Unveiling the Cerium(III)/(IV) Structures and Charge-Transfer Mechanism in Sulfuric Acid

The Ce3+/Ce4+ redox couple has a charge transfer (CT) with extreme asymmetry and a large shift in redox potential depending on electrolyte composition. The redox potential shift and CT behavior are difficult to understand because neither the cerium structures nor the CT mechanism are well understood, limiting efforts to improve the Ce3+/Ce4+ redox kinetics in applications such as energy storage. Herein, we identify the Ce3+ and Ce4+ structures and CT mechanism in sulfuric acid via extended X-ray absorption fine structure spectroscopy (EXAFS), kinetic measurements, and density functional theory (DFT) calculations. We show EXAFS evidence that confirms that Ce3+ is coordinated by nine water molecules and suggests that Ce4+ is complexed by water and three bisulfates in sulfuric acid. Despite the change in complexation within the first coordination shell between Ce3+ and Ce4+, we show that the kinetics are independent of the electrode, suggesting outer-sphere electron-transfer behavior. We identify a two-step mechanism where Ce4+ exchanges the bisulfate anions with water in a chemical step followed by a rate-determining electron transfer step that follows Marcus theory (MT). This mechanism is consistent with all experimentally observed structural and kinetic data. The asymmetry of the Ce3+/Ce4+ CT and the observed shift in the redox potential with acid is explained by the addition of the chemical step in the CT mechanism. The fitted parameters from this rate law qualitatively agree with DFT-predicted free energies and the reorganization energy. The combination of a two-step mechanism with MT should be considered for other metal ion CT reactions whose kinetics have not been appropriately described.

S-3 Tables   Table of  Contents  Abbreviated caption  Page  number   Table S1 Kinetics parameters for Ce 3+ /Ce 4+ charge transfer for different electrodes and electrolytes, through different measurement methods. S-5  Table S2 Compiled information for K-and L3-edge fits of CeCl3-7H2O standard. S-9  Table S3 Compiled information for K-and L3-edge fits of Ce 3+ . S-10    Table S5 Compiled information for K-and L3-edge fits of CeO2 standard. S-17    Table S7 Electrode effect on kinetic activity for outer-sphere redox couples S-26 Table S8 Symbols and their definitions used in kinetic rate law derivations. S-28    Section S1. Reported kinetic data for Ce 3+ /Ce 4+ redox couple on various electrodes and in different acids Table S1. Kinetics parameters for Ce 3+ /Ce 4+ charge transfer for different electrodes and electrolytes, through different measurement methods. Kinetic parameters reported include standard rate constant, 0 , exchange current density, 0 , the split between reduction and oxidation peaks, Δ , and anodic and cathodic charge transfer coefficients, and . In the main text we are referring to cathodic charge transfer when we report the charge transfer coefficient.   The fits in the k space and the magnitude component in the R space of the CeCl 3 -7H 2 O standard EXAFS at the Ce K-edge are shown in Fig. S1a and Fig. S1b, respectively. The fit of the CeCl 3 -7H 2 O standard EXAFS at the Ce L 3 -edge in both the R space and the k space is included in Fig. S2. From the fits of the CeCl 3 -7H 2 O standard at both edges, it is evident that both Ce-O and Ce-Cl scattering paths are necessary in the fit to capture the EXAFS spectra. The fitting results for both edges are included in Table S2. Although typically the amplitude reduction factor, S 0 2 , is not larger than 1.0, we see values greater than 1.0 for both edges here. One hypothesis for why we see S 0 2 values >1.0 here is that multi-electron excitation (MEE) effects were not corrected for, which are known to be prevalent for Ce samples. 29,30 Having accurate S 0 2 from the CeCl 3 -7H 2 O is necessary to get accurate coordination numbers for our fits of the cerium ions.   Table S2. K-and L3-edge fit parameters of CeCl3-7H2O standard. Fitting results for CeCl 3 -7H 2 O standard at the Ce L 3 -and K-edges from ARTEMIS, fitting with a Ce-O shell and a Ce-Cl shell. The following parameters were varied in the fit: amplitude reduction factor (S 0 2 ), the shift in threshold energy (ΔE 0 ), the shift in scattering distance ΔR, and the Debye-Waller factor (σ 2 ). The coordination number (CN) for each shell was set as a fixed global parameter. The scattering distance R included in the Table is calculated by adding the ΔR fitted value to the expected scattering  distance from the FEFF path (see Methods). Individual and co-fits of Ce K-edge and L 3 -edge for Ce 3+

Figures and
The co-fit of the Ce K-and L 3 -edge EXAFS of Ce 3+ in 2 M H 2 SO 4 at the Ce K-edge with the Ce-O shell is shown in the k space and the R space (magnitude) in Fig. S3a and S3b. The results of this co-fit at the Ce-L 3 edge are included in Fig. S4. In the fit, we used the edge-specific S 0 2 and shift in threshold energy, ΔE 0 , from the CeCl 3 -7H 2 O standard as global fixed parameters. We varied path specific parameters (coordination number, CN, shift in scattering distance, ΔR, and the Debye-Waller factor, σ 2 ) to obtain the best fit of the data. Fitting results are included in Table S3. It should be noted that the fit of the Ce 3+ EXAFS at the Ce L 3 -edge ( Fig. S4) does not visually capture the magnitude of the spectrum in the R space. We attribute this to artificial features that were not removed during the background removal and normalization procedure, as evidenced by the strong features occurring between 0 and 0.5 Å, wherein it is known that no real scattering event can be occurring. Additionally, it can be seen from the k data ( Fig. S4b) that the data exhibits poor quality after ~6.0 Å −1 , most likely due to MEE effects as well as the interference of the L 2edge.    (Table S2), and then were set as fixed global parameters for the fits below. Three parameters were varied in the fits for each scattering shell: the shift in scattering distance (ΔR) to obtain the scattering distance, coordination number (CN), and the Debye-Waller factor (σ 2 ). The scattering distance R included in the To confirm the trends that we observed in the co-fit shown above, we used an additional set of Ce L 3 -edge Ce 3+ in H 2 SO 4 EXAFS data, which we collected during a different beamline run than that shown in the main text. This data set demonstrated fewer quality issues in the k space, and when normalized, exhibited fewer background features. When we co-fit these data with the Ce K-edge data, the resulting CN, ΔR, and σ 2 were similar to those reported in Table S3, and the fit was visually in greater agreement with the EXAFS data. We could not use this set of L 3 -edge data in the final co-fit, however, because we did not collect EXAFS of a Ce 3+ standard for that specific run. Instead, to obtain a value for S 0 2 for this data set, we relied on a method similar to our process from our previous work, 31 in which we fit the Ce 3+ EXAFS with a CN set to 9 while guessing S 0 2 . We then fixed the S 0 2 value to the one obtained from the previous fit and allowed CN to vary. We acknolwedge that it is preferable to use a Ce 3+ standard to obtain accurate an S 0 2 value, and so we report the fit of the Ce 3+ EXAFS at the Ce L 3 -edge for which we also collected EXAFS of a standard, despite the poorer quality and resulting worse visual fit.
We show the results of co-fitting the Ce K-and L 3 -edge Ce 3+ in H 2 SO 4 EXAFS with Ce-O and Ce-S scattering paths in Fig. S5, and the fitting parameter values for this co-fit are included in Table S3. The resulting fit was unreasonable, with the CN for the Ce-S scattering path unreasonably large with an uncertainty that is larger than the actual fitted CN (7.3±7.4 Å). Thus, we conclude that there is no conclusive evidence that Ce 3+ is complexed by a sulfate or bisulfate in the first coordination shell. To further probe the influence of anions on Ce 3+ EXAFS, we compared the EXAFS of Ce 3+ at the Ce Kedge and fits with a Ce-O scattering path in three different acids: 2 M H 2 SO 4 ( Fig. S6a-c), 2 M CF 3 SO 3 H (TFSA, Fig. S6d-f), and 2 M CH 3 SO 3 H (MSA, Fig. S6g-i). The results of the fit at only the K-edge for each of these acids is included in Table S3. Although the CN obtained from the Ce-O fit for Ce 3+ in H 2 SO 4 is lower than that obtained from the co-fit, the uncertainty (±0.6) is large enough to suggest coordination with either 8 or 9 water molecules. Our DFT and EXAFS evidence of Ce 3+ coordination with 9 water molecules from our previous work, 31 however, resolves this uncertainty from the fit of the EXAFS at just the Ce K-edge. The Ce 3+ -O scattering distance in H 2 SO 4 is slightly larger than the scattering distance in either TFSA or MSA. As discussed in the main text, if Ce 3+ were complexed by sulfate or bisulfate in H 2 SO 4 , the Ce 3+ -O distance would be expected to be smaller than a Ce 3+ -O scattering distance in which the Ce 3+ was coordinated only by water, e.g., Ce 3+ in MSA or TFSA. This supports that Ce 3+ is not complexed by sulfate (or bisulfate) in sulfuric acid. We study the influence of electrolyte concentration on Ce 3+ coordination in Fig. S7. In Fig. S7a, the Ce L 3edge EXAFS peaks associated with Ce 3+ -O scattering demonstrate that there is no trend in peak location or magnitude with increasing H 2 SO 4 concentration. The lack of any trends in the spectra is also the case for increasing MSA concentration (Fig. S7b). The comparison of the Ce 3+ EXAFS Ce L 3 -edge in three different acids (TFSA, MSA, and H 2 SO 4 ) is included in Fig. S7c. Similar to the trend observed in the main text for the Ce 3+ EXAFS at the Ce K-edge, there is no shift in the peaks associated with Ce 3+ -O scattering as the electrolyte is switched from acids where Ce 3+ is known to solely coordinate with water (TFSA and MSA) to H 2 SO 4 . As demonstrated by the Ce L 3 -edge MD-EXAFS of Ce 3+ species (Fig. S7d), we expect there to S-13 be a shift to the left in both peaks if Ce 3+ is complexed by sulfate. Because we do not see this shift experimentally, we conclude Ce 3+ is coordinated solely by water in sulfuric acid.

Section S3. Additional EXAFS analysis of Ce 4+ species and DFT-predicted Ce 4+ anion complexation bond distances and free energies
Ce 4+ EXAFS at the Ce L 3

-edge
To complement the analysis of Ce 4+ EXAFS at the Ce K-edge in the main text (Fig. 2), we completed a similar analysis for Ce 4+ in H 2 SO 4 EXAFS at the Ce L 3 -edge, as shown in Fig. S8. In Fig. S8a, we compare the Ce 3+ EXAFS at the Ce L 3 -edge to the EXAFS of Ce 4+ in H 2 SO 4 that we reported in our prior work 32 but renormalized here. It is the co-fit between this L 3 -edge data and K edge data that is reported in the main text. In Fig. S8b, we compare additionally collected Ce L 3 -edge 0.1 M Ce 4+ 2 M H 2 SO 4 EXAFS (collected as part of a series in which H 2 SO 4 concentration was varied, labeled "Ce 4+ in varied [H 2 SO 4 ]") to the Ce 3+ EXAFS at the Ce L 3 -edge. From both comparisons, the peaks associated with the Ce 4+ -O scattering are shifted to the left of those associated with Ce 3+ -O scattering, and there are additional scattering peaks that occur for Ce 4+ between 2.5 Å and 3.25 Å that do not occur for Ce 3+ . Compared to the scattering peaks that we note for Ce 4+ at the Ce K-edge, these are shifted to the right by ~0.25 Å, which is expected because the (k) function of the L 3 -and K-edges are shifted 90° from each other, which will be translated into the R space. We see in Fig. S8c Fig. S8d and show that peaks in the same 2.5-3.25 Å region disappear when the Ce 4+ -sulfate scattering path is removed. Finally, we show the results of the co-fit of the Ce 4+ EXAFS at the Ce K-and L 3 -edges for the case when the Ce L 3 -edge Ce 4+ in H 2 SO 4 EXAFS data from our previous work 31 is used in Fig. S8e, and for the case when the "Ce 4+ in varied [H 2 SO 4 ]" Ce L 3 -edge EXAFS data is used in the co-fit in Fig. S8f. For both co-fits, we fit the EXAFS of a CeO 2 standard collected at the Ce L 3 -edge at the same time as the Ce 4+ in H 2 SO 4 data collection to obtain S 0 2 and ΔE 0 values. In both cases, the best fit of the Ce 4+ in H 2 SO 4 EXAFS was achieved with Ce-O and Ce-S scattering shells, with Ce-O CN around 9 and the Ce-S CN around 3. Fitting parameter results are included farther below in Table S6.

DFT-predicted Ce 4+ anion complexation bond distances
The averaged DFT-predicted Ce-O bond distances for different relevant Ce 3+ and Ce 4+ species are included in Table S4.

EXAFS standard for Ce 4+ at Ce K-edge and L 3 -edge
The fit of the CeO 2 standard EXAFS with three paths (2 Ce-O scattering paths and 1 Ce-Ce scattering path) at the Ce K-edge and L 3 -edge is shown in Fig. S10a-c and Fig. S10d-f, respectively. In these fits, we fixed the CN values for each path and varied the S 0 2 and ΔE 0 parameters to obtain the best fit. All fitting parameter results are included in Table S5. We then used the edge-specific values of S 0 2 and ΔE 0 in the co-fit of the Ce 4+ in H 2 SO 4 EXAFS as global fixed parameters.  Table S5. K-and L3-edge fitting parameters of CeO2 standard. Fitting results for CeO 2 standard at the Ce L 3 -and K-edges from ARTEMIS, fitting with two Ce-O shells and a Ce-Ce shell. The following parameters were varied in the fit: amplitude reduction factor (S 0 2 ), the shift in threshold energy (ΔE 0 ), the shift in scattering distance (ΔR), and the Debye-Waller factor (σ 2 ). The coordination number (CN) for each shell was set as a fixed global parameter. The scattering distance R included in the Table is calculated by adding the ΔR fitted value to the expected scattering distance from the FEFF path (see Methods).

Co-fits of Ce K-edge and L 3 -edge for Ce 4+ with one Ce-O shell
The co-fit of Ce 4+ in H 2 SO 4 EXAFS at the Ce K-and L 3 -edges using only a Ce-O shell is included in Fig.  S11, and the fitting parameter results are included in Table S6. The Ce-O co-fit overpredicts Ce 4+ -O CN and does not capture the features that we have previously shown to be due to Ce 4+ -S scattering. Because the fit of the Ce 4+ in H 2 SO 4 EXAFS with only a Ce-O scattering path results in a worse fit than the fit that includes a Ce-S scattering path, we conclude that Ce 4+ is complexed by sulfate or bisulfate in the first coordination shell.  species. For the fits, S 0 2 (amplitude reduction factor) and ΔE 0 were obtained by fitting the CeO 2 standard at the appropriate edge (see Table S4), and then were set as fixed global parameters for the fits below. Four parameters were varied in the fits for each scattering shell: the shift in scattering distance (ΔR) to obtain the scattering distance, coordination number (CN), the Debye-Waller factor (σ 2 ), and the third cumulant (σ 3 ). The scattering distance R included in the Table is calculated by adding the ΔR fitted value to the expected scattering distance from the FEFF path (see Methods).

Ce-O shell Ce-S shell Edge
Electrolyte 3.719 ± 0.148 1 *As described in the Methods section, additional 0.1 M Ce 4+ 2 M H2SO4 EXAFS were collected as part of a series of EXAFS measurements in which H2SO4 concentration was varied. For the sake of completeness, this "Ce 4+ in varied H2SO4" EXAFS sample was also used in the co-fit and showed to yield very similar results to those reported in the main text.

DFT-predicted Ce 4+ anion complexation free energies
We consider the Gibbs free energy of Ce 4+ anion complexation with sulfate or bisulfate. We find that the [Ce IV (H 2 O) 8 SO 4 ] 2+ (Fig. S12a) and [Ce IV (H 2 O) 6 (HSO 4 ) 3 ] + (Fig. S12b) species are comparable and most energetically favorable. As discussed in the main text, the series of ligand exchange steps from [Ce IV (H 2 O) 6 (HSO 4 ) 3 ] + to [Ce IV (H 2 O) 9 ] 4+ is most likely quasi-equilibrated, based on the agreement of the kinetic data with the CE, E RDS rate law. Of these quasi-equilibrated steps, we expect in the reduction direction that the replacement of the last bisulfate to form [Ce IV (H 2 O) 9 ] 4+ will most likely be the slowest, based on its unfavorable free energy of 38.3 kJ/mol (Fig. S12b). For the oxidation direction, the slowest ligand exchange step would most likely be to go from [Ce IV (H 2 O) 8

Kinetics on Pt
The averaged values of the exchange current densities obtained from the charge transfer resistance method on Pt are shown in Fig. S13a as a function of [Ce 4+ ] and in Fig. S13b as a function of temperature. We obtained charge transfer resistances, R ct , using Electrochemical Impedance Spectroscopy (EIS) and collected the charge transfer resistance data either immediately before or after we collected the steady state current responses to complete the Tafel analysis. The magnitudes of the exchange current densities agree within 31% of the values using the Tafel method and demonstrate similar dependence on both [Ce 4+ ] and temperature, providing additional confidence in the reported kinetic activity of Pt.

Fig
. S13. Exchange current densities for the Pt rotating disk electrode from charge transfer resistance (Rct) method. Exchange current densities as a function of (a) Ce 4+ concentration (based on titration) and (b) temperature, in 2 M H 2 SO 4 solution with total cerium concentration of 0.05 M. Reported values are averaged from three runs, with error bars representing one standard deviation from the average value in both horizontal and vertical directions. An Ag/AgCl reference electrode and graphite rod counter electrode were used. The solutions were sparged with nitrogen gas for at least 15 minutes before measurements were collected and blanketing with nitrogen was continued throughout the measurement collection. All reported measurements were done at 2000 rpm of the RDE.
The kinetic trends on Pt were similar from 0.01-0.05 M but differed at high concentrations of cerium. We obtained rate constants and cathodic charge transfer coefficients at 0.01 M cerium and 0.03 M cerium (Fig.  S14) that were similar to the values obtained at 0.05 M cerium, indicating that neither the cerium ion nor the electrode surface structures are changing from 0.01-0.05 M of cerium. At a total cerium concentration of 0.15 M (Fig. S15), reproducible data at intermediate concentrations of Ce 4+ could not be obtained. Based on our EXAFS measurements, cerium dimers do not form in the concentration range we tested (up to 0.1 M Ce), so we hypothesize instead that the inconsistent exchange current densities collected at 0.15 M total cerium are because of the effect of cerium on oxide formation on Pt, as discussed below. ). An Ag/AgCl reference electrode and graphite rod counter electrode were used. The solutions were sparged with nitrogen gas for at least 15 minutes before measurements were collected and blanketing with nitrogen was continued throughout the measurement collection. All reported measurements were collected at 2000 rpm of the RDE. The solid lines for each data set in (a) represent the predicted exchange current density for each total cerium concentration, using the Butler-Volmer rate law and the rate constant 0 and cathodic charge transfer coefficient reported in the main text for Pt at 0.05 M Ce. . An Ag/AgCl reference electrode and graphite rod counter electrode were used. The solutions were sparged with nitrogen gas for at least 15 minutes before measurements were collected and blanketing with nitrogen was continued throughout the measurement collection. All data were collected at 2000 rpm of the RDE. The solid lines for each data set in (a) represent the predicted exchange current density for each total cerium concentration, using the Butler-Volmer rate law and the rate constant 0 and cathodic charge transfer coefficient reported in the main text for Pt at 0.05 M Ce.
Pt nanoparticles in sulfuric acid in the presence of Ce are significantly more oxidized than Pt in the absence of Ce at the same potential based on X-ray absorption spectroscopy (Fig. S16). We hypothesize that at higher concentrations of total Ce, the Pt is more oxidized, which decreases the activity as evidenced by the unexpectedly low exchange current densities observed for a system of 0.15 M total cerium (Fig. S15). Decreased kinetic activity on an oxidized platinum surface have been reported previously for other outersphere redox couples 33 as well as the cerium redox couple. 20,34 Thus, when operating at 0.15 M, more consistent pre-treatment of the Pt RDE such as the scheme suggested by Randle and Kuhn 16 would be required to ensure that the Pt oxide that forms is consistently reproduced.  Table S14 for XAFS normalization parameters used.

S-24
The fit of the cathodic Tafel slope data as a function of temperature using the BV rate law is included in Fig. S17. It is clear that the BV rate law is able to model an increase in cathodic Tafel slope with temperature, although the slope is not entirely captured. In the BV rate law, cathodic Tafel slopes are a function of both temperature and the charge transfer coefficient, . In the BV formulism of kinetics, is assumed to be constant with temperature, thus explaining why the increase in cathodic Tafel slopes with temperature is only partially captured (through the direct dependence of Tafel slope on temperature but not through any variation of with temperature). . Reported values are averaged from three runs, with error bars representing one standard deviation from the average value in both horizontal and vertical directions. An Ag/AgCl reference electrode and graphite rod counter electrode were used. The solutions were sparged with nitrogen gas for at least 15 minutes before measurements were collected and blanketing with nitrogen was continued throughout the measurement collection. All data were collected at 2000 rpm of the RDE. Solid line is the fit to the data from the Butler-Volmer equation. The cathodic charge transfer coefficient was obtained through minimizing the normalized mean square error of the data.

Kinetics on glassy carbon
The exchange current densities extracted from the charge transfer resistance method for the GC electrode as a function of [Ce 4+ ] and temperature are shown in Fig. S18a and Fig. S18b, respectively. The trends are similar to those observed for exchange current densities extracted from the Tafel method, i.e., exchange current densities demonstrate a maximum in value at intermediate [Ce 4+ ] and they increase with temperature. The magnitude of the exchange current densities is larger for the charge transfer resistance method than the Tafel method, although they are still lower than the kinetic activity observed for the Pt electrode. We use the exchange current densities obtained from the Tafel method in the kinetic activity analysis, given the smaller uncertainty as compared to those observed in the exchange current densities from the charge transfer resistance method. The cathodic Tafel slopes as a function of temperature for the GC RDE are shown in Fig. S19, along with the fit of the BV rate law to the Tafel slope data. The general trend is the cathodic Tafel slope increases with temperature, which is similar to the trend for Pt, however the average data point at 298 K is lower than those at 288 K or 293 K. The data point at 298 K has a larger uncertainty than any of the other data points, which could explain this anomaly. The BV rate law does model the increase in cathodic Tafel slope with increasing temperature using a charge transfer value of 0.23. Reported values are averaged from three runs, with error bars representing one standard deviation from the average value in both horizontal and vertical directions. An Ag/AgCl reference electrode and graphite rod counter electrode were used. The solutions were sparged with nitrogen gas for at least 15 minutes before measurements were collected and blanketing with nitrogen was continued throughout the measurement collection. All data were collected at 2000 rpm of the RDE. Solid line is the fit to the data from the Butler-Volmer equation. The cathodic charge transfer coefficient was obtained through minimizing the normalized mean square error of the data.  6 ] 4− . The kinetic activity for the different electrode conditions are calculated by taking the ratio of the reported standard rate constants. The ratio for the standard rate constants obtained from the fit of the BV rate law to the Ce 3+ /Ce 4+ kinetic data in Fig. 3 are included in Table S7 for comparison. The factor of difference for these known outer-sphere redox couples range between 1.3−8.6, suggesting that the Ce 3+ /Ce 4+ , which has a factor of 4.7 between Pt and GC, also behaves as if outer-sphere. a) Ratio calculated using Ce 4+ reduction data from Fig. 6 in Kuhn and Randle (1985) paper for oxide coverage of 1 relative to oxide coverage of 4.5 for the electrode labeled "Electrode 1," which had been exposed to a longer period of phase oxide formation than other electrodes tested; 20 b) Calculated from reported data; c) 2,6anthraquinonedisulfonate; d) methylene blue; e) bis(4-methylstyryl)benzene; f) Other metal electrodes studied as well, including Pt, Pd, Au, and Cu, with exchange current densities ranging between 97 A/cm 2 and 116 A/cm 2 , while exchange current densities for Ag and Hg were 120 A/cm 2 and 50-79 A/cm 2

Section S5. Rate laws considered for cerium charge transfer
A summary of all symbols used in the derivation of the rate laws is in Table S8. Following the list of symbols, the derivation of each relevant rate law is described. The relevant rate laws are the Butler Volmer (BV) and Marcus theory (MT) rate laws, derived for a one-step charge transfer, and the rate laws derived for the CE mechanism (CE, E RDS and CE, C RDS) and the EC mechanism (EC, E RDS and EC, C RDS). Reaction rate of C step in EC mechanism 4 , −4 Forward and backward rate constants for C step in EC mechanism 4 Equilibrium constant for C step in CE mechanism ∆ 4 Free energy of C step in CE mechanism 3 Preexponential factor used in finalized rate law for EC mechanism with E RDS, equal to factor of , , 3+ or , 4+ , and , , Kinetic current for rate law for EC mechanism with C RDS 3 Reaction rate of C step in EC mechanism

BV rate law derivation
To derive the BV rate law, the overall reaction mechanism is assumed to be the following (Eq. S1), in which the electron transfer and a chemical step occur in the same step.
The transition state energies can be described using Eq. S5-S6, where Δ 0, ǂ is the redox transition state free energy, for the case in which the electrode potential, , is equal to the standard equilibrium potential of the redox species, °, and is the charge transfer coefficient. 41 Plugging Eq. S5-S6 into Eq. S3-S4, we obtain the following expressions for the oxidation and reduction rate constants (Eq. S7-S8).
Plugging Eq. S9-S10 into the overall rate law Eq. S2 and letting overpotential = − , where is the equilibrium potential of the reaction and defined in Eq. S11, we obtain the final rate law describing the kinetic current as a function of Ce 3+ and Ce 4+ bulk species, overpotential, and 0 and (Eq. S12). This is the Butler-Volmer equation and is the same as Eq. 1 in the main text. The standard rate constant 0 is a function of temperature (Eq. S13) through the Arrhenius equation, where is the activation energy.

=°− ln [Ce 3+ ] [Ce 4+ ]
(S11) )) (S13) Eq. S12 can be used to model the exchange current density and cathodic Tafel slopes as a function of both Ce 4+ concentration and temperature, by applying a Tafel analysis to the modeled current between the overpotential range of −0.250 V and −0.118 V, as was done for the experimentally collected data.

MT rate law derivation
To derive the MT rate law, the possible overall reaction mechanisms are shown in Eq. S14-S15. Note that MT does not allow for an inner-sphere structural change, failing Criterion 1 in Table 1, and meaning we must either model the E between the bisulfate-complexed (Eq. S14) or water-coordinated Ce species (Eq. S15). For the sake of simplicity, we will derive the rate law assuming Eq. S15 is the appropriate reaction mechanism, but the derivation would be analogous for Eq. S14.
[ (S15) The kinetic current for Eq. S15 would then be Eq. S16, where the reduction and oxidation rate constants and − are described through Eq. S17-18, using standard Marcus theory expressions. 41 In Eq. S17-S18, is the electronic transmission coefficient, which is related to the probability of electron tunneling, , 4+ and , 3+ are the precursor equilibrium constants for the Ce 4+ and Ce 3+ species, respectively, and is the nuclear frequency factor. 41 Δ ǂ and Δ − ǂ are the free energies of the transition state for the reduction and oxidation rate constants defined using MT, as shown in Eq. S19-S20, where is the S-31 reorganization energy and ∆ represents the free energy of the E reaction. ∆ is related to the electrode potential through Eq. S21.
Plugging these expressions into Eq. S16, and again substituting = − , where is defined in Eq. S11, and letting = , 3+ = , 4+ , we obtain the MT rate law (Eq. S22) that can be used to fit the exchange current densities and cathodic Tafel slopes as a function of [Ce 4+ ] and temperature using Tafel analysis in the same overpotential range as used in the experimental data collection process. Note that in the fitting process of the experimental data with the MT rate law, the concentration of the [Ce IV (H 2 O) 9 ] 4+ species in Eq. S22 was assumed to be equal to the bulk Ce 4+ species, which we in fact know to be [Ce IV (H 2 O) 6

CE Mechanism, E RDS rate law derivation
The kinetic current for the CE mechanism (Eq. 2a-b), assuming the E step is the RDS, is defined in Eq. S23, where the reduction and oxidation rate constants of the E step, 2 and −2 , are defined using MT in Eq. S24-S25. The meanings of , , 3+ , , 4+ , and are all the same as described previously. The free energies of the reduction and oxidation transition states for the E step, 2 ǂ and −2 ǂ , are defined using MT in Eq. S26-S27, with ∆ 2 defined in Eq. S28 as the reaction energy of the E step, and related to the difference between the potential of the electrode, , and ,2°, the equilibrium electrode potential of the species involved in the E step i.e., [ 9 ] 4+ must be defined in terms of known bulk species concentration, which can be done using the equilibrium constant of the C step (Eq. 2a). As mentioned in the main text, we assume that the ligand exchange occurs in one step rather than a series of C steps on the principle of choosing the simplest mechanism that fits the data. Assuming the C step is quasi-equilibrated, then the C step reaction rate, 1 , would be equal to zero (Eq. S29). 1 and −1 are the forward and backward rates of the C step and are related to the equilibrium constant of the C step 1 through Eq. S30. ∆ 1 in Eq. S30 is the reaction energy of the C step and is defined in Eq. S31, where ° is the standard equilibrium electrode potential of the overall reaction, as defined earlier.
Thus, in the instance that =°, ∆ 1 = ∆ 2 as would be expected from Scheme 1.

CE Mechanism, C RDS rate law derivation
The rate law for the case when the C step of the CE mechanism is rate-determining is defined below in Eq. S36, where again, the ligand exchange is assumed to occur in one C step instead of a series of C steps for simplicity.  9 ] 4+ is determined based on the assumption that the E step is quasi-equilibrated, and so 2 , the reaction rate of the E step, is equal to zero (Eq. S37). The reduction and oxidation rate constants are defined as earlier using MT in Eq. S24-S25. 2  Assuming a rate-determining step exists, the proposed mechanism in Eq. S40 has two possible rate laws.
Considering the case when the E step is the RDS, the kinetic current would be described as in Eq. S41 using MT. The reduction and oxidation rate constants of the E step are defined in Eq. S42-S43, where , , 3+ , , 4+ , and are all the same as described previously. in Eq. S44-S45, where 3 is the reorganization energy of the E step and ∆ 3 is the reaction energy of the E step, which is related to electrode potential through Eq. S46. In Eq. S46, ,3° is the standard equilibrium potential of the species involved in the E step, i.e., [ The  9 ] 3+ , and the equilibrium constant of the C step. Assuming the C step is quasi-equilibrated, then the reaction rate of the C step, 4 , is equal to zero (Eq. S47). In Eq. S47, the forward and backward rates of the C step, 4 and −4 , are related to the equilibrium constant 4 , as shown in Eq. S48. 4 can also be related to the free energy of the C step, ∆ 4 , as shown in Eq. S48. ∆ 4 is related to the electrode potential through Eq. S49, where, as defined earlier, ° is the standard equilibrium potential. When =°, ∆ 3 = ∆ 4 .
Briefly, although Eq. S53 does capture the experimentally observed behavior, the parameters from fitting are physically unrealistic as discussed below.

EC Mechanism, C RDS rate law derivation
The rate law for the EC mechanism when the C step is the RDS is shown in Eq. S54. The ligand exchange is assumed to occur as one C step.

Fig. S20
shows the fit of the Pt and GC kinetic data to the MT rate law, with fitting parameters and optimized to minimize the mean squared error between the experimental and modeled values. As discussed in the main text, the MT rate law is unable to capture the asymmetry of the observed data points, with the maximum in exchange current density predicted to occur at 50% [Ce 4+ ] instead of the observed ~70% [Ce 4+ ], as shown in Fig. S20a for Pt and Fig. S20e for GC. The modeled cathodic Tafel slope behavior as a function of [Ce 4+ ] for both Pt and GC (Fig. S20c, Fig. S20g) also does not agree with the observed behavior due to the forced symmetry of the MT rate law. The increase in exchange current density and cathodic Tafel slope as a function of temperature is only partially captured for both Pt and GC, most likely due to the low fitted reorganization energy values for both Pt and GC that are approximately equal at 30 kJ/mol (Fig. S20b,d for Pt and Fig. S20f,h for GC). S-37

S-38
The fit of the CE, E RDS derived rate law to the cathodic Tafel slope as a function of temperature is shown in Fig. S21. The model successfully predicts an increase in cathodic Tafel slope with increasing temperature, although the slope is not completely captured. This is most likely due to the value of the reorganization energy of the E step, 2 , being lower than what would be required to achieve a faster increase in cathodic Tafel slope with temperature. The fitted value of 2 is obtained through the fit of not only the data shown in Figure S21, but also the data in Figure 4b-d shown in the main text. The best fit was found for a value of 2 of 100.2 kJ/mol to satisfactorily fit the rest of the data in addition to the Tafel slope data. Figure S21. Fit of Pt cathodic Tafel slope as a function of temperature with CE, E RDS rate law. Tafel slopes for the Pt RDE (blue circles) extracted from Tafel plots as a function temperature at Ce 4+ concentration of 0.026 M and total Ce concentration of 0.05 M and fit using the CE, E RDS rate law (blue line). Reported values are averaged from three runs, with error bars representing one standard deviation from the average value in both horizontal and vertical directions. An Ag/AgCl reference electrode and graphite rod counter electrode were used. The solutions were sparged with nitrogen gas for at least 15 minutes before measurements were collected and blanketing with nitrogen was continued throughout the measurement collection. All data were collected at 2000 rpm of the RDE. The reorganization energy 2 and equilibrium constant 1 were obtained through minimizing the NMSE of the Pt data (NMSE = 0.049).
The fit of the GC data to the CE, E RDS rate law is shown in Fig. S22. Similar to the fit of the Pt data ( Fig.  4b-d), the CE, E RDS rate law successfully predicts a maximum in exchange current density as a function of [Ce 4+ ] at ~70% [Ce 4+ ] (Fig. S22a), an increase in exchange current density with temperature (Fig. S22b), a decrease in cathodic Tafel slope with increasing [Ce 4+ ], (Fig. S22c) and an increase in Tafel slope with temperature (Fig. S22d). The optimal fitting parameters for the fit of the GC data to the CE, E RDS rate law are 1 = 6.5 × 10 −5 , 2 = 71.8 kJ/mol, and 2 = 8.64. The differences between the fitted values for GC and Pt are discussed in the main text. The fits of the CE, C RDS rate law to the Pt and GC data are shown in Fig. S23. Although the rate law satisfies Criterion 6, which asserts that the exchange current density will increase with temperature, as shown in Fig. S23b for Pt and Fig. S23f for GC, it fails to capture the maximum in exchange current density at ~70% Ce 4+ concentration (Criterion 5, Fig. S23a for Pt and Fig. S23e for GC). This is because the anodic and cathodic branches of the current expression (Eq. S36) are only dependent on either [Ce 3+ ] or [Ce 4+ ], and never are a factor of both. From Eq. S39, the anodic current increases exponentially with positive overpotential, and the cathodic branch current is essentially constant with overpotential. As a result, Tafel slopes are unrealistically large at magnitudes of greater than 18,000 mV/decade, which is why they do not appear in Fig. S23c-d for Pt or Fig. S23g-h for GC (Criteria 7-9).  Fig. S24 shows the fits of the EC, C RDS rate law to the Pt and GC data. Similar to the CE, C RDS rate law discussed above, the kinetic current rate law (Eq. S57) predicts a linear dependence of exchange current density with [Ce 4+ ], as shown in Fig. S24a for Pt and Fig. S24e for GC. The behavior of the Tafel slopes is not appropriately captured in the case of the EC, C RDS rate law (Fig. S24c-d for Pt, and Fig. S24g-h for GC) because the cathodic current is predicted to increase exponentially with negative overpotential, making Tafel slopes too small compared to experimentally reported Tafel slopes (failing Criterion 7) and essentially constant with changes in [Ce 4+ ] or temperature (failing Criteria 8-9).

S-43
The fit of the EC, E RDS rate law to the Pt and GC data is included in Fig. S25. While all kinetic behavior (Criteria 5-9) is technically met through the fit, some of the optimized fitting parameters for Pt were unreasonable, i.e., the reorganization energy was unreasonably large at 663 kJ/mol, and for both Pt and GC, the predicted equilibrium constant of the C step suggested that Ce 3+ was preferentially complexed by sulfate, not coordinated by water. As discussed in the main text, our EXAFS results and DFT-predicted free energy of Ce 3+ water coordination contradict this finding, and thus we can reject the EC, E RDS rate law. It can also be argued that while the cathodic Tafel slopes do decrease slightly with the increase in [Ce 4+ ] for Pt (Fig. S25c), the slope of the predicted behavior is not steep enough, possibly due to the unreasonably large, predicted, reorganization energy value.

Section S6. DFT-predicted reorganization energies
The computed reorganization energies relevant to the cerium redox reaction in sulfuric acid, i.e., the watercoordinated Ce 3+ /Ce 4+ species, are included in Table S10. Reorganization energies were calculated as described in the Methods section. Considering the CE mechanism from which 2 is determined by fitting experimental kinetic data of the Ce 3+ /Ce 4+ redox reaction, the relevant Ce 3+ /Ce 4+ 9 ] 4+ species (87.2 kJ/mol) is lower than the fitted 2 value for Pt (100.2 kJ/mol) but higher than the value for GC (71.8 kJ/mol). Although the computed reorganization energies in the oxidation and reduction direction are not perfectly symmetric, they are not sufficiently different to explain the large asymmetry in the charge transfer. The 2 value that we obtain from fitting the Pt and GC data assumes that the reorganization energy of the oxidation step is equal to the reorganization energy of the reduction step, i.e., the reorganization energies are symmetric. We see from the calculations (  9 ] 4+ species, the reorganization energy associated with the reduction ( ( ) − ( ) + − = 108.4 kJ/mol) is not equal to the oxidation reorganization energy ( ( ) − ( ) + − = 65.9 kJ/mol), and as is convention, we instead report the average of these two values. While this analysis suggests that we may have an asymmetric Marcus theory electron transfer, we find that a factor of two difference between oxidation and reduction reorganization energies does not capture the extreme asymmetry observed in our kinetics. To demonstrate this, we show the free energy curves for a system with symmetric reorganization energies in Fig. S26a, for a system where the oxidation and reduction reorganization energies are different by a factor of two in Fig. S26b, and for a system where the reorganizations energies differ by a factor of nine in Fig. S26c. In the symmetric case, the intersection of the parabolas occurs at a reaction coordinate of 0.5, corresponding to a symmetric charge transfer coefficient of 0.5. In the asymmetric case where the reorganization energies differ by a factor of two, the intersection of the Marcus theory parabolas are only slightly shifted from the value of 0.5. For the intersection to shift to a reaction coordinate value of 0.25, which would align with our observed value of , the reorganization energies would have to differ by a factor of nine (Fig. S26c), which is unrealistic. This finding agrees with a previous report that the cerium redox reaction's unusual asymmetry cannot be described with an asymmetric model of Marcus theory alone. 3

Fig. 26.
Free energy parabolas using Marcus theory for two species, with the reorganization energy of the reduction, , fixed to 100 kJ/mol, and the oxidation reorganization energy, , set to (a) 100 kJ/mol, representing a symmetric system, (b) 200 kJ/mol, and (c) 900 kJ/mol. From our predicted rate law for a CE, E RDS mechanism, in acids other than H 2 SO 4 , the Ce 3+ /Ce 4+ kinetics would depend on the complexation energy of Ce 4+ , i.e., 1 . The rate law for this mechanism in Eq. S35 (Eq. 3 in main text) is dependent on the preexponential factor, 2 , ligand exchange equilibrium constant, and reorganization energy. If other acids follow the CE mechanism, the reorganization energy would be the same in all acids, since the species in the E step would still be [Ce III (H 2 O) 9 ] 3+ and [Ce IV (H 2 O) 9 ] 4+ . The ligand exchange equilibrium constant values in different acids from our DFT-predicted anion complexation energies 31 follow the experimental trend in redox potential shift from non-complexing HClO 4 to other acids. The preexponential factor may differ depending on the acid, but assuming these changes are minor compared to the effect of 1 , the rate would primarily depend on the value of 1 , with a higher 1 (weaker anion complexation) resulting in faster kinetics. The order in DFT-predicted 1 is HClO 4 > CH 3 SO 3 H > HNO 3 ~ CF 3 SO 3 H > H 2 SO 4 > HCl, so the Ce 3+ /Ce 4+ kinetic rate activity is expected to follow this order as well if the CE mechanism is dominant in these acids. As shown in Table S1, there is a large range in the values of standard rate constants reported for the Ce 3+ /Ce 4+ redox couple as a function of both electrode and electrolyte. While trends on Au and Pt specifically are inconclusive because of the large variation reported in Ce 3+ /Ce 4+ kinetic activity in H 2 SO 4 , kinetic trends on BDD and GC do appear to follow the expected trend, with kinetic activity following the order HClO 4 ~ HNO 3 > H 2 SO 4 on BDD and HNO 3 > H 2 SO 4 on GC. The expected trend is not followed on graphite, however, on which a mixture of MSA and H 2 SO 4 has faster kinetics than MSA. However, it is entirely possible an EC mechanism may dominate in other electrolytes, in which case the reorganization energy and ligand exchange equilibrium constant would both be different in each acid. While further analysis of the effect of acid on cerium redox kinetics is beyond the scope of this work, this simple analysis demonstrates the utility of the rate laws developed for the CE and EC mechanisms to understand trends in kinetic performance.

S7.1 EXAFS spectra comparing cerium(III) carbonate and cerium(III) trifluoromethanesulfonate precursors
To confirm our previous UV-Vis spectroscopy findings that the coordination of Ce 3+ does not change in 2 M TFSA (CF 3 SO 3 H) as the precursor salt is changed, 31 here we compare the Ce K-edge EXAFS of a 0.05 M Ce 3+ solution in 2 M TFSA prepared from 0.025 M Ce 2 (CO 3 ) 3 to the EXAFS of a 0.05 M Ce(CF 3 SO 3 ) 3 in 2 M TFSA solution in Fig. S27a-c. The spectra do not change significantly with precursor salt. For reference, the Ce K-edge EXAFS of 0.025 Ce 2 (CO 3 ) 3 in 2 M H 2 SO 4 , 2 M TFSA, and 2 M MSA are compared in Fig. S27d-f. The differences in EXAFS spectra in Fig. S27a-c are less significant than those observed in Fig. S27d-f, for which we have already concluded that there exist no differences in Ce 3+ coordination (Fig. S6). Thus, this EXAFS comparison presents additional evidence that the use of Ce 2 (CO 3 ) 3 as a precursor salt does not result in appreciable Ce 3+ -carbonate complexation in acidic media following sparging with nitrogen.

S7.2 EXAFS spectra comparing cerium(III) carbonate and cerium(IV) sulfate precursors
To confirm that the carbonate from the Ce 2 (CO 3 ) 3   . An Ag/AgCl reference electrode and graphite rod counter electrode were used. The solutions were sparged with nitrogen gas for at least 15 minutes before measurements were collected and blanketing with nitrogen was continued throughout the measurement collection. All data were collected at 2000 rpm of the RDE.

S7.3 Crossover measurements
Crossover of cerium ions through the ion exchange membrane can occur throughout the electrochemical oxidation procedure and kinetic measurements, resulting in a decrease in the total amount of cerium ions present in solution at the time of titration. To ensure the total cerium ion concentrations were consistent for each kinetic measurement, a sample of the solution in both the working and counter electrode compartments of the electrochemical cell was collected and UV-Vis measurements of these samples were collected. The UV-Vis spectra of the samples were compared to ascertain how much cerium was present in the counter electrode compartment. Fig. S29 shows an example of this comparison, with Fig. S29a showing the spectra of cerium solutions that had been oxidized in the working electrode compartment of the electrochemical cell to different amounts of Ce 4+ over the course of an experimental run that lasted for a total of 48 hours. Throughout the oxidation process, the peaks shift from those expected for pure Ce 3+ solutions (between 190 nm and 290 nm) to those expected for Ce 4+ in H 2 SO 4 (centered at 320 nm). Fig. S29b shows the spectra of the corresponding counter electrode solutions for each oxidation, along with the initial Ce 3+ solution spectrum from the working electrode compartment for comparison. The solution in the counter electrode after Oxidation 1 is shown in the inset in Fig. S29b, from which there is a negligible amount of cerium present after the first oxidation. From Fig. S29b, after 48 hours (Oxidation 3), the amount of Ce 3+ present in the counter electrode was approximately 8% of the total cerium added to the working electrode compartment initially, based on the intensity of the peaks at 255 nm. The amount of Ce 3+ present would account for all the Ce ions present in the solution, because the counter electrode compartment would be undergoing a reduction reaction to counter the oxidation reaction occurring in the working electrode compartment. Based on this analysis, no kinetic measurements were collected using solutions that had been in the electrochemical cell for more than 48 hours. S-49

S7.4 XAFS methods
We compare the XANES of the Ce 4+ in H 2 SO 4 solutions that had been shipped overnight in dry ice and stored in a refrigerator to CeO 2 standard XANES at both the Ce K-and L 3 -edges to confirm a 100% Ce 4+ oxidation state. As demonstrated in Fig. S30, the Ce 4+ XANES does not shift to the left of the CeO 2 standard XANES significantly for either edge, which would be expected if Ce 4+ had reduced back to Ce 3+ . Thus, we conclude that our samples were appropriately managed to maintain 100% Ce 4+ oxidation state. The masses of CeCl 3 and CeO 2 mixed with BN in the preparation of the Ce 3+ and Ce 4+ standard pellets, respectively, are included in Table S11. We varied the mass of CeCl 3 , CeO 2 , and BN to optimize absorption lengths and edge step of the samples. Although the Ce 3+ standard was CeCl 3 -7H 2 O, it was purchased as CeCl 3 , but mixed in air and the CeCl 3 hydrolyzed. The XAFS normalization parameters used for all Ce samples studied are included in Table S12. The EXAFS fitting parameters used for all Ce samples are included in Table S13. Whenever possible, the normalization parameters were kept consistent for all Ce 3+ and all Ce 4+ samples, and only varied when necessary to obtain appropriate normalization of the spectra.  The normalization parameters for the Pt L 3 -edge measurements included in Fig. S16 are included in Table  S14. S-52

S7.5 Kinetic measurements
The exchange current densities from the Tafel method were measured as a function of rotation rate for several Ce 4+ concentrations (with total Ce ion concentration fixed at 0.05 M) for the Pt and GC RDEs, as shown in Fig. S31a and Fig. S31b, respectively. For all [Ce 4+ ] tested for both Pt and GC, the magnitude of the exchange current density is not dependent on rotation rate once 2000 rpm is reached. Thus, we concluded that 2000 rpm was an appropriate rotation rate to use for all kinetic measurements to ensure mass transfer was non-limiting. In addition to plotting the exchange current density as a function of rotation rate to identify any significant mass transport effects, a Koutecký-Levich analysis was conducted for the Pt RDE for a solution of 0.02 M Ce 4+ /2 M H 2 SO 4 (total Ce ion concentration was 0.05 M) to confirm measured currents are independent of mass transport. In this analysis, steady state currents were collected at a broad range of potentials at the following rotation rates : 100, 500, 1000, 2000, and 3000 rpm. The inverse of the observed steady state current density was then plotted as a function of rotation rate −1/2 (converted to rad/s), so that the kineticlimited current can be extracted from the y-intercept, according to the Koutecký-Levich equation, Eq. S58, where is the measured steady state current density, is the kinetically-limited current density, is the Levich constant, and is rotation rate, in rad/s. used to identify exchange current density from the kinetically limited current densities and the measured steady state current densities, respectively. It should be noted that the current densities in Fig. S32 were extracted from a Tafel plot by fitting a linear trendline in an overpotential range of −0.250 V to −0.050 V and extrapolating to = 0 to obtain 0 . This larger overpotential range was used to ensure that there were at least three data points for the linear trendline fitting. As can be seen from Fig. S32, the exchange current density determined from the Tafel plot that uses the measured steady state current densities is 0.372 mA/cm 2 , while the exchange current density determined from the Tafel plot that uses the kinetically limited current densities extracted from the Koutecký-Levich plot is 0.375 mA/cm 2 , representing a less than 1% difference. Thus, it can be concluded that the activity of the Pt RDE in 2 M H 2 SO 4 with 0.05 M total cerium ions is not mass-transport limited. Another method to confirm that there are no mass transfer limitations is to measure kinetics in different reactors with different mass transport conditions. To confirm our kinetic activities are independent of electrochemical reactor design, exchange current densities (Tafel method) of the Pt RDE (2000 rpm) were compared to those of a Pt wire for a 0.05 M Ce/2 M H 2 SO 4 solution with stir bar rotation at several Ce 4+ concentrations. For the platinum wire electrode, a magnetic stir plate was used to operate outside of the mass transfer limited regime, as confirmed by the unchanging exchange current density with increased stirring speeds. As can be seen from Fig. S33, similar exchange current densities were obtained with the Pt RDE as were obtained with the Pt wire as a function of Ce 4+ concentration, confirming that both mass transfer and surface area effects were successfully controlled. S-55

Section S8. Computational methods
In Fig. S34 we show the modeled Ce 3+ and Ce 4+ species that were used as starting configurations to generate the MD-EXAFS. The MD-EXAFS was calculated using an 8 Å periodic box after confirming that the spectra did not change significantly compared to a 10 Å periodic box for the [Ce IV (H 2 O) 9 ] 4+ species (Fig.  S35).  S-56 We determined that the reorganization energies for the [Ce III (H 2 O) 9 ] 3+ /[Ce IV (H 2 O) 9 ] 4+ redox reaction converged with the Stuttgart RSC 1997 ECP basis set 42 for the cerium ion and the 6-31+G* basis set, 43 as seen in Fig. S36 for the outer-sphere reorganization energies. The inner-sphere reorganization energies were found to similarly converge in a series of calculations without using COnductor-like Screening MOdel (COSMO). Although the final inner-sphere reorganization energies reported in Table S10 were calculated using COSMO, we expect that the inner-sphere reorganization energies calculated from each basis set would experience a near constant shift (by the energy of the implicit solvent), and so we conclude that the inner-sphere reorganization energies converge with the Stuttgart RSC 1997 ECP and 6-31+G* basis sets as well. Figure S36. Outer-sphere reorganization energy converges with increasing basis set size for [Ce(H 2 O) 9 ] 3+ ⇌Ce(H 2 O) 9 ] 4+ + eredox couple. Mixed basis sets utilize the first basis set and its associated ECP on the cerium atom, and the 6-31+G* basis set on all non-cerium atoms.