Introduction

Incumbent societal challenges as the electrification of mass-transportation or aviation require a rush beyond the state-of-the-art in the way energy is stored and released1,2. This transition imposes a leap in battery performance and safety without jeopardizing the recent efforts for providing environmental-friendly technologies3,4. Since the last decades of the XX century, and pushed by an extraordinary collective effort, battery science is experiencing an astounding era of exploration through which an unprecedented variety of electrochemical systems has been proposed and studied in the last years5,6,7,8,9,10,11,12. The future success of novel battery paradigms roots in these fundamental studies, that are paving the way for technological breakthroughs and novel energy storage devices.

Aprotic metal-air batteries have theoretical figures that are one order of magnitude larger than those based on Li-ion/Na-ion paradigms13,14,15,16,17. However, all Li–O2, Na–O2, K–O2, Mg–O2, and Ca–O2 systems are still in their infancy, being their respective fundamental electrochemistry still under debate under many aspects.

Parasitic chemistries at positive electrodes in all metal-O2 batteries are the foremost obstacle that hinders the reversible operation of the redox system and thus of the entire battery. The current understanding of all alkaline-Me-O2 systems (Me = Li, Na, K) suggests a common pattern in the degradation mechanism, originating from (a) the nucleophilic attacks of reduced reactive oxygen species, i.e., MeO2, \({{{\mathrm{O}}}}_2^ -\), \({{{\mathrm{MeO}}}}_2^ -\), \({{{\mathrm{O}}}}_2^{2 - }\)18,19,20,21,22 (b) the formation of hydrogen peroxide starting from protons or water molecules23,24, or (c) the high reactivity of singlet oxygen molecules25 generated either by oxidation of peroxides26,27,28 or chemical disproportion of superoxide species mediated by weak Lewis acids24,26,29,30. Overall, the experimental evidence of singlet oxygen release in all alkaline metal-O2 systems24 poses serious fundamental questions concerning the thermodynamics and kinetics of these reactions. Apparently, the reactive mechanism that drives/hinders the singlet oxygen release is only drafted and poorly rationalized for alkaline metals beyond lithium.

Here we tackle the challenge to analyze the thermodynamics of the competitive disproportionation reaction channels of sodium and potassium superoxide in close comparison with LiO2, using multireference ab initio methods. The aim of this work is to shed light on the fundamental thermodynamics of two competitive paths that are involved in the onset of parasitic chemistries in metal-air batteries.

Results and discussion

The superoxide disproportionation reaction

$${{{\mathrm{O}}}}_2^ - + {{{\mathrm{O}}}}_2^ - \to {{{\mathrm{O}}}}_2^{{{{\mathrm{2 - }}}}} + {{{\mathrm{O}}}}_2$$
(1)

is a key ingredient of the chemistry of metal-air batteries. As it is widely known, and as we have shown in a previous paper23 this reaction evolves toward two possible products depending on the multiplicity of the resulting neutral oxygen molecule.

Reaction (1) is highly unlikely to take place in its pure anionic form without a cation, because the two superoxides repel each other via Coulomb forces. The presence of positive ions such as H+, Li+, Na+, and K+ weakens the Coulomb repulsion and allows the reaction to proceed. The most effective ion in catalyzing the reaction is the proton which makes it exothermic7, while the alkali metals are less efficient, and their reaction profiles are endothermic. In a set of previous calculations23, we have unequivocally shown that the reaction is dominated by thermodynamics and that kinetic plays only a limited role. The scheme of reaction (1) in the presence of metal is shown in Fig. 1.

Fig. 1: Scheme of reaction (1).
figure 1

a The \({{{\mathrm{MO}}}}_4^ -\) superoxide complex evolves into a peroxide \({{{\mathrm{MO}}}}_2^ -\) and a neutral oxygen molecule. b The typical energetic profile is endoergic and the simplified electron arrangements in the [O—O] π orbitals are shown.

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The \({{{\mathrm{MO}}}}_4^ -\) superoxide complex on the left of Fig. 1a is made by two superoxide anions that, each being a doublet, can arrange themselves in a singlet or triplet state. In the reactants channel, the two multiplicities have very similar energies. The system evolves toward the right forming a peroxide \({{{\mathrm{MO}}}}_2^ -\) plus an O2 neutral molecule either in its 3Σ or 1Δ states. In the product region, the energy difference between the singlet and the triplet state is significant and about 1 eV. The detailed path of the reaction has been explored using correlated multiconfigurational methods by us in ref. 23 for the H+ and Li+ ions. The former has an exoergic path, the latter an endothermic one. In other words, it turned out that, in the absence of protons, the reaction leading to singlet oxygen in the Li+ catalyzed reaction requires energy and is not spontaneous.

While our calculations clearly pointed in favor of a strong catalytic action of protons, the role of H+ (and ultimately that of water) in the disproportion (1) is still poorly understood and contrasting evidence has arisen in time27,31,32 as recently summarized by Hong and Byon33. While the role played by proton sources such as water is still controversial, the larger amounts of singlet oxygen arising in Na–O2 cells is a somewhat surprising result since reaction (1) with Na+ is expected to be more endothermic than that with Li+.

The difficulties in explaining the amount of singlet oxygen release by reconciling what is seen by experiments and the calculated reaction paths clearly points to the fact that the mechanisms of reaction (1) may not be quite as straightforward as initially thought.

In our opinion, the subtleties in the mechanism, are due to the intrinsic multiconfigurational nature of the electronic configurations involved. Single reference approaches (including DFT or MP2) naturally fails to correctly evaluate the simultaneous presence of more than one determinant (that roughly corresponds to a single Lewis structure) and to correctly account for the possible existence of multiple schemes of electronic occupations. The issue does not simply reduce to the presence of different multiplicities, but rather to the possibility of overlooking certain electronic configurations which might exist at a given geometry of the nuclei.

For this reason, we have carried out tailored calculations using a multiconfigurational space which should be large enough to include all the possible electronic occupation schemes of the metal and oxide moieties. This space stems from the p orbitals of the oxygen atoms that are responsible for the electronic rearrangements (see Fig. 1), and the valence s orbital on the metal. This multiconfigurational approach in treating reaction (1) should ensure that all possible low-lying energetic configurations, hence all possible reactive channels are included, even those that defy obvious chemical expectations.

To keep the presentation simple, and following our previous results23, we have decided to analyze reaction (1) exclusively from the point of view of thermodynamic. First of all, the initial formation of the superoxide complex \({{{\mathrm{MO}}}}_4^ -\) is an exothermic reaction with no kinetic barrier. Therefore, we consider the superoxide complex as the initial reference point of our investigation. We remark that the triplet-singlet energy gap in \({{{\mathrm{MO}}}}_4^ -\) is negligible and irrelevant for the following discussion. Secondly, we know from our previous calculations that the evolution of the complex toward the peroxide follows a path where the kinetic barriers are lower than or substantially equal to the final energies of the products, hence they can hardly be the source of any relevant obstacle to reaction beyond thermodynamics.

For these reasons, to sample the thermodynamic of reaction (1), we can limit ourselves to consider the following dissociation process:

$${{{\mathrm{MO}}}}_4^ - \to {{{\mathrm{MO}}}}_2^ - + {{{\mathrm{O}}}}_2$$
(2)

The energies of the three species have been evaluated separately using the correlated multiconfiguration approach mentioned above and suitable MP2 geometries (see Methods section). The energy variation along reaction (2) has been simply obtained as

$$\Delta E = E\left( {{{{\mathrm{O}}}}_2} \right) + E({{{\mathrm{MO}}}}_2^ - ) - E({{{\mathrm{MO}}}}_4^ - )$$
(3)

where the energies involved depend on the multiplicity of O2 and on the electronic state of \({{{\mathrm{MO}}}}_2^ -\).

The geometries of the \({{{\mathrm{LiO}}}}_4^ -\) and \({{{\mathrm{LiO}}}}_2^ -\) species at the MP2 level with the relevant geometric information are shown in Fig. 2. The respective ones for Na+ and K+ are displayed in Supplementary Figs. 1 and 2. The \({{{\mathrm{LiO}}}}_4^ -\) complex has a symmetric shape with the two superoxides [O—O]− groups bound to the central ion and oriented at 90° with respect to each other. The M—O distances in \({{{\mathrm{MO}}}}_4^ -\) increases as the ionic radius of the alkali increases, moving from 1.89 Å for Li+ to 2.25 Å for Na+ and 2.62 Å for K+. The [O—O]− distance is constant at 1.35 Å. The \({{{\mathrm{MO}}}}_2^ -\) peroxide is a triangle with the [O—O]2− distance varying only slightly with the accompanying ion from 1.57 to 1.59 Å. The M—O distances are 1.69 Å for Li+, 2.06 Å for Na+, and 2.34 Å for K+.

Fig. 2: Geometries of the Li superoxide and peroxide complexes.
figure 2

a The \({{{\mathrm{LiO}}}}_4^ -\) superoxide complex. b The \({{{\mathrm{LiO}}}}_2^ -\) peroxide complex. The M—O distances are reported to highlight the symmetry. The CASSCF(18,13) electronic density is reported on the xy plane as contours.

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The path to the peroxide

As shown in detail in ref. 23, the path to the peroxide formation is straightforward and follows closely the scheme in Fig. 1. The metal retains its positive charge all along the dissociation reaction (2) which is driven by the transfer of the π unpaired electron of one of the superoxide doublets into the hole of the π orbital of the other. This electron transfer produces a closed shell peroxide ion and a neutral oxygen molecule. Along this reaction path, the [O—O] distance is 1.35 Å in \({{{\mathrm{MO}}}}_4^ -\) and becomes 1.6 Å in the \({{{\mathrm{MO}}}}_2^ -\) product. The latter value is the key to recognize the formation of a peroxide. The energies involved in the transformation (2) for the formation of the superoxide depend on the multiplicity of the neutral O2 molecule. They are reported in Table 1.

Table 1 NEVPT2 energies (in eV) of the transformation of \({{{\mathrm{MO}}}}_4^ -\) in peroxide \({{{\mathrm{MO}}}}_2^ - + {{{\mathrm{O}}}}_2\). The values refer to ground state O2(3Σ). The values in parenthesis pertains to O2(1Δ). Approximate values for H+ have also been reported for comparison.
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All reactions involving an alkali metal are endothermic and require a substantial amount of energy for singlet oxygen formation (4.1–4.8 eV in vacuo). The trend for reaction energies in the gas phase is H+ « Li+ < Na+ ~ K+.

The inclusion of a solvent (albeit in an approximate fashion through the SMD model) alters this trend only slightly. The general effect of the solvent is that of stabilization of the \({{{\mathrm{MO}}}}_2^ - + {{{\mathrm{O}}}}_2\) products due to the presence of two solvation shells, hence greater solvation energy with respect to the single molecular complex \({{{\mathrm{MO}}}}_4^ -\). The stabilization is greater for the solvent with a larger dielectric constant (acetonitrile), but, despite this, the reaction leading to singlet oxygen remains endothermic at about 2.9–3.5 eV for all metals.

The path to a reduced metal superoxide

The triangular structure reported in Fig. 2b is a peroxide with the [O—O]2− distance around 1.6 Å. The electronic multiplicity is a singlet, and the valence electrons are essentially occupying the π orbitals of the peroxide ion. The s orbital on the metal is empty. However, we have found that another electronic state of this complex exists and that its energy can be lower than the peroxide depending on the value of the O—O distance. We noticed that, if this distance is reduced to around 1.35 Å, i.e., the typical distance of a superoxide ion, the lowest energy electronic configuration corresponds to an electron in the s orbital of the metal and to the [O—O] moiety in a doublet state with a partially occupied π shell. In other words, there are geometries on the potential energy surface where the reduction of the metal (by oxidation of one of the superoxide ions) becomes accessible and less energetic than the disproportionation of the two superoxides. This process, whose likelihood depends on the O—O distance leads to the generation of a \({{{\mathrm{M}}}}^0{{{\mathrm{O}}}}_2^ -\) complex. The latter is essentially a global singlet state made of two doublets with unpaired electrons occupying the π orbital of the [O—O]− superoxide and the s orbital on the metal. An updated version of the scheme of Fig. 2, now including both the peroxide and the superoxide is shown in Fig. 3.

Fig. 3: Scheme of reaction (1) in the triplet state including the metal reduction channel.
figure 3

The \({{{\mathrm{MO}}}}_4^ -\) superoxide complex evolves into: a A peroxide \({{{\mathrm{MO}}}}_2^ -\) and a neutral triplet oxygen molecule (the same as in Fig. 1). b An \({{{\mathrm{M}}}}^0{{{\mathrm{O}}}}_2^ -\) complex through a reduction of the metal. The simplified electron arrangements in the involved π and s orbitals are shown.

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The optimal geometries of the reduced metal complexes \({{{\mathrm{M}}}}^0{{{\mathrm{O}}}}_2^ -\) are difficult to obtain using single reference methods (this is probably the reason why they did not appear in the literature before). In order to make the SCF procedure converge to them it is necessary to use a broken symmetry approach whereby the unrestricted orbitals are forced to be asymmetric. When this is done the resulting geometries retain the typical [O—O] distance of the superoxide and the metal acquires an electron. The geometry of the \({{{\mathrm{Li}}}}^0{{{\mathrm{O}}}}_2^ -\) is reported in Fig. 4 along with the two half-filled CASSCF orbitals.

Fig. 4: Minimum geometry and orbitals of the superoxide complex \({{{\mathrm{Li}}}}^0{{{\mathrm{O}}}}_2^ -\).
figure 4

The M—O distances are reported to highlight the symmetry. The CASSCF electronic density is reported on the molecular plane as contours. The two singly CASSCF occupied orbitals are also reported.

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The metal reduction channel leading to \({{{\mathrm{M}}}}^0{{{\mathrm{O}}}}_2^ -\) appears, from our calculations, to compete in energy with the disproportion leading to the peroxide. The relevant energy values are reported in Table 2.

Table 2 NEVPT2 energies (in eV) of the transformation of \({{{\mathrm{MO}}}}_4^ -\) into \({{{\mathrm{M}}}}^0{{{\mathrm{O}}}}_2^ - + {{{\mathrm{O}}}}_2\). The values refer to ground state O2(3Σ). The values in parenthesis pertains to O2(1Δ).
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When in vacuo, the reaction leading to triplet oxygen appears at values ranging from 2.6 to 1.8 eV with a trend that is Li+ > Na+ > K+ which is opposite to the one found for the peroxide channel. In other words, when moving from Li+ to K+ the metal reduction channel is more and more energetically favored while the superoxide formation is less and less likely.

The reduction channel leading to \({{{\mathrm{M}}}}^0{{{\mathrm{O}}}}_2^ -\), contrarily to the disproportion one is not favored by the presence of a solvent, but instead its energy increases when increasing the polarity and dielectric constant of the medium. This can be understood by considering that the \({{{\mathrm{M}}}}^0{{{\mathrm{O}}}}_2^ -\) complex has a small charge separation and, hence its solvation energy decreases with solvent dielectric constant.

A pictorial way to represent the situation is reported in Fig. 5, where we report the energies of the disproportionation to the peroxide (blue bars) vs the energies of the metal reduction channel (red circles), both for the formation of the triplet oxygen.

Fig. 5: Superoxide reduction vs metal reduction.
figure 5

Energies of the peroxide channel (blue bars) vs energies of the metal reduction channel (red circles) with respect to the \({{{\mathrm{MO}}}}_4^ -\) complex: a Li, b Na and c K.

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The energies required to form singlet oxygen are larger than ~1 eV in both channels. For each ion we show the results in vacuo and in the two solvents. We stress that the results in the solvent are only approximate since the solvent in the calculation is rendered as a continuum with a molecule-containing cavity. The real solvent might produce additional effects due to the presence of real molecules which are not accounted for in these calculations.

We clearly see that the reduction channel is always competitive with the disproportionation and that the energies required to form singlet or triplet oxygen, if available, would necessarily also open the reduction process at the expense of the metal ion.

For Li+ (panel a of Fig. 5), the reduction channel is the preferred reaction pathway in vacuo (by ~0.5 eV), while it is destabilized (by ~1 eV) in the presence of a solvent. The likelihood of reducing the metal (in solution) is very low for this ion. Hence, the production of singlet oxygen must come from the disproportionation channel that requires 2.9–3.3 eV depending on the solvent polarity (Table 1, first row).

Moving to Na+, the reduction of the metal is by far the most likely process in vacuo since it turns out to lie ~1.7 eV below the peroxide one. When moving to diethyl ether which has a low dielectric constant the reduction is at par with the disproportionation while its energy is slightly above the latter when the dielectric constant increases (acetonitrile). It is remarkable that, if we take the energy values in vacuo for Na+, then the threshold to produce singlet oxygen by metal reduction lowers to only 3 eV instead of the 4.8 eV needed if the reaction followed the peroxide formation path. A similar huge energy gain can be seen for K+ where the reaction involving the metal reduction needs 2 eV less than the disproportionation to peroxide.

Our computational model has been kept purposedly simple to allow for the tracing of the correct electronic state of each molecule involved in the reaction mechanisms. Nevertheless, it is possible to explore the effect due to the additional presence of a second metal cation that binds the \({{{\mathrm{O}}}}_2^ -\) reagent and the \({{{\mathrm{MO}}}}_2^ -\) product. This has been done and the results are presented in the Supplementary Discussion and in Supplementary Table 3.

Impact on Li–O2, Na–O2, and K–O2 battery operation

The competition between the disproportionation and the metal reduction channels starting from the superoxide dimers unavoidably alters the consolidated reaction landscape in aprotic Li-, Na- and K-O2 batteries. In Fig. 6 we summarize the potential energy surfaces of the formation and evolution of the negatively charged lithium, sodium, and potassium superoxide dimers in the model solvent for diethyl-ether. There we highlighted the energy required to give the most stable products able to initiate a degradation/parasitic chemistry in batteries (onset degradation reactant, ODR) either it being singlet oxygen or neutral alkali metals. Neutral metal atoms can act as ODR or as agents promoting a parasitic chemistry because of their well-known reducing properties and of their propensity to attack protic impurities and oxygen molecules.

Fig. 6: Disproportionation/reduction reaction thermodynamics.
figure 6

Competitive reaction paths for a Li, b Na, and c K superoxide dimers calculated in diethyl ether. The number in eV indicates the energetic barriers to the first available channel that leads to ODR (i.e., singlet oxygen or neutral metal see text for details).

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The reaction paths start from the bimolecular dimerization (which is exoergic) and proceeds by dissociating into a metal complex and a free oxygen molecule. The latter step is highly endoergic and, in this respect, one may safely assume that it represents an insurmountable energetic barrier at room temperature. Remarkably, for Na and K metal ions, the lowest-lying dissociation channel is an ODR one leading to the reduction of the metal atom. Overall, the energies required to activate ODR paths are larger than 1 eV, and therefore none of these reactive channels are active in equilibrium conditions at room temperature34.

In all batteries, overpotentials are required to allow ionic transport in the electrolyte (ohmic drop) and to overcome the activation energy (electron transfer overpotential) at the electrode/electrolyte interphase to initiate any redox reaction35,36. Electron transfer phenomena in electrochemical systems have been scouted by Markus and collaborators in many landmarking papers35,37,38,39,40,41,42. Under the assumption of a direct energy transfer from the polarization overpotential to the internal energy of redox reagents35,39, the application of large overpotentials can provide the superoxide species enough kinetic energy to climb the activation energies for disproportionation or metal reduction.

This assumption is not fully accurate for all electrochemical systems37,38, as it neglects possible nonadiabatic effects, but approximates to a reasonable extent the overall energetics of an out-of-equilibrium polarized electrode with flowing current. In this respect in all Li- Na- and K-O2 batteries the application of small or moderate overpotentials can easily initiate degradation paths originated from the reduction of metal ions to isolated metal atoms. In particular the O2− + LiO2 → 3O2 + Li0(O2−) reaction is possibly activated during the discharge of a Li-O2 battery by the application of an overpotential >0.4 V in respect to the E°(O2/LiO2) redox potential. The same reaction channels require overpotentials as small as >0.1 and >0.2 V, in respect to the E°(O2/NaO2) and E°(O2/KO2) redox potential, respectively, in the case of Na-O2 and K-O2 batteries. Such small overpotentials are unavoidable to operate aprotic alkaline-metal-oxygen batteries and are commonly reported in the experimental literature13,16,17,22,25,43,44,45. On the contrary, the release of singlet oxygen requires larger overpotentials (i.e., >0.7, >1.1, >1.2 V for Li, Na, and K, respectively in respect to the corresponding E°O2/MeO2)13,16,17,22,25,43,44,45.

These thermodynamic data suggest that, during the discharge of aprotic metal air batteries, 1O2 is unlikely to form, unless specific conditions are met. In particular, the adoption of cathodic potential cut-offs below 2.3 V vs Li in the case of Li-O2 cells, the presence of proton contaminations or weak Lewis acids can initiate 1O2 release, as demonstrated by us in refs. 23,24,46. The thermodynamic data also indicate that degradation reactions initiated by dimerization of superoxides species and their evolution to the metal reduction channel are energetically accessible under moderate overpotential regimes in the discharge of all aprotic Li-, Na-, K-O2 batteries. This reaction channel could be responsible for the release of ODR in the discharge of metal-air cells, thanks to its most favorable thermodynamics compared to the singlet oxygen release.

A direct experimental evidence is mandatory to prove the occurrence of this unexpected reaction channel: this goal is beyond the scope of this work, but some comments are appropriate. A possible validation protocol can exploit operando or ex situ nuclear magnetic resonance spectroscopy (NMR) on electrochemical cells or analog chemical environments (e.g., KO2 disproportion catalyzed by other metals following a strategy adopted by us recently in ref. 46). NMR has been exploited in battery science in recent years to decipher complex reaction mechanisms for Li and Na electrochemistry as well as O2 redox reactions in aprotic metal-air cells47,48,49,50. In particular, the ability of NMR to discriminate magnetic centers in specific bonding environments can be the key to prove the reduction of metal ions to isolated metal atoms during disproportionation reaction in comparison to the expected formation of peroxides.

In Conclusion, we tackled the analysis of the disproportion mechanism of dimeric superoxide species coordinated by sodium, lithium, and potassium ions to mimic the reaction environment of Li-, Na-, and K-O2 batteries. Our electronic structure calculations and the related thermodynamic evaluations unequivocally show the existence of an additional parasitic redox process besides the expected disproportion to peroxide and molecular oxygen. This additional reaction channel leads to the production of solvated neutral metal atoms. The formation of these species, even in small amounts, might have a negative impact on the performance of aprotic metal-O2 batteries given their well-known reducing activity and high reactivity towards protic impurities and oxygen molecules.

Methods

Multireference calculations

All the calculations have been carried out with the ORCA code51 and with the augmented triple-ζ ma-def2-TZVP52 basis set. Where possible, the resolution of identity (RI) approximation has also been used. The geometries of the molecules were optimized using the spin-component-scaled version of MP2 perturbation theory (SCS-MP2)53. The superoxide complex (the reagent) converges to its triplet ground state, while the singlet triatomic molecule (the product) converges to two different geometries depending on the electronic state. The optimizations, in their restricted variant, converge to the most stable geometry of the peroxide \({{{\mathrm{M}}}}^ + \cdots \left[ {{{{\mathrm{O}}}}_2} \right]^{{{{\mathrm{2 - }}}}}\) which is a closed shell singlet. When SCS-MP2 is used in its unrestricted broken symmetry formalisms, the optimizations converge instead to the superoxide geometry \({{{\mathrm{M}}}}^0 \cdots \left[ {{{{\mathrm{O}}}}_2} \right]^ -\) which is an open shell diradical singlet. The two resulting complexes differ not only by their electronic occupations but also by the [O—O] distance value. Since a diradical singlet is intrinsically non mono-determinantal in character, one should be cautious in using broken symmetry MP2 for its minimum geometry. We have indeed repeated the optimization of one of the trimers using the fully correlated NEVPT2 energies to check the accuracy of the approach. The geometries that we obtained with NEVPT2 and MP2 were essentially identical (see Supplementary Table 1).

To check the convergence of the correlation energy in the chosen basis set, we have repeated one of the calculations using the ma-def2-QZVPP one. The differences in energy were minimal and the relevant data are reported in section Supplementary Methods and in Supplementary Table 2.

In order to properly account for the diradical nature of the species and to determine accurate energies of the electronic states (without relying on inappropriate single-reference ansatz), multi-reference correlated n-electron valence state perturbation theory (NEVPT2)54,55 has been used at the MP2 geometries. The strongly-contracted NEVPT2 lowest singlet energies were calculated on top of a CASSCF(10,7) wave-function for the \({{{\mathrm{MO}}}}_2^ -\) triatomics and a CASSCF(18,13) one for the \({{{\mathrm{MO}}}}_4^ -\) complexes. The active spaces include all the p valence electrons and orbitals from the oxygen atoms and the valence s orbital of the alkali.

All the NEVPT2 energies were checked against a multi-reference configuration interaction (MRCI) with single and double excitations and Davidson corrections, but the former data have been preferred in this context because of their size consistency.

In order to obtain consistent thermodynamic data, the energy of an isolated O2(3Σ) or O2(1Δ) molecule at the same computational level was added to the \({{{\mathrm{MO}}}}_2^ -\) triatomic molecules energies.

Solvent models

These calculations were first performed in vacuum, and then repeated including approximate solvation effects by mean of the implicit solvation model based on solute electron density (SMD) by Cramer and Truhlar56, using two different parametrized solvents (acetonitrile and diethyl ether). The optimized MP2 geometries were relaxed in the SMD solvents. While it would seem advisable to explore the outcomes of our study using computational setups that include an explicit solvent, we must stress here that this would render the present calculation hardly feasible. CASSCF calculations require a clear-cut configuration space made by a small number of active orbitals. The presence of additional molecules in the system would inevitably lead to severe performance degradation, more importantly, to active orbital space selection issues and, ultimately to convergence problems.