New Strategies for Direct Methane-to-Methanol Conversion from Active Learning Exploration of 16 Million Catalysts

Abstract

Despite decades of effort, no earth-abundant homogeneous catalysts have been discovered that can selectively oxidize methane to methanol. We exploit active learning to simultaneously optimize methane activation and methanol release calculated with machine learning-accelerated density functional theory in a space of 16 M candidate catalysts including novel macrocycles. By constructing macrocycles from fragments inspired by synthesized compounds, we ensure synthetic realism in our computational search. Our large-scale search reveals that low-spin Fe(II) compounds paired with strong-field (e.g., P or S-coordinating) ligands have among the best energetic tradeoffs between hydrogen atom transfer (HAT) and methanol release. This observation contrasts with prior efforts that have focused on high-spin Fe(II) with weak-field ligands. By decoupling equatorial and axial ligand effects, we determine that negatively charged axial ligands are critical for more rapid release of methanol and that higher-valency metals [i.e., M(III) vs M(II)] are likely to be rate-limited by slow methanol release. With full characterization of barrier heights, we confirm that optimizing for HAT does not lead to large oxo formation barriers. Energetic span analysis reveals designs for an intermediate-spin Mn(II) catalyst and a low-spin Fe(II) catalyst that are predicted to have good turnover frequencies. Our active learning approach to optimize two distinct reaction energies with efficient global optimization is expected to be beneficial for the search of large catalyst spaces where no prior designs have been identified and where linear scaling relationships between reaction energies or barriers may be limited or unknown.

1. Introduction

Direct methane-to-methanol conversion with high selectivity remains a challenge¹ in unlocking natural gas as a feedstock.² Catalysts that can readily convert methane into methanol must be potent enough to activate the strong C–H bonds of methane but also release methanol to facilitate catalyst turnover and avoid catalyst poisoning or methanol overoxidation.^3,4 Thus, the design of optimal catalysts for methane-to-methanol or similarly challenging reactions requires balancing inherent tradeoffs between activity, selectivity, and stability, motivating an exhaustive search over chemical space.⁵ Enzymes with mononuclear Fe active sites (e.g., TauD⁶⁻⁸ or P450^9,10) have demonstrated the capability to perform selective partial oxidation of substrates with strong C–H bonds, including methane. These enzymes have thus motivated the design of ligands¹¹⁻¹⁴ for synthetic systems, including both homogeneous¹⁵⁻¹⁸ and heterogeneous¹⁹⁻²² catalysts. Nevertheless, to date, no earth-abundant molecular catalyst has been identified that meets all criteria.

Compounding the challenges of the need to search a large space, high-valent metal-oxo moieties that are frequently invoked for C–H bond activation²³ in transition-metal complexes,^14,24−28 heterogeneous catalysts,²⁹⁻³⁵ or enzymes^6,7,10 are challenging to isolate and characterize experimentally.³⁶⁻⁴¹ Instead, first-principles computation with density functional theory (DFT) has filled this gap⁴²⁻⁴⁶ in understanding the electronic structure⁴⁷⁻⁵⁰ needed for C–H bond activation. First-principles modeling has revealed the role of spin state in reactivity⁵¹⁻⁵⁵ and the importance of multistate reactivity⁵⁶⁻⁵⁹ that is difficult to study experimentally.⁶⁰ To reduce the computational cost of catalyst screening needed to explore a large chemical space, it is appealing to extend to homogeneous catalysis^61,62 linear free energy relationships (LFERs)^29,63−67 between thermodynamic steps or Brønsted–Evans–Polanyi (BEP)⁶⁸⁻⁷¹ relationships between barrier heights and reaction energies. Nevertheless, in open-shell transition-metal catalysis, the nature of the LFERs or BEPs is rarely known beforehand.⁷² These relationships are also easily disrupted by changes in the metal-local structure⁷³⁻⁷⁵ or spin state.^24,76 Although these disruptions in thermodynamic or kinetic scaling increase the computational cost by requiring full characterization of the catalytic cycle, they simultaneously provide the opportunity for overcoming kinetic or thermodynamic limitations observed in catalysts that obey these scaling relations, potentially providing paths to overcome challenges in direct methane-to-methanol conversion.

Thus far, an Edisonian approach has been unsuccessful in identifying effective molecular catalysts for direct methane-to-methanol conversion, in part because the variations in chemistry that can be explored in a single study are fairly limited (e.g., to Hammett tuning^26,77,78). As an alternative approach, the absence of universal scaling relations between intermediate energetics provides an opportunity for nonlinear machine learning (ML) models that can be used over a larger space. Rather than relying on linear relationships between quantities or small variations in chemistry, ML models can be trained to directly predict catalyst reactivity on the basis of chemical composition and applied to thousands of compounds.⁷⁹ In recent years, this strategy has reduced the time to property prediction to seconds, which would otherwise take days using DFT, and has led to accurate predictions of reaction energetics,⁸⁰⁻⁸³ redox and ionization potentials,⁸⁴⁻⁸⁷ and frontier orbital energetics^88,89 that pave the way for catalyst discovery over large chemical spaces. Importantly, they have demonstrated the limits of conventional descriptor-based screening, highlighting where quantities such as the frontier orbital energies of reactive intermediates are poor predictors of reaction energies.⁸⁰

A key trade-off for the ML approach is that sufficient data must be acquired to make the models predictive. When aiming to search a large space of catalysts or materials, active learning⁹⁰⁻⁹⁴ is preferred as a strategy to acquire data^95,96 where models are most uncertain.^97,98 For example, efficient global optimization^99,100 (EGO) was employed to search a space of 2.8 M redox couples⁸⁴ to reveal design principles in weeks instead of decades. It is thus attractive to exploit similar approaches to search for C–H activation catalysts, given that no known mid-row 3d transition-metal complexes efficiently convert methane into methanol.⁶⁹

In this work, we construct a 16 M compound space of realistic Mn and Fe catalysts with novel tetradentate macrocycles and coordinating axial ligands. We demonstrate that our active learning approach enables the discovery of optimal catalysts in the design space where strong thermodynamic or kinetic scaling relations do not hold. We decouple the roles of axial and equatorial ligands on tuning reaction energetics and determine that Hammett tuning of catalysts has a modest effect relative to optimal macrocycle design or axial ligand selection. The discovered lead compounds provide alternative designs with a novel spin state and ligand chemistry in comparison to known best-in-class catalysts.

2. Reaction Mechanism

We calculate the reaction energies for the radical rebound mechanism¹⁰¹ for methane-to-methanol conversion by mononuclear Mn and Fe catalysts. For these catalysts, we consider two resting state oxidation states, M(II) and M(III), in their corresponding spin states (Supporting Information, Table S1). We do not study Cr and Co catalysts because Cr catalysts form terminal metal-oxo moieties that are too stable and cannot activate C–H bonds,^24,102 while Co metal-oxo intermediates are rarely stable (i.e., are past the “oxo wall”).¹⁰³ From a resting state structure (1), we form a high-valent terminal metal-oxo (2) upon two-electron metal oxidation by nitrous oxide (Figure 1). An alternate oxidant choice (i.e., triplet O₂) will rigidly shift reaction energetics but does not affect relative energetics (Supporting Information, Table S2). We compute the oxo formation energy, ΔE(oxo), as

Figure 1.

Radical rebound mechanism for the partial oxidation of methane to methanol. The cycle proceeds clockwise from the resting state (1) in oxidation state n = II/III to the metal-oxo intermediate (2) formed by two-electron oxidation with N₂O, followed by HAT to form a metal-hydroxo intermediate (3) and rebound to form a metal-bound methanol intermediate (4). A representative catalyst is shown, with the metal (M) shown in brown corresponding to Mn or Fe in this work. All catalysts have a tetradentate equatorial ligand (L₁) and a monodentate axial ligand (L₂). We color the arrows of steps that have been observed to be turnover-determining, with potential turnover-determining transition states (TDTS) in blue and the turnover-determining intermediate (TDI) in red.

Upon oxo formation, the metal formal oxidation state changes from M(II/III) to M(IV/V). The high-valent M(IV/V)=O intermediate then undergoes hydrogen atom transfer (HAT) from a methane substrate to form a M(III/IV)–OH intermediate (3), leaving a methyl radical (Figure 1). We compute the reaction energy for the HAT step, ΔE(HAT), as

Following HAT, the methyl radical rebounds with the M(III/IV)–OH intermediate to form a metal-bound methanol intermediate (4, Figure 1). We compute the ΔE(rebound) step as

The catalyst then returns to its resting state (1) upon methanol release, ΔE(release),

Although both oxo formation^73,104 and HAT^105,106 can be turnover-determining, methanol release is believed to be a universal thermodynamic sink (i.e., its release is rate-limiting) in the radical rebound mechanism.^24,102

3. Design Space and Objectives

Molecular complexes that have been studied for C–H activation typically consist of Fe(II) centers coordinated by nitrogen atoms in analogy to enzymes.^11,107,108 This means many chemical environments, such as those with O, P, and S coordinating atoms, have not been thoroughly examined for C–H activation, motivating their inclusion in a wider search. To create an expanded space of macrocycles that are likely to be synthetically accessible, we design realistic ligands by recombining fragments of known ligands into new tetradentate macrocycles, ultimately producing spaces as large as 16 M catalysts. From this space, we aim to discover ligand chemistry that optimally tunes catalyst energetics to reach the “utopia point”¹⁰⁹ where a catalyst will simultaneously activate methane and release methanol.

As an example of our fragment-based approach, the pyrrole subunits that comprise a porphyrin and the dimethylamine subunits that comprise a cyclam can be combined to produce new macrocycles (Figure 2). We also include 3p equivalents of more well studied 2p fragments (e.g., P-coordinating phosphole in analogy to N-coordinating pyrrole, Figure 2). Many of these fragments have been a part of synthesized macrocycles^110,111 but not necessarily those that have been studied for C–H activation. In select cases, fragments may have multiple accessible charge states (e.g., pyrrole and phosphole), and in those cases, we consider all possibilities to construct macrocycles (Supporting Information, Figures S1 and S2). We join fragments together with compatible bridge atoms that determine the macrocycle ring size and aromaticity (Supporting Information, Tables S3 and S4). These combinations result in 16,986 candidate tetradentate macrocycles of varying coordinating atom identities, size, ring size, charge, and aromaticity (Supporting Information, Table S4 and Figure S3).

Figure 2.

15 fragments (top) and 9 bridges (middle) that are used to construct tetradentate macrocycles in transition-metal complexes for light alkane oxidation. All metal-coordinating atoms are highlighted by gray circles, and X is used to indicate the possibility of multiple metal-coordinating element types. Two fragment types in cis orientation are combined with up to three distinct bridges to construct a macrocycle. (bottom) An example macrocycle is shown that is constructed from two dimethylamine fragments and two pyrrole fragments joined via one phosphorus bridge, one methylene bridge, and two bridges with no additional atoms.

Our ligand design strategy reproduces existing chemistry such as porphyrins, corroles, and cyclams but also extends beyond these compounds to other candidate macrocycles (Supporting Information, Figure S4). Quantitatively, comparison of our database to tetradentate ligands present in the Cambridge Structural Database¹¹² (CSD) demonstrates that our candidate ligand space reproduces existing local chemical environments and introduces new environments that are yet to be reported in the experimental literature (Supporting Information, Figure S5). We combine these tetradentate macrocycles with eight representative neutral and anionic axial ligands and nine metal oxidation- and spin-state combinations [i.e., M(II/III) where M = Mn or Fe in LS, IS, and HS states] to create an initial design space of over 1.2 M candidate complexes (Supporting Information, Figure S6). The addition of functional groups to these macrocycles leads to a space of 16 M candidate complexes. For complexes that have distinct connectivity but are duplicates in revised autocorrelation⁸⁵ (RAC) descriptor space (i.e., are isomers), we systematically retain one case, after controlled studies that demonstrated only minor differences in catalyst thermodynamics between the two isomers (Supporting Information, Figure S7).

We use EGO¹¹³ with a 2D expected improvement (2D-EI) criterion¹⁰⁰ to simultaneously optimize ΔE(HAT) and ΔE(release) because a catalyst must simultaneously be able to activate methane and release methanol to prevent overoxidation.^3,5 We also chose these two objectives to optimize because the existence of moderate scaling between ΔE(oxo) and ΔE(HAT) within a single metal/oxidation state suggests²⁴ greater opportunities to independently optimize HAT and methanol release rather than HAT and oxo formation. Although we optimized these two reaction energies with EGO, we computed all four reaction steps when generating DFT data (see Section 2 and the Supporting Information). The ML-predicted ΔE(HAT) and ΔE(release) values for a new complex, x, are determined as

where μ̂_ΔE(HAT) and μ̂_ΔE(release) are the ML-predicted mean values and σ̂_ΔE(HAT)² and σ̂_ΔE(release)² are the effective variances from the model uncertainty.¹⁰⁰ The protocol we employ largely follows prior work,⁸⁴ where we use artificial neural networks (ANNs) with a calibrated latent space uncertainty metric^84,97 as our surrogate models, and we employ k-medoids sampling at each generation to generate new DFT data and retrain our models (Supporting Information, Figure S8 and Text S1).⁸⁴

4. Results and Discussion

4.1. Design Outcomes in Initial Macrocycle Space

To start our exploration of the catalyst space, we performed k-medoids sampling over the 1.2 M compound design space to generate an initial set of reaction energies for ML model training (Supporting Information, Figure S9). We then trained independent ANN models to predict ΔE(HAT) and ΔE(release) (Supporting Information, Figure S10). For the first generation (i.e., generation 0), we collected 516 pairs of ΔE(HAT) and ΔE(release) reaction energies and used them to train ANNs with RACs as input features (Supporting Information, Tables S5–S7). Although EGO is typically carried out with Gaussian process (GP) models, we selected independent ANNs because they have superior performance on the generation 0 test data (Supporting Information, Table S7 and Figure S11). As in prior work,⁸⁴ we used the 10-nearest-neighbor ANN latent space distances⁹⁷ to training data as an uncertainty quantification (UQ) metric (Supporting Information, Figure S12). The latent space distance provides an intuitive measure of UQ and brings to an ANN-based approach the benefits of realistic uncertainties usually associated with GPs.

We then applied the trained ANN models to predict reaction energies for the initial 1.2 M compound design space and select new catalysts for DFT characterization based on their 2D-EI scores (Supporting Information, Table S8). This allows us to improve ANN model performance at the Pareto front while discovering new Pareto-optimal catalysts. At each generation, we selected the top 10,000 catalysts by their E[I] scores and performed k-medoids sampling to initiate 400 representative metal-oxo DFT calculations for model retraining in subsequent generations (Supporting Information, Table S9 and Figures S13 and S14). To quantify model improvement by generation, we use lookahead errors⁸⁴ that assess the ability of models to predict properties for subsequent generations. Over three full generations of 2D-EI, ΔE(HAT) lookahead errors reduce threefold (i.e., from 15 to 5 kcal/mol), and ΔE(release) lookahead errors are reduced twofold (i.e., from 8 to 4 kcal/mol). Model errors on set-aside test data at each generation also improve to a lesser degree (Figure 3 and Supporting Information, Figures S15 and S16). We thus stopped the search for Pareto-optimal catalysts after three generations, as lookahead errors approach test set errors by this point and average E[I] scores decrease markedly (Figure 3 and Supporting Information, Figure S17).

Figure 3.

(left) E[I] over the initial ligand space as predicted by ANN models during generation 0 (top) and generation 1 (bottom). The Pareto front after each generation is shown in gray. (right) Lookahead and test set mean absolute errors (MAEs) for ΔE(HAT) (top, in kcal/mol) and ΔE(release) (bottom) for the two single-task ANNs. Each bar is colored by the generation at which it is trained, as indicated in the top inset. Lookahead MAEs are reported on data sets (1–3, as indicated on axis) generated in each relevant subsequent generation. The MAEs on a representative test set for each generation are reported.

Studies of bio-inspired Fe(II) catalysts have typically targeted high-spin (HS) Fe(II) because most enzymes capable of C–H activation are believed to be HS Fe(II) stabilized by weak field N- and O-coordinating species. At odds with these expectations, we observe Fe(II) compounds in low spin (LS) states to have the best tradeoff between ΔE(HAT) and ΔE(release) on the Pareto front, having both the most favorable HAT thermodynamics and not binding methanol too tightly (Figure 4). This observation was made possible by the EGO exploration despite the fact that our initial sampling of the design space primarily favored intermediate-spin (IS) Mn(II) compounds (Figure 4 and Supporting Information, Figures S18 and S19). After three generations of EGO, only a single IS Mn(II) catalyst is Pareto optimal and Fe(II) catalysts in LS states with weak-field oxygen coordinating (e.g., dimethylether, 4H-pyran, and furan) fragments occupy the majority of the Pareto front, although these are generally not in their ground spin states (Figure 4 and Supporting Information, Table S10). The relative position of a catalyst on the Pareto front is governed by the equatorial ligand field strength: stronger ligand fields with donating equatorial ligands have relatively better ΔE(release) in comparison to ΔE(HAT). The compounds that have the most favorable HAT thermodynamics at the cost of binding methanol more tightly primarily contain oxygen-coordinating macrocycles, whereas those that release methanol more readily but do not favor HAT contain 3p-coordinating macrocycles (Figure 4 and Supporting Information, Figure S18). Indeed, exothermic HAT thermodynamics with a methane substrate have been observed on HS Fe(II) catalysts⁷³ with weak-field oxygen-coordinating ligands and on LS Fe(II) SACs with O, N, or S coordination.¹¹⁴ Similarly, HAT thermodynamics from larger substrates (e.g., ethane, propylene, or 2-propanol) are near thermoneutral⁴⁵ or negative³⁵ for HS Fe(II) metal–organic framework catalysts. Over this initial 1.2 M catalyst design space, EGO reveals ligands that have been understudied (e.g., 3p-coordinating ligands) combined with LS Fe(II) will lead to catalysts with a good balance between ΔE(HAT) and ΔE(release), as long as the LS state is the ground state for the complex (see Section 4.3).

Figure 4.

(top) Pareto-optimal compounds from the initial ligand space simulated during three generations of the design algorithm shown in ball-and-stick representation, with Fe in brown, Mn in purple, C in gray, N in blue, O in red, P in orange, and S in yellow. (middle) Fragments that make up Pareto-optimal compounds in the initial ligand space. (bottom) Compounds simulated during three generations of the design algorithm, colored by generation and with unique symbols for each metal center (as indicated in inset legend). The range of values sampled in each generation is indicated by a convex hull. For generation 2, four outlier points that expand the convex hull are truncated from the plot. A final Pareto front is indicated by letters A–F.

4.2. Exploring Hammett Tuning Effects on Macrocycles

A frequently pursued synthetic approach to fine-tune catalyst reaction energetics is to functionalize macrocycles with electron-withdrawing or -donating groups. To evaluate this strategy, we performed controlled studies on functionalized porphyrins, which suggested the possibility to tune macrocycle energetics by 5–15 kcal/mol (Supporting Information, Figure S20). Due to our observation of strong variations in reaction energies by functional group tuning, we take a two-pronged approach, where we first consider all functionalized variants of the initial 1.2 M macrocycles and we next use energetic cutoffs to limit the set of compounds to functionalize (see Section 4.3). We perform this second parallel search due to observations that stronger equatorial ligand fields have a more balanced tradeoff of HAT and release energetics. We thus first functionalized the initial set of 1.2 M macrocycles with common functional groups used in Hammett tuning, enlarging our catalyst space to 16 M compounds (Figure 5). We functionalize all C–H bonds on bridges and C–H bonds on select fragments (Figure 5 and Supporting Information, Figure S21). This approach reproduces well-studied macrocyclic ligands, such as phthalocyanine or tetraphenylporphyrin but also introduces new chemistry. Functionalization of porphyrins has different effects on ΔE(HAT) and ΔE(release) (Supporting Information, Figures S22 and S23). Thus, functional group addition can be expected to alter the set of catalysts at the Pareto front and change the identity of the tetradentate macrocycles that comprise them.

Figure 5.

Functional groups used to perform Hammett tuning on macrocycles, ranging from electron-donating groups to electron-withdrawing groups. C–H bonds on select fragments and bridges (left) can be functionalized, while remaining fragments and bridges (right) are not considered for functionalization.

Although functional group addition tunes energetics of characterized reaction steps by 5–15 kcal/mol for porphyrins, these functional groups are distant from the metal center on the molecular graph. The RAC featurization we use has a default cutoff of correlating atoms only three bond paths (i.e., d = 3) apart. Thus, we revisited our models and examined the potential benefit of using higher-depth (d = 4) RACs to ensure we adequately capture changes in functional groups (Supporting Information, Figure S24). As a test of whether or not it is beneficial to utilize higher-depth RACs, we sampled the 16 M compound space of functionalized catalysts with k-medoids sampling to select 1800 new data points and trained new ML models in generation 4 using the higher-depth RACs (Supporting Information, Tables S11 and S12 and Figure S25). The revised RAC representation improves model accuracy, reducing test set MAEs for both HAT and release to below 5 kcal/mol. This performance is superior to the best-performing models we obtained using the original RACs on data for functionalized macrocycles (Supporting Information, Figure S25).

To attempt to find functionalized compounds better than those along the Pareto front of unfunctionalized catalysts, we carried out additional generations of EGO in the 16 M compound functionalized macrocycle space. Because the functional groups expand our space 10-fold, we increased our selection of catalysts to the top 100,000 (i.e., as judged by E[I] scores) over which we carried out k-medoids sampling (Supporting Information, Figure S26). Over this expanded search in the 16 M compound space, within three additional generations lookahead errors again reduce to test set errors and E[I] values reduce significantly (Supporting Information, Figures S27 and S28). For the final Pareto set, four out of the five Pareto-optimal catalysts are those obtained after functionalization, while the one catalyst from the initial space contained no sites compatible with functionalization (Figures 4 and 6). The functional groups present in the resulting set of catalysts range from electron donating (e.g., methyl and phenyl) to electron withdrawing (e.g., cyano and fluoro) on N or O coordinating equatorial ligands (Figure 6). Three out of the four functionalized Pareto-optimal catalysts have functional groups on bridge C–H bonds, as opposed to on fragments, suggesting the greater importance of steric bulk in our functionalization strategy in comparison to through-bond, electronic effects. For instance, functionalizing a bridge carbon with two phenyl groups promotes distortion of the metal-coordinating nitrogen atoms (Figure 6 and Supporting Information, Figure S29). We observe that functional groups do indeed shift catalysts to the Pareto front, as none of the optimal functionalized macrocycles had their unfunctionalized forms on the Pareto front of the initial design space.

Figure 6.

(top) Fragments that make up Pareto-optimal compounds in the global functionalized ligand space simulated during three additional generations of the design algorithm after introducing functional groups. Functional groups are highlighted in cyan boxes, and metal-coordinating atoms are indicated by translucent circles. (bottom) Compounds simulated during three additional generations of the design algorithm after introducing functional groups, colored by generation and with unique symbols for each metal center (as indicated in inset legend). The Pareto front prior to the introduction of functional groups is shown as a black dotted line. A final Pareto front is shown as a solid black line. New functionalized compounds that reach the Pareto front are indicated by letters A–D. A black dashed line indicates a value of 0 for ΔE(HAT).

4.3. Identifying Design Principles for Catalysts with Optimal Reaction Energetics

Good catalysts for methane-to-methanol conversion should have a balanced tradeoff between ΔE(HAT) and ΔE(release) while also being in their ground spin states, so that their resting states are the reactive states.¹¹⁵ However, the 2D-EI criterion treats advances in any direction beyond the current Pareto front as equally important, although many such directions are catalytically unimportant because they do not satisfy this balance. Thus, in the second prong of our approach, we applied relative energy cutoffs of ΔE(HAT) < 10 kcal/mol and ΔE(release) < 30 kcal/mol, as predicted by the generation 3 ML models (e.g., the optimal models for catalyst skeletons prior to functional groups) to identify the catalyst designs most likely to fulfill this balance between ΔE(HAT) and ΔE(release) in our search space. This reduces the theoretical space of catalysts to 30,095 compounds that represent 2.5% of the initial design space (Figure 7 and Supporting Information, Figure S31). Small changes to the cutoff values do not alter conclusions about the compounds favored in this the space (Figure 7 and Supporting Information, Figure S32). Examining the generation 3 Pareto-optimal compounds, we observe that these catalysts have 15- or 16-membered rings (i.e., the same as corroles, porphyrins, and phthalocyanines). Within our energetic cutoffs, however, the distribution of macrocycle ring sizes for compounds resembles that of the design space (Supporting Information, Table S10 and Figure S33). A clearer preference is established for the axial ligand: we observe a strong preference for anionic axial ligands with 3p coordinating atoms within the energetic cutoffs (Figure 7). The selection of these anionic axial ligands appears to have an overriding effect on energetics with respect to the equatorial macrocycle ligand chemistry (Figure 7). A focused search starting from these catalyst scaffolds is motivated because compounds that are within our balanced energetic zone are unlikely to be prioritized by the global search as a result of lower E[I] scores relative to compounds with thermodynamically favorable ΔE(HAT) and highly endothermic ΔE(release) (i.e., with generation 4 models) (Supporting Information, Figure S30).

Figure 7.

(top) Axial ligands present within our energetic cutoff zone of 30,095 catalysts, colored by their metal-coordinating atom with oxygen in red, nitrogen in blue, sulfur in yellow, and phosphorus in orange. (bottom) Absolute frequencies of fragments that comprise equatorial ligands for catalysts in the cutoff zone, with chord thickness representing increased frequency of the pairs of fragments. The circles containing the fragments are colored by the metal-coordinating atom identity, with carbon in gray, oxygen in red, nitrogen in blue, sulfur in yellow, and phosphorus in orange.

The source of this preference for anionic axial ligands in good catalyst designs could either be due to the lower ligand field strength or alteration of the charge on the metal center. To identify changes in metal charge, we computed the partial charges for resting state catalysts from generations 0 to 3 and grouped them by axial ligand type (Figure 8 and Supporting Information, Figure S34). Indeed, compounds with axial anionic ligands have metal centers with a lower partial charge (i.e., are more neutral) due to increased charge transfer from the axial ligand. A series of one-tailed Welch’s t-tests¹¹⁶ confirms that for all coordinating atom identities in our set (i.e., N, O, P, or S), compounds with negative axial ligands have less oxidized metal charges than compounds with neutral axial ligands at a 5% significance level (Supporting Information, Table S13). This lower positive partial charge on the metal should be expected to reduce electrostatic attraction to methanol, favoring its release. The axial ligand has a larger effect on release energetics than it does on HAT, and we observe that ΔE(HAT) distributions are unchanged between neutral and anionic axial ligands (Supporting Information, Figure S35). Analysis of the metal-oxo HOMO level, which has been used as a descriptor for ΔE(HAT) reactivity,¹⁰⁶ reveals it to be more sensitive to equatorial ligand charge than to axial ligand charge (Supporting Information, Figure S36). The lack of anionic axial atoms in best-in-class ligands such as Me₃NTB¹¹⁷ {Me₃NTB = tris-[(N-methyl-benzimidazol-2-yl)methyl]amine} and TQA¹³ [TQA = tris(2-quinolylmethyl)amine] highlights the need to incorporate negatively charged axial atoms as a ligand design principle. Furthermore, the preference for 3p coordinating atoms over 2p coordinating atoms emphasizes design opportunities over the underexplored chemical space of strong-field, anionic axial ligands.

Figure 8.

Metal Hirshfeld charge (in units of e) for the resting states of catalysts from generations 0 to 3 with different axial ligand coordinating atoms (N, O, P, and S from top to bottom). Axial ligands are categorized into anionic (green) and neutral (blue) forms. The average metal Hirshfeld charge for complexes with anionic and neutral axial ligands is shown as green and blue lines, respectively. A dotted black line indicates a metal Hirshfeld charge of 0.

As in the first search strategy, we use Hammett tuning to design improved catalysts. However, here, we functionalize only the subset of compounds in our energetic zone to ask if we can find catalysts with balanced ΔE(HAT) and ΔE(release) that were overlooked during our global search. Adding functional groups to the subset of catalysts that fall within our energetic cutoffs increases the number of compounds to over 412,000. Over this subspace of compounds, we are able to train models that have even smaller test set errors (<4 kcal/mol) than when we studied the full space, enabling us to predict the effects of functional group tuning more robustly (Supporting Information, Table S14 and Figures S37–S39). Over three additional generations of EGO using models trained only on the set of compounds that fall within our energy window and their functionalized variants, we find two Pareto-optimal catalysts that outperform any prior designs. These two catalysts are also both in their ground spin state. One catalyst has a LS Fe(II) center with strong-field phosphole and trimethylphosphine fragments and trifluoromethyl functional groups (Supporting Information, Figure S39 and Table S15). Although 3p-coordinating ligands may be sensitive to oxidation, they have been used for comparable reactions such as oxidative C–H bond insertions.¹¹⁸ The other complex consists of an IS Mn(II) center with a weak-field equatorial macrocycle that is constructed from flexible oxygen fragments and amino-functionalized bridges (Supporting Information, Figure S39 and Table S15). For this compound, the metal-oxo bond in the oxo intermediate has a strong tilt, which has been hypothesized to promote increased reactivity for Fe(IV)=O complexes⁴¹ (Supporting Information, Figure S40).

For both of the strategies of our two-pronged approach, we observe more modest changes in the composition of the Pareto front than originally anticipated based the magnitude of functional group effects on the reaction energetics for modified porphyrins. Because Hammett tuning is predicated on the electron-donating and -withdrawing nature of functional groups, we hypothesize that the effects of functional groups were likely smaller because the design space was enriched with nonaromatic compounds. To determine the relative percentage of aromatic compounds in our focused design space, we estimated aromaticity in the equatorial macrocycles from their canonical SMILES strings (Supporting Information, Figure S41). Indeed, almost all (97%) compounds that are within the cutoff zone (i.e., the 30,095 compounds) have little-to-no aromaticity. This suggests that within our set of macrocycles nonaromatic compounds exhibit better reaction energetics than their aromatic counterparts despite the fact that porphyrinoid compounds¹¹⁹ are frequently studied for C–H activation. Although some aromatic macrocycles fall within the cutoff zone, the best cases still have ΔE(HAT) and ΔE(release) energetics that are predicted by the ML models to be far (i.e., 8–10 kcal/mol) from the Pareto front in comparison to nonaromatic compounds (Supporting Information, Figure S42). Thus, functional group tuning can be expected to alter the energetics of aromatic macrocycles by a significant 5–15 kcal/mol margin, but most are too far from the Pareto front to surpass nonaromatic compounds after functional group tuning. These observations further strengthen the case for ML-accelerated search of a wide macrocycle space rather than a focus on functional group tuning within a fixed macrocycle structure.

4.4. Catalytic Cycles of Pareto-optimal Catalysts

To validate our best-case catalyst designs, we completed the radical rebound catalytic cycle by computing additional reaction energies and barrier heights for all Pareto-optimal catalysts that were identified by our two-pronged search. Here, we focus our analysis on the two catalysts identified in our search that have the catalytically active spin state as the ground state. Because we optimized the catalysts for HAT and methanol release thermodynamics, we already know the reaction energetics of these steps are favorable for HAT, and near thermoneutral for release. For HAT, a strong BEP relation means that favorable reaction energetics also correspond to favorable HAT kinetics, forgoing the need for a HAT transition state (TS) calculation. For methanol release, we model it as an unassisted dissociation and so we neglect any kinetic barrier. Thus, to complete the catalytic cycle, we next computed properties related to oxo formation and radical rebound, which require explicit calculations of the TSs. We obtained oxo formation barrier heights and approximate TSs [i.e., with nudged elastic band (NEB), see Computational Details] for the Pareto-optimal catalysts (Figure 9 and Supporting Information, Table S16 and Figure S43). We also computed the kinetic barrier for the rebound step using potential energy scans in which the distance between the methyl radical and oxygen atom was constrained and all other degrees of freedom were relaxed (see Computational Details).

Figure 9.

Full energy landscape of the two ground state Pareto-optimal complexes with the Mn(II) lead complex (top), and Fe(II) lead complex (bottom) in green and red, respectively. We draw the reaction coordinate from reactants (R) to products (P) through a metal–N₂O bound intermediate, the oxo formation TS (TS1), the metal-oxo intermediate (=O), the HAT TS (TS2), the metal-hydroxyl intermediate (−OH), the rebound TS (TS3), and the methanol-bound intermediate. The turnover-determining TS (TDTS), and turnover-determining intermediate (TDI) are shown inset, along with the energy span (δE) that governs efficient catalysis. HAT TSs were found to be barrierless and are omitted, with neighboring steps connected by a dotted line.

Although strong BEP relations between reaction energies and barrier heights have been invoked,^68,71 we observe these to only hold for HAT and not for oxo formation, consistent with prior work^68,73,120 (Supporting Information, Figures S44 and S45). Because thermodynamic scaling between oxo formation and HAT can be disrupted in molecular complexes,²⁴ we chose to use EGO to optimize our catalysts for HAT and methanol release with the expectation that oxo formation could still be favorable. Therefore, we explicitly evaluate oxo formation kinetics for the nine Pareto-optimal catalysts. Indeed, all nine Pareto-optimal catalysts, two of which are in their ground spin states, have favorable oxo formation reaction energetics relative to the resting state as well (Supporting Information, Table S16). We calculate a low barrier height (i.e., 5 kcal/mol) for oxo formation with our Pareto-optimal IS Mn(II) catalyst that has a weak-field oxygen-coordinating macrocycle (Figure 9 and Supporting Information, Table S16 and Figure S43). This low barrier height is correlated to the metal-oxo tilt we observed in the Mn(IV)=O intermediate (Supporting Information, Figure S40). The oxo formation barrier for the Pareto-optimal ground state LS Fe(II) catalyst with strong-field ligands is significantly higher (i.e., 25 kcal/mol) but still lower than what we had previously observed (>30 kcal/mol) for Fe(II) model catalysts (Supporting Information, Figure S43).

Finally, we observe that the radical rebound step that completes the catalytic cycle is nearly barrierless in the Pareto-optimal catalysts and thus cannot govern turnover for these catalysts (Supporting Information, Figure S46). For the Pareto-optimal IS Mn(II) catalyst, we find that the more stable oxo corresponds to a higher (e.g., 10 kcal/mol) rebound barrier height, although this step still does not become a rate-determining step (Figure 9 and Supporting Information, Table S16 and Figure S46).

To identify the best catalyst of our two ground state Pareto-optimal catalysts, we use the energetic span model¹²¹ to approximate catalyst turnover frequencies. In all nine Pareto-optimal catalysts, oxo formation is the TDTS, and the methanol-bound intermediate is the TDI. Despite the moderate kinetic barriers for oxo formation from N₂O for all Pareto-optimal catalysts, the improvements for methanol release thermodynamics relative to oxo formation kinetics make the expected catalytic performance of these identified complexes better than any HS Fe^IV=O catalysts that we have previously investigated⁷³ (Supporting Information, Figure S43). Comparison of these systems to the model catalytic systems from prior work⁷³ indicates that one of Pareto-optimal catalysts reduces the energy span by 15 kcal/mol, which corresponds to 11 orders of magnitude increase in computed catalyst turnover frequency, while the other resembles the best-performing minimal model from prior work (Supporting Information, Table S17 and Figure S47). While higher levels of theory or explicit experiments would be necessary to validate such a large predicted increase in turnover frequency, we anticipate that this evidence of improvement is beyond the uncertainty from DFT functional choice and basis set.

Additionally, we find that the relative energetics and energy spans for methane oxidation by our Pareto-optimal catalysts are comparable to the computed energy spans reported for multimetallic metal–organic framework nodes^45,46 (Supporting Information, Table S18). This comparison is only qualitative due to the sensitivity of the TDTS and TDI energetics to functional choice.¹²² The lowered oxo formation activation barriers and improved thermodynamics for methanol release energetics that correspond to smaller energy spans for our macrocyclic catalysts demonstrate that our ML-accelerated EGO strategy can uncover new and more efficient catalysts. One limitation of studying all spin states is that the discovered Pareto-optimal catalysts are not necessarily in their ground spin state, as we found in our global search. This may have prevented us from finding more realistic lead candidate catalysts in nearby regions of chemical space. Another challenge is that the ground spin state may differ across the catalytic cycle (i.e., the reaction may not be spin-conserved). Current catalyst designs focus on the orthogonal tuning of equatorial and axial ligands for improved reaction energetics. Future catalyst studies will focus on designing pentadentate scaffolds that can simultaneously incorporate effects of increased out-of-plane distortion that can further reduce oxo formation barriers and promote ground state reactivity with a fully spin-conserved reaction coordinate. Subsequent catalyst design efforts will also quantify the effect of oxidant (i.e., N₂O vs O₂) choice on barrier heights, which can affect conclusions on catalyst design when considering the TDTS.

5. Conclusions

Catalyst design requires consideration of tradeoffs between different steps of the catalyst energy landscape that are difficult to optimize by trial and error alone. As LFERs and BEP relations seldom hold strongly in single-site catalysis, these trade-offs are not readily captured by descriptor-based screening. While this observation complicates screening, it suggests further opportunities to overcome present limitations in the design of active methane-to-methanol catalysts. To overcome the limitations of prior approaches, we used multi-objective EGO to optimize reaction energies for two steps in the radical rebound mechanism for direct methane-to-methanol conversion. We constructed a nearly 2 M compound space of catalysts comprising oft-studied Mn and Fe centers combined with equatorial ligands constructed from fragments inspired by synthesized macrocycles. By using EGO in combination with iteratively retrained ANN models, we identified that novel low-spin Fe(II) compounds were often paired with strong-field (e.g., P or S-coordinating) axial ligands that differed from more commonly studied HS Fe(II) catalysts with weak-field ligands.

To mimic Hammett tuning commonly employed in catalyst screening, we added functional groups to fragments and bridges of our macrocycles, thereby expanding our candidate space to nearly 16 M compounds. Because the most favorable macrocycles lack aromaticity, the improvement of reaction energetics achieved through functional group addition was smaller than improvement achieved by changing the macrocycle or axial ligand. Because a global search prioritized catalysts with unbalanced HAT and release, we enforced a balance between these two objectives by selecting a subset of catalyst scaffolds prior to functionalization. Over this set, we both improved the model performance and observed that the best tradeoff between HAT and methanol release occurred when catalysts had negatively charged axial ligands that correlated to facile methanol release.

Finally, we computed the kinetic barriers alongside reaction thermodynamics for the radical rebound mechanism in methane-to-methanol conversion of the Pareto-optimal catalysts. As EGO was used to optimize catalysts for HAT and methanol release, we focused on determining if this led to any deleterious effect on oxo formation or radical rebound. Analysis of these steps revealed that all Pareto-optimal catalysts form metal-oxos favorably and have modest barrier heights for oxo formation (here, with N₂O as the oxidant) relative to previously studied catalysts. At the same time, the radical rebound step is never rate-determining. Thus, our catalyst screening strategy captured the key steps to optimize for this reaction mechanism. Energetic span analysis on the two lead compounds in their ground state spin, an IS Mn(II) catalyst and a LS Fe(II) catalyst, revealed both had favorable energetics that would lead to reasonable turnover frequencies. Our 2D-EI approach applied to in silico synthesized macrocycles represents a promising strategy for rapidly optimizing decoupled reaction steps and is expected to be general to other reaction systems in homogeneous catalysis. The search strategy could potentially be improved by requiring that a catalyst be in its ground spin state or that the entire catalytic cycle is spin-conserved, for example, by incorporating ground state classification into the optimization algorithm. Another area of future research is to alter the acquisition function to better focus on catalytically relevant regions of the Pareto front instead of exploring unproductive regions of chemical space. Lastly, future efforts will focus on the role of the terminal oxidant in reducing oxo formation barrier heights, which can dictate catalyst turnover frequencies.

6. Methods

Gas-phase geometry optimizations and single-point energy calculations were performed using DFT with a development version of TeraChem v1.9.¹²³ The B3LYP^124−126 global hybrid functional with the empirical D3 dispersion correction¹²⁷ using Becke–Johnson damping¹²⁸ was employed for all calculations. The LACVP* composite basis set was employed throughout this work, which consists of a LANL2DZ effective core potential^129,130 for Mn, Fe, Br, and I and the 6-31G* basis¹³¹ for all other atoms. As in prior work, we focus on relative energetics over a large data set, and we neglect solvent corrections and zero-point vibrational energy or entropic corrections to avoid a significant increase in the computational cost.²⁴

Singlet calculations were carried out in a spin-restricted formalism following prior work,²⁴ whereas all other spin states were performed as unrestricted calculations. The convention of majority-spin addition of radicals is employed throughout. Level shifting¹³² of 0.25 Ha was applied to both majority- and minority-spin virtual orbitals to aid self-consistent field convergence to an unrestricted solution. Geometry optimizations were carried out with the translation rotation internal coordinate optimizer¹³³ using the L-BFGS algorithm. Default tolerances in the convergence criteria were employed for the maximum energy gradient of 4.5 × 10^–4 hartree/bohr and the energy difference between steps of 10^–6 hartree. The initial geometries for metal-oxo species were constructed using molSimplify,¹³⁴ which uses OpenBabel^135,136 as a backend to interpret SMILES strings. Tetradentate macrocycle SMILES strings were constructed using custom algorithms available in molSimplify and validated using RDKit version 2020.03.2.²⁶ We oriented any methyl groups on metal-coordinating atoms syn relative to the metal-oxo to follow the most common isomer in experimentally characterized metal-oxo compounds.^137−139

Job submission was automated by molSimplify with a 24 h wall time per run with up to five resubmissions. Geometry optimizations were carried out with geometry checks¹⁴⁰ prior to each resubmission and structures that failed any check were eliminated (Supporting Information, Table S19). Open-shell structures were also removed from the data set following established protocols^80,88,140 if the expectation value of the S² operator deviated from its expected value of S(S + 1) by >1 μ_B² or the combined Mulliken spin density on the metal and oxygen differed from the spin multiplicity by >1 μ_B. We employed an ML strategy to predict calculation failure from the electronic structure¹⁴⁰ using a multitask neural network classifier¹⁴¹ applied up to the first forty steps of the geometry optimization (Supporting Information, Tables S19–S21 and Figure S48). We used this classifier to terminate metal-oxo calculations that were confidently predicted to be unproductive¹⁴² (Supporting Information, Table S22). After a successful metal-oxo calculation, all other intermediates were calculated without using the dynamic classifier to terminate calculations.

In addition to metal-oxo intermediates, other radical rebound intermediates were generated in the following sequence. All metal-hydroxo geometries were generated by adding an H atom to the optimized metal-oxo structure, and all methanol-bound intermediates were generated by adding a methyl group to the optimized metal-hydroxo structures using a custom script in molSimplify, as in prior work²⁴ (Supporting Information, Figures S49 and S50). Resting state catalyst structures were obtained as single-point energies after the removal of the methanol molecule from methanol-bound intermediates. The workflow starts by optimizing the metal-oxo geometry, and if this or a subsequent intermediate does not succeed, downstream intermediate optimizations are not attempted.

Approximate TSs for N₂O activation were modeled with the NEB method with climbing image^143,144 as implemented in the TeraChem^123,145 interface to DL-FIND.¹⁴⁶ Approximate TSs for the radical rebound step were obtained via a series of constrained optimizations in which the metal-oxo oxygen and methyl radical carbon distance was scanned from 2.6 to 1.4 Å in 0.1 Å increments while letting all other atoms relax.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacsau.2c00176.

• Structures for all reactive intermediates for all generations; structures for approximate TSs; raw electronic energies and reaction energies for all catalysts in all generations; and train-validation-test data splits at each generation (ZIP)

• Spin and oxidation state definitions; reference energetics for all small molecules used for reaction energies; criteria used for geometry and electronic structure checks; dynamic classification latent space entropy cutoffs and comparisons; details of workflows for reactive intermediate functionalization; examples of macrocycle construction; macrocycles eliminated due to repeats; counts of ring sizes in macrocycles; statistics on macrocycles with different coordinating atoms; duplicate complexes in the RAC space; comparisons of hypothetical compounds to synthesized compounds; visual and mathematical explanation of 2D-expected improvement; relationship between HAT and methanol release energetics; failure rate statistics and k-medoids sampling for generation 0; hyperparameters for all ANN models; comparison between ANN and GP model performance; timing for one full generation of EGO; demonstration of EI by generation; train-validation-test set splits; ANN performance by generation; evolution of Pareto optimal catalysts over time; rules for functionalizing macrocycles on fragments and bridges; analyses on functionalization effects on metal charges and HOMO levels; SHAP analysis on ANNs with functionalized macrocycles; changes in chemical space diversity while subsampling ligand space; statistical analyses of metal Hirshfeld charges; effects of net charge on frontier orbital energies; analyses of spin splitting energies for lead complexes; identification of aromaticity via SMILES; verification of BEP principle for oxo formation and HAT; full energy landscapes for Pareto optimal catalysts; TS structures for N₂O activation; kinetic barriers for radical rebound; and comparison to existing catalysts for light alkane activation (PDF)

The authors declare no competing financial interest.