Computational Design of an Electro-Organocatalyst for Conversion of CO2 into Long Chain Aldehydes

Density functional theory calculations employing a hybrid implicit/explicit solvation method were used to demonstrate that an electro-organocatalyst designed in our previous work for reducing CO2 to formaldehyde could also be capable of coupling formaldehyde to form long chain aldehydes. The catalytic activity is enabled by an electron-rich vicinal enediamine (>N–C=C–N<) backbone that activates formaldehyde by reversing the polarity on the carbon atom, enabling it to act as a nucleophile in the subsequent aldol addition step. The catalyst then enables reductive dehydroxylation of the aldol addition product by facilitating outer-sphere electron transfer. The optimal pH as well as the limiting potential and formaldehyde concentration are identified and related to the kinetic balance between several rate limiting steps. Finally, the optimal conditions for coupling with the CO2 reduction cycle are discussed, demonstrating that the electro-organocatalyst is capable of efficiently converting CO2 into aldehyde products with a turnover frequency (per carbon atom) on the order of 0.1–1 s–1.


INTRODUCTION
One of the most significant obstacles toward a societal transition to renewable energy is the lack of an efficient and cost-effective means of storing this energy during times of surplus production and releasing it during times of surplus demand. 1,2Electrocatalytic reduction of CO 2 to liquid transportation fuels or chemical feedstocks could serve this purpose if the process could be improved in terms of cost and efficiency. 3−6 Existing electrocatalysts exhibit limitations such as low Faradaic efficiency, low current densities, high overpotentials, or insufficient selectivity toward carbon products other than CO, hindering their practical implementation. 7ecently, we demonstrated in a computational study that an electro-organocatalyst containing a vicinal enediamine backbone is capable of electrochemically reducing CO 2 to formaldehyde at a potential of −0.85 V vs RHE with a turnover frequency on the order of 0.1−1 s −1 . 8The catalytic cycle occurs by the mechanism depicted in Scheme 1 and corresponds to the overall four-electron reduction, It was found that the electron-rich π system of the vicinal enediamine backbone is able to activate CO 2 by formation of a C−C bond and then facilitate outer sphere electron transfer from an inert cathode.Unlike transition metal electrocatalysts, the electro-organocatalyst is not capable of producing undesired products such as CO and H 2 since no mechanism exists for elimination of these species from an organic molecule.Additionally, we postulated a mechanism by which this electro-organocatalyst could be capable of forming long chain aldehydes by carrying out reductive aldol condensation to grow an alkyl chain one carbon at a time.One iteration of the chain growth process involves reductive addition of formaldehyde to a second aldehyde to produce a product aldehyde that is longer by one carbon atom, Coupled with the CO 2 reduction cycle, the chain growth process effectively corresponds to chain growth by reductive addition of CO 2 , producing long chain aldehydes from nothing but CO 2 , protons, and electrons, Not only are these long chain aldehydes a more valuable product than formaldehyde, but our calculations also show that in the absence of this chain growth process the electroorganocatalyst would become inactivated by buildup of the formaldehyde product to concentrations in excess of a few mmol/L.Thus, the chain growth process is actually an integral part of the reduction of CO 2 by this electro-organocatalyst.
Formation of the C−C bond in the chain growth cycle occurs by an "umpolung" mechanism that involves reversal of polarity on one of the two carbon atoms forming the bond.Organocatalysts such as N-heterocyclic carbenes (NHCs) have been extensively studied for their ability to carry out such reactions. 9,10The formose reaction occurs by a similar mechanism in which glycoaldehyde functions as both the product and the organocatalyst. 11The formoin reaction is a juxtaposition of these two types of reactions, whereby thiazolium NHCs catalyze the benzoin condensation of formaldehyde into carbohydrates. 12In all three cases, the organocatalyst activates an aldehyde by attack of a nucleophilic carbon center on the carbonyl group of the aldehyde, Tautomerization leads to the formation of a Breslow intermediate in which the "carbonyl" carbon is rendered nucleophilic by conjugation with electron donating groups on the organocatalyst. 13,14This carbon can then function as a nucleophile in an aldol addition step to an electrophilic center on a second reactant molecule in reactions such as Benzoin condensation, 15 formoin condensation, 12 and the Stetter reaction. 16While the vicinal enediamine electro-organocatalyst carries out the C−C bond formation step by a similar mechanism, it is also able to facilitate electron transfer to the resulting product leading to reductive dehydroxylation.
In the current manuscript, we computationally explore the chain growth process by an electro-organocatalyst containing a vicinal enediamine backbone.Using density functional theory, we show that it is indeed feasible as we previously hypothesized and that it operates at a comparable turnover frequency to the CO 2 reduction cycle even at low concentrations of formaldehyde.We find that the chain growth cycle has many mechanistic similarities to the CO 2 reduction cycle, with both being facilitated by the special catalytic properties of the vicinal enediamine motif.

OVERVIEW OF THE CO 2 REDUCTION CYCLE CATALYZED BY A VICINAL ENEDIAMINE ELECTRO-ORGANOCATALYST
Previously, we showed that an electro-organocatalyst with a vicinal enediamine functionality is capable of electrochemically reducing CO 2 to formaldehyde by the catalytic cycle depicted in Scheme 1. 8 This mechanism was based on a representative electro-organocatalyst 1,3-dimethyl-4-imidazoline (DM4Im) containing the vicinal enediamine backbone.The cycle begins with CO 2 activation occurring by an electrophilic substitution mechanism, which is enabled by the electron-rich π system of the vicinal enediamine backbone, Deprotonation of the C5 position in S 2 to form S 3 occurs with the aid of a proton transfer mediator (PTM) that could take the form of a molecule, polymer, or surface having a pK a close to the operating pH.Following CO 2 activation, a sequence of proton coupled electron transfer (PCET) steps occurs that leads to a two-electron reduction of S 3 to S 6 , The PCET is initiated by protonation of S 3 at the C4 position to give S 4 , with the proton being donated by the PTM.This is the rate limiting step of the catalytic cycle and has a rate that is independent of the cathode potential since it is a nonelectrochemical step.Following protonation, S 4 readily undergoes alternating transfer of two electrons from the cathode and two protons from the PTM to form S 6 .This is then followed by tautomerization and dehydration of the resulting geminal diol to give S 9 in which the carboxylate group formed from CO 2 addition is transformed into an aldehyde.The aldehyde then undergoes a second PCET sequence followed by tautomerization that converts the aldehyde in S 9 into a hydroxymethyl group in S 13 .The hydroxymethyl is finally eliminated as formaldehyde to complete the catalytic cycle and return the catalyst to the initial state S 1 .
There are several key features of the vicinal enediamine electro-organocatalyst that enable it to carry out the CO 2 reduction cycle, which will be shown to also have importance in the chain growth cycle.The addition of CO 2 and elimination of formaldehyde proceed through zwitterionic intermediates (e.g., S 2 ) that are stabilized by favorable placement of the resulting positive charge on the N3 nitrogen atom.This property also enables facile dehydration of the geminal diol in the step S 8 → S 9 .Likewise, placement of positive charge on the N1 nitrogen atom facilitates rapid The Journal of Physical Chemistry A tautomerization in steps S 6 → S 8 and S 11 → S 13 even at neutral pH.
The electron transfers involved in the two PCET sequences are favorable at modest overpotentials because the initial and final states avoid unfavorable placement of formal positive or negative charge on any carbon atom due to the unique structure of the electro-organocatalyst.Instead, positive charge is placed on one of the nitrogen atoms in the initial state (e.g., S 4 → S 4* ) or negative charge is placed on an oxygen atom in the final state (e.g., S 5* → S 5 ).This feature of the electroorganocatalyst is discussed further in Section 6.5 for an analogous PCET step that occurs during the chain growth cycle.

DENSITY FUNCTIONAL THEORY CALCULATIONS
We make use of a free energy profile to visualize the kinetics of the elementary steps and the overall chain growth cycle discussed in the previous section.Full details of the DFT calculations carried out to construct this profile are given in the Supporting Information and in ref. 8 These calculations were performed using the Vienna Ab initio Simulation Package 17 (VASP) along with the VASPsol extension 18,19 that allows for implicitly modeling the electrolyte by a continuum electrostatic description.The Bayesian error estimation functional with van der Waals correlation (BEEF-vdW) 20 was used in all calculations.This functional was chosen based on its balanced accuracy for describing a wide range of energetic quantities, ranging from molecular formation, reaction, and activation energies to cohesive energies of solids and chemisorption on solid surfaces.In addition, the BEEF-vdW functional includes dispersion interactions such as those that would occur between molecules and surfaces.While surfaces are not included in the present study, we envision combining this system with a solid proton transfer mediator such as a metal oxide surface in the future.To account for errors in the description of certain functional groups by semilocal exchange-correlation functionals such as BEEF-vdW, empirical corrections are applied to the electronic energy of CO 2 and any molecule with a carboxylate, carbonyl, or geminal diol group.
Transition states for most steps were found by performing a roughly converged nudged elastic band calculation 21,22 to obtain an initial guess, followed by a dimer calculation 23 to refine the transition state.We neglect activation barriers for steps involving proton transfer between the PTM and an oxygen atom (S 15 → S 14 , S 18 → S 19 , S 24 → S 25 , S 29 → S 30 ) as well as steps involving electron transfer (S 25 → S 25* , S 26* → S 26 ) since the energetic barriers of such steps were found to be negligible in our previous work. 8s generalized gradient approximation (GGA) functionals such as BEEF-vdW are known to underestimate activation barriers for proton transfer reactions, 24 we have also performed benchmarking calculations using hybrid DFT functionals on two of the key proton transfer steps, S 3 → S 4 in the CO 2 reduction cycle and S 19 → S 20 in the chain growth cycle.These calculations were performed in NWChem 25 using the PBE0 26 and B3LYP 27 functionals.We find that the activation barriers calculated using these hybrid functionals are no more than 0.08 eV higher than those calculated using the BEEF-vdW functional.This is due to the fact that the BEEF-vdW functional underpredicts these barriers less severely than more widely used GGA functionals such as PBE, 28 consistent with reported results for other nonlocal van der Waals GGA functionals. 24Further details of the benchmarking calculations are discussed in the Supporting Information.
Transition states for steps involving proton transfer between the proton transfer mediator (PTM) and a carbon atom were computing using formic acid or formate as a model PTM.The barriers were then extrapolated to a PTM having a pK a equal to the pH using the approach developed in our previous work on the CO 2 reduction cycle. 8The extrapolation is based on an analogy with Marcus theory for electron transfer reactions. 29or a protonation step S i → S j , the effective activation barrier with a PTM having a pK a equal to the pH is given by, max(pH p , 0) where λ i→j is the reorganization energy and pK a corresponds to the model PTM (formic acid/formate).The parameters β HA and β A − are constants that account for the free energy to form a precursor complex between the reactant or product and the PTM.The intrinsic reaction free energy is given by, where pK a,j is the pK a of the product S j .A similar extrapolation is used for a deprotonation step S i → S j , where the intrinsic reaction free energy is given by (pK a,i is the pK a of the reactant S i ), Further details of this approach, including the parameter values, are given in the Supporting Information.
To account for interactions between the catalytic intermediates and the electrolyte, we employ a hybrid implicitexplicit solvation method that was developed in our previous manuscript. 8Including certain water molecules explicitly in the DFT calculations allows for an appropriate description of strong hydrogen bonds that can form between the solute and solvating water molecules, particularly in charged intermediates and transition states.This hybrid solvation method differs from other similar methods (c.f.refs 30 and 31 ) by applying an empirical hydrogen bonding correction to account for the entropy loss associated with formation of hydrogen bonds, rather than computing this contribution explicitly from vibrational frequency calculations.Doing so avoids the difficulties and errors associated with applying the harmonic approximation to lose vibrational modes associated with hydrogen bonding interactions.The correction, G W,corr , is determined by fitting the self-solvation free energy of water to the experimental value.
Free energies of intermediates and transition states were determined by adding translational, rotational, and vibrational contributions to the electronic energy E 0,aq computed by VASP with implicit solvation, The last term accounts for the effective chemical potential of the n W molecules of explicitly hydrogen bonded water,

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where E W,aq is the electronic energy of a water molecule implicitly "solvated" in the electrolyte.The number of explicit water molecules is determined based on the commonly used "variational" approach, 30 by selecting the configuration with the lowest free energy according to eq 5.

MECHANISM OF THE CHAIN GROWTH CATALYTIC CYCLE
The chain growth catalytic cycle is depicted in Scheme 2 and is based on the same representative electro-organocatalyst  13,14 Once formed, it undergoes aldol addition of an aldehyde to the C6 position to give S 19 .The C6 carbon of S 11 is rendered nucleophilic by polarity reversal from conjugation with the lone pair on N1, enabling it to attack the electrophilic carbonyl of the aldehyde.As with aldehyde addition/elimination and CO 2 addition, the aldol addition proceeds through a zwitterionic intermediate S 18 in which positive charge is formally placed on the N1 nitrogen.Intermediate S 19 undergoes deprotonation of C6 followed dehydration of C7 (the carbonyl position of the aldehyde) to give S 21 .The deprotonation step involves proton transfer to the PTM to give S 20 , which then dehydrates by a concerted mechanism in which a proton is shuttled from the hydroxyl on C6 to the hydroxyl on C7 followed by water elimination.Intermediate S 21 then tautomerizes to S 24 by transfer of a proton from the C4 position to the C7 position via the cationic intermediate S 22 .
Intermediate S 24 resembles S 9 of the CO 2 reduction cycle except that the hydrogen on C6 is replaced by an alkyl group.The remainder of the chain growth cycle proceeds by an analogous mechanism to the second half of the CO 2 reduction cycle (from S 9 ); S 24 undergoes a PCET sequence to yield S 26 which then tautomerizes to S 28 and eliminates an aldehyde to complete the catalytic cycle.
A similar mechanism for C−C bond formation occurs in the formose reaction, 11 whereby two successive aldol additions of formaldehyde to glycolaldehyde followed by retro-aldol elimination produces a second molecule of glycoaldehyde.The reaction is autocatalytic, with glycolaldehyde acting as both a product and an organocatalyst.The catalytic activity of glycolaldehyde results from deprotonation to its enediolate form, which can then undergo aldol addition with formaldehyde.This is followed by tautomerization and aldol addition of a second molecule of formaldehyde, with the cycle completed by retro-aldol elimination of glycoaldehyde.The mechanism is depicted in Scheme 3, which has been drawn in such a way to highlight the similarities between the formose reaction and the chain growth cycle in Scheme 2. It can be seen that the vicinal enediamine electro-organocatalyst functions in an analogous way, carrying out two successive additions of formaldehyde.However, the electro-organocatalyst is also capable of facilitating dehydration and electron transfer so that acetaldehyde is formed as the product instead of glycoaldehyde.

FREE ENERGY PROFILE OF THE CHAIN GROWTH CYCLE
As detailed in our previous work, 8 the free energy profile is constructed from the free energies of each catalytic intermediate (ΔG i • ) and transition state (ΔG i→j ‡• ) relative to the initial state S 1 of the catalytic cycle.A reactive intermediate S i formed from S 1 by a process involving protons, electrons, and any number of other reactants and products A k , a Adapted with permission from Ref. 11 Copyright 1959 Elsevier Ltd.

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has a relative free energy given by, where G i • and G 1 • are the absolute free energies computed by eq 5 of S i and S 1 , respectively.In this expression, q = n H + -n e − is the net charge of S i .The set of reactants and products A k consists of CO 2 , H 2 O, formic acid, and formaldehyde.Expressions for the chemical potentials μ Ad k of these reactants and products as well as the hydrogen atom and proton chemical potentials μ H + and μ H are given in the Supporting Information.
The relative free energy of the transition state for the reaction S i → S j is given by, The activation barrier Δ a G i→j • is defined as the free energy difference between the transition state and the preceding intermediate S i .A special form is used for reactions involving proton transfer, which is discussed in the Supporting Information.For any other step converting S i to S j , possibly involving one or more reactant molecule A k , the activation barrier is given by, where G i→j ‡• is the absolute free energy of the transition state computed by eq 5.The stoichiometric coefficients ν Ad k represent the number of each molecule A k involved in formation of the transition state.
The free energy profile for the chain growth cycle is presented in Figure 1 for the representative electro-organocatalyst 1,3-dimethyl-4-imidazoline (DM4Im).This molecule was chosen as a model catalyst in our previous work 8 since it is one of the simplest structures containing the vicinal enediamine backbone.The profile corresponds to a temperature of 80 °C, a pH of 6.7, an electrode potential of −0.89 V vs RHE, and a formaldehyde concentration of 4.3 mmol/L (log[CH 2 O] =−2.4).The temperature was chosen to represent an upper limit of the operating temperature for an anion exchange membrane electrolyzer while the other operating conditions were chosen to maximize the TOF of the chain growth cycle as discussed in Section 7.
As we discussed in our previous work, 8 the free energy diagram can be related to the TOF of the catalytic cycle by the "waterfall analogy".To see this, one visualizes water cascading from left to right down the free energy profile and pooling up behind all of the barriers corresponding to transition states.The surface of the water behind all of the barriers is indicated by the upper dashed line on the diagram and corresponds to the transition state free energies of all irreversible steps.Likewise, all of the quasi-equilibrated transition states are below this line, i.e. "under water".The lower dashed line is identical to the upper dashed line shifted down by an amount equal to the global barrier for the catalytic cycle ΔG ‡ .We refer to this lower dashed line as the resting f ree energy prof ile, while we refer to the free energy profile comprised from the standard free energies of the intermediates and transition states as the standard f ree energy prof ile.The former depicts the free energies of the states at the steady state concentrations while the latter depicts the free energies of the states at the standard concentration of 1 mol/L.Note that the resting free energy profile is monotonically decreasing, consistent with the second law of thermodynamics; thus, the irreversibility of each step can be characterized by the drop in the resting free energy profile associated with that step.Graphically, the global barrier is the minimum amount one must shift down the resting free energy profile so that it lies below the standard free energy profile.Kinetically, it is related to the TOF of the catalytic cycle by, i k j j j j j y

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Any state lying on the resting free energy profile is a global resting state of the catalytic cycle and will represent a dominant state of the catalyst at steady state.In general, there is only one global resting state but under special conditions there can be two or more as is the case here.
Using the waterfall analogy, one can also identify the rate limiting steps of the catalytic cycle.First, we define the resting state associated with an irreversible step as the lowest free energy state to its left (or "upstream") on the free energy diagram.A formal rate limiting step is an irreversible step for which the associated resting state is also a global resting state − in other words, the standard free energy of the resting state lies on the resting free energy profile.Looking at the free energy diagram for the chain growth cycle in Figure 1 It is no coincidence that multiple rate limiting steps exist; this occurs because the free energy profile has been calculated at the optimal conditions for operating the catalytic cycle.In particular, the precise values chosen for the three operating conditions (pH, electrode potential, and formaldehyde concentration) lead to the occurrence of the three additional rate limiting steps.To aid in understanding why this occurs, we define the total barrier for an irreversible step in the catalytic cycle as the difference in free energy between its transition state and its resting state.For a formal rate limiting step, this latter state will be a global resting state and its total barrier will be equal to the global barrier.In Section 7, we discuss how the total barriers of various steps are affected by pH, electrode potential, and formaldehyde concentration and how this leads to changes in the formal rate limiting step under different conditions.

ELEMENTARY STEPS IN THE CHAIN GROWTH CYCLE
We now discuss the energetic and mechanistic details of the individual elementary steps in the chain growth cycle.The energies discussed will be those corresponding to the optimal conditions for operating the isolated chain growth cycle (pH = 6.7,U = −0.89V vs RHE, log[CH 2 O] = −2.4),which is discussed in detail in Section 7. Most of these energies will be somewhat different at the optimal conditions for coupling with the CO 2 reduction cycle (pH = 7.8, U = −0.85V vs RHE, log[CH 2 O] = −3.1),which is discussed in Section 8. 6.1.Formaldehyde Activation.The initial state of the organocatalyst is in equilibrium between the active state (S 1 ) and an inactive state (S 0 ) in which the C5 position is protonated, The calculated pK a of S 0 is 6.8 so that the protonated state is slightly favored by −0.01 eV at the optimal pH for the isolated chain growth cycle of 6.7.In contrast, the deprotonated state is thermodynamically favored at the optimal pH (7.8) for coupling the chain growth cycle with the CO 2 reduction cycle.
The catalytic cycle begins from the deprotonated state S 1 by activation of formaldehyde, which occurs by electrophilic substitution at the C5 position in S 1 to give S 13 .This occurs by the reverse of the formaldehyde elimination mechanism in the CO 2 reduction cycle and is similar to how S 1 activates CO 2 .Formaldehyde addition occurs in three elementary steps, the first being electrophilic addition to the C�C π bond at the C5 position to give the zwitterionic intermediate S 15 , In this and other reaction schemes, the reaction free energies and activation barriers are indicated below each reaction arrow in eV, with the activation barriers enclosed by parentheses.The activation barrier for the electrophilic addition step is 0.73 eV with respect to S 1 while the total barrier with respect to the global resting state S 0 is 0.74 eV.At the limiting formaldehyde concentration (log[CH 2 O] = −2.4),this step is formally reversible since the transition state is lower in free energy than the transition states of two subsequent steps, S 12 → S 11 and S 19 → S 20 .The pK a associated with protonating S 15 on the oxygen is 10.4 which readily occurs at the optimal pH of 6.7 to give S 14 .In our previous study on the CO 2 reduction cycle, 8 it was found that proton transfer steps to and from oxygen atoms do not have a transition state on the potential energy surface.The proton transfer occurs spontaneously in the energetically favorable direction when the proton transfer mediator (PTM) approaches the intermediate.Therefore, the barrier for this step is purely entropic and assumed to be negligible.
As shown in Figure 2, the transition state for S 1 → S 15 involves rehybridization of both carbon atoms forming the new C−C bond from sp 2 to sp 3 , with a C−C bond distance of 2.28 Å.This is associated with an energy barrier of only 0.17 eV; however, the entropic penalty for bringing formaldehyde from the bulk electrolyte raises the free energy barrier to 0.73 eV with respect to S 1 .The additional thermodynamic barrier of 0.01 eV to deprotonate the resting state S 0 leads to a total The Journal of Physical Chemistry A barrier of 0.74 eV.Both formaldehyde addition intermediates S 15 and S 14 are stabilized by placement of the formal positive charge on N3, making the electron-rich π system in S 1 exceptionally nucleophilic.As a result, the cationic S 14 is only 0.05 eV higher in free energy than the neutral S 1 .This is the same property that allows S 1 to activate CO 2 in the CO 2 reduction cycle.
Once S 14 is formed, it deprotonates at the C5 position to complete the formaldehyde addition process.The proton is transferred to the basic form of the PTM to give intermediate S 13 , which involves rehybridization of C5 from sp 3 to sp 2 .This leads to an activation barrier of 0.71 eV and a total barrier of 0.77 eV with respect to the local resting state S 0 , making the step partially rate limiting.
Comparing to Scheme 3, it can be seen that this mechanism for formaldehyde activation is analogous to the mechanism by which formaldehyde addition occurs to the enediolate form of glycoaldehyde during the formose reaction. 11In the formose reaction, the α carbon of the enediolate functions as the nucleophile during this step.The enediolate is expected to be more nucleophilic than the vicinal enediamine S 1 but requires a significantly higher pH to exist in the deprotonated state (>14 for the enediolate vs ∼7 for S 1 ).As such, this step should be intrinsically slower than the initial step of the formose reaction but does not require high pH conditions that are incompatible with the presence of CO 2 .
6.2.Tautomerization.The chain growth cycle contains three tautomerization steps in which a proton is transferred between two carbon atoms.All three occur by a similar mechanism through a cationic intermediate in which the formal positive charge is placed on N1.The pK a values of the carbon atoms being protonated range from 5.7 to 8.3 so that the intermediates readily form at neutral pH conditions.The first tautomerization converts S 13 to S 11 and is the reverse of the final tautomerization in the CO 2 reduction cycle, The intermediate S 12 forms by protonation of the C4 position, having a pK a of 5.7 so it is slightly uphill in free energy at the optimal pH of 6.7.As with other proton transfer steps, the proton is donated from the acidic form of the PTM and involves a barrier of 0.76 eV with respect to the resting state S 13 , making the step partially rate limiting.Intermediate S 12 then deprotonates from the C6 position to give S 11 .The proton has a pK a of 8.2 so that the deprotonation step is also uphill in free energy at the optimal pH.The deprotonation step is formally rate limiting, having a total barrier equal to the global barrier of 0.83 eV with respect to the resting state S 13 .
The second tautomerization step occurs following dehydration and involves the transfer of a proton from the C4 position to the C7 position in S 21 to yield S 24 , Protonation at C7 to give the S 22 intermediate is associated with a pK a of 6.4 and thus is only slightly higher in free energy than the resting state S 21 at the optimal pH of 6.7.Deprotonation of S 22 at C4 yields S 24 and is associated with a pK a of −4.0, making it highly favorable at the optimal pH.The unusually low pK a value arises from stabilization in S 24 due to conjugation of the lone pair on N3 with the carbonyl group.A similar conjugated interaction was observed in the product of the dehydration step in the CO 2 reduction cycle, making this step highly irreversible also.The total barriers for the protonation and deprotonation steps are 0.71 and 0.65 eV, significantly lower than the global barrier of 0.83 eV; thus, neither step is rate limiting.The protonation step is slightly irreversible at the optimal pH, while the deprotonation step is highly irreversible.Together, both steps account for 22% of the irreversibility of the catalytic cycle.Both transition states are depicted in Figure 3, where it can be seen that the carbon positions involved in the proton transfer are rehybridizing between sp 2 and sp 3 , while the proton is in transit between the PTM and the carbon atom.
The final tautomerization step is analogous to step S 11 → S 12 → S 13 in the CO 2 reduction cycle, or the reverse of this process at the beginning of the chain grown cycle.A proton is transferred from C4 to C6 and is the last process that occurs before the aldehyde is eliminated to complete the catalytic cycle, Protonation of C6 leads to the intermediate S 27 with an associated pK a of 8.3, while subsequent deprotonation of C4 yields S 28 with an associated pK a of 6.0.The protonation step is quasi-equilibrated, having an activation barrier of 0.66 eV, while the deprotonation step is formally rate limiting with a total barrier equal to the global barrier of 0.83 eV.

Aldol Addition.
The aldol addition step results in formation of the C−C bond and occurs immediately after tautomerization of S 13 to S 11 .Conjugation of the enolic C�C double bond in S 11 with the lone pair on N1 reverses the polarity of the C6 position to allow the normally electrophilic carbon to function as a nucleophile instead.This enables electrophilic addition of the aldehyde at this position to give the zwitterionic intermediate S 18 ,

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An initial precursor state depicted in Figure 4 was found in which the formaldehyde appears to form a charge transfer complex with S 11 , having an interaction energy of −0.12 eV below the energy of separated S 11 and formaldehyde.The C− C distance is fairly short in this complex at 2.40 Å, well below the van der Waals distance of 3.40 Å.The energy of the nudged elastic band calculation between the initial and final states decreased monotonically and did not indicate any transition state, although the initial state did not spontaneously optimize to the final state.For this reason, we were unable to find a transitions state and therefore assumed that it lies very close to the initial state.We therefore used the initial precursor complex to calculate the free energy barrier for this step.Although the precursor complex is lower in energy than separated formaldehyde and S 11 , the entropy penalty for transferring the formaldehyde from the bulk electrolyte makes the free energy barrier 0.43 eV.The total barrier is 0.61 eV with respect to the resting state S 13 .The C−C bond formation step is then followed by protonation of the oxygen in the zwitterionic S 18 intermediate to give S 19 , with an associated pK a of 8.4 making it favorable at the optimal pH of 6.7.
The polarity reversal of C6 that enables aldol addition is a key feature of how NHCs carry out "umpolung" chemistry in reactions like benzoin condensation and the Stetter reaction. 9ntermediate S 11 is analogous to the Breslow intermediate that is formed in these reactions, whereby conjugation of the C�C double bond with one or more nitrogen atoms in the NHC engenders nucleophilicity to the otherwise electrophilic carbon. 13,14This step is fundamentally identical to the first C−C bond formation reaction in the formoin reaction, in which formaldehyde is activated and rendered nucleophilic by addition to a thiazolium salt and subsequently undergoes aldol addition with a second formaldehyde. 12he aldol addition step is also analogous to the second C−C bond formation step that occurs in the formose reaction between formaldehyde and the enediolate form of dihydroxyacetone. 11Similar to the function of NHCs, conjugation with O − makes the C�C double bond in the enediolate nucleophilic, allowing it to attack the electrophilic carbon of formaldehyde.As was the case when comparing formaldehyde activation by S 1 to formaldehyde addition to glycoaldehyde during the formose reaction, the enediolate form of dihydroxyacetone is expected to be more nucleophilic than S 11 while requiring a higher pH to exist in the deprotonated form (>14 for the enolate vs 8.2 for S 11 ).Thus, addition to S 11 should be intrinsically slower than the analogous step during the formose reaction but can occur at a significantly lower pH that is compatible with the presence of CO 2 .
6.4.Dehydration.Dehydration of the aldol addition product at the C7 position yields the zwitterionic intermediate S 21 .This process occurs in two steps, First, the C6 position of S 19 is deprotonated to give the S 20 intermediate.The transition state, depicted in Figure 5, is similar to those of other proton transfer reactions between carbon atoms and the PTM, particularly S 12 → S 11 and S 26 → S 27 .The associated pK a is 10.0, so the free energy is uphill by 0.23 eV at the optimal pH of 6.7.The total barrier is 0.83 eV with respect to the resting state S 13 , making this step formally rate limiting and irreversible.
Following deprotonation, S 20 undergoes water elimination via a transition state in which the C−OH bond at the C7 position elongates to 2.26 Å, as seen in Figure 5.The oxygen in the hydroxide leaving group is strongly hydrogen bonded to three explicit water molecules, lowering the free energy of the transition state by 0.79 eV compared to the case with only implicit solvation.One of these water molecules is bridged by a fourth explicit water molecule to create an uninterrupted hydrogen bonding network between the OH group at the C6 position and the hydroxide.This provides additional stabilization compared to the structure where the hydrogen bonding bridge is not formed.The intrinsic barrier with respect to S 20 is 0.41 eV, giving a total barrier of 0.67 eV with respect to the resting state S 13 .Consequently, this step is not rate limiting, although it is highly irreversible.Together, both steps involved in dehydration account for 18% of the total irreversibility of the cycle.As in several other steps, the zwitterionic dehydration product S 21 is stabilized by formal placement of the positive charge on N1.The process commences with protonation of the oxygen in S 24 , which is associated with a pK a of 2.4.The pK a is significantly higher than for a typical carbonyl oxygen due to delocalization of the positive charge onto N3 via conjugation.As with S 15 → S 14 , no transition state was explicitly calculated since we have previously found that the transition state is purely entropic for proton transfer to or from oxygen.The proton transfer is then followed by an electron transfer from a chemically inert cathode to give the radical intermediate S 25* .This electron transfer requires a potential of −1.57V vs SHE to be thermodynamically favorable, making the process uphill in free energy at the operating potential of −1.33 V vs SHE.Nonetheless, the thermodynamic potential is comparatively low for electron transfer to a singlet organic molecule due to the unique structure of the electro-organocatalyst.Specifically, the unpaired electron is delocalized over the six atoms forming the conjugated π system.The reduction potential of the combined PCET step S 24 → S 25* is −1.40 V vs RHE, leading to an overall thermodynamic barrier for this process of 0.52 eV at the limiting potential of −0.89 V vs RHE.
After S 25* is formed, the C4 position protonates to give S 26* via the transition state depicted in Figure 4.The associated pK a is 12.6 making the proton transfer highly favorable at the optimal pH of 6.7.This leads to a relatively low barrier of 0.32 eV with respect to S 25* , giving a total barrier of 0.83 eV with respect to the local resting state S 24 .This makes the step formally rate limiting at the limiting potential of −0.89 V vs RHE and is in fact the condition that defines the limiting potential as discussed in Section 7.2.The protonated intermediate S 26* then undergoes a highly irreversible electron transfer accounting for 51% of the total irreversibility of the cycle.
6.6.Aldehyde Elimination.Aldehyde elimination follows the third tautomerization step S 26 → S 28 to complete the catalytic cycle and return the catalyst to the initial state S 1 .The process follows a mechanism analogous to formaldehyde elimination in the CO 2 reduction cycle, or equivalently the reverse of formaldehyde addition at the beginning of the chain growth cycle, It begins with protonation of the C5 position in S 28 to give S 29 by the transition state depicted in Figure 6.This step has a barrier of 0.77 eV with respect to the local resting state S 28 , making it partially rate limiting.As will be seen in Section 7.1, the optimal pH of 6.7 is actually defined by the pH where this step has the same barrier as the preceding tautomerization step S 27 → S 28 .The associated pK a is 5.2, making the step slightly uphill in free energy at the optimal pH.The protonation step is then followed by deprotonation of the hydroxyl on C6 to form the zwitterionic intermediate S 30 .This step has a pK a of 10.0 so that it is fairly uphill in free energy.As with the other steps involving proton transfer to or from oxygen, we assume there is no energetic transition state for this step; only an entropic transition state that is considered not to be kinetically relevant.
The final step involves elimination of the product aldehyde from S 30 .This is enabled by the charge separation in the zwitterionic intermediate, whereby a lone pair on the deprotonated oxygen makes the aldehyde an excellent electrophilic leaving group.Likewise, the iminium C�N + bond between N1 and C4 is able to accept the electrons from the dissociating C−C bond.These effects combine to give a low intrinsic barrier of 0.27 eV with respect to S 30 and a total barrier of 0.60 eV with respect to the local resting state S 28 .

KINETIC DEPENDENCE OF THE CHAIN GROWTH CYCLE
In the previous sections, we discussed the mechanism and free energy profile of the chain growth cycle calculated at the optimal pH of 6.7, the limiting potential of −0.89 V vs RHE, and the limiting formaldehyde concentration of log[CH 2 O] = −2.4.In this section, we explore how the barriers of different steps in the cycle depend on these conditions and ultimately how these conditions are determined by kinetic trade-offs between various steps.7.1.Effect of pH on the Free Energy Profile.The total barriers (defined in Section 5) for several steps have been plotted in Figure 7 as a function of the electrolyte pH, where it can be seen that all of them exhibit kinks at particular pH values where the resting state for that step changes from one state to another.By definition, the global barrier for a given pH is equal to the highest total barrier at that pH, and the corresponding step is the formal rate limiting step.Across the entire pH range, one can identify three steps that are formally The lowest global barrier occurs at the optimal pH of 6.7 where the formal rate limiting step changes from S 27 → S 28 to S 12 → S 11 .As will be discussed in more detail below, both steps are deprotonation steps but S 12 → S 11 proceeds from an unprotonated resting state (S 13 ) while S 27 → S 28 proceeds from a protonated resting state (S 27 ).Therefore, the total barrier for S 12 → S 11 increases with pH due to the dependence of the S 13 → S 12 quasi-equilibrium on pH, while the total barrier for S 27 → S 28 decreases with pH.The point where the total barriers of these two steps cross defines the optimal pH of 6.7.
To better understand the reason underlying the changes in rate limiting at pH values of 3.7 and 6.7, it is first necessary to discuss the somewhat counterintuitive behavior exhibited by the total barriers of protonation steps at low pH and of deprotonation steps at high pH.Naively, one might expect the total barrier of a protonation step to continuously decrease as pH decreases.This is true as long as the resting state of the protonation step is the preceding unprotonated species.However, when the pH is low enough that the resting state shifts to an earlier protonated species the trend in the total barrier reverses and it starts to increase with decreasing pH.This occurs because the transition state for the protonation step is actually "less" protonated than the resting state since the resting state is fully protonated while the transition state is only partially protonated (as the proton is in transit).Therefore, the free energy of the resting state decreases faster than the free energy of the transition state as the pH decreases, leading to an increase in the total barrier.This behavior is seen S 28 → S 29 when the pH is lower than 6.0.When the pH is above this value, the resting state is the unprotonated state S 28 so that the total barrier decreases as pH decreases.Below this pH though, the resting state switches to the protonated state S 27 so that the total barrier now starts to increase as pH decreases further.
An analogous argument can be made for the increase in total barrier of a deprotonation step at sufficiently high pH.As pH increases, all three deprotonation steps depicted in Figure 7 initially exhibit decreasing barriers because the corresponding resting states are protonated: S 27 for S 27 → S 28 , S 19 for S 19 → S 20 , and S 14 for S 12 → S 11 (although the free energy of S 12 is only 0.01 eV higher than S 14 ).As pH increases, the resting state for S 12 → S 11 switches from the protonated S 14 to the unprotonated S 13 at a pH of 5.8.Further increasing the pH to 8.3 switches the resting state for S 27 → S 28 from the protonated S 27 to the unprotonated S 26 .Finally, increasing the pH past 8.4 switches the resting state for S 19 → S 20 from the protonated S 19 to the unprotonated S 18 .Once the resting state switches to an unprotonated state, the total barrier begins to increase with pH rather than decreasing since the transition state of the deprotonation step is now "more" protonated than the fully unprotonated resting state.
The pH dependence of the total barriers of all other steps in the catalytic cycle are reported in Figure S3 in the Supporting Information.All of the proton transfer steps exhibit the same "inverted volcano" behavior where the "peak" is associated with a change from a protonated to an unprotonated resting state.The steps that do not involve proton transfer also exhibit a slope discontinuity in the pH dependence, whereby the total barrier decreases with pH at low pH before becoming independent of pH at higher pH.This is also caused by a change from a protonated to an unprotonated local resting state, but the shape is different because the reaction free energy of the step itself (and thus the intrinsic barrier) does not depend on pH since the step does not directly involve proton transfer.All of these steps proceed directly from an unprotonated state; therefore, when this state happens to be the resting state for the reaction step, the total barrier is equal to the intrinsic barrier of the step, which is independent of pH.At low pH when the resting state switches to a protonated state earlier in the cycle, there is an additional free energy penalty associated with deprotonating the resting state before the reaction step can occur.This thermodynamic penalty increases as pH decreases leading to an increase in the total barrier of ln 10 k B T ≈ 0.07 eV per unit decrease in pH.
We can now better understand the change in the formal rate limiting step from S 27 → S 28 to S 12 → S 11 at a pH of 6.7.Despite both being deprotonation steps, S 12 → S 11 proceeds from the unprotonated resting state S 13 so that the total barrier increases with pH, while S 27 → S 28 proceeds from the protonated resting state S 27 so that the total barrier decreases with increasing pH.Therefore, S 27 → S 28 has a higher total barrier below a pH of 6.7 while S 12 → S 11 has a higher total barrier above this pH.
The reason for the change in formal rate limiting step from S 19 → S 20 to S 27 → S 28 at a pH of 3.7 is more subtle.Since both steps involve deprotonation proceeding from an unprotonated resting state, the total barriers of both steps increase as pH decreases; however, the barrier for S 19 → S 20 increases faster than the barrier for S 27 → S 28 .This occurs because S 19 → S 20 is thermodynamically less favorable than S 27 → S 28 and thus has a transition state that is closer to the product state.Therefore, the barrier of S 19 → S 20 is more sensitive to changes in the reaction free energy caused by changes in pH.As the pH decreases, the total barrier for S 19 → S 20 eventually becomes higher than the total barrier for S 27 → S 28 at a pH of 3.7, making the former step rate limiting at pH values lower than this.
7.2.Effect of the Electrode Potential on the Free Energy Profile.The electrode potential affects both the The potential of −0.89 V vs RHE is thus not an "optimal" potential per se, but a limiting potential below which the TOF becomes independent of potential.A more cathodic potential will not increase the TOF since electron transfer is not kinetically involved in any of the rate limiting steps at these potentials.
7.3.Effect of Formaldehyde Concentration on the Free Energy Profile.We finally examine the effect of formaldehyde concentration on the kinetics of the isolated chain growth cycle.Technically, we are referring to the concentration of methanediol since this is the predominant form of formaldehyde in aqueous solution.The formaldehyde concentration can potentially affect the total barriers of all steps in the cycle up to dehydration (S 20 → S 21 ).This latter step is sufficiently irreversible that the total barriers of all subsequent steps are independent of the formaldehyde concentration.
The total barriers for all steps up to dehydration are depicted in Figure 9a with respect to formaldehyde concentration.The key point is that there are two ways in which the total barrier of  The Journal of Physical Chemistry A a step depends on formaldehyde concentration−either through direct involvement of formaldehyde in the step or through quasi-equilibrium of a preceding step that involved formaldehyde addition.As [CH 2 O] increases, the latter dependence vanishes as the equilibria of all such preceding steps shifts toward the product.Thus, at high [CH 2 O], steps that directly involve formaldehyde addition (S 1 → S 15 and S 11 → S 18 ) exhibit a first order dependence on formaldehyde while all other steps exhibit zero order dependence.
To further explore the kinetic dependence of individual steps on formaldehyde concentration, the steps in Figure 9a have been divided into four groups depending on which resting state they proceed from S 1 → S 14 , S 14 → S 13 , steps between S 13 and S 19 , and steps between S 19 and S 21 (dehydration).Additionally, the plot can be divided into four regimes based on the kinetic dependence of each group of steps with respect to [CH 2 O].In the leftmost regime, corresponding to low [CH 2 O], all steps proceed from the resting state S 0 .The steps between S 1 and S 11 exhibit first order dependence on [CH 2 O] while the steps between S 11 and S 21 exhibit second order dependence.This dependence arises from the fact that one molecule of formaldehyde is transferred from the bulk electrolyte (in the formaldehyde addition step) to obtain the transition states for steps between S 1 and S 11 from the resting state S 0 , while two molecules of formaldehyde are transferred (one in formaldehyde addition and one in aldol addition) for the steps between S 11 and S 21 .
At log[CH 2 O] of −2.4,intermediate S 13 becomes lower in free energy than S 0 so that the resting state for all steps between S 13 and S 21 shifts to S 13 .As a result, formation of the transition states for these steps from the new resting state S 13 no longer requires transfer of formaldehyde from the bulk electrolyte in the formaldehyde addition step.Thus, the tautomerization steps between S 13 and S O] = −2.4 that the resting state switches from S 0 to S 13 .This is independent of pH since both steps are deprotonation and proceed from unprotonated resting states.

COUPLING OF THE CHAIN GROWTH AND CO 2 REDUCTION CYCLES
We now discuss the kinetic coupling between the CO 2 reduction cycle and the chain growth cycle.These two cycles are coupled by the formaldehyde concentration since the kinetics of both potentially depend on it.When the formaldehyde concentration is at steady state, the TOF of the CO 2 reduction cycle will be twice the TOF of the chain growth cycle since the former produces one molecule of formaldehyde while the latter consumes two molecules.Also, when the two cycles are coupled they must operate at the same pH.The limiting global barrier of each cycle, defined as the global barrier when the respective cycle is not kinetically limited by either potential or formaldehyde concentration, is plotted in Figure 10 with respect to pH.At all values of pH, the limiting global barrier of the CO 2 reduction cycle is higher than the limiting global barrier of the chain growth cycle.Therefore, the optimal pH for the coupled cycle is determined by the optimal pH of the CO 2 reduction cycle, which was determined to be 7.8 in our previous study. 8t this pH, the CO 2 reduction cycle is zero order in formaldehyde when log[CH 2 O] is lower than −2.5.Above this concentration, the global resting state shifts from S 1 to S 13 so that the cycle is inhibited to first order by formaldehyde.The kinetic dependence of the chain growth cycle on formaldehyde concentration at this same pH is depicted in Figure S4 in the Supporting Information, where it can be seen that the behavior is qualitatively similar to the behavior at a pH of 6.7 (the optimal pH of the isolated chain growth cycle).One can see from this figure that the chain growth cycle is second order in formaldehyde when log[CH 2 O] is less than −2.5 and the rate The Journal of Physical Chemistry A limiting step is the deprotonation step S 19 → S 20 proceeding from the resting state S 1 .Above this concentration, the rate limiting step shifts to the deprotonation step S 12 → S 11 proceeding from resting state S 13 and is thus kinetically independent of formaldehyde concentration.As mentioned in Section 7.3, it is purely a coincidence that the rate limiting step and the resting state switch at exactly the same value of log[CH 2 O].
When log[CH 2 O] is equal to −2.5 neither cycle is limited by [CH 2 O] since S 1 and S 13 have the same free energy.At these conditions, the chain growth cycle has a TOF that is about 3.5 times higher (1.2 s −1 ) than the CO 2 reduction cycle (0.34 s −1 ).Steady state is reached when this ratio is 0.5 and occurs at log[CH 2 O] of −3.1.At this formaldehyde concentration, the CO 2 reduction cycle is still operating at its limiting TOF of 0.34 s −1 , while the chain growth cycle has slowed to a TOF of 0.17 s −1 .The free energy diagrams for both the CO 2 reduction cycle and the chain growth cycle at these conditions are depicted in Figure S5 in the Supporting Information.

CONCLUSIONS
In summary, we have computationally demonstrated that the electro-organocatalyst designed in our previous work for carrying out electrochemical CO 2 reduction to formaldehyde should also be capable of electrochemically coupling the formaldehyde into long chain aldehydes.The key feature of the catalyst is a vicinal enediamine (>N−C�C−N<) catalytic motif that activates formaldehyde to function as a nucleophile in aldol addition with a second aldehyde while subsequently allowing for efficient electron transfer from a chemically inert cathode along with facile tautomerization.
The mechanism begins with addition of formaldehyde to the vicinal enediamine.The electron-rich π system that results from having two nitrogen atoms adjacent to the C�C bond reverses the polarity of the carbon atom in formaldehyde, allowing it to act as a nucleophile in the subsequent aldol addition condensation step.This is then followed by a sequence of two proton coupled electron transfer (PCET) steps where the electron transfers occur by an outer sphere mechanism from a chemically inert cathode.The PCET mechanism is analogous to the second PCET step in the CO 2 reduction cycle, being facilitated by the ability of the >N + � C−CH�O π system to avoid unfavorable placement of formal charge on either of the carbon atoms; the two electrons instead formally transfer to the nitrogen and oxygen atoms.Tautomerization and dehydration steps are also facilitated by the ability of the electron-rich nitrogen atoms to accommodate formal positive charge.The cycle ends with elimination of an aldehyde that occurs analogously to the initial formaldehyde addition step.
The catalytic cycle is found to give the highest turnover frequency of 9.3 s −1 at a pH of 6.7 and a limiting electrode potential of −0.89 V vs RHE.The optimal pH is determined by a balance between rate limiting protonation and deprotonation steps while the limiting potential is determined by a balance between rate limiting electrochemical and nonelectrochemical steps.The chain growth cycle, which consumes formaldehyde, is found to be kinetically faster than the CO 2 reduction cycle that produces the formaldehyde.Therefore, the optimal pH for coupling the two cycles is 7.8 which is the optimal pH for operating the CO 2 reduction cycle.When both cycles are coupled under steady state conditions, the turnover frequency of the chain growth cycle is predicted to be 0.17 s −1 .
Technical details of the DFT calculations, description of the hybrid solvation method, method for finding electron transfer transition states, corrections to DFT energies, free energy relation for extrapolating proton transfer barriers, details of benchmarking calculations in NWChem for hybrid DFT functionals, free energy contributions of intermediates and transition states, the pH dependence of the total barriers of all other steps in the catalytic cycle, the kinetic dependence of the chain growth cycle on formaldehyde concentration at pH of 7.8, the free energy diagrams for both the CO 2 reduction cycle and the chain growth cycle under optimal coupling conditions, structures and hydrogen bonding free energies of intermediates and transition states, free energy contributions for intermediates and transition states (PDF) ■

Figure 1 .
Figure 1.Free energy diagram of the chain growth cycle from formaldehyde to acetaldehyde.The relative free energy of each intermediate and transition state is labled in eV.The electron transfer steps are represented as vertical steps in the diagram (S 25 → S 25* , S 26* → S 26 ).The upper dashed line indicates the "surface" in the "waterfall" analogy for interpreting the diagram, while the lower dashed line indicates the resting free energy along the reaction path.The four vertical arrows indicate the global barriers associated with the formal rate limiting steps S 12 → S 11 , S 19 → S 20 , S 25* → S 26* , and S 27 → S 28 .Free energies are computed at 80 °C, the catalytically optimal pH of 6.7, the limiting potential of −0.89 V vs RHE, and the limiting formaldehyde concentration of log[CH 2 O] = −2.4.
, one can see that there are four formal rate limiting steps, S 12 → S 11 , S 19 → S 20 , S 25* → S 26* , and S 27 → S 28 .A global barrier of 0.83 eV is associated with these four steps, corresponding to a turnover frequency of 9.3 s −1 .In addition to these, there are three partially rate limiting steps, S 14 → S 13 , S 13 → S 12 , and S 28 → S 29 , that have total barriers within ln 10 × k B T (0.07 eV) of the global barrier.The formal rate limiting steps are associated with four global resting states, S 0 , S 13 , S 24 , and S 27 .Five other states (S 1 , S 14 , S 12 , S 19 , and S 29 ) are close to being global resting states, lying within ln 10 × k B T (0.07 eV) of the resting free energy profile.The four global resting states along with the five states that are nearly global resting states would all be expected to have appreciable steady state concentrations.

Figure 2 .
Figure 2. Transition states for formaldehyde addition to S 1 and deprotonation of S 14 by the PTM (formate) involved in formaldehyde activation (S 1 → S 15 → S 14 → S 13 ).Relevant bond distances are labeled in Å.

Figure 3 .
Figure 3. Transition states for protonation of the C7 position and deprotonation of the C4 position by the PTM (formic acid and formate, repsectively) as occur during tautomerization (S 21 → S 22 → S 24 ).Relevant bond distances are labeled in Å.

Figure 5 .
Figure 5. Transition states for deprotonation of the C6 position (S 19 → S 20 ) by the PTM (formate) and dehydration of the C7 position (S 20 → S 21 ) by a concerted proton shuttling mechanism.Relevant bond distances are labeled in Å.

Figure 6 .
Figure 6.Transition states for protonation of the C5 position by the PTM (formic acid) and subsequent elimination of acetaldehyde.Relevant bond distances are labeled in Å.

Figure 7 .
Figure 7.Total barriers for possible rate limiting steps with respect to electrolyte pH, calculated using [CH 2 O] = 1 mol/L.Protonation steps (S 28 → S 29 ) are indicated by short dashed lines and deprotonation steps (S 12 → S 11 , S 19 → S 20 , and S 27 → S 28 ) are indicated by long dashed lines.The dashed vertical lines indicate the boundaries between the kinetic regimes where different steps are formally rate limiting.The optimal pH (6.7) corresponds to the boundary where the formal rate limiting step shifts from S 27 → S 28 to S 28 → S 29 .

Figure 8 .
Figure 8.Total barriers for the possible rate limiting electrochemical (S 25* → S 26* ) and nonelectrochemical (S 12 → S 11 and S 27 → S 28 ) steps with respect to electrode potential.The limiting potential is indicated by the vertical dashed line and corresponds to the potential where the electrochemical step is on the cusp of becoming formally rate limiting.

Figure 9 .
Figure 9.Total barriers for different steps in the chain growth cycle with respect to log[CH 2 O].(a) The top panel indicates kinetic regimes defined by the kinetic dependence of these steps with respect to [CH 2 O].Changes in the kinetic dependence of a step arise from changes in the resting state of that step as [CH 2 O] varies.(b) The bottom panel indicates kinetic regimes defined by the rate limiting steps, which are S 19 → S 20 at low [CH 2 O] (red) and S 12 → S 11 /S 27 → S 28 at high [CH 2 O] (blue).The limiting log[CH 2 O] is defined as the lowest value at which the TOF is zero order in [CH 2 O].In both panels, dotted lines indicate steps between S 0 and S 14 , short dashed lines indicate the step S 14 → S 13 , medium dashed lines indicate steps between S 13 and S 19 , long dashed lines indicate steps between S 19 and S 21 , and solid lines indicate steps after S 21 (so that longer dashes correspond to later steps in the cycle).The steps are grouped according to the resting state they proceed from, as indicated at the top of the top panel.

Figure 10 .
Figure 10.Limiting global barriers for the CO 2 reduction and chain growth cycles with respect to electrolyte pH.The limiting global barriers correspond to electrode potentials more cathodic than the limiting potential and formaldehyde concentrations below (CO 2 reduction) or above (chain growth) the limiting concentration.
11 become zero order in [CH 2 O], while the aldol addition and dehydration steps between S 11 and S 21 become first order.At log[CH 2 O] of −2.2, S 19 becomes lower in free energy than S 13 so that the resting state for the dehydration steps between S 19 and S 21 shifts from S 13 to S 19 .Consequently, the dehydration steps no longer require transfer of formaldehyde from the bulk electrolyte to form their transition state from the resting state S 19 , eliminating their kinetic dependence on [CH 2 O].Lastly at log[CH 2 O] of −1.5, S 14 becomes lower in free energy than S 0 so that the resting state for S 14 → S 13 shifts from S 0 to S 14 and this step also becomes zero order in [CH 2 O]. Figure 9b examines how the rate limiting step changes with respect to [CH 2 O].At low [CH 2 O], the deprotonation step involved in dehydration, S 19 → S 20 , is rate limiting but switches to S 12 → S 11 and S 27 → S 28 (the deprotonation steps involved in tautomerizations) at high [CH 2 O].In the low [CH 2 O] regime, the rate limiting step (S 19 → S 20 ) requires the transfer of two molecules of formaldehyde from the bulk electrolyte to form its transition state from the resting state S 0 , leading to a second order overall dependence on [CH 2 O].At the formaldehyde concentration increases, the total barrier for this step rapidly decreases until the tautomerization steps (S 12 → S 11 and S 27 → S 28 ) become rate limiting at the limiting log[CH 2 O] of −2.4.Since both of these steps proceed from their respective resting states (S 13 and S 27 ) without involving formaldehyde, they have no kinetic dependence on [CH 2 O] so that the overall kinetics are zero order in formaldehyde with a constant global barrier of 0.83 eV.This defines the limiting value of log[CH 2 O] of −2.4, which represents the lowest formaldehyde concentration before the TOF begins to decrease, analogous to how the limiting potential represents the highest potential before the TOF begins to decrease.Coincidentally, the rate limiting step switches from S 19 → S 20 to S 12 → S 11 at the exact same value of log[CH 2