Raw catalytic CV data

All experiments were carried out in buffered aqueous solutions. Based on previously reported data,^{5c, 5d} a temperature of 40 °C was selected to measure catalytic currents that are both sizeable at low scan rates (down to 0.05 V s⁻¹) and low enough to be overcome at fast scan rate (5 V s⁻¹ and upwards). CV catalytic currents are indeed decreasing functions of the dimensionless parameter where is the scan rate, k the first order or pseudo-first order rate constant of the catalytic reaction, T the temperature, R the gas constant and F the Faraday constant.^9b

The CVs are recorded at a glassy carbon electrode (surface area S =0.02 cm²) in 0.05 M 2-(N-morpholino)ethanosulfonate (MES) buffer added with 0.05 M Na₂SO₄ (unless stated otherwise) or in 0.1 M sulfuric acid solutions, at a series of pHs varying from 1 to 9 in the absence of H₂ or in the presence of 1 atm. H₂. The moderately low water solubility of the catalyst, and hence the small concentration used (50 or 100 μM), results in CV responses showing a substantial double layer capacitive charging current on top of which appears the faradaic responses of interest. Raw data had thus to be corrected from the double layer charging current before going to the kinetic analysis via subtraction of a blank CV recorded in the absence of catalyst (examples are given in Figure S1). Another consequence of the poor water solubility of the catalyst is its facile adsorption onto the electrode surface, which had to be carefully polished before each run.¹² Typical background corrected CV responses obtained at low scan rate are shown in Figure 2, showing archetypal steady-state S-shaped responses characteristic of canonical kinetic conditions.^9c

Catalysis being revealed by the S-shape of the CV response and the height of the plateau current, the experiments under H₂ indicate in the first place that catalysis is bidirectional: the same catalyst is indeed able to catalyze the oxidation of H₂ and the reduction of proton from the solution. Within this framework, it is observed that, in the presence of saturated H₂ atmosphere (Figure 2b), H₂ oxidation catalysis is much more pronounced than H₂ evolution catalysis. In the latter case, significant catalysis only appears at the lowest pH values, both in the absence and presence of H₂. Conversely, the catalytic plateau currents for H₂ oxidation are significant at all pH values and increase between low and intermediate pH values.

It is also remarkable that the potentials at which the CV responses stand are practically the same for H₂ oxidation and evolution, which reflects reversible catalytic behavior. The equilibrium potentials experimentally measured from CVs (Figure 2b) at i=0 at various pH values data (theoretically corresponding to 0 V vs. RHE) are plotted in Figure S2 and overlaid with the theoretical curve. It is seen that the agreement is fair but a substantial deviation is observed at pH above 7 indicating that the equilibrium is maybe more difficult to reach. This resonates with the fact that both reductive and oxidative catalytic currents decrease at pH values above 7 concomitantly with a flattening of the catalytic response. We nonetheless observe at pH<6 a slight anodic shift (of ca. 10–35 mV depending on the pH value) of the experimental value compared to the theoretical value (Figure S2). We do not have a definite explanation for this shift that might be due to some junction potential. This shift was added in simulations described below to get a match with the experimental data.

Finally, the HER currents observed in the absence of H₂ (Figure 2a) are significantly higher (with a two-fold factor at pH=1) than when H₂ is present (Figure 2b), a typical behavior for product-hindered catalytic kinetics that has been already reported for such reversible catalysts^{5, 6c} and hydrogenase enzymes.^1d

Thermodynamics characteristic of the Ni^II/Ni^I system

Pourbaix diagrams can be very useful to understand the catalytic properties of such bidirectional systems, as exemplified by Muckerman and coworkers a decade ago.¹³ To establish such a diagram, we first recorded a set of CV responses over the whole pH range at 40 °C and at a scan rate high enough to overcome catalysis.^{9b, 9c} Reversible signals could be measured at all pH values and are shown in Figure 3a.

Figure 3b compares the peak current of CVs, recorded at 5 V s⁻¹ after correction from the capacitive current, to the classical Randles-Sevcik relationship for a one electron stoichiometry:¹⁴

(1)

relating, in the case of a fast, diffusion controlled electron transfer, the peak current, i_p, to the surface area S (0.02 cm²), the catalyst concentration C⁰ (50 μM), the scan rate v, the temperature T (313.16 K) and the diffusion coefficient D (in cm² s⁻¹). For the latter we used 2.8×10⁻⁶ cm² s⁻¹, a value previously determined by Shaw and co-workers using NMR pulsed field gradient methods.^{5d, 15} From these data, we can conclude that the electron stoichiometry is 1.

The Pourbaix diagram (Figure 4a) positioning the Ni^II and Ni^I species of the system in their different protonation states can then be obtained by relating the pH value and the apparent standard potential¹⁶ derived from the midpoint between the anodic and cathodic one-electron peaks (Figure 3a). It is worth noting that the slope of the central descending straight-line (62 mV per pH unit) in Figure 4a corresponds to the value expected for a one-electron/one-proton process at the operating temperature (40 °C). The ensuing zones of thermodynamic stability of the various reactants are as shown in Figure 4b. In the following, we will represent by the subscript letter “_r” a properly positioned amine acting as proton relay, “⁺H_r” representing a protonated form of the relay.

This set of experiments allowed evaluating four thermodynamics values corresponding to a proton-coupled electron transfer square scheme shown in Figure 5 (square scheme 1) via a fitting of the experimental data with:

(2)

We obtain:

and

The Pourbaix diagram (Figure 4) confirms that _rNi^II gets protonated at low pH values, as previously demonstrated by NMR measurements,^{5a, 5d} and in line with previous determination of pK_a values for pendant amines of the same complex (pK_a=0.64 in water for the _rNi^II/ ⁺H_rNi^II).¹⁷ The pK_a value at which the Ni^I species is protonated is significantly higher, around 7, in line with the increased basicity of a complex with a reduced overall charge following reduction of the Ni center.

Insights into the nickel-hydride species and overall mechanistic scheme

As highlighted by Muckerman and coworkers,¹³ getting a comprehensive understanding of the catalytic system requires gaining data for the redox couples involving metal-hydride intermediates. To that aim, we recognize that, under 1 atm. H₂, _rNi^II is fully converted into the doubly protonated/doubly reduced species ⁺H_rNi^IIH⁻ as previously demonstrated by UV-visible spectroscopy¹⁸ and ³¹P NMR measurements.^{5a, 5d, 19}

In an attempt to get the thermodynamic values characterizing a second square scheme involving such metal-hydride species, CVs were recorded at 5 V s⁻¹ under 1 atm. H₂ (Figure S3). At such high scan rate, we were expecting to characterize solely the one-electron oxidation of Ni^II-H into Ni^III-H species. Unfortunately, we observe that, at all pH values, the oxidation wave is bielectronic and that the return reduction wave is smaller, likely monoelectronic. This indicates that, starting from ⁺H_rNi^IIH⁻ (or _rNi^IIH⁻ at pH values > 7), it is not possible to isolate the lower square scheme 2, shown in Figure 5, only involving Ni^II-H and Ni^III-H species. In other words, the Ni^III-H species reacts fast in a way allowing their further oxidation. This resonates with the key property of this family of catalysts²⁰ and with the expected role of a proton relay,^{3c, 7, 21} allowing for fast intramolecular proton transfer generating ⁺H_rNi^I from _rNi^IIIH⁻ via the concentration effect discussed in the introduction.⁷ Hence, the bielectronic oxidation wave observed at 5 V s⁻¹ starting from ⁺H_rNi^IIH⁻ (or _rNi^IIH⁻ at pH values > 7)¹⁷ captures its proton-coupled oxidation to _rNi^IIH⁻, intramolecular deprotonation of the hydride species by the pendant base generating the ⁺H_rNi^I species (characterized by the k_-1 constant in Figure 5) and oxidation of the former into a Ni^II species. Consistently with the measurements shown in Figure 3, the return reduction wave is monoelectronic and corresponds to the formation of a Ni^I species. Here the protonation of the Ni^I to form the Ni^III-H species actually involved in catalysis (characterized by the k₁ rate constant in Figure 5 and thus observed at lower scan rate) is too slow to be observed at 5 V s⁻¹.

Figure 5 gathers all the mechanistic elements assembled so far with the two square schemes (only the upper one, i.e. square scheme 1 is fully characterized at the moment) connected on the right by the previously reported hydrogenation equilibrium (hereafter named C₂ with the K₂=k₂/k₋₂ thermodynamic constant)^{5a, 19} and on the left by the intramolecular proton transfer step (hereafter named C₁ with a K₁=k₁/k_-1 thermodynamic constant) discussed just above.

In the H₂ evolution process, fast proton-coupled reduction of the _rNi^II species generates ⁺H_rNi^I (square scheme 1). Then intramolecular proton transfer converts ⁺H_rNi^I into _rNi^IIIH⁻; this step is characterized by the equilibrium constant K₁, and, although it is fast in the H₂ oxidation direction (see above), it may be rate determining for H₂ evolution. _rNi^IIIH⁻ is in equilibrium via proton coupled electron transfer (PCET) with ⁺H_rNi^IIH⁻ (square scheme 2). From this species, dihydrogen can evolve through a second potentially rate-determining chemical step, characterized by the equilibrium constant K₂, closing the loop between both one electron/one proton square schemes and enabling catalysis. H₂ oxidation can proceed through the reverse pathway thanks to activation of H₂ by the frustrated Lewis pair formed between the pendant proton relay and the Ni^II ion, to generate ⁺H_rNi^IIH⁻. Here again fast PCET generates _rNi^IIIH⁻ (square scheme 2), which undergoes rapid intramolecular hydride deprotonation by the pendant base to reach the upper square scheme 1.

Although introducing a new doubly-branched structure, the mechanistic scheme shown in Figure 5 actually captures most of the catalytic sequences proposed so far,^{3c, 5a, 5c, 5d, 17} with the exclusion of those initially proposed and involving a non-protonated Ni⁰ intermediate, that we know from the Pourbaix diagram that it is not accessible under the experimental conditions used here. The introduction of the two square schemes in the model allows including pathways avoiding high energy intermediates over the whole range of pH values. Importantly, this scheme allows to differentiate between the intermolecular proton transfer steps (within the two square schemes) that are diffusion-controlled and vital to regenerate the catalytic species and maintain catalysis⁷ and the intramolecular proton transfer steps, involving the formation or the cleavage of metal-hydride and H−H bonds positioned vertically on the left and right sides. These chemical steps, C₁ and C₂, involving changes in the formal oxidation state and coordination sphere of the central nickel ion, are likely to be rate determining in one or the other direction.

Thermodynamics of the chemical steps

The intramolecular proton transfer between _rNi^IIIH⁻ and ⁺H_rNi^I likely remains at equilibrium under catalytic conditions where CV responses have S-shaped canonical forms as represented in Figure 2b. Obviously, this equilibrium is strongly displaced with k_-1≫k₁ and thus but no direct evaluation of is possible at this stage. Still, as shown later on, it will be possible to extract kinetic information from the data gathered at 5 V s⁻¹ under H₂ atmosphere once the thermodynamics of the intramolecular proton transfer has been indirectly evaluated from the analysis of the catalytic response at pH>7.

The hydrogenation equilibrium involving ⁺H_rNi^IIH⁻ and _rNi^II can be characterized by UV/Visible spectroscopy, similarly to other DuBois/Shaw catalysts.¹⁸ Measurements carried out under various partial pressures of H₂ allowed the evaluation of 20 (see Supporting Information for details, Figure S4) when =1 atm. In other words, with = 20 and =1 atm. Stable spectra are obtained within seconds after mixing and the observation of an isosbestic point is a strong indication that the equilibrium conditions were met, although this value certainly captures reversible isomerization of ⁺H_rNi^IIH⁻ into other doubly protonated Ni⁰ species.^5c

Then, simple thermodynamic relationships corresponding to the scheme shown in Figure 5 lead to:

and

with .

Because the pendant amine is not electronically linked to the nickel active site, we can make the assumption that its pK_a is only sensitive to electrostatic effects from the active site. Hence we make the assumption that complexes with the same overall charge, i.e. and on the one side and and on the other side have similar pK_a’s, which leads to and . This hypothesis is further supported by previous estimation of the value between 6.8 and 7.9.¹⁷

Thus, we obtain:

(3)

and

(4)

As already mentioned, the determination of is not possible at this stage. Nonetheless, with the thermodynamic parameters obtained so far, we can first attempt to interpret the catalytic CV responses shown in Figure 3b considering, as a first approach, that the two squares schemes in Figure 5 are at equilibrium, i.e., assuming fast protonation/deprotonation of the pendant amine over a large range of pH from acid/base couples in the solution, either MES or sulfate-based buffer or water acid/base couples.

Simplified reformulation of the reaction mechanism and analytical treatment

Building on previous analysis of two electron reversible catalytic systems,¹¹ we demonstrate in the Supporting Information that instead of considering the time and space variations of the concentrations of the whole set of eight forms of the catalyst, one may consider the four two-by-two sums of the species that have been assumed to be under unconditional proton transfer equilibria, i.e., successively,

The first square scheme may thus be represented by the two first sums as if they were two redox reactants giving rise to a single Nernst law, after taking into account the various PCET equilibria. The same applies to the second square scheme, introducing C and D, We consider the case where the solution is buffered so as to maintain proton concentration, [H⁺], constant. In total, the whole reaction scheme in Figure 5 may be replaced by the simpler equivalent EC_rEC_r mechanism shown in Figure 6. (and ) is a pseudo first-order rate constant (in s⁻¹) including the concentration of H₂; hence we can write: where is the partial pressure of H₂ and is 1 atm (similarly, we will define ). At equilibrium (i.e. at current nil), we have:

(5)

The corresponding kinetics relationships are (see Supporting Information):

(6)

(7)

The analytical expression of the CV corresponding to this equivalent EC_rEC_r mechanism is given in the Supporting Information and is equivalent to the equations previously reported by Fourmond and Léger.¹¹ The limit corresponding to a fast and equilibrated intramolecular proton transfer corresponding to the conversion between and the nickel hydride is simply obtained by considering and in the general expression thus leading to:

A first outcome of the above analysis is the determination of the rate constants for the H₂ addition/evolution steps. As seen from equation (8), both apparent rate constants and can be easily obtained from a combination of the plateau currents:

(9)

and

(10)

Considering the plateau currents obtained from experiments performed with 50 μM of catalyst with 100 %, 50 %, 25 % and 12.5 % H₂ at pH = 1 (Figure 7) and knowing that , application of equation (9) and (10) gives and . Then, by application of equation (7), we obtain the averaged values:

Analysis of the catalytic data for pH=1

In principle, simulations of the CVs according to the analytical expression (equation 8) could provide an evaluation of and hence of because it is the only unknown parameter. However, simulations show that any value corresponding to < 0.1 leads to satisfactory fitting (see Figure S5). Using equation 5, only an upper value of can thus be obtained from this set of data: < 0.2. Simulations considering and taking into account the experimental deviation of (Figure S2) are shown in Figure 7 together with the experimental data. Excepted for one experimental condition (upper right panel) where the anodic current is clearly off,²² a fair agreement is obtained between experiments and simulations showing the consistency of the analysis as the shape of CVs in between the two plateaus is correctly reproduced. Remarkably, a single set of determined parameters allows reproducing the decrease of the anodic plateau current and the increase of the cathodic plateau current as H₂ partial pressure is decreased. This is a strong validation of the proposed model and its analytical expressions when taking into account the explicit linear dependence of with .

Analysis of the catalytic data for 1<pH<6

With all required parameters in hand, we can now calculate the plateau currents at different pH values (taking into account the pH dependence of , and ). As shown in Figure 8, the model predicts that the anodic plateau current increases as pH is increased and it is indeed observed experimentally. The reason for this increase is that the relay on ⁺H_rNi^II gets fully deprotonated at pH>2 therefore increasing the amount of _rNi^II available in solution for H₂ uptake. Consequently, the observed increase of the anodic plateau current from pH=1 to pH=4 is a confirmation that the pK_a of ⁺H_rNi^II ( ) is indeed close to 1 as previously determined¹⁷ and guessed from the Pourbaix diagram where the inflexion of the apparent standard potential is barely seen (Figure 4a).

Analysis of catalytic data measured at pH >7 and determination of remaining thermodynamic data

At pH>7, we observe two striking features calling for a thorough review of the assumptions corresponding to the framework allowing the satisfactory analysis at acidic pH (Figure 2b): (i) CVs are distorted: the experimental anodic current is much more sluggish than predicted by the analytic expression and (ii) plateau currents are smaller than predicted considering that H₂ addition is the only rate-determining step. Given the pK_a’s of the pendant amine acting as a proton relay, we first recall that the dominant sequence at pH>7 is, in the H₂ oxidation direction: _rNi^II+H₂ ⁺H_rNi^IIH⁻ _rNi^IIH⁻ _rNi^IIIH^{− +}H_rNi^I _rNi^I _rNi^II (and vice versa for H₂ evolution). Here deprotonations (PT) of ⁺H_rNi^IIH⁻ and ⁺H_rNi^I have the same moderate driving force because the pK_a of ⁺H_rNi^IIH⁻ and ⁺H_rNi^I are close to 7 while the pK_a of the buffer base (MES) is 6.15. Plateau currents being related to the characteristics of the chemical steps (PT, C₁ and C₂) and C₁ and C₂ being pH independent, the observed deviation of the plateau current from predictions given by equation (8) necessarily implies that at least one proton transfer step should not be considered at equilibrium, the effect being larger for the PT step not following or preceding the slow C₂ chemical step. In other words, the simple equivalent EC_rEC_r mechanism shown in Figure 6 has now to be replaced by another simple four elements scheme corresponding to an EP_rEC_r sequence with the thermodynamic and kinetic parameters defined in Figure 9 together with:

(11)

and

taking into account that and with (see Supporting Information for details).

Application of equation (12) to the experimental anodic plateau currents at pH=8 we obtain 10 s⁻¹ and 63 s⁻¹ for rate-determining protonation and deprotonation steps. These pseudo-first order protonation/deprotonation rate constants can be rationalized as follows. Protonation can arise from AH=H₃O⁺ with a rate constant equal to 10⁹ M⁻¹s⁻¹, i.e. close to diffusion limit in line with a reaction with a large driving force. Alternatively the buffer (MES) acid form can be the source of proton with a second order rate constant being only 1.4 10⁴ M⁻¹s⁻¹ as a consequence of the small driving force unable to compensate the fact that the pendant amine is buried into the ligand sphere. Regarding deprotonation, if hydroxide ion is considered as proton acceptor, the deprotonation rate constant is 6 10⁷ M⁻¹s⁻¹ whereas it would be only 1.25 10³ M⁻¹ s⁻¹ with the buffer base. Again these values are in agreement with the corresponding driving forces and accessibility of the solution base to the proton relay.

We now analyze the shape of the CVs at pH>7. As already mentioned, the variation of the current between the two plateaus is more sluggish than predicted by the analytic expression resulting from replacement of the reaction pathway by a simpler equivalent EP_rEC_r scheme in which true and apparent electron transfer are assumed to be fast. We thus infer that the observed sluggishness is due to interference of electron transfer kinetics. At pHs 5–7, we note that the peak separation (78 mV) of the reversible wave under N₂ at 5 V s⁻¹ (Figure 3) is larger than the canonical Nernstian value (62 mV at 40 °C) although this pH range corresponds to the buffering capacity of MES buffer.²³ From the 78 mV anodic-to-cathodic peak separation at 5 V s⁻¹ under N₂ at pH>7 and 40 °C, it is possible to evaluate the standard rate constants 0.05 cm s⁻¹ corresponding to the redox couple _rNi^II/ _rNi^I.⁹ Fitting the CV at pH=8 considering the simplified reaction scheme depicted in Figure 9 and the already evaluated thermodynamic and kinetic parameters , , , , , and ) gives the adjusted parameter 2×10⁻⁴ cm s⁻¹ (see Figure S6 and details for simulations in the Supporting Information). As shown in the Supporting Information, assuming a Butler–Volmer type kinetic law for the _rNi^IIIH⁻/ _rNi^IIH⁻ couple with a transfer coefficient equal to 0.5, the apparent standard rate constant is:

If we assume that both redox couples _rNi^II/ _rNi^I and _rNi^IIIH⁻/ _rNi^IIH⁻ have similar reorganization energy associated to electron transfer, then and therefore allows evaluating 2 10⁻⁵ via equation (13). As a consequence, from the relationships previously derived linking , and (equations (3) and (4)) we obtain and thus providing a complete thermodynamic characterization of the reaction scheme shown in Figure 5.

Such a positive positioning of the standard potential of the Ni^IIIH/Ni^IIH couples (in both protonation states of the relay) compared to the Ni^II/Ni^I couple is in full agreement with previous calculation on cobalt-based H₂ evolution catalysts: the standard potentials of the Co^IIIH/Co^IIH couples of cobaloximes²⁴ and cobalt diimine-dioxime complexes²⁵ were also found positive by 100 and 130 mV to the respective Co^II/Co^I couples.

Hence the Pourbaix diagram corresponding to square scheme 2 can be drawn according to equation (14) (red line in Figure 4a and Figure 4c).

(14)

We can now come back to the data gathered at 5 V s⁻¹. under 1 atm. H₂ (Figure S3) to evaluate the kinetics of the fast, energetically unbalanced, intramolecular proton transfer step C₁. The scheme to be considered is an ECE process corresponding to the D/C/B/A sequence in Figure 6 as if there were no chemical connection between A and D because hydrogen uptake/evolution is too slow to be efficient at 5 V s⁻¹. Simulations of the CV taking into account values of , and show that the position of the anodic peak requires that is in the order of 5×10⁹ s⁻¹ (Figure S7) and thus 10⁵ s⁻¹. This is in line with a very fast intramolecular proton transfer in both direction and remaining at equilibrium in catalytic conditions. Still, according to the theoretical investigation of the ECE process,²⁶ the contribution of the second electron transfer remains small in the cathodic peak provided is small, typically smaller than 0.1. At 5 V s⁻¹ and considering the evaluated values of and , this condition is achieved for pH>1, in line with the observation of the monoelectronic waves described earlier in the absence of H₂ (Figure 3a).

All parameters corresponding to the reaction scheme in Figure 5, and their compacted version in Figure 6 (valid at pH<7) and in Figure 9 (valid at pH>7) have now been evaluated. Simulation of the whole CVs with the analytical expression (equation (8)) are shown in Figure 10b for pH=1 to 6. For pH=7 to 9, numerical simulations were performed (see Supporting Information for details) according to the reaction scheme in Figure 10. As shown in Figures 10 and S6, this single set of parameters allow to obtain a good agreement of the proposed model with the experiments performed over the whole range of pH values. This is also true when catalysts concentrations and partial pressures of H₂ are varied as previously shown in Figure 7. Importantly the data at pH>7 illustrate the critical role of equilibrated proton transfers to exogenous acid/base couples to maintain efficient catalysis. The HOR turnover frequency is indeed partially limited by proton transfer in basic conditions.²⁷

Turnover frequencies, catalytic bias and reversibility

We have shown that, at pH<7, the critical chemical step for both HER and HOR is H₂ uptake and evolution. Chemical step C₂ is thus the one to be optimized to get a better catalyst. Indeed, the proton relay is responsible for the high local concentration allowing fast intramolecular proton transfers in both directions for chemical step C₁.

It also appears that the catalyst is catalytically biased,²⁸ being much more efficient for HOR than HER at all pH values. The bidirectionality of the catalyst is made possible because the slowest process in both directions, H₂ uptake and evolution, is fast enough (respectively 10 s⁻¹ at 1 atm H₂ and 0.5 s⁻¹) for a substantial current to flow at moderate scan rate. This study confirms that the presence of the proton relay installed inside the catalyst molecule close to its metallic active site, thus forming a frustrated Lewis pair to capture and activate H₂ during HOR and allowing for optimal hydride proton coupling during HER, is key for bidirectionality.

However, H₂ uptake and evolution steps are convoluted with proton transfers equilibrium constants between the proton relay and exogeneous acid or base that affect the solution concentration of the species able to react with or evolve H₂, as expressed by and (equation 7). Hence the pH variation of the apparent rate constants (Figure 11) shows the effect of relays protonation/deprotonation on the kinetic of both HER and HOR catalysis. Because the intramolecular proton transfer is fast, the maximal turnover frequencies correspond to and , as shown in reference:²⁹

and

At low pH, H₂ uptake competes with the protonation of the relay in _rNi^II leading to a small decrease of the catalytic HOR current. The same applies at high pH, but at the expense of catalytic HER current which is almost shut-down due to an unfavorable protonation of the relay preceding H₂ evolution. We note here that the pH variation of maximal turnover frequencies (Figure 11) and of plateau currents (Figure 8) are not identical because the plateau currents are also related to the size of the diffusion reaction layer that are pH dependent.²⁹ Nonetheless, from equations 15 and 16 (or 9 and 10), the catalytic bias seems to be controlled by, and only by, the K₂ value convoluted with protonation equilibria, which favors, in the present case, H₂ uptake over evolution. This is at variance with previous models, that were relating the catalytic bias to the positioning of the H⁺/H₂ equilibrium with the Pourbaix diagram of the catalytic system.³⁰

Besides bidirectionality, one remarkable feature of the present system is its reversibility. As discussed by Fourmond et al.,¹¹ there is no need for every chemical step to be thermoneutral (equilibrium constant close to unity) to achieve reversibility. It is indeed illustrated experimentally by the present example (Figure 12). Despite the mismatch between the pK_a of the proton relay functionality ( ) and the pK_a of the active site hydride ( ) estimated from the value of pK₁=4.7, the system is reversible over a large range of pH. Actually, this unbalanced equilibrium ( ≪1) is compensated by the Pourbaix diagram corresponding to the second square scheme in Figure 5 being ca. 250 mV more positive than that of the first square scheme (Figure 4), so that Eq. 5 is fulfilled with E_eq=0 V vs RHE. Such a positioning of the two Pourbaix diagrams correspond to the inverted configuration making the second PCET step always easier than the first one whatever the direction (HER or HOR) of catalysis. This configuration is the only one that allows for reversibility, with the limit being a superposition of the two Pourbaix diagrams. Noteworthy, this condition applies to the apparent E⁰_sq potentials values, taking into account the various protonation equilibra. Note that it is even not required for the two Pourbaix diagrams of the catalyst to overlap the Pourbaix diagram of the catalyzed reaction.

An important finding is that large deviations from a flat energy landscape may still be detrimental to the rate of catalysis as observed here for pH>6. The unbalanced energetics of chemical step C₁ compensates the large deviation of the second Pourbaix diagram from the equilibrium potential of the H⁺/H₂ reaction (and vice-versa) (Figure 12b) but this has consequences on the reversibility of the system. Indeed, for catalysis to take place at small overpotential (i.e. with substantial current at potentials close to the equilibrium potential), an electron transfer must occur far away from its standard potential. This is illustrated in Figure 12b where the driving force for the electron transfer _rNi^IIH⁻ _rNi^IIIH⁻ has a driving force as large as 0.33 eV at pH=8 (Note that the driving force is only 0.24 eV at pH=0). This results in an apparent slowness of the electron transfer when this electron transfer is directly coupled to the energetically unbalanced intramolecular proton transfer as it is the case in the dominant pathway at pH>7. This effect does not modify the bidirectionality nor the catalytic bias, i.e. the turnover frequencies, but it slows down the apparent reversibility corresponding to a sluggish increase of the current around the equilibrium potential.