1 Introduction

Peroxides are compounds with the general formula ROOR\('\), where R and \(R'\) correspond to any (generally organic) chemical group bonded through a covalent bond. These are naturally ubiquitous in the environment mainly because of atmospheric \(\hbox {O}_2\), which can react, for instance, with water to form hydrogen peroxide (\(\hbox {H}_2\hbox {O}_2\)) [1]. In biological systems, \(\hbox {H}_2\hbox {O}_2\) is widely known as a side product of the cellular reduction of \(\hbox {O}_2\), which is coupled to the oxidation of nutrients [2,3,4]. In presence of reduced metallic cations as \(\hbox {Fe}^{2+}\), the oxygen–oxygen bond cleavage in hydrogen peroxide is facilitated by the so-called Fenton reaction [5, 6], where the subsequent formation of the strong oxidizing hydroxyl radical (\(^{\cdot }\)OH) results in the destruction of cellular components as lipids, proteins and nucleic acids, leading to inflammatory processes, carcinogenesis and aging [7,8,9,10]. Peroxides have also been used in a wide range of industrial processes as the synthesis of polymeric resins [11, 12] and the removal of organic contaminants from water bodies [13, 14].

The biological and industrial role of peroxides is directly related to the low stability of the O–O bond in comparison with other compounds in which oxygen is bonded to elements of the second period [1]. Keeping this in mind, the theoretical characterization of the chemistry of peroxides should include a comprehensive study of the homolytic peroxide bond breaking (PBB). In our previous work, we performed a systematic study defining numerical descriptors associated to the PBB process in hydrogen peroxide and carrying out a benchmark study of the performance of DFT methods [15, 16] (64 functional-basis set combinations; Table 1) [17]. We first considered the bond dissociation energy (BDE) as an energetic descriptor, which is defined in the following way for a ROOH peroxide:

$$\begin{aligned} \hbox {BDE} =\Delta _{f}\hbox {H}_{298.15}^0(\hbox {RO}{^{\cdot }}) + \Delta _{f}\hbox {H}_{298.15}^0({^{\cdot }}\hbox {OH}) -\Delta _{f}\hbox {H}_{298.15}^0(\hbox {ROOH}) \end{aligned}$$
(1)

A validation based on the BDE takes advantage of the availability of experimental reference data [18]. However, it is not easy to identify sources of error in its calculation from first principles because deviation from experimental data could be originated from the electronic structure and frequency calculations as well as from the inability of computational methods to describe other experimental conditions such as deviations from the ideal behavior. As an alternative, extrapolated CCSD calculations at the complete basis set (CBS) limit provide additional benchmark reference data reflected in the electronic energy (including internuclear repulsion) associated to the PBB process (electronic BDE, eBDE), which is defined as:

$$\begin{aligned} \hbox {eBDE}=E(\hbox {RO}{^{\cdot }})+E({^{\cdot }}\hbox {OH})-E(\hbox {ROOH}) \end{aligned}$$
(2)

In our recent study, we observed good agreement of the BDE value of hydrogen peroxide calculated with several DFT-based methods compared to its experimental reference value of \({50.35}\,{\hbox {kcal mol}^{-1}}\) [18]. The unsigned errors did not exceed \({5.0}\,\hbox { kcal mol}^{-1}\) for almost all the tested functional-basis set combinations (Table 1). A basis set dependence explained in terms of qualitative arguments based on the features of each functional-basis set combination was also observed. Same trend was verified for the calculation of eBDE, whose reference value was obtained by performing highly correlated ab initio calculations. These results enabled us to conclude that the main source of error in the prediction of BDE is mainly dominated by the electronic structure description. The importance of a correct description of the electron correlation energy was evidenced from high-level ab initio calculations and supported the feasibility of using DFT methods to computationally describe the PBB process in hydrogen peroxide [17].

Here, we study whether the conclusions for hydrogen peroxide can be extended to a total of 14 peroxides with the general formula ROOH, where the R group ranges from alkyl and alkoxy groups to halogen atoms. Criteria for the choice of ROOH peroxides considered their chemical variability in terms of the number of atoms and elements present in the R group, the availability of experimental reference values for BDE and the feasibility of the calculation of a reference for eBDE. Their structure is shown in Table 2 together with a syntax used to differentiate them and their BDE experimental references. The hydrogen peroxide radical anion \(\hbox {H}_2\hbox {O}_2^-\) was also included as a model system for the reaction in presence of metals as in the Fenton reaction, where the metallic center acts as a reducing agent transfering an electron to the peroxide bond [19].

For applications, choice of a peroxide within a group of candidates is often required instead of the knowledge of specific features of only one of them [6]. Therefore, rather than focusing only on the reproduction of the exact value of BDE we also analyzed whether the different functional-basis set combinations are able to reproduce BDE’s trend among peroxides accounting for the experimental error.

Table 1 Chosen \(E_\mathrm{xc}[\rho ]\) functionals and their features (UE and MUE values in \(\hbox {kcal mol}^{-1}\))
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Table 2 Chosen peroxides, their experimental reference values of BDE [18], syntax of the labels used in this manuscript for their R group and their corresponding ball-and-sticks models (obtained by using the software Chemcraft [35])
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2 Methods

All peroxides listed in Table 2 together with hydrogen peroxide anion (\(\hbox {H}_2\hbox {O}_2^-\)) were considered in this work. Reference values for eBDE were obtained using highly correlated ab initio calculations for the peroxides containing the R groups H (reference peroxide), Cl (electron-withdrawing group), Et and CMeO (bulky aliphatic and electron-withdrawing group). Calculations were carried out considering Unrestricted Hartree–Fock determinants (UHF) at the CCSD(T) level using the complete basis set (CBS) extrapolation scheme of Halkier et al.[36] combined with Dunning’s basis sets [37,38,39] up to triple-\(\zeta\) quality. Geometries optimized at B3LYP/6-31+G(d, p) level of theory were used for the four ROOH peroxides (for a justification, see section S.1 in supporting information). All these calculations were performed using the PSI 4 program [40] coupled to the MRCC suite [41].

Assessment of DFT-based methods was performed in the same way as in our previous work [17]: Geometry optimization and frequency calculations of ROOH peroxides, \(^{\cdot }\)OR and \(^{\cdot }\)OH radicals were carried out in order to confirm minimum energy structures on the potential energy surfaces (zero imaginary frequencies) and to obtain formation enthalpies of these species. BDE and eBDE values (Eqs. 1 and 2) were calculated for each functional-basis set combination from the obtained structures. All electron condition and the command \(grid=ultrafine\) for the numerical computation of integrals were considered in all the calculations, which were carried out using the Gaussian 09 package of programs [42].

Exchange–correlation functionals (\(E_\mathrm{xc}\)) used in the calculation of eBDE and BDE values and their main features are shown in Table 1. This choice is the same as in our previous work [17]; these are covering a wide range in the so-called Jacob’s ladder and reproducing BDE values in the database ABDE12 used by Peverati and Truhlar in 2014 [34]. Moreover, inclusion of range separated hybrid functionals and empirical parameters in the design of functionals were also selection criteria.

The effect of the basis set was tested by comparing four Pople’s basis sets, which may determine the flexibility of the electron density (i.e. its homogeneity vs. non-homogeneity) as previously described by us [17]: MG3S basis set (6-311+G(2df, 2p)) [43] was the starting point because this was used in the databases of Peverati and Truhlar in 2014 [34]. To reduce the computational cost the 6-31G(d,p) basis set was considered and then by successively adding diffuse functions and splitting the valence one arrives to the 6-31+G(d,p) and 6-311+G(d,p) basis sets, respectively. The MG3S basis set is obtained by adding polarization functions to the lastly considered set. Only basis sets including diffuse functions were considered to ensure a correct description of the hydroxyl anion [44].

For a particular ROOH peroxide, performance of each functional-basis set combination was firstly given by the computation of signed and unsigned error (SE and UE, respectively):

$$\begin{aligned} \begin{aligned} \hbox {SE}&=\text {Calculated value} - \text {Reference value}, \\ \hbox {UE}&=|\hbox {SE}| \end{aligned} \end{aligned}$$
(3)

Mean signed and unsigned errors (MSE and MUE, respectively) were also computed as stated in Eq. 4:

$$\begin{aligned} \begin{aligned} \hbox {MSE}= \frac{\sum _{\text {peroxides}}\mathrm{SE}}{n_{\text {peroxides}}} \end{aligned}&\begin{aligned} \quad \hbox {MUE}= \frac{\sum _{\text {peroxides}}\mathrm{UE}}{n_{\text {peroxides}}} \end{aligned} \end{aligned}$$
(4)

Statement of an order relation among the experimental reference BDE values of the peroxides in Table 2 was made by considering their general form:

$$\begin{aligned} \hbox {BDE}(X) = \overline{\hbox {BDE}(X)} \pm \hbox {err}(X) \end{aligned}$$
(5)

where \(\overline{\hbox {BDE}(X)}\) and \(\hbox {err}(X)\) correspond respectively to the average experimental value and its associated error for an arbitrary ROOH peroxide X. Considering the value of \({1.0}\hbox { kcal mol}^{-1}\) as a suitable bound of chemical accuracy, the following order relation on the set of the experimental reference values of BDE is defined in this work:

$$\begin{aligned} \begin{aligned}&\hbox {BDE}(B)> \hbox {BDE}(A) \\&\quad \Leftrightarrow \big [\overline{\hbox {BDE}(B)} -\hbox {err}(B) - {1.0} \hbox { kcal mol}^{-1}\big ] \\&\quad > \big [\overline{\hbox {BDE}(A)} + \hbox {err}(A)+ {1.0} \hbox { kcal mol}^{-1}\big ] \end{aligned} \end{aligned}$$
(6)

From Eq. 6, the useful parameter \(\Delta\) can be defined, where the relation \(\overline{\hbox {BDE}(B)}>\overline{\hbox {BDE}(A)}\) is kept:

$$\begin{aligned} \Delta =\big [\overline{\hbox {BDE}(B)}-\hbox {err}(B)\big ]-\big [\overline{\hbox {BDE}(A)}+\hbox {err}(A)\big ] \end{aligned}$$
(7)

and therefore:

$$\begin{aligned} \begin{aligned}&\hbox {BDE}(B)> \hbox {BDE}(A) \\&\quad \Leftrightarrow \Delta >{2.0} \hbox { kcal mol}^{-1} \end{aligned} \end{aligned}$$
(8)

With this order relation, diagram in Fig. 2a summarizes all possible order relations among the experimental BDE reference values. For the validation of DFT methods from this point of view, any wrongly reproduced order relation is defined as a “penalty” and the total number of penalties per DFT-based method was used as an additional performance criterion.

3 Results and discussion

Table 3 Ab initio eBDE and predicted BDE values for the selected set of five peroxides
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To exclude that deviations from experimental BDE reference values are due to factors different from the correct description of the electronic structure of the involved species, the calculation of eBDE was considered. As reference, we took CCSD(T) calculations extrapolated to the complete basis set (CBS) limit as described in Methods section. This analysis encompassed only five peroxides: H, Et, Cl, CMeO and the anion of hydrogen peroxide (\(\hbox {H}_2\hbox {O}_2^-\)) due to the computational cost. The geometries for these calculations were obtained at the B3LYP/6-31+G(d,p) level because this combination presented a deviation of only \({0.01}\,\hbox { kcal mol}^{-1}\) in the eBDE value of hydrogen peroxide in comparison with that calculated with the experimental geometry (see section S.1 in Supporting Information). For the neutral species H, Et, Cl and CMeO, our results show the same trend for eBDE as in BDE calculations (see sections S.2, S.3 and S.4 in Supporting Information). These facts lead us to establish a necessary condition to ensure that different performance among methods arises from the electronic structure part of the calculation as previously observed for the hydrogen peroxide [17]. Moreover, this supports that CCSD(T)/CBS calculations in combination with DFT geometries and thermal corrections are able to predict the experimental BDE values within chemical accuracy at least for these neutral peroxides (Table 3). In the case of the anion of hydrogen peroxide (\(\hbox {H}_2\hbox {O}_2^{-}\)) a reference eBDE of \({7.75}\,\hbox {kcal mol}^{-1}\) was obtained. As it was expected, this value is smaller compared to the other peroxides by at least \({20.0}\,\hbox {kcal mol}^{-1}\), although ZPE and thermal contribution is practically constant among methods, which allowed us to anticipate a BDE reference value for the anion (Table 3). However, eBDE calculated using DFT methods for this species deviates by at least \({20.0}\,\hbox { kcal mol}^{-1}\) from the ab initio reference. Additionally, these methods did not always predict a smaller eBDE value for \(\hbox {H}_2\hbox {O}_2^-\) compared to the neutral peroxides (see section S.5 in Supporting Information). Wrong description of parameters including anionic species had just been described in our previous work [17], where a non-systematic trend was obtained for the electron affinity of hydroxyl radical by using the same set of functional-basis set combinations considered here.

Having found enough evidence for the feasibility of explaining differences among DFT methods in the reproduction of BDE values through arguments based on the electronic structure of the systems, we performed our further analysis based on deviations from experimental BDE reference data. Figure 1a summarizes the mean unsigned error (MUE) and mean signed error (MSE) for bond dissociation energies (BDE) calculated for the peroxide database with different functionals and basis sets compared to the results of hydrogen peroxide reported recently by us (Fig. 1b) [17]. For hydrogen peroxide, we established that the signed error (see inset) decreases for all functionals when the basis set is augmented from 6-31G(d,p) to 6-31+G(d,p) and 6-311+G(d, p) but increases with the inclusion of polarization functions (MG3S basis set [43]). This trend is confirmed for all peroxides as the insets of SE on Figs. 1a and S.1 (supporting information) reveal. The inclusion of diffuse functions and more functions for the valence shells leads to smaller BDE values because of the electronic energy of the radicals \(^{\cdot } \hbox {OH}\) and \(^{\cdot } \hbox {OR}\) becomes more negative faster compared to the peroxides ROOH when the basis set is enhanced, probably due to the increased flexibility of the larger basis sets, i.e., the so-called size-extensivity problem. The opposite is observed when passing from 6-311+G(d, p) to 6-311+G(2df, 2p) (MG3S) because the addition of polarization functions implies the inclusion of d and f atomic orbitals, which generates a displacement of the electronic density of a nature different from such observed for the addition of s and p atomic orbitals. Therefore, it is not surprising that the behavior of density functionals changes when these modifications on the basis set are made, specifically in terms of the energy variation of the reactant ROOH in comparison with such observed for the products \(^{\cdot } \hbox {OH}\) and \(^{\cdot } \hbox {OR}\) (see insets on Fig. 1 and Figure S.1 in supporting information).

For GGA functionals (N12, BLYP and PBE), all SE were positive for hydrogen peroxide but this trend is not confirmed for all peroxides. Comparing the SE of the different peroxides two groups are distinguished: peroxides with R consisting of less than two atoms (R = H ,Cl, F, rad and Br) present the same behavior as hydrogen peroxide, and the other ones, where the R group consists of more than one atom, present negative SE for these three functionals (see Figure S.1 in Supporting Information). The SE is most negative for R-groups containing an oxygen atom as the ketones. The dissimilar behavior of these two groups of peroxides explains the small values of MSE when the SEs are averaged over all peroxides. For the remaining functionals (meta-GGA and hybrids), the behavior is approximately uniform among peroxides.

One has to consider that the approximate form of \(E_\mathrm{xc}\) functionals is often parameterized on energy differences rather than on the absolute electronic energy. Therefore, the unsigned error is more appropriate to assess their performance for peroxides. For GGA functionals, basis set dependence observed for UE in HO–OH (Fig. 1b) is attenuated when MUE values are calculated (Fig. 1a). This fact is a numerical consequence that the basis set dependence on UE for peroxides with positive SE values is inverse to the other ones (for example, compare \(\hbox {CH}_2\hbox {CMe}_3\) and F in Figures S.1d and S.1e in Supporting Information). This is ultimately originated by the just commented basis set dependence on SE, which is present in all the here considered peroxides, as pointed in the second paragraph of this section. Non-local functionals (meta-GGA and hybrid ones) present a basis set dependence where the MUE increases in the order 6-31G(d, p) to 6-31+G(d, p) and 6-311+G(d, p) and decreases for MG3S in accordance with the result of hydrogen peroxide. This is a direct consequence that these functionals do not present the dependence on the R group as observed for GGA functionals. This could indicate that inclusion of certain amount of exact non-local HF exchange improves the agreement with reference values for the peroxides with larger R group consisting of more than one atom. Non-local range separated functionals as the \(\omega\)B97 family and M11 show smaller MUE, specifically for the combination M11/MG3S, which had just been reported by us as a good method for hydrogen peroxide [17]. The only functional including dispersion effects on this set is \(\omega\)B97X-D, whose good performance can not be attributed exclusively to this feature, since this is not the only difference with other functionals, even in the case of \(\omega\)B97X, because the correlation term was also reparameterized [30]. To support even more this argument, we assayed the effect of including dispersion by calculating eBDE values for the small reference peroxide HO–OH with the small basis set 6-31+G(d,p) and for the large peroxide \(\hbox {CH}_2\hbox {CMe}_3\) with the largest basis set used here (MG3S). We performed these calculations using the functional B3LYP including the D3 dispersion correction. Differences with regard to the B3LYP functional ranged only from \({0.25}\,\hbox {kcal mol}^{-1}\) (for HO–OH) to \({1.00}\,\hbox { kcal mol}^{-1}\) for \(\hbox {CH}_2\hbox {CMe}_3\). Yet, in general the MUE of all peroxides are larger than the values for hydrogen peroxide probably because of the dependence of the error on the number of electrons in the system. This is mainly the case for peroxides with R groups consisting of more than one atom, for which the values of MUE can exceed 7.0 or even \({10.0}\,\hbox {kcal mol}^{-1}\) for hybrid functionals as B3LYP or PBE0 (see Figure S.1 in Supporting Information). Because of its dependence on the cardinality of the basis set and therefore on the number of electrons, we also computed BSSE corrections for the small reference peroxide HO–OH with the small basis set 6-31+G(d,p), and for the large peroxide \(\hbox {CH}_2 \hbox{C}\hbox {Me}_3\) with the largest basis set used here (MG3S). These calculations were performed with the robust functional B3LYP. Difference in BSSE was only \({0.11}\,\hbox {kcal mol}^{-1}\) (\({1.00}\,\hbox {kcal mol}^{-1}\) for HO–OH and \({0.89} \hskip 2pt \hbox {kcal mol}^{-1}\) for \(\hbox {CH}_2\hbox {CMe}_3\)), which allow us to argue BSSE is not a quantity whose consideration affects the interpretation of the obtained results through the set of peroxides.

Fig. 1
figure 1

a Mean unsigned error (MUE) of the calculated values of BDE with different DFT methods for the peroxide database with regard to the experimental reference values. Inserted graph shows the corresponding mean signed error (MSE) values. b Unsigned (UE) and signed error (SE) for hydrogen peroxide as comparison

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For some applications, it is more important to establish an order relation among the BDE reference values of the different peroxides rather than the reproduction of absolute bond dissociation energies to anticipate which peroxide would dissociate more easily. The reactivity order map shown in Fig. 2a relates the experimental values and is used as reference to quantify what functional-basis set combination reproduces the experimental reactivity order. If a DFT method does not reproduce correctly one of the 50 order relations established in Fig. 2a, then a penalty is assigned. The overall number of penalties for the tested DFT methods are summarized in Fig. 2b. The \(\omega\)B97X-D [30] functional presents the lowest number of penalties besides the just described low value of MUE. However, the M11 Minnesota functional [33], which showed small values of MUE, presents a large number of penalties. This suggest that this functional might reduce MUE possibly due to the number of parameters it contains, but it is less robust in the reproduction of the BDE order among different peroxides. Therefore, if the order of BDE is of interest the \(\omega\)B97 family of functionals seems to be more appropriate probably due to the inclusion of a range separation function, which is an effect not depending on the empirical fit that differentiate them from other similar functionals like B3LYP [29].

Fig. 2
figure 2

a Map showing the order relations among the experimental reference BDE values for the peroxides considered in this study. Arrows point to peroxide with higher BDE value, as indicated in the inset. b Penalties committed by each DFT method in the reproduction of the 50 order relations shown in the map of part a

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To further analyze the performance of DFT functionals, we show in Table 4 order relations for which at least one penalty was detected, their number of penalties and the associated \(\Delta\) values. The fact that there is not a correlation between the \(\Delta\) value and the amount of penalties allows us to conclude that the main source of penalties relies on the features of the functional-basis set combinations (Table 4). To study the chemical nature of the penalties we analyzed their distribution among different R groups. Table 4 shows that peroxides with R groups \(\hbox {CH}_2\hbox {CMe}_3\) and CMeO clearly possess larger number of penalties with regard to the total order relations in which they are involved (Fig. 3). This fact could be derived from the larger fluctuations in the computed BDE values for these species as discussed before for the MSE, mainly in the case of GGA functionals.

Table 4 BDE order relations for which penalties were commited by the DFT methods considered here
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Fig. 3
figure 3

Percentage of penalties committed by all considered DFT methods for peroxides with different R groups

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4 Conclusions

In this work, we have extended our previous validation of 64 systematically chosen DFT functional-basis set combinations in the description of BDE and eBDE values for the hydrogen peroxide [17] to a set of 14 ROOH peroxides for which experimental data of BDE are available. In general terms, error trends observed in hydrogen peroxide are conserved in the case of non-local functionals (meta-GGA and hybrid ones): Same basis set dependence is verified for each functional in the reproduction of eBDE and BDE, which suggests that variation among results mainly arises from the electronic structure part of the calculation. This fact allows us to give an explanation of these differences through arguments based on the electronic structure and at the same time to make a rational choice of the functional-basis set combination when computational calculations of BDE or eBDE are required. Within these non-local functionals, the \(\omega \hbox{B97}\) family present smaller absolute deviations for all the peroxides. Local GGA functionals show a specie-depending behavior that can be rationalized in terms of the amount of atoms and elements present in the R group. Therefore, this validation does not only constitute a report of error values, but it also provides a rational strategy for the choice of functional-basis sets based on the features of the functional, the basis set and the specific peroxide to be described.

The performance was not only based on the calculation of deviations from reference values, it also considered the establishment of a reactivity order to evaluate the extent on which DFT methods are capable to reproduce it. Although in general, all the DFT methods reproduced the order relations in at least a 90%, this analysis evidenced some differences between the \(\omega\)B97 family and M11 functionals, which both present the smallest absolute deviation from the reference values: the order is only well reproduced by the first one. We conclude that the establishment of a reactivity order is a valuable strategy for a subsequent validation of DFT methods, for example, in applications where the choice of a certain material is required instead of predicting a specific value. The outcome of this validation enables us to recommend DFT for the study of bond dissociation energy order among peroxides.

Finally, we have also obtained a reference value for the anion \(\hbox {H}_2\hbox {O}_2^-\) from first principles. This value could serve as reference to study the dissociation process of this anion as a limit case of the peroxide bond breaking of the reduced hydrogen peroxide by metallic centers as in the Fenton reaction [19]. However, this work also emphasizes caution in the use of DFT methods when anionic systems are intended to be described, as we have noted in our previous work for the electron affinity of hydroxyl radical [17].