The Redox Potentials of Small Inorganic Radicals and Hexa-Aquo

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry

The Redox Potentials of Small Inorganic Radicals and HexaAquo Complexes of First Row Transition Metals in Water. A DFT Study Based on the Grand Canonical Ensemble Federica Arrigoni, Raffaella Breglia, Luca de Gioia, Maurizio Bruschi, and Piercarlo Fantucci J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b01783 • Publication Date (Web): 18 Jul 2019 Downloaded from pubs.acs.org on July 22, 2019

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The Journal of Physical Chemistry

The Redox Potentials of Small Inorganic Radicals and Hexa-Aquo Complexes of First Row Transition Metals in Water. A DFT Study Based on the Grand Canonical Ensemble Federica Arrigoni,§a Raffaella Breglia,§b Luca De Gioia,b Maurizio Bruschi,b* Piercarlo Fantuccia a

Department of Biotechnology and Biosciences, University of Milano-Bicocca, Piazza della Scienza 2, 20126 Milan, Italy.

b

Department of Earth and Environmental Sciences, University of Milano-Bicocca, Piazza della Scienza 1, 20126 Milan, Italy.

Corresponding Author:

Prof. Maurizio Bruschi e-mail [email protected] tel.

+39-0264482816

fax

+39-0264482890

§ These authors contributed equally to the work

1 ACS Paragon Plus Environment

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Abstract The potentials of redox systems involving nitrogen, oxygen and metal ions of the first row transition series have been computed according to the general approach of the grand canonical ensemble which leads to the equilibrium value of the reduction potential via a (complete) sampling of configuration space at a given temperature. The approach is a single configuration approach in the sense that identical molecular structures are sampled for both the oxidized and reduced species, considered in water solution. In the present study the solute and a cluster of 11-12 water molecules are treated explicitly at the same level of theory, and embedded in a continuum solvent. The molecular energies are computed in the framework of the Density Functional Theory (DFT). Our approach is basically different from the approach based on the thermodynamic cycle involving gas phase calculation of the electron affinity of the oxidized species, corrected by the differential hydration energy (obtained from continuum solvent models only) between oxidized and reduced forms. The calculated redox potentials are in agreement with the available experimental data much closer than other results so far presented in the literature. Our results are very satisfactory also in the case of the 3+/2+ redox states of the first row transition metals, i.e. systems with a high positive charge for which enhanced effects of the solvent are expected.


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1. Introduction The role played by the redox potentials of molecules, radicals and metal ions, both in chemistry and biochemistry, has been emphasized in the literature in several reviews.1–4 From a theoretical point of view, recent studies gave evidence that acceptable values of the redox potentials can be obtained (with an average error of 200-300 mV), provided that the solute species are treated at high level of theory and that a suitable representation of the solvent is adopted.5,6 As pointed out in our previous overview,7 the problem of the solvent representation is particularly severe just in the case of water, because water molecules can interact directly with the solutes, accepting or donating H-bonds and forming first and second coordination shells in the case of metal ions, in particular those characterized by high charge. In this context, the approaches aiming at describing the solvent effects only via an implicit model seem to be exposed to difficulties, which can be overcome (at least partially) by adopting a mixed cluster-continuum solvent model in which a few water molecules surrounding the solute are treated explicitly.8 This model is expected to give improved values of the redox potentials. Recent reviews discuss the theoretical methods that can be used to evaluate redox potentials.7,9 The approach based on the ThermoDynamic Cycle (TDCyc) includes the evaluation of the electron affinity in gas phase (ox + e −  → red) (where ox and red are the oxidized and reduced forms, respectively) corrected by the differential solvation energy of the two species, obtained using one of the continuum solvent models. Among the methods aiming at a direct evaluation of the reduction free energy, we can mention the direct evaluation of free energies in solution within the SMD solvation model,10 the Thermodynamic Integration (TI) method,11 which is applied to the process (1 − λ)ox + λred, where λ (0 ≤ 𝜆 ≤ 1) is the integration parameter, and the Grand Canonical Ensemble (GCE) approach,12,13 which considers the chemical potentials of the ox and red species as well as that of the electron. The chemical potential of the electron (supplied by an external reservoir) is considered as constant: for each assigned value of the electron potential, a different equilibrium 3 ACS Paragon Plus Environment


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condition can be reached. The inclusion of the solvent effects via a fully explicit or combined explicit-implicit approach enter the TI or GCE methods in a quite natural manner. The GCE

approach has been used in the present study, combined to a mixed cluster-

continuum solvent model, for the calculation of the redox potential of small molecules and inorganic radicals, as well as of the full series of the hexa-aquo complexes of the first row transition metals. Our mixed cluster-continuum solvation model has been inspired by the study of Bryantsev et al.14 on theoretical calculation of solvation free energy of charged solutes. A general strategy for multistep definitions of solvation shells has been also developed by Sterling and Bjornsson.15 The paper is organized as follows. In the next section our computational method will be described, and technical details of the DFT calculations and of GCE will be provided. In Sect. 3 the results of calculations will be presented and discussed in comparison with other computational studies and experimental data for the following systems: NO/NO−, NO+H+/HNO, NO2+H+/HNO2,

NO3/NO3−,

O2/O2−,

OH/OH−,

[M(H2O)6] 3+/2+

(M=Sc,

NO2/NO2−,

Ti,....Cu)

and

[Cu(H2O)6] 2+/1+.

2. Methods We recognized7 the two-states method of Tavernelli et al.12 based on GCE as a very attractive method, because it describes in a very elegant way how the system composed by ox in the presence of red can be driven to equilibrium. Here we summarize the general outline of the method. For a simple one-electron reduction reaction 𝑜𝑥 + 𝑒 − → 𝑟𝑒𝑑

(1)

the absolute redox potential is defined as ∆𝐸 = −∆𝐺/𝐹

(2)

where ΔG is the variation of Gibbs free energy and F is the Faraday constant. In the following we will assume that ∆𝐺 can be substituted by the ∆𝐴, the Helmholtz free energy, which is directly


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accessible as a statistical average of the sampling of the configuration space. The redox equilibrium can be expressed as follows in terms of the chemical potentials: 𝜇𝑜𝑥 − 𝜇𝑟𝑒𝑑 + 𝜇𝑒 = 0

(3)

where the chemical potential of the electron µe (coming from an external reservoir) is constant. Eq. 3 can be cast also in a form involving the mole fractions of the ox and red species 1

𝑥𝑜𝑥 = 𝑥𝑟𝑒𝑑 = 2

(4)

where 𝑥𝑜𝑥 = (1 + exp(𝑤𝛽) )−1 ;

(5)

and β is the inverse of Boltzmann factor (𝛽 = 1/𝑘𝑇). The quantity w is defined as follows: 𝑤 = 𝐸𝑜𝑥 (𝑹) − 𝐸𝑟𝑒𝑑 (𝑹) + 𝜇𝑒 ,

(6)

where E ox and E red are the molecular energies computed at the same nuclear configuration R for the N-atom system (R is the vector collecting the 3N nuclear coordinates). Therefore, according to the formulation of Ref. 12, a unique configuration space is searched for ox and red forms in order to find the equilibrium xox value at a prefixed value of µe by propagating a MD trajectory using instantaneous averaged forces defined as −𝒇(𝑹) = 𝑥𝑜𝑥

𝜕𝐸𝑜𝑥 (𝑹) 𝜕𝑹

+ 𝑥𝑟𝑒𝑑

𝜕𝐸𝑟𝑒𝑑 (𝑹) 𝝏𝑹

,

(7)

which are always different from zero, also at the equilibrium. In the present study we followed a different way to search for the equilibrium condition. We started from the obvious observation that Eq. 5 gives the value 𝑥𝑜𝑥   =  1/2 when ∣w∣= 0. The condition ∣w∣= 0 can be approached using a Monte Carlo (MC) (Metropolis-Hasting16) sampling at fixed values of T and µe. At each accepted MC step, the computed value of w gives the mol fraction x ox which is accumulated to produce the average value , which is the final result of the simulation. Moves are performed with a random sampling of the space of R components along the direction of the gradient of ∣w∣:



𝝏|𝑤| 𝝏𝑹

= 𝑠𝑔𝑛(𝑤)(

𝜕𝐸𝑜𝑥 (𝑹) 𝜕𝑹

−

𝜕𝐸𝑟𝑒𝑑 (𝑹) 𝜕𝑹

)

Page 6 of 31

(8)

In contrast to Eq. 7, the gradient of |w| can be a null vector when, at a given configuration R, both ox and red forms experience the same distortion from their optimal structure. We have verified in several numerical experiments that the MC sampling here proposed converges to a nearly stationary value of more rapidly than a MD propagation using the forces of Eq. 7. Due to the logistic shape of Eq. 5, the best µe can also be obtained by linear 1

interpolations in the region 𝑥𝑜𝑥 ≈ 2. The convergence of the MC sampling can be determined by the convergence of the computed quantity in successive blocks of 500 accepted moves. The simulation is stopped when the variation of is smaller than 10-4. In Figure 1 it can be seen that such a convergence can be reached in about 5000 accepted points, even starting from a molecular structure (the optimized structure of the reduced form) far from the final one. In general, a better convergence rate can be obtained starting from a better initial structure. We have implemented the preliminary search of such a molecular geometry by minimizing the |w| quantity (Eq. 6) considered as a function of the λ parameter, controlling the transformation ox → red: (1 − 𝜆)𝑜𝑥 + (𝜆)𝑟𝑒𝑑. Therefore, the best starting geometry is defined as 𝑹 = (1 − λ𝑚𝑖𝑛 )𝑹𝒐𝒙 + λ𝑚𝑖𝑛 𝑹𝒓𝒆𝒅

(9)

where 𝑹𝒐𝒙 and 𝑹𝒓𝒆𝒅 are the best structures of ox and red, respectively. In this study we evaluated the redox potentials using the method outlined above (hereafter denoted GCE-MC), with the transition energies ox → red computed at Quantum Chemical (QC) level, according to the Density Functional Theory (DFT). QC calculations (energy and energy gradients) are performed using the TURBOMOLE set of programs,17 and an ad hoc constructed driver is used for MC sampling and statistics. A few calculations on small molecules have been done also using the Gaussian16 set of programs.18 The solvent effects have been considered via a mix cluster-continuum model,8 which makes the calculations much more computer demanding than the 6 ACS Paragon Plus Environment

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TDCyc method, because the presence of the explicit water molecules makes the size of the system treated at QC level bigger. In addition, the MC sampling requires that energy and energy-gradient calculations are repeated thousands of times. The DFT calculations have been done using the TPSS19 functional and the TZVP basis set.20 The dispersion energy is accounted for using the DISP3 method21–23 and the effect of the continuum solvent is simulated by the COSMO model.24,25 The choice of the TPSS functional is motivated by the consideration that a functional free from the Hartree-Fock exchange contribution is required in order to apply the Resolution of Identity (RI) approximation,26 particularly convenient in large scale calculations. In addition, several tests (not reported here for brevity) proved that, for redox potentials, TPSS works better than other pure functionals like BP86.27,28 In the case of the NO radical (considered as a test-case for the reliability of the DFT scheme) some calculations have been performed using also the TPSS/QZVPP scheme,29 and the CBS-APNO method.30,31

Figure 1: Variation of xox and along the MC simulation for the redox pair NO 2  ⁄ NO 2 −  (see text).

An example of evolution of xox and during a typical GCE-MC simulation is reported in Figure 1 for the redox pair NO 2  ⁄ NO 2 −  (see below). In this case the MC searching is initiated from a



Page 8 of 31

nuclear configuration corresponding to the optimal geometry of the [NO 2 (W 11)] − (plus COSMO), at which point xox = 0 and xred = 1. The value converges rapidly toward the value ½ , while xox shows frequent oscillations (room temperature effect) in the range [0.3, 0.7] (or even larger), which correspond to oscillation of w (Eq. 6) in the interval [-0.8, 0.8 mH]. Therefore,

due to the

characteristics of Eq. 5, even very small variations of the transition energy ox→ red may produce relatively large variations in xox.

3. Results and discussion 3.1 The redox properties of small inorganic molecules and radicals In this section we present results obtained from DFT calculations of the reduction potentials of a few nitrogen and oxygen redox pairs. However, before going into a detailed analysis of the redox properties, the level of accuracy of our DFT calculations will be discussed for the case of NO ⁄ NO −  system, chosen because of its importance in biochemical processes,32,33 and because it has been extensively studied also recently by advanced computational approaches.5,6 Table 1 reports the results obtained for 2NO, 1NO +, 3NO − and 1NO−, using the DFT TPSS/TZVP and TPSS/QZVPP schemes, and the CBS-APNO complete basis set method. As for the molecular constants of the neutral species NO, our DFT results show a general, acceptable agreement both with the experiment34 and with the most accurate calculations so far presented (Ref. 5, 6, 35 and present work). Table 1 shows the adiabatic values (corrected for Zero Point Energy (ZPE)) for the Electron Affinity (EA) and the Ionization Energy (IE) of NO. The predicted EA value is small and negative (−0.027eV) when TPSS/TZVP is used instead of being small and positive, as predicted by TPSS/QZVPP (0.050eV) and the high level CBS-APNO method (0.107eV). These values can be compared with the very accurate couple-cluster result equal to 0.023eV reported in Ref. 35, and with the experimental value equal to 0.026eV.36


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Table 1. DFT values for molecular constants of NO,  NO + and NO −.a re

ωe

μ

NO

1.160 / 1.156 / 1.159 (1.151)

1887.5 / 1889.3 / 2233.5 (1904.0)

0.141 / 0.165 / 0.256 (0.153)

NO + 

1.069 / 1.066 / 1.067

2391.1 / 2380.8 / 2866.01

0.366 / 0.345 / 0.659

3

NO − 

1.285 / 1.278 / 1.275 (1.258)

1355.3 / 1359.4 / 1623.4 (1363.3)

0.790 / 0.850 / 0.142

1

NO − 

1.279 / 1.270 / 1.272

1362.3 / 1372.3 / 1700.1

1.101 / 1.248 / 0.878

IE

EA

9.612 / 9.562 / 9.274 / (9.264)

-0.027 / 0.035/ 0.107 (0.026)

NO a

The three values in each entry refer to TPSS/TZVP, TPSS/QZVP and CBS-APNO calculations, respectively, whereas in parenthesis are the experimental values (when available). r e (A),  ω e (cm  − 1),  and μ(D) are equilibrium distance, harmonic frequency and dipole moment, respectively. IE and EA are adiabatic ionization energies and electron affinity (eV) corrected by ZPE. Experimental data from Refs. 34,36–38,41.

The best DFT computed IE value (TPSS/QZVPP) is in error by about 0.30eV with respect to the CBS-APNO value (present work, 9.274eV) and the experiment (9.264eV).37 Finally, as for singlet-triplet(ground state) energy separation in NO − , experiment gives a value of about 17 kcal/mol,38,39 while the TPSS/TZVP, TPSS/QZVPP and CBS-APNO values are equal to 32.70, 31.38 and 18.46 kcal/mol, respectively. In comparison, the results of very accurate calculations are 23.52 kcal/mol (CCSD(T) method with aug-cc-pV5Z basis35), 20.5 and 18.0 kcal/mol (CASSCF (8e6o) and QCISD(T)40), respectively. Therefore, as for the singlet-triplet separation, the DFT results are affected by a quite severe overestimation. However, as it will be discussed in the context of the redox potential of the NO  /1NO  − couple (see below), the singlet-triplet separation in gas phase is a quantity of little relevance for the redox process occurring in water. In gas phase, at 298.15K, the ΔG value for the reaction NO + e− → 3NO −  has been computed as equal to 0.25 (TPSS/TZVP) and −1.2 kcal/mol (TPSS/QZVPP), while −2.38 kcal/mol is the value



Page 10 of 31

of the best electron potential (Eq. 3) found in GCE-MC sampling. This latter value will be commented further in connection with the redox potential of NO evaluated in water solution. In order to conclude the analysis of our results on the NO ⁄ 3NO −  system, we comment its redox properties in water solution. As described above, the results, reported in Table 2, refer to a single molecule (A) embedded in cluster of 11 water molecules (W 11); the A(W 11) complex is surrounded by the bulk solvent simulated by the COSMO model. All values are obtained at TPSS/TZVP/DISP3/COSMO level by means of the GCE-MC simulations, but the same redox system has been also computed using the TPSS/TZVP/PCM scheme. The computed absolute potentials are shifted by -4.44V, which is the recommended value for the absolute potential of SHE.42 The corrected values of Eo calculated for the system NO ⁄ 3NO −  are equal to −0.799V (COSMO) and −0.793V (PCM), a result proving that the particular model adopted to describe the solvent effects outside the solute-water cluster has a negligible effect. Our values compare very well with the most recent experimental value −0.81V43 or −0.8±0.2V,6 and with the theoretical results of Bartenberger et al (−0.76V)6 and Dutton et al (−0.90V)5 obtained using the CBS-QB3 level of theory and the TDCyc (PCM) method, respectively. Note, however, that in Refs. 5 and 6 the computed values are either shifted by a constant quantity determined using the O 2  ⁄ O 2 − system as the standard reference, or obtained via linear regression of theory vs experiment. Our computed Eo potential of NO ⁄ 3NO −  corresponds to an equilibrium μ e value of −0.1338H (−83.96 kcal/mol), much more negative than that obtained for the corresponding gas phase reduction (see above), and proving that the reduction of NO is much favored in water solution, certainly due to the larger hydration free energy of the anion. In Table 2 we also report the E0 value for the NO ⁄ 1NO − couple, equal to −1.169V, more negative than reduction to 3NO −  but higher than the value quoted by Bartenberger et al. (−1.67V).6 This latter estimate was obtained by scaling the computed value for the NO ⁄ 3NO −  couple by the singlet-triplet energy separation in gas phase, computed at CASPT2 level.40 In order to explain such 10 ACS Paragon Plus Environment

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a large disagreement (about 500mV) between our estimate and that of Ref. 6, we performed a more detailed investigation of the singlet-triplet energy separation of NO − in the presence of water. The reactions considered are x NO −(g) → x NO −(aq) (x = 1, 3). The free energy values were computed at TPSS/TZVP level for isolated anions in gas phase x NO −(g), for a W 11 cluster of water molecules (plus PCM) and for the [x NO(W11)] − species (plus PCM). In the gas phase, the free energy difference for the singlet-triplet separation is equal to 33.46 kcal/mol, while in water solution it reduces to only 7.54 kcal/mol (0.327eV). This latter value allows to correct our computed TPSS/TZVP/PCM GCE-MC E0 for the NO/3NO− couple (−0.793V) to the new estimate of E0 for the NO/1NO− couple as equal to −1.126V, a value close to the GCE-MC result (see Table 2), which is therefore confirmed.

Table 2. Computed redox potentials for inorganic molecules and radicals.a redox half reaction

best μ e b

E 0 theory ⁄ V

E 0 exp. ⁄ V c

NO + e −  → 3NO − 

−0.1338

−0.799

−0.81

NO + e −  → 3NO − d  

−0.1340

−0.793

−0.81

NO + e −  → 1NO − 

−0.1202

−1.169

—

NO + H 3O +  + e −  → HNO + H 2O

−0.1612

−0.053

−0.11±0.03

NO 2 + e −  → NO 2− 

−0.2014

1.038

1.04±0.02

NO 2 + H 3O +  + e −  → HNO 2 + H 2O

−0.2030

1.274

—

NO 3 + e −  → NO 3 − 

−0.2535

2.458

2.67

O 2 + e −  → O 2− 

−0.1569

−0.171

−0.18±0.02

OH + e −  → OH − 

−0.2365

1.994

1.98

The computed absolute redox potentials are scaled by −4.44V which is the recommended value for the absolute potential of standard hydrogen electrode.42 All the results are obtained from TPSS/TZVP/DISP3/COSMO calculations, if not otherwise noted (see text); b μ e is the equilibrium electron potential (in Hartree); c The experimental values are taken from the References 6, 40, 43, 45–52. d Results from TPSS/TZVP/PCM (see text). a



Page 12 of 31

A question that can be raised concerns the origin of a so large stabilization of the singlet anion in water solution. The answer came from a careful investigation of the final optimized structure of [1 NO(W11)] − (plus PCM) which actually corresponds to a spontaneous evolution of such a system toward the protonated HNO species, characterized by the hydrogen atom at a relatively short distance (about 1.6Å) from the oxygen atom of a OH− group, which in turn is strongly involved in a H-bond network. We will denote such an occurrence with the notation [ONH⋯OH] − and its molecular structure is illustrated in Figure 2. In addition, it is important to note that formation of [ONH⋯OH] − is specific for the singlet anion: all the efforts to obtain a similar structure for the triplet anion (i.e. optimizations started from different points, included the final structure of [ONH⋯OH] −) were unsuccessful. This means that the triplet anion has no tendency to be protonated from water to give the 3HNO acid, a result fully consistent with the observation that 3HNO is a strong acid with pKA  −1.8.44 Considering all these results, we can conclude that the strong stabilization of the singlet anion of NO in water is certainly due to its tendency to be easily protonated. In other words, 1HNO is a very week acid, as it will be illustrated in the following.

Figure 2: Optimized TPSS/TZVP/PCM structure of [1NO(W11)] − system with the spontaneous evolution toward a HNO − OH− form. Selected distances in Å. Atoms legend: O = red; N = blue; H = white.


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In order to theoretically investigate the pKA of HNO we simulated the reduction of NO in the presence of a H 3O + ion to give the reduced and protonated HNO species in its ground state singlet form, obtaining Eo = −0.053V (see Table 2). Such a value is in error by about 100mV when compared with the estimated value equal to −0.14V.44 Nevertheless, our result is interesting because by combining our μ e values for the two half-reactions NO + e− →  3NO −  and NO + H3O+ + e− → 1

HNO + H2O we can derive the pK A value of 1HNO as equal to 12.60, in good agreement with the

values proposed 11.6±3.435 or ~11.4.44 We also evaluated the pK A of 1HNO using a completely different approach, that is the direct evaluation of free energy values at TZVP/TPSS/PCM level for the species involved in the reaction 1HNO + H 2O → 3NO − + H 3O  + . Each species A is considered separately as an hydrated species A(W 23) (plus PCM) . The resulting pK A value is 8.70, considerably smaller than that quoted above. In spite of expected differences due to different basic methodological assumptions, our results are completely consistent with the view that 3NO − exists in water solution, at a physiological pH, almost completely in the protonated singlet ground state form, thus confirming all theoretical data and experimental results so far obtained.6,40,44 Another nitrogen redox system we studied in detail is NO 2 ⁄ NO 2− and the associated reduction in the presence of an hydronium ion: NO 2 + H 3O +  + e −  → HNO 2 + H 2O for which we predict values of Eo equal to 1.038V, extremely close to the experimental value (1.03V)44,47–50, and 1.274V, respectively (see Table 2). Again, by combining the two reactions, we estimate a pK A value for HNO 2 equal to 3.94, in acceptable agreement with experimental values 3.1651 or 3.25.52 Finally, the last nitrogen redox pair reported in Table 2 is NO 3 + e −  → NO 3− , for which our theretical E0 (2.458V) is almost coincident with the experimental one (2.466±0.019).45 Table 2 also reports two examples of redox systems involving oxygen: O 2 ⁄O 2−  and OH ⁄OH −, for which, again, our computed results compare very well with the available experimental values and with other theoretical estimates. Indeed, for the O 2 ⁄O 2−  couple the computed E0 equal to −0.171V



Page 14 of 31

can be compared with the experimental value −0.16V,51 whereas for the OH ⁄OH −  couple the calculated value of 1.994V matches nearly perfectly the experimental E0 equal to 1.98V.46 The results for such simple radical and molecular systems are very encouraging, in particular when considering that they are obtained without any ad hoc scaling vs experiment correction. This has motivated the extension of our study also to the full series of hexa-aquo complexes of the first row transition metals ions, as it will be discussed in the following section. Clearly the method proposed in this work may be much more computational expensive with respect to approaches based on the ThermoDynamic Cycle (TDCyc) or the direct evaluation of free energies in solution within the SMD solvation model.10 The latter, in particular, has been shown to predict redox potentials of a quite large series of polyatomic organic compounds with a mean absolute error (MAE) of about 0.25V.10 However, such method seems to be less reliable for the evaluation of the redox potentials of red/ox couples involving small strongly polarizing species such as some of those considered in the present work. Indeed, the redox potentials of the OH/OH−  and NO2/NO2−  systems, calculated at the G3MP2-RAD(+)/SMD level of theory according to the procedure described in Ref. 10 (and using a value of 4.28V for the absolute potential of the hydrogen electrode, as suggested by Ho10 and Marenich et al9) are equal to 1.54V and 0.76V, respectively; values about 0.4V and 0.3V smaller than the experimental ones (see Table 2). The situation is partly alleviated for less polarizing systems (at least considering the neutral species) such as NO and O2 for which the G3MP2-RAD(+)/SMD calculated redox potentials are equal to −0.88V and −0.27V, differing by less than 0,1V with respect to the experimental values (see Table 2). Nevertheless, it is important to note that redox potentials calculated with the method presented in this work are always in closer agreement with the experiment.


Page 15 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.2 The redox properties of the hexa-aquo complexes of the first row transition metals. In this section we present results of DFT-MC calculations on the series of aquo complexes [M(H2O)6] 3+ ⁄ 2+  (M = Sc-Cu; for Cu also the system [Cu(H 2O)6] 2+ ⁄1+ has been considered), with the transition metals always considered in their high spin state, even in the case of Co3+ and Ni3+ ions where the low spin ground state cannot be excluded.53 As for the experimental values with which our results can be compared, we will make reference mainly to Uudsemaa and Tamm paper53 which is a detailed review of experimental and computational results on this class of compounds. The hexa-aquo complexes considered here are challenging systems because the high positive charge of the ions induces a strong polarization of the first shell water molecules, thus enhancing the role of the solvation. On the other hand, the systems are computationally relatively easy because, in several cases, the oxidized and reduced forms have very similar molecular structures, an ideal situation for an approach based on a single configuration for both redox species, like our GCE-MC method. Uudsemann and Tamm53 calculated the redox potentials of the hexa-aquo complexes of the first row transition metals using the TDCyc method combined with a mixed cluster-continuum solvent model, with a shell of 12 water molecules surrounding the hexa-aquo ion. Indeed, they clearly showed that either the presence of only the continuum solvent or the presence of only the (small) cluster of water molecules is an incomplete physical model, leading to computed redox potentials always far from the experiment. The importance of considering a second shell of water molecules outside the coordination sphere has been also discussed in detail by Radoń et al. 54 in the context of advanced calculations on Fe3+ and Ru3+ aquo complexes. The physical model adopted in the present study (the hexa-aquo complex surrounded by 12 water molecules, with an external continuum solvent simulated by the COSMO method) is similar to the model of Ref. 53, but it differs because we evaluate the redox potential from a statistical average at room temperature instead of adopting a TDCyc procedure. Finally, we should stress also the point that Uudsemann and 15 ACS Paragon Plus Environment


Page 16 of 31

Tamm53 found that all attempts of computing entropy contributions from vibrational and rotational energies of the full complex+water cluster system is completely unsuccessful, to the point that they preferred to use the entropy contributions derived from experimentally measured temperature dependence of the redox potentials. Our results must be considered as very satisfactory when compared with the most accurate experimental data available and with the results of Ref. 53, and this confirms the general reliability of the computational approach. A comment is needed as for the Sc3+/2+ system for which our CGEMC simulation predicts E0 = −1.718V, significantly different from the experimental value quoted in Ref. 55, but close to the theoretically computed value reported in Ref. 53 (−1.61V). The trend of the computed values of the present study, and of Ref. 53, is very parallel to the experimental values for the series Ti-Cu, while the case of Sc seems to be an outlier so significant to drive us to the conclusion that this experimental value should be reconsidered. Such a conclusion was already drawn by Uudsemann and Tamm53 and now it seems to be confirmed by the results of a different theoretical approach. In Table 3 we also report the computed E° value for Cu2+/Cu+, which is in agreement with the experimental one. Such a case has been considered because both coordination geometry and coordination number of Cu2+ are markedly different from that of Cu+. This can be easily seen from Figure 3, where we report also the average structure 𝑹 = 𝑥𝑜𝑥 𝑹𝑜𝑥 + 𝑥𝑟𝑒𝑑 𝑹𝑟𝑒𝑑 (referred to as Cu*) at a point characterized by xox = 0.502. In order to appreciate the degree of reconstruction of the best geometry of both Cu2+ and Cu+ complexes, we evaluated the RMS of the Cu2+/Cu* (0.402 Å) and Cu+/Cu* (0.410 Å) geometries, which show that the “intermediate” structure Cu* is indeed very close to the average of the structure of the reduced and oxidized species.


Page 17 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 3. Computed redox potentials for hexa-aquo complexes of the first row transition metals ions M 3 +⁄ M 2+. a

a

b

c d

redox half reaction

E 0 theory ⁄ V

E 0 theory ⁄ V b

E 0 exp. ⁄ V c

Sc3+ + e −  →  Sc2+

−1.718

−1.61

−2.3

Ti3+ + e −  →  Ti2+

−0.845

−0.75

−0.9

V3+ + e −  →  V2+

−0.224

−0.17

−0.255

Cr3+ + e −  →  Cr2+

−0.408

−0.50

−0.407, −0.41, −0.42d

Mn3+ + e −  →  Mn2+

1.542

1.21

1.51, 1.56, 1.5415d

Fe3+ + e −  →  Fe2+

0.808

0.73

0.771

Co3+ + e −  →  Co2+

1.941

1.75

1.92, 1.81d

Ni3+ + e −  →  Ni2+

2.150

1.94

2.26±0.12

Cu3+ + e −  →  Cu2+

2.399

2.32

2.4

Cu2+ + e −  →  Cu+

0.158

---

0.153

The computed absolute redox potentials are scaled by −4.44V which is the recommended value for the absolute potential of standard hydrogen electrode.42 Theoretical values from Ref. 53 in which calculations were carried out using the TDCyc method combined with a mixed cluster-continuum solvent model. Experimental values from Ref. 53 and references therein. In bold are the experimental values used in the linear regression discussed in the text.

Figure 3: Best geometry of Cu2+ (left), Cu+ (right) complexes and of the “intermediate” Cu* form (middle). Selected distances in Å. Atoms legend: Cu = light blue; O = red; H = white.



Page 18 of 31

The agreement between the theoretical data for Sc-Cu of our study (X) and that of Ref. 46 (Y) is proved by the following regression line Y = 0.9267X – 0.0028; R = 0.9971. A similar regression line between our data (X) and the experimental values (Y) (see Table 3) is Y = 1.0953X – 0.121; R = 0.9960 and further highlights the very good agreement between calculated and experimental redox potentials. All E0 values reported above have been obtained by scaling the computed absolute potentials (CAPs) by 4.44V, taken as reference absolute potential of the hydrogen electrode (APHE).42 We are aware that the choice of APHE = 4.44V may be subjected to some criticism because other values for APHE, ranging from 4.2V to 4.7V, have been also reported in the literature.56–65 Clearly, a different choice of APHE can affect the agreement of our results with the experiment. Nevertheless, it is worth noting that a different choice of APHE could affect the comparison of the absolute values, while the linear correlation will remain exactly unchanged. In order to give to our results a stronger internal consistency, we decided to scale the CAPs of our ox(aq) + e − → red(aq) systems by the CAP of the Fe3+/Fe2+ couple. This choice is motivated by the observation that several accurate measurements of E0(Fe3+/Fe2+) agree on the value of 0.771V.66–69 Therefore, the new derived values E0calc(vs Fe3+/Fe2+) are compared with the experimental values E0exp(vs Fe3+/Fe2+). The results, reported in Table 4, are still very satisfactory. The largest absolute error (excluding the cases of Sc3+/Sc2+ (see above) and Fe3+/Fe2+) for a set of 14 redox systems is about 250mV (for the NO3/NO3− system), and the average absolute error is 61 mV. Therefore, the use of a reference CAP internal to the computational method adopted (i. e. obtained with the same explicit-implicit model for the solvent) does not change the quality of our computed results. In addition, this result indicates that the effect of the discontinuity of potential at the explicit/implicit solvent interface may be accounted for just using the internal reference potential or, in other words, that the effect of such a discontinuity seems to be transferable across the considered systems.


Page 19 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 4. Comparison of the computed and experimental redox potentials vs Fe3+/Fe2+ potential. redox half reaction

E0calc(vs Fe3+/Fe2+) ⁄ V

E0exp(vs Fe3+/Fe2+) ⁄ V

NO + e −  → 1NO  − 

−1.471

−1.58

NO + H 3O  +  + e −  → HNO + H 2O

−0.862

−0.88

NO 2 + e −  → NO 2 − 

0.232

0.27

NO 3 + e −  → NO 3 − 

1.650

1.90

O 2 + e −  → O 2 − 

−0.979

−0.95

OH + e −  → OH  − 

1.188

1.21

Sc3+ + e −  →  Sc2+

−2.527

−3.1

Ti3+ + e −  →  Ti2+

−1.653

−1.7

V3+ + e −  →  V2+

−1.033

−1.026

Cr3+ + e −  →  Cr2+

−1.216

−1.178, −1.18, −1.19a

Mn3+ + e −  →  Mn2+

0.734

0.74, 0.79, 0.770a

Fe3+ + e −  →  Fe2+

0.000

0.000

Co3+ + e −  →  Co2+

1.133

1.15, 1.04a

Ni3+ + e −  →  Ni2+

1.342

1.49

Cu3+ + e  −  →  Cu2+

1.591

1.6

Cu2+ + e  −  →  Cu+

−0.641

−0.618

a

In bold are the experimental values used in the linear regression of Fig. 4.

The very good agreement between the theoretical E0calc(vs Fe3+/Fe2+) values (X) and the experimental E0exp(vs Fe3+/Fe2+) values (Y) referred to the internally consistent E0(Fe3+/Fe2+) scale is further evidenced by the regression line calculated for the set of the redox systems of Table 4 (with the exclusion of the Sc3+/Sc2+ and Fe3+/Fe2+ systems) and reported in Fig. 4.



Page 20 of 31

Figure 4: Correlation between theoretical and experimental E0 values reported in Table 3 referred to the E0(vs Fe3+/Fe2+) scale.

4. Discussion and Conclusions We performed a computational study of the redox potentials of solutes considered in water solution, where the solvent is represented by a small number of explicit water molecules embedded in a continuum solvent modeled according to COSMO (or PCM). DFT calculations using the TPSS/TZVP scheme proved to be a suitable basis for GCE-MC simulations. The results of the present study are of the same (or often of significantly better) accuracy of several other results reported in the literature, including those based at very high level of theory but essentially based on TDCyc scheme and continuum solvent models. Such approaches give satisfactory correlations with the experimental values, but they do not seem to be able to give acceptable estimate of absolute redox potentials. A further step toward a more realistic physical model is therefore the consideration of a little cluster of water molecules, which act as first solvation shell in the case of small inorganic molecules and radicals, or as the external solvation shell in the case of the hexa-aquo metal


Page 21 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


complexes. This is the model adopted in our approach for solvation effects, which is close to that used by Uudsemaa and Tamm.53 In contrast to the TDCyc method, the GCE-MC scheme includes all the thermal effects in a very natural manner, via the MC simulation carried out at fixed temperature. Finally, our study shows that very accurate results for redox potentials can be obtained without any ad hoc scaling of data adjusted to the experiment, and working within a well balanced DFT. On the other hand, our approach seems to suffer, in its present implementation, of a strong limitation represented by the number of solvent molecules which can be treated in an explicit way. We worked with 11-12 water molecules and such a number may be increased perhaps up to 20-30 without reducing the size of the basis set employed for the whole system. The question can be raised if the size of the water cluster that we have used is large enough to efficiently hydrate the solute. We have checked that solutes with pronounced polarity (or charged) always occupy central position in the cluster. This is very clearly shown in Figure 3 as an example for metal aquo complexes. On the contrary, in the case of a solute with null or very small polarity (O2, NO) the solute tends to occupy a peripheral position of the W11 cluster, with no evident hydrogen bonds formed with the water molecules. We have further checked this point in the case of the NO/3NO− which has been studied using a water cluster of 30 water molecules (W30). The results of the calculations done at the same level of theory (TPSS/TZVP/COSMO) showed that the neutral NO species occupies a central position of the cluster but it is able to produce a large cavity in which the shortest distance with water molecules is about 3.0 Å, thus confirming the null tendency of NO to be hydrated. In the case of NO− the solute forms some hydrogen bonds with the water molecules as in the case of the W11 cluster. Finally, the computed redox potential for the NO(W30)/NO−(W30) system is equal to 0.766V (vs SHE), only 33 mV different from the result obtained with a W11 cluster (see Table 2). Similar considerations hold for the OH/OH− couple solvated in a cluster of 20 water molecules (W20). The computed value of E0/(SHE) for the OH(W20)/OH−(W20) system is equal to 2.050V, only about 50mV higher than that reported in Table 2, for the OH(W11)/OH−(W11) system. These results should



Page 22 of 31

suggest that the increase of the size of the cluster of water molecules does not dramatically affect their accuracy. However, it is evident that for strictly non polar solutes for which experimental values are available only in non-water solvents due to solubility problems, the computational cost of our approach could became very high (or prohibitive), because one would need to include into the QC part a number of multi-atom solvent molecules. In this case, in order to attain a more general applicability, the present theoretical scheme can take advantage from one of the linearly scaling DFT procedures.70–74 In a recent series of papers,75,76 a new improved methodology for the evaluation of the redox potentials in water solutions (named “computational SHE”) has been introduced with the aim of considering the redox solute and the hydronium ion in a continuous representation of explicit water solvent, thus overcoming the need of an external SHE refence potential coming from (uncertain) experiment. The new approach has in common with the present the following point: the basic ingredient to the free energy value for the reaction ox(aq) + e −  →  red(aq) is the integration along the TI path of the vertical electron detachment energy of the red species, using the DFTMD (ab initio Molecular Dynamics based on DFT) method. The vertical excitation energy of red form along the sampling path is present also in our method, in the definition of our w quantity (see Eq. 6). However, our method is still based on finite size water cluster representation of the solvent, an approach affected by the discontinuity of the potential internal to the solvent. In this respect, we believe that the adoption of an internal reference absolute potential (CAP; see above) derived for a redox couple computed with the same solvent representation, can help to minimize the inconsistencies affecting the finite size cluster representation. As for the accuracy reached by the methods based on DFTMD,75 we should observe that it is not definitely higher than the accuracy of the present results, which are obtained with a simpler (but more approximate) procedure.


Page 23 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5. Acknowledgment We acknowledge CINECA for the availability of high performance computing resources as part of the agreement with the University of Milano-Bicocca.

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TOC Graphic


The Redox Potentials of Small Inorganic Radicals and Hexa-Aquo

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