the basic structure is 1.3 V vs. SHE in ACN [114].The effect of electron-withdrawing and electron-donating groups was demonstrated for this class of molecules in a screening approach at the B3LYP/6-31+G(d) (PCM, water) level of theory [115].
Other materials containing a benzyl group are homobenzylic ethers.Doan et al systematically studied 1400 homobenzylic ether derivatives in one electron oxidation reactions (figure 14(b)) [116] at the B3LYP/6-31+G(d,p) level of theory using the PCM solvent model simulating the solvent acetonitrile.The Bayesian Optimization (BO) machine-learning tool was then used to search for favorable materials and find the best structure.The research demonstrates the modern trend of using machine-learning tools in combination with quantum chemical data.Significant results were obtained from a large-scale screening of 112 000 structures analyzed by active learning.As a result, oxidation potentials of around 0.75 V were predicted, but unreliably high potentials (⩾3.5 V vs. SHE) were obtained for several electron-deficient systems with strong withdrawing substituents such as -NO 2 .The unreliable results may be due to the  [120,121], CO, CH2 for (a) [118]).incorrect description of nitro compounds at the DFT level of theory with a corresponding overestimation of the free energies of the reactions.Reorganization energies and the reaction energetics of deprotonation and demethylation of charged species are proposed as the theoretical relative stability criteria for this compound class [117].

Verdazyl radicals
Verdazyl radicals are another well-established class of stable organic radicals.The two most common variants of this class are (1) the so-called Kuhn-verdazyl radicals with X = CH 2 and (2) the oxo-verdazyl radicals with X = CO (figure 15) [118].However, the thio-verdazyl radicals have also been reported [119].
The basic structure of the verdazyl radicals consists of a cyclic 1,2,4,5-tetrazine.Unlike other classes of stable radicals, verdazyl radicals have a sterically unprotected radical moiety [120].The high radical stability is attributed to the delocalization of the unpaired electron between the N1-N2 and N4-N5 atoms [122].There is no spin delocalization at the C3 positions found because the SOMO has a nodal plane there [122,123].Additional stabilizing resonances are obtained for the oxo-and thio-verdazyl radicals by the carbonyl and thiocarbonyl groups, respectively.
In organic electrolytes, verdazyl radicals have three redox states: the oxidized, the neutral (radical) and the reduced state (figure 15(a)).This makes them a potential candidate for symmetric organic RFBs.Cell voltages of ≈1 V (Kuhn-verdazyl radical), and ≈1.6 V (oxo-verdazyl radicals) are expected based on experiments [118].Unfortunately, the stability of the verdazyl radicals is not yet satisfactory.The capacity rapidly faded in symmetrical batteries over 50 cycles (Kuhn-verdazyl radical) [124] and 150 cycles (oxo-verdazyl radical) [125], respectively.While the reduced form was speculated to be responsible for the degradation in battery tests [124], a recent study by Steen et al attributes this to decomposition of the verdazyl radical [126].In aqueous acidic electrolytes, the verdazyl radical shows a very different behaviour.The verdazyl radical disproportionates into the oxidized (verdazyl cation) and reduced (leuco-verdazyl) states (figure 15(b)) [120].A first detailed study of the verdazyl species after this disproportionation was recently performed by Kunz et al [121].It was shown that the reversible redox reaction from the oxidized to the reduced species, is a single reaction involving two electrons as well as two protons (figure 15(c)).At pH = 0, the midpoint potential of the redox reaction is 0.146 V vs. SHE.Furthermore, the redox reaction has a large rate constant (k 0 ≈ 2 × 10 −2 cm s −1 ) compared to other organic compounds that are discussed as potential redox active materials.The one-step reaction process means that the acidic verdazyl solutions cannot be used for symmetrical RFBs.However, the ability to store two electrons, the use of aqueous electrolytes, and the fast reaction kinetics are currently motivating further research into electrochemistry of verdazyl species in aqueous acidic electrolytes.
Koivisto and Hicks [122] speculated that the lack of spin density at the C3 group would dampen the effect of substituents at C3 on the spin distribution.However, a combined experimental and theoretical study by Tanaseichuk et al [123] showed that substituents at C3 have strong effects on the SOMO energy and reaction barriers.The calculations were performed at B3LYP/6-31G(d) level of theory and focused on the determination of orbital energies and spin densities.The reaction pathway of the thermolysis of the verdazyl radical in organic electrolytes was further unraveled by Steen et al [126] using DFT calculations.The reaction pathway was calculated at the B3LYP/def2-TZVP level of theory.The solvent (ACN) was simulated with the SMD model in subsequent single point calculations.The results were verified by experimental NMR studies.Theoretical investigations were also carried out by Kunz et al [121] to gain further insight into the thermodynamics and protonation sites of the verdazyl in aqueous acidic electrolytes.A protonation site screening was performed by combining the semiempirical tight binding method xTB (GFN2-xTB) in the CREST program package with DFT.The DFT calculations were performed at the B3LYP-D4/def2-TZVPD COSMO level of theory with COSMO simulating water.The pH-value was included by adding a proton concentration dependence on the correction of the Gibbs energy of the proton.Furthermore, the thermodynamics of the disproportionation of the verdazly radical in water was calculated (figure 15(b)).It was shown that disproportionation into the oxidized and reduced verdazyl species is only possible at low pH-values.

Summary of theoretical studies of radicals for RFBs
Stable organic radicals represent a large and diverse class of molecular compounds with potential as redox active materials for RFBs.The systematic theoretical consideration of radicals in the ground state and in redox reactions has in most cases been successfully carried out at the DFT level of theory and in agreement with experimental results of redox potentials, solvation and magnetic properties [96,97,99,106].However, the results require careful analysis and interpretation of the electronic structure and states.Open shell restricted and unrestricted methods were used in the calculations discussed.For most systems, DFT with PBE or B3LYP density functionals provide results with an accuracy comparable to post-Hartree-Fock methods.When using unrestricted methods, spin contamination is a possible source of potential problems associated with the correct description of the electronic structure.For the investigation of reaction mechanisms with conical intersections of PESs including excited states (figure 16) and systems with multi-reference ground states, the description by single reference methods such as Kohn-Sham DFT becomes problematic.In these cases, possible errors in spin state definitions, incorrect electronic structure predictions and reaction energy errors are expected [98].Only some specific types of density functionals with exact exchange can correctly describe such situations in model systems as presented in the work of Liu for simple diatomic systems [127,128].
In addition, DFT is known to be deficient in the description of radical reactions, even compared to the MP2 method, and this can be critical for investigations of side reactions and degradation [129].Different types of electron correlations lead to inconsistent results for different molecules and complicate the description of reactions such as degradation mechanisms, and the calculation of experimentally measurable observables such as optical spectra at the commonly used DFT/PBE and DFT/B3LYP levels of theory.The use of wave function based multi-reference methods based on Hartree-Fock or in combination with DFT is problematic due to the system size.However, it is applicable to potentially interesting molecules of limited size or model systems.Typically these molecules are not larger than a few tens of atoms (figure 3) and the size of the active orbital space is not larger than 16 with a corresponding number of reference wave function configurations as discussed in the review by Szalay et al [130,131].Differences in method efficiency may become a major challenge for future screening research with large structural diversity, as there is no universal theoretical method for all types of systems and all research questions.

Quinones
The first reported organic redox active materials for RFBs were quinone derivatives [7,132,133].Quinones are used in aqueous and non-aqueous electrolytes [31,134,135].The reduction reaction of a quinone in  aqueous medium occurs with a two-electron and two-proton transfer under acidic conditions (figure 17(a)).Hence, the arising aromatic conjugation in the benzene ring of the product stabilizes the system [134,135].There is a large class of quinone-based redox active materials for RFBs with different carbon basis structures [136,137].The first compound in this series is the benzoquinone with substituents R 1 = R 2 = R 3 = R 4 = H, while the best studied redox-active materials are the anthraquinones (figure 17(b)) [138].
The orbital representation of the electronic structure exhibits a shift of HOMOs and LUMOs after electron transfer.The LUMO of the quinone becomes the HOMO of the dianion and stabilizes the bonds in the aromatic structure.The HOMO of the oxidized form does not change to become an energetically lower lying orbital in the reduced form.The LUMO+1 of the oxidized form remains the LUMO of the reduced form (figure 18) [139][140][141].The electronic structure of the orbitals in figure 21 suggests a strong electronic delocalization in the aromatic framework.
The unsubstituted anthraquinone core structure as an anolyte has relatively low reduction potentials of −0.68 and −1.43 V (vs.SHE) in DMF [142].Anthraquinone-2,6-disulfonate in aqueous solution with 1 M H 2 SO 4 has a reduction potential of 0.21 V vs. SHE [132].The first reduction step in non-aqueous solution produces the anion-radical.Nevertheless, it is possible to synthetically modify this class of compounds to vary the properties such as the redox potentials or solubility [143,144].Solubility in water is usually controlled with polar active Brønsted acic groups such as -SO 3 H [145][146][147].
Theoretical calculations of the electrochemical properties of benzoquinone in acidic environment have been performed at the B3LYP(PCM, water)/6-31+G(d,p) level of theory.The electrostatic potential maps (EPM) contours presented in the investigations show that the most likely sites for nucleophilic reactions on carbon atoms are near electronegative atoms such as oxygen.The molecular orbital energies and HOMO-LUMO gaps indicate trends in the electrochemical behaviour of molecules in reduction and oxidation reactions.There is a tendency for the redox potentials of the molecules to decrease linearly with the LUMO energy.Moreover, a correlation was found between high solubility and most negative and positive redox potentials: molecules with low redox potentials are more soluble [138].A correlation between electron affinity and redox potentials in quinone based systems was also confirmed at PBE0/6-31+G(d,p) level of theory for naphthaquinones and anthraquinones [148].Furthermore, a strong influence of hydroxyl substitution on the redox potential was described.Modification of the molecular structures to improve the chemical and electrochemical properties of various quinone cores such as benzene, naphthalene, and anthracene opened up a large class of active materials and a large area for theoretical screening [149][150][151][152].The latter was carried out with DFT using PBE and B3LYP density functionals.Variation of the quinone core structures and side groups in theoretical studies provide redox potential changes for different systems with quinone groups in the range of −0.8 up to 1.6 V vs. SHE.The effect of electron-withdrawing and electron-donating groups is most important in influencing the redox potential during structural changes.The screening studies demonstrate, as expected, that electron donating groups such as -OMe, -SH, -OH, -NH 2 decrease the redox potential, while electron withdrawing groups such as -F, -CN, -NO 2 , -COOH increase it.Due to the π-conjugation of the aromatic core structure, the mesomeric effect has a much greater influence than the inductive effect.Furthermore, a non-trivial effect of the -OH groups was confirmed in the screening study (figure 19) by Schwan et al [150].This study found that intramolecular hydrogen bonds between the quinone group and the hydroxyl group lead to an anomalous increase in the redox potential [150].
Modification of anthraquinones with amino groups changes the redox properties, the electrochemical reactions become reversible by oxidation due to electron donating effect of amino groups.The four-electron oxidation of tetra-aminoantraquinone was experimentally demonstrated in the work of Pahlevaninezhad and calculations were provided at B3LYP(PCM, DMSO)/6-31+G(d,p) level of theory [134].
In addition to the redox potential and solubility, it is important to estimate the stability of the screened molecules.This was done for quinone-based materials with the substituents of carboxylic, phosphonic and sulfonic acid.For these, the specific side reactions of Michael addition and the formation of gem-diols were taken into account.To perform the more than 1000 000 calculations, Tabor et al combined their DFT study with the application of semi-empirical methods [137].The theoretical results demonstrate good agreement with experimental observations.The data analysis suggests that redox potentials themselves can serve as stability criteria, since molecules with too high redox potentials are typically unstable.The instability of systems with redox potentials higher than 1.0 V occurs due to Michael addition reactions [137].
Another approach to estimate the stability of the compounds was introduced by Schwan et al [150].They developed and discussed structural descriptors for quinone-based systems based on the change in dihedral angles of the planar ring structures.The deformation of the planar structure was used to estimate the stability.Large changes in dihedral angle were associated with instability [150].The basis of this approach goes back to the work of Zhigalko et al, who found that non-planar geometries for benzene, naphthalene and anthracene are only possible up to some limiting dihedral angles [153].The search for stability descriptors is an important challenge for screening applications and may be beneficial for the development of RFBs active materials.
In other screening research, the solubility of 1,400 molecules was studied by the analysis of hydration energies using the Hartree-Fock method for the structural calculations in combination with MM constant density calculations DFT-GAFF and a simple point charge (SPC/E) solvent model for water.Higher hydration energies were found for quinones with the functional groups -CHO, -COOCH 3 , -COOH, -PO 3 H 2 and -SO 3 H [154].
Waas et al showed that the commonly used DFT/B3LYP approach to calculate the geometric structures and redox potentials of quinones predicts the same trends as obtained at the CCSD(T) and perturbation theory levels of theory [155].The accuracy of the MP2 method is comparable to that of DFT/B3LYP but has slower convergence of results with basis set size.MP2 overestimates the Gibbs free reaction energy by ∼20% compared to CCSD(T) and B3LYP.
Apart from the method, the modeling of the solvent environment is also an important aspect.Kim et al showed that by employing the solvent model PCM, the redox potentials of anthraquinone derivatives with ether groups at low pH value have a large mean absolute deviation of 194 mV compared to values from the corresponding experiments.They showed that the application of the QM/MM approach with an explicit description of the solvent effect leads to reliable results and corrects the errors of the implicit models [140].Despite possible errors, implicit solvent models such as SMD and COSMO are the most common way to include environmental effects into quantum chemical calculations and in most cases provide satisfactory results compared to explicit solvent models [140,156,157].
Screening studies of quinone derivatives were already successfully carried out to investigate trends in the redox potentials and solubility.Challenges remain in the screening of stability against side reactions or synthetic availability.The former can be partially solved by applying semi-empirical methods or by developing descriptors [30].
Naphthaquinones and antraquinones can be substituted with heteroatoms in the aromatic core.An example studied in the context of RFBs are phthalazine, aza-anthraquinones and their derivatives.The molecules have positive potential shifts in comparison to unsubstituted quinones due to a stronger electronegativity of the nitrogen compared to carbon [141,158].Theoretical calculations of the compounds at PBE-D3(BJ) (PCM, water)/aug-cc-pVDZ level of theory for phthalazine (figure 20(a)) and at B86-D3(BJ) (PCM, water)/aug-cc-pVDZ for 2,3-diaza-anthraquinone (figure 20(b)) result in very good agreement with experimental redox potentials in acidic medium.Nevertheless, there are significant deviations for potentials in basic medium [141].A possible reason for this difference could be that the redox reaction follows a different mechanism in basic medium.
The HOMOs and LUMOs of the 2,3-diaza-anthraquinone are given in figure 21.At the electronic level of theory, the reaction mechanism is the same as for anthraquinones.The LUMO of the molecule becomes the HOMO and stabilizes the central ring of the aromatic structure after a two-electron and two-proton transfer in acidic medium.The difference to the anthraquinone without heteroatoms is the small anti-symmetry of the electron density distribution with a shift towards the more electronegative nitrogen atoms.This is clearly visible for the HOMO of the oxidized form.

Nitrogen-containing molecules
Nitrogen-containing molecules are a large class of promising redox active materials for RFBs.This class overlaps with the classes of radicals and substituted quinones and represents such molecules modified with nitrogen-containing groups, and with nitrogen-containing core structures.

Pyridinium-based systems
Pyridine-based cations form an important class of anolyte-active materials [159].These cations react to radicals in a one-electron reduction reaction, see figure 22.The electron is transferred to the LUMO (figure 23) and delocalized over the heterocyclic core.For the theoretical description of these compounds at the DFT level of theory, the B1B95 density functional was recommended for one-electron redox reactions.The calculated redox potential differs from the experimental results by less than 0.02 V [160].Most of the known materials in this group are dipyridyl systems with a viologen core structure in which two-electron reductions occur.Here, the cation radical is formed in a one-electron reduction and can be further reduced to a neutral molecule (figure 24).The first and second reduction potentials for methyl viologen in aqueous medium are E 0 = −0.45V and E 0 = −0.76V vs. SHE (R 2 = Me) [161,162].Korshunov et al experimentally obtained 9% variation in the redox potential for the viologen-based molecules and their structural design using host-guest type supramolecular complexes [163].The substituents R 1 and R 2 affect the redox potentials.The corresponding HOMO-LUMO energies were experimentally and theoretically studied.Theoretical calculations were performed at the M06-2X (SMD, water)/6-31+g(d,p) level of theory and confirm the electronic donor and acceptor effects for the viologen-containing systems as well as for other systems discussed.Smaller HOMO-LUMO splitting leads to higher reduction potentials [165,166].
Another cationic bispyridinylidene anolyte (figure 25) with a high specific capacity and a reduction potential of −1.00 V vs. SHE was experimentally studied by Alkhayri and Dyker [167].In addition, uncharged pyridine derivatives were reported as compounds with low redox potentials in the range of −2.03 to −0.28 V vs. SHE [168].A bipolar system in combination with a TEMPO structural core was also synthesized and investigated.The compound combines the advantages of viologen and TEMPO.It has an energy density of over 16 Wh l -1 and a high retention rate [94].The electronic structure calculated at the DFT-B3LYP/6-31G * level of theory with an implicit solvation model (SMD) shows that the LUMO of the molecule is almost identical to the corresponding orbital of TEMPO and the HOMO contains a larger contribution of the viologen molecular fragment [94].Kannappan et al studied the redox chemistry and structure of viologen derivatives that contain two linked viologen structural fragments by experiment and theoretical calculations at DFT-B3LYP and MP2 level of theories.The results of x-ray structural information and measured electronic structure properties demonstrate agreement with theoretical calculations.The calculations performed allowed to obtain structural information on the singlet-and triplet-state π-dimerized bis-viologen species [169].

The succinimid structural core
The next nitrogen-containing anolyte material discussed is N-Methylphthalimide, which is reduced to an anion radical [E 0 = −1.06V and −1.34 V vs. SHE [approximated by Ag + /Ag (0.1 M AgNO 3 ACN) in acetonitrile reference electrode potential), ionic liquid] [170].The reduction is achieved by an one-electron transfer with formation of an anion radical, see figure 26.N-Methylphthalimide was used as the catholyte in the combination with a TEMPO-based molecule.Stable charging and discharging with high coulombic efficiency (90%) was achieved for the first few cycles [171].
The system was found to be a potentially interesting redox active material even with additional quinone groups.This is based on the DFT calculations at PBE-D3/aug-cc-pVDZ level of theory in water (described with the PCM), which predicted a high redox potential of 1.14 V vs. SHE for the quinone structure with the N-Methylphthalimide core [denoted as dihydroxyphthalazine(sumi) (DHP(sumi))], see figure 27 (R = H), in a screening study with 42 molecules [158].At the same time, the stability based on the deformation of the cyclic planar system was estimated to be low for some substituted DHP derivatives due to high dihedral angles.However, synthetic availability cannot simply be verified by the structural deformation analysis.The   same redox activity as for the quinones of DHP(sumi) has been considered by Hoffman et al [158].However, reactivity at the succinimide moiety is also possible and can be considered as a possible side reaction.The results of the screening efforts are presented in figure 28.

Phenazine-based structures
The next major and very promising class of nitrogen-containing heterocyclic systems are the phenazine-based redox active materials [172][173][174][175][176]. The reduction reaction of phenazine in aprotic medium is shown in figure 29.During the reduction of the phenazine in aqueous media, the nitrogen atom is protonated [177].There is a two-electron transfer to the LUMO and destruction of the aromatic electronic structure of the central ring.As a result, the molecule in the reduced form is no longer flat, and the electronic structure at the N-atoms acquires an electronic configuration with sp 3 hybrid orbitals.These structural changes result in out of plane deformation and a loss of the π-conjugation throughout the molecule.However, a recent study reports a planar structure of a disubstituted reduced form of the phenazine structural core [178].The redox potential of the reaction is −0.94 V in ACN [179] and −0.99 V in DME [180] vs. SHE.The main challenge with phenazine-based systems is usually the low solubility, while the system has good electrochemical stability [30].The solubility issue can be addressed by molecular modification with additional functional groups such as sulphonic (-SO 3 H) or carboxylic acid (-COOH) [91,176].
A detailed theoretical study of the solvation issue and redox potentials of various phenazine derivatives was presented by Zhang at B3LYP(COSMO, water)/DNP level of theory [181].Solvation energies and redox potentials for reactions with two-electron and two-proton transfer were calculated.Correlation of LUMO energies and redox potentials shows that the redox potential decreases mostly linearly with increasing LUMO energy.The calculated redox potentials are in very good agreement with the experimental results.The correlation of the solvation energies with experimental solubility was also determined.Screening studies of the redox potentials of phenazine derivatives were presented by de la Cruz et al [182].The electron withdrawing effect was confirmed and is much stronger than for anthraquinones (potential change 1.3 V compared to 0.5 V for anthraquinones).Reorganization energies and root mean square deviations were used for comparison of stability.A more negative shift of the redox potential was discovered for electron donating groups in position R 2 compared to position R 1 .As with quinones, the high stability of the molecules in electrochemical reactions opens up great application prospects.Tuning of the redox properties by modification and molecular design can further improve this redox active material.In the fully reduced and oxidized forms, the molecules are closed-shell electronic systems that can be efficiently explored by DFT-based methods.This was already done, for example, in a screening study at the B3LYP(SMD, water)/6-31+G(d) level of theory presented by Assary et al [183].In any case, in the theoretical study of phenazine derivatives, the correct protonation of the reduced and oxidized forms must be considered, as this affects the thermodynamic energy and influences the prediction of the redox potentials.Therefore, the correct inclusion of solvation effects with implicit solvent models and the consideration of different possible protonation types is necessary for reliable and correct results.
The combination of nitrogen-containing heterocyclic systems with quinones was investigated as potential electrolytes for symmetrical RFBs in the theoretical screening research of Fornary et al at the DFT-B3LYP/6-311G(d,p) (PCM, water) level of theory [184].In this study, the dependency of the redox potential on molecular structures and heteroatom positions and solubility was studied.Structure-property relationships were predicted and analyzed for potentially interesting phenazine-based redox active materials.The study considered different substitutions at the aromatic system with heteroatoms containing several redox units in one molecule.Interactions at the electronic level between different redox units and intramolecular hydrogen bonding were identified as factors influencing the redox potential.The solubility differences between the compounds studied can be up to four orders of magnitude, and the redox potentials are in the range from −0.5 V for reduction up to 1.4 V for oxidation (SHE).
The redox potential shift due to the modification of the phenazine structure with electron donating and electron withdrawing groups was also confirmed for phenazine based structures at LC-wPBE/6-311+G(d,p), B3LYP/6-31G+(d) (PCM, water) and at PW91(COSMO, water)/DNP levels of theory [185][186][187].The protonation effect on the redox potential of single functionalised phenazine derivatives is not large, with changes in the range of 200 mV between the amino and carboxylic groups.This indicates that the destruction of aromaticity during the reaction reduces the influence of the mesomeric effect on the redox potential shift due to the loss of conjugation in the reduced form.Variation of the phenazine structural core with additional pyridine rings could change the redox potential more significantly by a stronger influence of the electronic structure on the redox centers.The addition of extra pyridine rings to the phenazine structure decreases the redox potential by 0.2 V [187].
ML methods using four regression models were applied to DFT calculated results of redox potentials by Ghule et al [40].A small data set with 151 phenazine derivatives containing 20 unique functional groups was used to train the models based on automatic relevance determination regression, Gaussian process regression, kernel ridge regression and support vector regression.The trained models achieved high accuracy with a coefficient of determination R 2 > 0.74 compared to DFT reference calculations for all used models.

Azobenzene
A non-aqueous high capacity RFB based on azobenzene (AB) as anolyte was reported [35,188].A two-electron reduction to the corresponding anion occurs during the reaction, see figure 30.This leads to a stronger delocalization of the electron density (figure 31) and electrostatic stretching of the central N-N bond.Azobenzene has a redox potential of −1.22 V vs. SHE in ACN [188,189].Unfortunately, azobenzene is not stable in protic systems and shows decomposition tendencies such as dimerization reactions or disproportionation [36].
Zu et al studied the solubility of different azobenzene derivatives at the DFT level of theory.By modifying the basic structure with strong hydrophilic groups, several derivatives such as 4-amino-1,1 ′ -azobenzene-3,4 ′ -disulphonic acid monosodium salts represent promising anolytes with   high-performance for aqueous systems [35].The geometric structure of azobenzene was studied in detail at CC2 and MP2/cc-pVTZ levels of theory and compared to DFT (BP86, B3LYP) and to experimental results by Fliegl et al [190].DFT provides reliable results for structural parameters and vibrational frequencies compared to experimental values, but not for electronic structure and excitation energies.Vibrational frequency spectra calculated by MP2 show a tendency to blue-shift, while DFT frequencies are usually slightly red-shifted.In conclusion, wave function-based methods are recommended by the authors as the methods of choice for investigating substituent effects on the structure and optical properties of azobenzenes [190].

Tetracyanoethylene
Molecules with delocalized electrons in combination with cyano-groups show promising redox activity.For example, Wang et al studied the redox activity of tetracyanoethylene (TCNE) (see figure 32) as an anolyte [191].A reduction reaction was observed with two peaks at 0.49 and −0.54 V vs. SHE in ACN (recalculated for Ag/AgNO 3 (0.01 M, ACN)).
The reduction reaction involves a two-electron transfer in which the LUMO of the TCNE molecule becomes occupied and becomes the HOMO of the corresponding anion.
Theoretical calculations showed that the molecular orbital energies increase from the neutral form to the fully anionic form, see figure 33.The C=C bonding contributions in the HOMO and the C=C antibonding contributions in the LUMO indicate the nodal plane of the wave functions perpendicular to the bond.There is also a strong delocalisation in the LUMO for the fully reduced TCNE 2− compared to the single charged molecule.

Summary of theoretical studies of nitrogen-containing molecules for RFB
Nitrogen-containing molecules are probably the largest class of promising redox active materials for organic RFBs [37].The systems have a wide range of applications and therefore many theoretical studies were performed.Calculations of the electrochemical properties provide useful information on the structure, design, and the applicability of different theoretical methods.In most cases, DFT calculations give at least qualitatively correct results.The DFT calculations are successfully used to predict various experimental Reproduced from [191] with permission from the Royal Society of Chemistry.Color coding: C (grey), N (blue).trends and in screening research.Not all systems in this class were sufficiently theoretically studied.This leaves room for screening studies of solvation energies and redox potentials as well as the calculation of explicit electrochemical reactions and the exploration of degradation mechanisms with theoretical methods.However, the quantum chemical description of ion-radical systems or intermediates such as the azo radical anion ArNNAr − in reduction reactions needs to be carefully analyzed at the DFT level of theory.The DFT calculations may contain localization-delocalization errors and spin contamination.Such systems may exhibit multi-reference character and require multi-reference approaches for accurate description.
The combination of nitrogen-containing fragments with other redox active moieties such as quinones leads to a variety of possible redox reaction products.These complex molecules with multiple redox active centers require careful analysis as different redox reaction pathways may occur at different reaction centers.In this case, the stability and reaction mechanism should be considered for different possible reaction products.
In addition, the interaction of redox-active molecules with their environment is not correctly described by implicit solvent models when reactions with the solvent occur.For example, the protonation of nitrogen atoms and hydrogen bonding have a strong influence on the redox potential and electrochemical properties.Therefore, different reaction mechanisms must be considered, including different degrees of protonation.In addition, the correctness of the model systems should be carefully analyzed with chemical intuition before the theoretical calculations are performed.

Sulphur-nitrogen-containing heterocyclic molecules 3.6.1. Phenothiazine-based structures
Other classes of redox-active molecules with a heterocyclic core are represented by molecules in which the nitrogen atom in the basic nitrogen-containing compound is replaced by sulphur or oxygen.A typical example is the class of phenothiazine-based structures that can be used as catholytes in RFBs, see figure 34 [36].These are phenazine-like structures in which one nitrogen atom is replaced by a sulphur atom.The redox potentials of the two-electron oxidations (figure 35) (with R 1 = -CH 3 , R 2 = H) are 0.95 V and 1.55 V vs. SHE in ACN [192].Figure 39 shows the change in the HOMOs during the electron transitions.After the single-electron oxidation, the HOMO of phenothiazine becomes more localized in the aromatic core.After the second oxidation, an energetically lower orbital becomes the new HOMO.This leads to aromatization of the central ring and high stability of the cation.Reprinted with permission from [193].Copyright (2021) American Chemical Society.Redox potential calculations of phenothiazine and N-methylphenothiazine in acetonitrile (PCM) at the DFT (B3LYP, B3PW91) and Hartree-Fock levels of theory were performed by Rawashdeh with Pople basis sets.While the Hartree-Fock method correctly describes the geometric structure, the redox potentials are strongly underestimated.The best agreement with experimental results (100 mV deviation) was obtained using the B3PW91/6-311+G * method [194].
The influence of electron-donating and withdrawing groups on the orbital energies of these molecules was studied in several investigations at DFT/B3LYP-D3/TZ2P, TD-DFT:M052x/6-31G(d), B3LYP/6-311G(d,p) levels of theory.The results follow the trend of experimental redox potential measurements.A decrease in the HOMO energy due to electron withdrawing substituents leads to higher redox potentials of the phenothiazine derivatives.Electron donating groups decrease the redox potential [192,[195][196][197][198][199].Redox potential variations of 0.84 V-1.08 V vs. SHE in the first oxidation step and 1.45 V-1.66 V vs. SHE in the second oxidation step were reported for methyl groups and chlorine atoms at R 2 position [192].Thus, phenothiazine and phenazine exhibit the same tendencies.

Benzothiadiazole based molecules
Another sulphur-containing structure of interest for RFBs is 2,1,3-benzothiadiazole (BzNSN) [87].The molecule transforms into an anion-radical on reduction with one electron and has a similar geometric structure as the benzo-1,2,3-dithiazolyle radical discussed in section 3.3.4(figure 36).BzNSN has a low reduction potential of −1.05 V vs. SHE in ACN and is relatively stable in non-aqueous solvents [200].Further modification of the basic structure with the replacement of the sulphur atom by nitrogen was reported in experimental research as a new promising class of low potential anolytes [201].
The electronic properties of BzNSN derivatives have been considered in several studies at the DFT level of theory.The influence of electron donating and withdrawing groups on the redox potentials was demonstrated at the B3LYP/cc-pVDZ level of theory.Substituents containing heteroatoms with higher electron affinity than N increase the oxidation potential and decrease the HOMO energy compared to substituents containing electron-rich atoms such as Se [202].Moreover, the shift of optical absorption and emission for substituted molecules due to electrochemical reduction and oxidation has been confirmed by TDDFT calculations at the B3LYP/cc-pVDZ, B3LYP/def2-TZVP and M08-HX/6-311++G(d,p) levels of theory [202][203][204].Despite the deviations in the predicted values, the theoretical results describe the tendencies of substitution effects on the redox potential qualitatively correctly in comparison to experimental results.
The adsorption of BzNSN on a graphene electrode was modeled at the PBE-D3/pw level of theory by Howard et al.They demonstrate a strong interaction of the molecules with surface defects [205].These results predict passivation of reactive defects on graphene electrodes in battery technologies.

Summary on theoretical studies of sulfur-nitrogen-containing molecules for RFB
Sulphur atoms as heteroatoms in aromatic molecules shift the electronic density compared to nitrogen-only systems.This usually causes a shift in the redox potentials, with values decreasing when sulphur is introduced.This was confirmed in a theoretical screening study by replacing heteroatoms in cyclic structures [206].A detailed theoretical consideration of the electronic structure and redox activity in the context of RFBs is not yet available.An exception represents the group of the phenothiazine-based structures.

Pyrylium ions-containing systems
Antoni reported about the synthesis of a potentially useful compound containing pyrylium ions (figure 37) that could be used in RFBs [207].The molecular structure is very similar to pyridine-based active materials, with nitrogen replaced by oxygen.The positive charge in the oxygen is delocalized across the aromatic system, but there is also a tendency to concentrate on neighboring carbon atoms, leading to a cationic character [208].The redox reaction of isolated pyrylium and thiopyrylium structures was described by Saeva.It can be understood as a one-electron process with reduction to a radical or a two electron process with reduction to a corresponding anion with charge localization at the oxygen atom (figure 37) [209].Derivatives such as pyrylenes can have high reduction potentials (E 0 = 0.77 V vs. SHE) and be strong oxidants [207].However, the stability of such organic ions is questionable as various degradation reactions such as dimerization of the radical form can take place [210].The stabilization of pyrylium systems by functionalization with carbenes was discussed in this context.But here the reduction mechanism becomes more complicated because of electronic conjugation [207].
The coordination of counter ions such as halogen anions to the pyrylium structure was investigated at MP2/6-311++G * * and B3LYP/6-311++G(d,p)(6d,10f) levels of theory in DMSO and benzene (PCM) by Milov et al [211].The cationic nature of the pyrylium was confirmed by the coordination of the counterions to the carbon atoms.They also demonstrated the importance of counterions in stabilizing the pyrylium structure due to the electron density distribution and a covalent bond formed between the counter ion and pyrylium.DFT and MP2 methods provide similar structural information and zero-point frequencies for the pyrylium ion [211].Optical spectra (UV-Vis) of pyrylium based structures calculated by TDDFT at the B3LYP/6-31G(d) level of theory are in only approximate qualitative agreement with experimentally measured spectra [212].A detailed theoretical description of pyrylium-water complexes at the B3LYP/6-31+G(d,p) level of theory and analysis of molecular orbitals was carried out by Parreira and Galembeck [213].No significant change in the pyrylium structure was found as a result of coordination to water.This is due to the weak hydrogen bonding of the pyrylium C-H group with the water molecules.The coordination of the water molecules was confirmed to take place on carbon atoms at position 2 and 6 of the pyrylium ring.Orbital analysis shows s-and p-character of the bonding orbitals within the coordination.
Pyrylium-based structures have the potential to be used as active materials for RFBs, although studies are so far rare.Oxygen atoms as heteroatoms in aromatic molecules shift the electronic density to higher redox potentials.Detailed theoretical consideration of the electronic structure and redox activity of these molecules in the context of RFBs active materials are not investigated or presented.

Organometallic complexes
Another interesting class of active materials, both from a practical and theoretical point of view, is the combination of organic materials with metal atoms.The combination of fine-tuned organic materials with stable properties and metal ions also represents a RFB class of redox active materials.Here we give only a brief overview of this very large class of active materials and discuss some representative computational investigations.
Organometallic compounds utilize metal ions in non-aqueous systems with fast kinetics.A classic example of organometallic active materials is metallocene complexes (figure 38).The redox reaction in these compounds is accompanied by a change in the oxidation state of the central ions.Battery systems with these active materials usually contain charge carriers such as Li ions [214].HOMOs and LUMOs of ferrocene (Fc) in the oxidation reaction are presented as an example in figure 39.The qualitative change of the d yz orbital due to the superposition with p orbitals is clearly visible.
Properties can be varied by additional functionalization of the cyclopentadienyl rings of the metallocene with one or a few organic substituents.The same molecular design by substitution is possible as in the case of purely organic active material modifications (R groups in figure 38).The redox potentials of one-electron reactions M + /M 0 can thus be varied over a wide range from −1.23 V to 0.79 V vs SHE [216].
In addition to metallocenes, a large number of other complexes were used as active materials in RFBs.Typical examples are various complexes of bipyridine (bpy) derivatives and metal acetylacetonate derivatives  The Royal Swedish Academy of Sciences.All rights reserved.[217].Some complexes with a heavy atomic core such as U or Ru were also explored for use as electrolytes.Several complexes such as [Ru(bpy) 3 (BF 4 ) 2 ] have high cell voltages of up to 2.6 V and thus great potential for future application [217].
The theoretical calculations of systems with bipyridine ligands and different metallic cores require more accurate approaches than standard DFT due to the breakdown of hybrid and double hybrid functionals in the electronic structure calculations.DFT calculations lead to the prediction of an incorrect spin-state.Therefore, even the energetics of the redox reactions cannot be reliable estimated.The theoretical prediction of other measurable parameters of the molecules at the DFT level of theory is also questionable.Thus, these calculations require more powerful electronic structure theory methods.The DFT failure was investigated e.g. by Drosou et al [218] and Milko and Iron [219] at CASSCF/NEVPT2/def2-TZVPP level of theory and described as being caused by the multi-reference nature of the underlying structures.However, several of the GGA functionals considered such as τ-HCTH and HCTH can also describe the electronic states with relatively small errors [219].The problem of the identification of coordination complexes with multi-referent ground states at the DFT level of theory was also discussed in the work of Shee et al [220].The successful application of auxiliary field quantum Monte-Carlo methods with cc-pVXZ-DKH basis sets was demonstrated with high accuracy for the calculation of redox potentials of metallocenes with multi-configurational wave functions.This methodological approach has a scaling factor of O(N 4 ) with Gaussian basis sets, and looks very promising for challenges involving the description of multi-referent character, symmetry breaking states and similar problems [80].
The theoretical exploration of other organometallic systems with cores such as Cr, V, Mn, Fe, Ru showed that for screening of various parameters such as redox potentials and solubilities, DFT (PCM, acetonitrile)-B3LYP/6-31+G(d) can provide reliable results.In the study by Kucharyson, the calculations were performed for ferrocene and acetylacetonates-based complexes [221].A correlation was found between the electron density on the metal ions and electrochemical stability.This is a potentially interesting descriptor of stability.The orbital energies of ferrocene in one-electron oxidation reactions in vacuum were studied at DFT-PBE level of theory.A shift of the HOMO-LUMO gap to lower energies after oxidation and conversely a shift to higher energies upon reduction back to the neutral state were identified [215].The previously discussed change in the orbital overlap due to one-electron oxidation is an interesting phenomenon, which may indicate changes in chemical bonding in the underlying system (see figure 39).
Theoretical investigations of organometallic complexes are also important for understanding the complex mechanisms of electrochemical reactions of the material class.In this context, the stability of lithium complexes with 1,2,3,4-tetrahydro-6,7-dimethoxy-1,1,4,4-tetramethylnaphthalene (TDT) in different oxidation states was successfully explained at the B3LYP/6-31+G(d) level of theory with the SMD solvent model by Carino et al.Using a combined DFT and in situ spectroscopic approach, the coordination of neutral TDT molecules with lithium ions was determined and confirmed [222].
In summary, organometallic complexes show possible advantages of the organic system compared to metal-based electrolytes in terms of structural flexibility, and advantages of the metal ions in terms of high oxidation state stability.Moreover, complexes can be important objects of investigation as intermediates in mechanisms of redox reactions in battery systems with metal ion charge carriers.The theoretical description of organometallics in screening studies and the understanding of reactions and degradation mechanisms can be more difficult due to the complex electronic structure, which can have a multi-reference character already for ground states.Due to the size of the systems and the complexity of the electronic structure, systematic methodological approaches for screening studies are an important challenge for quantum chemistry in the investigation of organometallic materials for RFBs.

Conclusions
The central components for energy storage in RFBs are the active materials in the anolyte and catholyte.Many different potentially redox-active organic materials from different classes are known today, such as radicals, quinones, nitrogen-containing molecules, heterocyclic systems with nitrogen, oxygen and sulphur, and organometallic complexes, all of which were considered in this review.The redox active materials were discussed from a theoretical point of view in terms of quantum chemical screening and investigations of the redox reactions.Theoretical studies are currently available for most of the classes of the redox active materials considered.However, some of these considerations are only superficial and not systematic enough.In-depth analysis of molecular and electronic structure, reaction mechanisms, or the correlation between electronic structure, redox properties and stability is often lacking.In addition, detailed descriptions, illustrations of the electronic structure and orbitals of molecules are sometimes missing.
Otherwise, theoretical screening is particularly important and useful.It can support experimental synthesis, as it is practically impossible to experimentally explore the large number of potential candidates for redox-active materials with highly variable structures.With the help of computational screening studies, a pre-selection can be made.Moreover, the influence of synthetic modifications such as substitutions of already well investigated basic structures on redox potentials, can be predicted at least qualitatively with standard quantum chemical approaches.DFT-based theoretical screening methods are now recognized as an efficient tool for helping to solve the practical problem of searching and designing redox-active materials for RFBs.The DFT based methods provide reasonable qualitative results for redox potentials in most cases and allow calculations with acceptable computational time.The main difficulty in using DFT lies in the choice of the density functional for the molecule class considered.
In general, DFT-based calculations of single-molecule reactions are employed using an implicit solvent model (COSMO, PCM, SMD), and the redox potential is the most important parameter for the primary screening studies.For most redox reactions involving closed-shell systems, DFT calculations with density functionals such as PBE and B3LYP provide reliable results regarding redox activity, solubility, and corresponding trends of structural modifications.Nevertheless, the analysis of such results should be carefully carried out, as the DFT method can fail for specific molecules and reactions [129,218].Theoretical results for radicals and systems with charge delocalization with potential electronic state degeneration may contain larger discrepancies compared to molecules with closed shell non-degenerated states.These molecules require careful analysis of the nature of the electron correlations.The prediction of different correlation types for products and reagents can lead to inconsistent description of the reaction energies at a chosen level of theory and requires careful selection of an appropriate quantum chemical method for all molecular forms in the reaction.Outside of the electronic structure description, errors in the predicted experimental parameters arise from the limitations of the solvent models used [140].Additive errors in the absolute values of the redox potential can also arise from the chosen reference constants such as the absolute SHE potential.However, these deviations do not affect trends and are not linked to the correctness of quantum chemical calculations.
An important problem in the systematic screening is the prediction of the stability of potential organic redox active materials.This concerns molecules in all redox states.Due to the many types of degradation mechanisms, there is no universal descriptor of stability that is independent of the specific organic material.The search for both universal and powerful stability descriptors is probably one of the most important issues in computational screening methods.However, different stability descriptors have already been proposed.Examples of the different descriptors are reaction energies of possible side reactions [117], structure deformation energies based on dihedral angles [150], and electron density distributions at active centers.These represent useful descriptors for the selected basic structures but are not a universal approach for all compound classes [221].The different system classes and structural fragments are most efficiently described by different descriptors.
Beyond DFT screening studies, there are two important directions for further theoretical consideration.The first is to generalize the screening data using statistical methods and ML tools [206] to further accelerate predictions for molecules of similar nature and their properties such as redox potential and solubility based on statistics.The integration of these new approaches is currently an innovative and developing research area.The second is the detailed theoretical investigation of potentially interesting candidates using more precise theoretical approaches such as wavefunction-based correlation methods, especially multireference methods.These will be used to verify the electronic ground state, to calculate reaction and degradation mechanisms, and to more accurately investigate experimentally measured parameters [130].The detailed investigations provide information on reaction paths, kinetics and charge transfer in accordance with Marcus theory and transition state theory.
The future of the most efficient and thorough theoretical studies of active materials may therefore lie in combined hierarchical approaches (figure 40).First, the most interesting potential candidates are selected by calculations at lower levels of theories, such as DFT or yet more large-scale statistical analysis or semi empirical methods.Even at the basic DFT level, specific descriptors of molecular properties can be calculated to gain understanding of important trends: Orbital energies correlated with redox potentials, polarity indicating solubility, structural deformation energy as possible criteria of stability, etc.Second, a more detailed analysis using more expensive and in-depth theoretical methods such as CASSCF-based or quantum Monte-Carlo-based approaches will be used to confirm the electronic structure and clarify the values of interest as well as for a detailed analysis of reaction mechanisms or degradation mechanism.In addition, the synthetic availability will be assessed.In particular, the search for organic redox-active materials is a good example of applying methods from theoretical chemistry to solve practical technological problems.and D Mollenhauer acknowledge the fruitful discussion and support from CMBlu Energy AG. J Janek acknowledges funding by the German Research Foundation (DFG) under Project ID 390874152 (POLiS Cluster of Excellence).

Figure 1 .
Figure 1.Overview of molecular classes, properties and used methods in theoretical studies of this review.

Figure 4 .
Figure 4. Reduction (right path) and oxidation (left path) of a TEMPO-based molecule (center) to the aminoxyl anion (right) and to the oxoammonium cation (left).

Figure 8 .
Figure 8. Reduction and oxidation of the PTIO-radical (center) to the oxoammonium cation (left) and to the aminoxyl anion (right).

Figure 9 .
Figure 9. Reduction and oxidation of the Blatter radicals (center) to the Blatter cation (left) and the Blatter anion (right).

Figure 16 .
Figure 16.Schematic illustration of a conical intersection in a system.

Figure 17 .
Figure 17.(a) Reduction of a benzoquinone via a two-electron and two-proton transfer in aqueous acidic solution to a hydroquinone.(b) Reduction reaction of the anthraquinone basic structure via a two-electron and two-proton transfer in aqueous acidic solution to a 9,10-dihydroxyanthracene.

Figure 19 .
Figure 19.Calculated redox potentials of the AQ derivatives at the PBE-D3/aug-cc-pVDZ(PCM) level of theory (all values vs. SHE).The positions of the substituents are given on the x-axis.The black line indicates the redox potential of pristine AQ, the blue and white areas help to distinguish between the different numbers of substituents.The crossed-out data points represent molecules with a dihedral distortion within the ring system of more than 22 • , which are considered to have a low stability.Taken from[150] and modified, John Wiley & Sons.[© 2020 The Authors.Published by Wiley-VCH GmbH].

Figure 22 .
Figure 22.One-electron reduction of pyridinium cation-based anolyte core structure with formation of the pyridyl radical.

Figure 24 .
Figure 24.Two-electron reduction of the viologen dication (containing two pyridyl cores).The first reduction leads to a viologen cation radical and the second to the formation of an additional double bond.

Figure 26 .
Figure 26.Two-step reduction of N-Methylphthalimide to the corresponding dianion.

Figure 29 .
Figure 29.Two-electron reduction of phenazine without protonation at the two nitrogen atoms.The reduction leads to destruction of the π-conjugation in the central ring and deformation of the molecule.

Figure 30 .
Figure 30.Two-electron reduction reaction of azobenzene to azobenzenium dianion in an aprotic environment.

Figure 33 .
Figure 33.Molecular orbitals (HOMO and LUMO) of the neutral and reduced states of TCNE calculated at DFT level of theory.Reproduced from[191] with permission from the Royal Society of Chemistry.Color coding: C (grey), N (blue).

Figure 34 .
Figure 34.The core structure of phenothiazine and its oxidation in a two-step reaction to the aromatic phenothiazinium dication in an aprotic environment.

Figure 37 .
Figure 37. One-and two-electron reduction of an isolated pyrylium ion to the radical in the first step and to the pyrylate anion in the second step in an aprotic environment.

Figure 38 .
Figure 38.General structure of a metallocene complex where M = V, Cr, Mn, Fe, Co, Ni, Ru, Rh and R = any organic functional groups.

Figure 40 .
Figure 40.Combined approach for most fully theoretical analysis of active materials.

Table 1 .
Conversion data for the recalculations to SHE values.

Table 3 .
Experimental redox potentials (in mV vs. SHE) of dithiazolyle-and dithiadiazolyle radical in ACN.