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Theoretical Predictions of Redox Potentials of Fischer-Type Chromium Aminocarbene Complexes Hana Kvapilová,*,†,‡ Irena Hoskovcová,‡ Jiří Ludvík,† and Stanislav Záliš*,† †

J. Heyrovský Institute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejškova 3, 182 23 Prague 8, Czech Republic ‡ Department of Inorganic Chemistry, Institute of Chemical Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic S Supporting Information *

ABSTRACT: Redox potentials of series of chromium aminocarbene complexes with general formulas [(CO)5CrC(R)N(CH3)2] and [(CO)4CrC(R)N(CH2CHCH2)2] were calculated using DFT methods for both metal-localized oxidation and ligand-localized reduction processes. The electrostatic contribution of solvation was approximated by the polarizable continuum model (PCM); specific interactions of the complexes with counterions of supporting electrolyte were considered by explicitly including these ions in the model. The theoretical redox potentials were correlated with experimental values, and the qualities of the results of the approaches used were compared. It was shown that both sets of calculated redox potentials reproduce the experimental data well. The mean average error of the calculated redox potentials was 0.088 V with the counterions and 0.111 V without the counterions. The best results were obtained for oxidation processes, where the mean average error decreased from 0.110 to 0.059 V due to the inclusion of the counterions.

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INTRODUCTION Carbene complexes are a group of compounds formally derived from the carbene molecule CH2. According to their structure and properties, three types of carbene complexes are generally distinguished: Fischer, Schrock, and N-heterocyclic carbenes. The Fischer and Schrock complexes can be described by general formula LnMCR2, where the carbene moiety :CR2 binds to the metal atom M by a formal double bond. By mutual interaction of the carbene ligand substituents R and electron donating−withdrawing effect of the metal plus additional ligands L, the double bond can be polarized in both senses: Mδ+Cδ− in Schrock type carbenes or Mδ−Cδ+ in Fischer type carbenes. In the Fischer type carbene complexes, the partial negative charge on the metal atom is stabilized by coordination of π-accepting ligands such as CO or CN groups to the metal, while the electronic deficiency on the carbene carbon atom is stabilized by the presence of electron-donating groups such as −NR2 and −OR on the carbene carbon.1−3 Fischer-type carbene complexes are widely used in organic synthesis as both reactants4−10 and catalysts.11,12 The synthetic routes are based on redox properties of the complexes and include a variety of thermally and photochemically initiated redox reactions, often with unsaturated hydrocarbons, producing various types of heterocyclic compounds.13−18 Chromium Fischer carbenes take part in photochemically initiated enantioselective reactions.19−21 In addition to their role in synthesis, some carbene complexes have been studied for possible use in nonlinear optics22 or as electrochemical probes for biosensors.23 © XXXX American Chemical Society

An in-depth understanding of structure−reactivity relationships is essential in order to design complexes for specific use. The present paper deals with Fischer aminocarbene complexes of Cr with the general formulas [(CO)5CrC(R)N(CH3)2] and [(CO)4CrC(R)N(CH2CHCH2)2], which have been recently studied by electrochemical methods, and a relationship between their structural features and electrochemical behavior was established using the linear free energy relationship (LFER) approach.24−28 The experimental study was supported by theoretical calculations of electronic structures of the compounds, as the localization and energy levels of frontier molecular orbitals were correlated with their redox potentials.26−29 However, examples of Cr and Fe aminocarbene complexes where the LFER approach fails were reported,27 and recent results show that effect of the structural flexibility of the molecules plays an important role in certain cases.28,30 The relationship between molecular structure and redox properties thus becomes more complicated and the theoretical description based only on electronic structure and the HOMO and LUMO energies of the compounds in their electronic ground state is not sufficient. In order to design new systems with desired redox properties, a reliable theoretical model is needed to predict the values of their redox potentials before they are synthesized, Special Issue: Organometallic Electrochemistry Received: March 13, 2014

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dx.doi.org/10.1021/om500259u | Organometallics XXXX, XXX, XXX−XXX

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tions of redox potentials on irreversible systems are possible when the electron exchange itself is fast (reversible) and the irreversibility is caused by follow-up reactions of the oxidized or reduced species, which is the case for the aminocarbene complexes.28 The aim of this work was to create a model which would allow reliable predictions of the redox potentials of the metalloorganic compounds under study. In the course of this work, effects of solvation and the influence of interaction of the electroactive molecules with counterions of the supporting electrolyte on calculated redox potentials were investigated.

thus avoiding costly and unnecessary procedures. The aim of the present work is to test such a model by correlating calculated oxidation and reduction potentials with the recently obtained experimental data. A number of papers dealing with calculations of redox potentials have emerged in the past decade, using various theoretical approaches.31−52 A typical way to compute the redox potentials is based on the free energy thermodynamic (Born−Haber) cycle.32−39,52−54 Density functional theory (DFT) is usually the method of choice, as it provides precise results while being relatively inexpensive. Several systematic studies were conducted on series of organic molecules, metal ions, and organometallic compounds in order to find suitable combinations of functionals and basis sets. Roy et al.29,30 reported that the choice of the basis set has a minor influence in comparison to the functional used. In their studies on transition-metal complexes, PBE and BP86 functionals provided the best results and no systematic difference was found between the GGA and hybrid-GGA type functionals. Baik and Friesner31 pointed out that when the B3LYP functional was used to calculate redox potentials of organic compounds, metal complexes, and metallocenes, major improvement in the results was achieved when including diffuse functions in the basis sets. Solvation energy is now routinely estimated using methods based on the self-consistent reaction field (SCRF),55 where solvent is approximated as a polarizable dielectric continuum. A molecule of the solute is embedded in a polarized cavity built in the solvent according to a fixed set of parameters. The electrostatic contribution of the solvation energy obtained in this way is sufficient when there are no strong molecular interactions between particles of the solute and solvent. This is not the case, however, when considering charged particles. Uudsemaa and Tamm38 showed that two coordination spheres (18 water molecules) and the SCRF were required in order to estimate redox potentials of aqueous d-block metals with sufficient accuracy. Shimodaira et al.37 used up to six water molecules in their study on metal ions, small ligand clusters, and Ag complexes. Even though the performance of computational resources is continually increasing, good-quality quantum mechanical calculations on series of coordination or organometallic compounds with one or two full solvation spheres are not yet feasible. Moreover, solutions in electrochemical experiments contain additional particles/ions, which may interact with molecules/ions of the studied compounds together with the solvent molecules. Some ways were proposed to deal with the complexity of the systems. Cattenacci et al.50 calculated redox potentials of ferrocene in different solvents using a combination of molecular dynamics with perturbed matrix method calculations, which allowed them to include the solvent explicitly. Matsui et al.46 developed a model using a pseudocounterion, as they added a charge-dependent correction term for the counterion around the charged compounds, obtaining the best results using the B3LYP functional and including diffuse functions in the basis sets. Hughes and Friesner47 derived a method to correct the DFT-B3LYP calculated redox potentials of transition-metal complexes on the basis of a set of seven physical parameters. In the present work we calculated standard one-electron oxidation and reduction potentials of series of chromium aminocarbene complexes using DFT methodology. With scarce exceptions, the studied complexes undergo reversible oneelectron oxidations and irreversible reductions.25−28 Calcula-

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RESULTS AND DISCUSSION General Considerations. Structures of the Fischer-type aminocarbene complexes included in the study are summarized in Figure 1. All of the compounds contain hexacoordinated

Figure 1. Complexes under study. In the series of complexes 1 and 2 R = OCH3 (a), CH3 (b), H (c), Cl (d), COOCH3 (e), CF3 (f). In the series of complexes 3 and 4 X = O (a), S (b), N−CH3 (c).

Cr(0) as the central metal atom, and they are organized into four series according to their structural analogies: in series 1, the dimethylamino group is bound to the carbene carbon atom together with a phenyl residue, variously substituted at the para position. In series 2, the methyl groups on nitrogen are replaced by allyl residues, one of which creates a chelate ring by coordinating itself in a η2 fashion to the Cr atom. Compounds in series 1 and 2 are labeled a−f according to the phenyl substituent R: a, OCH3; b, CH3; c, H; d, Cl; e, COOCH3; f, CF3. Series 3 and 4 are analogous to series 1; the phenyl ring is replaced by a five-membered heterocycle, bound to the carbene carbon as a 2-hetaryl (series 3) or a 3-hetaryl (series 4) residue. The compounds in series 3 and 4 are labeled a−c according to the heterocycle: a, furan; b, thiophene; c, N-methylpyrrole. We have demonstrated that due to the polarization of the metal−carbene carbon bond the Fischer aminocarbene complexes have two well-separated redox centers, specified by localization of the frontier molecular orbitals.26−28 The oxidation center, represented by the HOMO, is located prevalently on the central metal and the reduction center, represented by the LUMO, is found mainly on the carbene carbon and the C−N bond. Visualizations of the frontier molecular orbitals of nonchelated complex 1c and chelated complex 2c are shown as examples in Figure 2. A more precise description of the redox centers is provided by the spin density distribution in the one-electron-oxidized and one-electronreduced states of the complexes, depicted in Figure 3. In the oxidized complexes the spin density distribution resembles entirely the HOMO in the neutral complex. However, in the B


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complexes range between 0.490 and 0.910 V and one-electron reduction potentials range from −2.036 to −1.601 V vs SCE. The calculations of the standard oxidation and reduction potentials (or “redox” in reference to both the oxidation and reduction processes) were based on the Born−Haber cycle presented in the Experimental Section (Scheme 1). For ΔG calculations we decided to only use one specific functional and basis function set of one type throughout the whole series, so that we could focus on the effects of solvation and interaction of the compounds with counterions of the supporting electrolyte. The B3LYP functional was chosen because our recent results have shown that this method describes the properties of the aminocarbene complexes well. Very good agreement was found between calculated and experimental data such as structural parameters30 and electronic absorption spectra.29 The B3LYP functional provided good-quality results also for structurally related carbene complexes in the work presented by several authors,18,56−61 including the very recent estimates of the ionization potentials and electron affinities by Landman et al.62−64 The choice of basis sets was also based on our previous work; however, we added the diffuse functions on all atoms as Baik and Friesner31 recommended when calculating redox potentials. The experimental redox potentials were referenced to the saturated calomel electrode (SCE). In contrast to this, absolute redox potentials were obtained from the calculations. In order to compare both sets of data, the calculated redox potentials had to be shifted to the SCE reference. The potential of SCE is well-defined with respect to the standard hydrogen electrode (SHE).65 Various procedures have been reported to determine the absolute potential of the SHE, and values in the range 4.1− 4.7 V were presented,66−71 depending on the method and conditions. We decided to use the value 4.44 ± 0.02 V reported by Trasatti,72 which is listed in the recommendations by the International Union of Pure and Applied Chemistry (IUPAC).73 Due to the uncertainty of the absolute potential of the SHE, we mainly paid attention to the parameters of the correlation between the calculated and measured data:slope and correlation coefficient of the regression line. The absolute errors and average absolute errors of the calculated potentials with respect to the experimental potentials serve to show consistency of the results throughout the series of complexes. To estimate the effect of the solvent and the counterion, the calculations were done in three ways: in vacuo, with only the PCM solvent correction, and with addition of a counterion while using the PCM. To analyze the results given by all three models, the calculated data were correlated with the experimental data. In all cases linear plots were obtained. Oxidation potentials and reduction potentials were correlated separately, and an additional correlation was done involving the whole set of redox potentials. The linearity of the overall graph indicates the reliability of calculations of the oxidation and reduction potentials as standard redox potentials according to Schemes 1 and 2 and eqs 1 and 2 (see Theoretical Procedures in the Experimental Section). In Vacuo Calculations. The redox potentials obtained without the solvent correction EVac are affected by large error, and they serve mainly to evaluate relative contributions of the other two models. Complete data for these calculations are given in Table S1 and Figure S1 in the Supporting Information. The correlations found for this case are of rather low quality. The calculated redox potentials are more positive with respect to the experimental values EExp in the case of oxidation

Figure 2. Visualizations of the redox centers of complexes 1c and 2c.

Figure 3. Visualizations of the spin density distributions in the oxidized and reduced forms of complexes 1c and 2c.

reduced complexes the unpaired electron is delocalized throughout the aromatic residue of the carbene ligand, in contrast to the localization of the LUMO in the neutral compounds. In addition, the reduction is followed by visible changes of the complexes’ geometry, causing steric effects to play an important role: rotation of the aromatic substituent is necessary in order to allow the extended electronic delocalization, and it is accompanied by rotation of the amino group.28,30 The geometry of the carbene ligand in the reduced complexes is sterically more demanding in comparison to the ground state structures, and the steric hindrance probably also causes the noticeable bending of the carbonyl groups in the reduced complexes. For these reasons, in the case of the reduction potentials the theoretical description based solely on the ground state LUMO energies is insufficient. Redox centers of the remaining complexes have similar character, slightly modified by substitution effects. Values of experimentally found one-electron-oxidation potentials of these C


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processes and more negative in the case of reduction processes. The lowest and highest absolute errors are 1.501 and 1.794 V for oxidations and 0.956 and 1.357 V for reductions. The mean average absolute error is 1.622 V for oxidations and 1.178 V for reductions. The total mean average absolute error is 1.394 V. Equations of the linear regression lines are Eox,Vac = 1.182Eox,Exp + 1.487 (R2 = 0.871) for oxidations, Ered,Vac = 1.196Ered,Exp − 0.815 (R2 = 0.736) for reductions, and EVac = 2.068EExp + 0.811 (R2 = 0.996) for oxidations and reductions combined. Calculations with PCM. The redox potentials Eox,PCM and Ered,PCM calculated with use of the PCM solvation model are compared to the experimental values Eox,Exp and Ered,Exp in Tables 1 and 2, respectively. All of the calculated redox

Table 2. Calculated Values of Standard Redox Potentials of the Reduction Processes and Their Comparison with Experimental Reduction Potentials redn potential (V) compd

PCM calculation Ered,PCM

counterion + PCM Ered,CI+PCM

exptl vs SCE Ered,expa

1a 1b 1c 1d 1e 1f 2a 2b 2c 2d 2e 2f 3a 3b 3c 4a 4b 4c

−1.833 −1.837 −1.795 −1.706 −1.518 −1.590 −1.960 −1.983 −1.926 −1.826 −1.544 −1.644 −1.450 −1.520 −1.781 −1.733 −1.774 −1.905

−1.791 −1.847 −1.788 −1.614 −1.516 −1.555 −1.973 −1.942 −1.858 −1.885 −1.524 −1.671 −1.424 −1.509 −1.908 −1.746 −1.733 −1.948

−2.005 −2.000 −1.955 −1.830 −1.630 −1.750 −1.975 −1.943 −1.928 −1.900 −1.621 −1.795 −1.591 −1.601 −1.894 −1.909 −1.915 −2.036

Table 1. Calculated Values of Standard Redox Potentials of the Oxidation Processes and Their Comparison with Experimental Oxidation Potentials oxidn potential (V) compd

PCM calculation Eox,PCM

counterion + PCM Eox,CI+PCM

exptl vs SCE Eox,expa

1a 1b 1c 1d 1e 1f 2a 2b 2c 2d 2e 2f 3a 3b 3c 4a 4b 4c

0.890 0.962 1.001 1.009 1.034 1.050 0.628 0.621 0.631 0.660 0.670 0.684 0.934 0.963 0.860 0.927 0.980 0.833

0.909 0.846 0.955 0.955 0.902 0.920 0.560 0.610 0.524 0.525 0.609 0.655 0.954 0.935 0.784 0.928 0.970 0.791

0.837 0.849 0.850 0.887 0.884 0.910 0.490 0.506 0.519 0.529 0.546 0.550 0.897 0.856 0.826 0.874 0.832 b

a

Experimental data for series 1 and 2 were measured in N,Ndimethylformamide and published in ref 27; data for series 3 and 4 were measured in acetonitrile and published in ref 28.

a

Experimental data for series 1 and 2 were measured in N,Ndimethylformamide and published in ref 27; data for series 3 and 4 were measured in acetonitrile and published in ref 28. bOxidation potential was not determined due to total irreversibility of the oxidation process.

potentials are more positive with respect to the experimental values, except for the reduction of compound 2b. In comparison with in vacuo calculations the PCM solvent model gives much better correlation with experimental potentials: the lowest and highest absolute errors decreased to 0.034 and 0.151 V in the case of oxidations and to 0.002 and 0.176 V in the case of reductions. The mean average absolute error is 0.110 V for oxidations and 0.113 V for reductions. The total mean average absolute error is 0.111 V. The total average error is lowered by 1.283 V, the improvement being larger for oxidations (average error decreased by 1.512 V) than for reductions (average error decreased by 1.065 V). Correlations of the calculated and measured data are presented in Figure 4 (inset graphs show correlations for oxidation and reduction processes separately). It appears that the model fits better when it is applied to the oxidation processes, where the slope of the regression line is 0.933 and R2 = 0.941, in comparison to the reduction processes, where the

Figure 4. Correlation diagrams for the calculated (EPCM) and experimental (EExp) redox potentials; calculations with PCM. The equation of the linear regression line for all of the redox potentials (red line) is EPCM = 1.000EExp + 0.109, R2 = 0.998. Inset graph A shows the correlation for oxidation processes: Eox,PCM = 0.933Eox,Exp + 0.159, R2 = 0.941. Inset graph B shows the correlation for reduction processes: Ered,PCM = 1.013Ered,Exp + 0.133, R2 = 0.845.

slope is 1.013 and R2 = 0.845. Closer analysis of the calculated reduction potentials shows that the results found for series 2, which include the chelated complexes with the allyl residues bound to the nitrogen atom, are inconsistent with the results for the rest of the compounds. The average absolute error of D


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the calculated reduction potentials of series 2 is 0.060 V; however, the correlation between calculated and measured reduction potentials of series 2 is rather poor: Ered,PCM = 1.305Ered,exp + 0.615 (R2 = 0.887). On the other hand, when the series 2 is omitted (only the nonchelated complexes are taken for correlation), the average error is 0.139 V and the equation of the linear regression line becomes Ered,PCM = 0.902Ered,exp − 0.041 with R2 = 0.972, which is comparable to the regression for the oxidation processes (correlation diagrams are presented in Figure S2 in the Supporting Information). The high slope of the regression line of series 2 and shift of the absolute values with respect to the nonchelated compounds imply that the results for the chelated complexes are afflicted by a different type of error than the results for the nonchelated compounds. Reduction of aminocarbene complexes is accompanied by major structural changes of the molecular geometry.25,28,30 When an electron is accepted into the LUMO orbital, πelectron delocalization is established throughout the whole carbene ligand. To allow such extended electronic communication, the phenyl (or heterocyclic) substituent is twisted to the position almost coplanar with the plane represented by the central metal, carbene carbon, and nitrogen atom. In addition, the dimethylamino group (series 1, 3, 4) rotates around the carbene carbon−nitrogen bond to allow the new position of the phenyl residue. The noncoordinated allyl group (series 2) interferes with the α position of the phenyl ring more than the methyl group (series 1, 3, 4). Various conformations of the allyl residue are possible, making it more difficult to specify the system present in the redox reaction. Correlation over the full set of redox potentials shows an excellent linear relationship between the calculated and measured redox potentials with slope 1.000 and correlation coefficient R2 = 0.998. The larger span of the scale and higher number of points compensate for the deviations in the calculated reduction potentials. Calculations with Counterions. As we have already noted, charged particles such as the one-electron-oxidized and -reduced forms of the studied complexes may interact not only with the solvent molecules but also with other ions present in the solution. In a typical electrochemical experiment such ions are provided by the supporting (inert) electrolyte, which is used to suppress migration-caused transport of the electroactive species to the electrode. Therefore, we examined the influence of counterions of the supporting electrolyte used on calculations of the redox potentials. In all of these calculations, the electrostatic contribution of solvation was also considered, approximated by PCM. To calculate the oxidation potentials, one hexafluorophosphate anion PF6− was added to both the neutral and oneelectron-oxidized states of the complexes. As an example, the optimized geometry of complex 1c with the PF6− counterion is shown in Figure 5. Interaction of the oxidation center with the counterion is stronger in case of the oxidized complex: the distance between Cr and P atoms is 7.40 Å in the case of neutral 1c and 5.17 Å in the case of oxidized 1c. In the experiments, tetrabutylammonium salt was used. However, we decided to lower the computational costs by using a less bulky alkylammonium cation. Therefore, to calculate the reduction potentials, one tetramethylammonium cation N(CH3)4+ was added to the neutral state, as well as to the oneelectron-reduced state of the complexes. Figure 6 shows the optimized structure of complex 1c with an N(CH 3 ) 4 + counterion as an example. The distance between the carbene

Figure 5. Optimized structures of the model of 1c with the PF6− counterion provided by the supporting electrolyte: (left) 1c in the ground state (singlet state, total charge −1); (right) 1c in the oneelectron-oxidized state (doublet state, total charge 0).

Figure 6. Optimized structures of the model of 1c with the N(CH3)4+ counterion provided by the supporting electrolyte: (left) 1c in the ground state (singlet state, total charge +1); (right) 1c in the oneelectron-reduced state (doublet state, total charge 0).

carbon (reduction center) and the nitrogen atom of N(CH3)4+ is 8.10 Å for the neutral form of 1c and 5.56 Å for the reduced form of 1c, pointing to the considerably stronger interaction in the latter case. Before calculating the redox potentials for the whole set of compounds, several models were created for the neutral, oxidized, and reduced species where the counterion was placed at various locations around the complex, and the geometries of these structures were fully optimized. The most stable structures were chosen as models for the remaining complexes. It should be noted that problems with convergence of the geometry optimization had to be overcome in some cases. Systems where the counterions were added to the neutral molecules were especially problematic due to the weak interaction between the two particles. Many local stationary states, very close in energy, were observed due to flat potential energy surfaces. No convergence problems were observed for the oxidized or reduced particles. Redox potentials Eox,CI+PCM and Ered,CI+PCM calculated including interaction with the counterion in addition to the PCM solvent model are summarized in Tables 1 and 2. Graphic correlations between the calculated and experimental redox potentials are presented in Figure 7. All of the calculated redox potentials are more positive than the experimental values, except for both redox potentials of 3c and oxidation potentials of 1b and 2d. Although the presence of the counterions does not significantly change the absolute values of the calculated redox potentials, it influences the correlations with the experimental values. There is improvement of the calculated redox potentials of the oxidation processes when the counterion is included explicitly in the model. A better slope of the regression line, E


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potentials in comparison to the model without the counterion may be explained this way as well. The fact that we used tetramethylammonium cation instead of tetrabutylammonium cation does not seem to matter in the case of series 2. However, the behavior of this series is different from that of the rest of the complexes, where exchange of the cation may have caused additional deviations. Correlation over the whole set of redox potentials shows good agreement between the calculated and experimental data. Considering the whole range of data compensated for higher errors in one area of the data set. The total mean average error is now 0.088 V, which is 0.023 V lower than in the case of the calculations that did not consider the counterion. The slope and R2 values of the linear regression are comparable, and the intercept is closer to the coordinate origin.

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CONCLUSION A model that employed explicit counterions of supporting electrolyte in addition to the solvation approximated by PCM was created to calculate redox potentials of series of Fischer type chromium aminocarbene complexes. This approach was successfully applied to metal-localized oxidation and ligandlocalized reduction processes. The linearity of the overall graphs, including oxidation and reduction processes, points to the overall consistency of the results provided by the model. Both methods (PCM only and PCM + counterion) produce results that are in good agreement with the experimental data. It is worth noting that the fact that the theoretical treatment works for the experimentally irreversible reduction potentials helps to prove that the electrode reaction itself is reversible and the irreversibility is caused by fast follow-up reactions. Stabilization by the counterion is the most useful in the calculations of reduction potentials of the chelated series 2, where the description of the reduced compounds is more complicated than in the case of the nonchelated complexes. Results obtained using the solvation model without the counterions are already satisfactory; nevertheless, addition of the counterion improves the slopes of the particular linear regressions. The mean average error of the calculated redox potentials was 0.088 V with the counterions and 0.111 V without the counterions. The best results were obtained for oxidation processes, where the mean average error decreased from 0.110 to 0.059 V due to inclusion of the counterions. The only inconsistency was observed in the case of the calculated reduction potentials of the chelated aminocarbene complexes (series 2). It was explained by a more complicated description of the reduced states of the chelated complexes caused by the floppy noncoordinated allyl residue. Both models have been proven to be reliable and usable to estimate theoretically redox properties of aminocarbene complexes proposed for synthesis, the model without the counterions being more affordable.

Figure 7. Correlation diagram for the calculated (ECI+PCM) and experimental (EExp) redox potentials of the oxidation processes: calculations with the counterions and PCM. The equation of the linear regression line for all of the redox potentials (red line) is ECI+PCM = 0.977EExp + 0.071, R2 = 0.998. Inset graph A shows the correlation for oxidation processes: Eox,CI+PCM = 0.977Eox,Exp + 0.069, R2 = 0.917. Inset graph B shows the correlation for reduction processes: Ered,CI+PCM = 1.064Ered,Exp + 0.232, R2 = 0.808.

0.977, is observed. The lowest absolute error of the calculated oxidation potentials is now 0.003 V (down from 0.034 V), and the highest absolute error is 0.138 V (down from 0.151 V without counterion). The mean average absolute error for the oxidation is 0.059 V. In the case of the reduction processes the lowest absolute error is now only 0.001 V, but the highest absolute error increased to 0.216 V. The mean average absolute error is 0.115 V. The overall correlation between the calculated and experimental reduction potentials is comparable with the case of calculations without the counterion, even though both the slope (1.064) and correlation coefficient (0.808) of the linear regression are rather less satisfactory. The average absolute error for series of chelate complexes 2 is 0.051 V, and that for the nonchelate complexes is 0.147 V. The linear regression equation for series 2 is Ered,CI+PCM = 1.294Ered,Exp + 0.598 (R2 = 0.948), and the linear regression for the remaining compounds is Ered,CI+PCM = 0.972Ered,Exp + 0.094 (R2 = 0.838), comparable to those for the oxidation processes (the correlation diagrams presented in Figure S3). This means that addition of the counterion improved the correlation in case of the chelated series 2, leading to better slope and R2. In the case of the nonchelated compounds there is also visible improvement of the slope of the regression line when adding the counterion; however, the correlation coefficient decreased due to the higher deviations between the calculated and measured potentials. We have mentioned that we had to overcome some convergence problems during the geometry optimizations. Most of them occurred in the case of the neutral complexes coupled with the N(CH3)4+ cation. The eventual convergence to the local energy minima may be the source of the relatively large range of absolute errors and consequently the lower value of R2. The larger span of the absolute errors of the oxidation

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EXPERIMENTAL SECTION

Experimental Data. Syntheses, structural characterizations, and experimental values of redox potentials of the complexes under study have been reported in the literature.25−28 Theoretical Procedures. The calculations of Cr aminocarbene complexes were carried out using the Gaussian 0974 software package and employed the density functional theory method. In all of the calculations Becke’s three-parameter hybrid functional B3LYP75,76 was used together with the 6-311++G(3df) basis set for the Cr atom and 6311++G(d) basis sets for the remaining atoms. The geometries of the compounds were fully optimized with no symmetry constraints. Vibrational analysis followed the geometry optimizations to verify the F


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character of the stationary states and to obtain the zero-point energy and thermal corrections to the free energy of the systems. Three states were optimized for each complex: neutral state (ground state, singlet), negatively charged one-electron-reduced state (−1, doublet), and positively charged one-electron-oxidized state (+1, doublet). In the calculations with the counterions, one particle of N(CH3)4+ was added to the neutral and reduced forms of the complexes and one particle of PF6− was added to the neutral and oxidized forms of the complexes, altering the total charge of the systems while keeping the respective redox states. The calculations on molecules without the counterions were carried out both in vacuo (i.e., without interaction with any other medium) and with solvent correction described by the polarizable continuum model (PCM)55,77 implemented in G09 using the integral equation formalism.78 Solvent parameters in the PCM were chosen according to the solvents used in experiments, N,N-dimethylformamide (ε = 37.219) or acetonitrile (ε = 35.688), as built into the G09 software. The systems with the counterions were fully optimized using the PCM model. Calculation of the redox potentials was based on the Born−Haber cycle (Scheme 1). The free energy of the one-electron redox half-

Scheme 2, where the formal notations lead from the oxidized forms to the reduced forms of the complexes. The free energies of the oxidized and reduced species calculated with consideration of solvation effects were corrected using the PCM. The theoretical values of standard redox potentials Etheor were obtained from the calculated free energies of redox reactions in solution ΔGs,redox, using eq 1. In that equation, F = 96485 C mol−1 is

Etheor = −ΔGs,redox /(zF )

(1)

the Faraday constant and, since we only consider one-electron-redox reactions, number of exchanged electrons z is 1. The free energy of the redox half-reaction ΔGs,redox (Scheme 2) was calculated as the difference between the absolute free energies of the reduced (Gs(red)) and oxidized (Gs(ox)) forms (eq 2).

ΔGs,redox = Gs(red) − Gs(ox)

(2)

Absolute theoretical redox potentials were obtained this way. The experimental redox potentials, which we used in the correlations, were given as relative potentials referenced to the saturated calomel electrode (SCE). The potential of the SCE is 0.2412 V more negative65 with respect to the standard hydrogen electrode (SHE), the absolute potential of which is 4.44 eV.72 To get the same reference state as in the experiment, 4.199 V was subtracted from the calculated absolute redox potentials. The correlation diagrams and linear regression relationships were constructed using OriginPro 8 software by OriginLab. The structures of the compounds shown in the figures were plotted using GaussView software.

Scheme 1. Free Energy Thermodynamic (Born−Haber) Cycle for Redox Half-Reaction

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ASSOCIATED CONTENT

S Supporting Information *

reaction in solution, ΔGs,redox, has three components: the free energy change in the gas phase, ΔGg,redox (which consists of the adiabatic ionization potential and the thermal contributions), and solvation energies of the oxidized and reduced species, ΔGsolv,ox and ΔGsolv,red. The particular oxidation and reduction potentials were calculated as standard redox potentials of the oxidation and reduction processes. In analogy to the general free energy cycle shown in Scheme 1, the redox half-reactions in solutions can be formally described according to

A table, figures, and xyz files giving redox potentials calculated in vacuo, correlation diagrams for calculations in vacuo and supplementary correlation diagrams for calculations with PCM and the counterions of the supporting electrolyte, and cartesian coordinates of the computed structures. This material is available free of charge via the Internet at http://pubs.acs.org.

Scheme 2. Formal Description of the Oxidation and Reduction of the Carbene Complexes for Calculations of the Standard Redox Potentials: Example for Series 1a

a

The top and bottom schemes represent the oxidation and reduction processes, respectively. G


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AUTHOR INFORMATION

Corresponding Authors

*E-mail for H.K.: [email protected]. *E-mail for S.Z.: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Ministry of Education of the Czech Republic (grant LD14129), the GAČ R (Czech Science Foundation, grant number 13-04630S) and the J. Heyrovský Institute RVO 61388955 fund. H.K. acknowledges financial support from specific university research (MSMT No 20/ 2014).

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