Unveiling mechanistic patterns of copper-catalyzed radical bond formation through linear free energy relationship

Domain: chemistry
Published: 2025-12-23
Editorial outcome: major_revision

Abstract
Copper-catalyzed radical transformations have seen extensive investigations over recent decades. Due to the complexity of controlling factors and the diversity of reaction components, accurately delineating the bond formation mechanisms—namely reductive elimination, outer-sphere radical substitution, and ion pair-type bond formation—poses a formidable challenge. This study reveals the linear free energy relationships (LFER) for the C–C bond formation of Cu-catalyzed radical transformations. Leveraging this discovery, we formulate an accurate LFER model for predicting radical bond formation mechanisms in this catalytic system. By establishing LFER scales for ligands, radicals, and nucleophiles, the bond formation mechanisms across 132,300 potential Cu-catalyzed radical transformations are revealed, offering a detailed mechanistic guide for future C–C bond designs. This research advances the mechanistic understanding of Cu-catalyzed radical transformations, providing systematic and quantitative insights into the mechanistic preference of radical bond formations.

Introduction
As central intermediates in synthetic chemistry, radicals are prominently recognized for their high reactivity and diverse generation pathways1–3. However, this reactivity also presents significant challenges in selectivity control4, prompting chemists to employ open-shell transition metal catalysis in radical chemistry for taming these reactive species4–8. By leveraging the diverse d-electron configurations and the rich functionalities of ligands, radical transformations have achieved remarkable advancements in synthetic control and application scope4–6,9. Copper catalysis, in particular, is celebrated for its exceptional reactivity, efficient selectivity control, and versatility in radical bond formation modes5,10–13. These innovations have led to a variety of effective radical transformation strategies, especially radical transformation reactions that have opened additional avenues for stereoselective functionalization14 and cross-coupling of C(sp3)–H bonds6,15, establishing copper catalysis as a representative system for radical transformation(Fig.1a).

The synthetic value of Cu-catalyzed radical transformations has sparked intense interest in unraveling their reaction mechanisms. Fruitful experimental and computational mechanistic studies have elucidated the molecular-level elementary processes of these catalytic systems(Fig.1b), particularly highlighting the Cu(II)-mediated radical bond formation processes that play a decisive role in stereoselectivity determination5,13,16,17. These studies have culminated in the identification of three widely accepted bond formation modes: reductive elimination (RE) of Cu(III) intermediates, outer-sphere radical substitution (RS), and ion pair-type (IP, also known as radical-polar crossover) bond formation via carbocation5,10. Interestingly, as ligands, radicals, and nucleophiles vary, all three bond formation modes have proven to be operative. The density functional theory (DFT) calculations from Liu, Stahl, and co-workers indicated that Cu/bisoxazoline(Cu/Box)-catalyzed C–H cyanation proceeds through a reductive elimination mechanism18, and Liu’s subsequent research on C–H trifluoromethylation provided mass spectrometric evidence of the transient Cu(III) intermediates19. Stahl et al. were able to capture the carbocation intermediates in Cu/bioxazoline(Cu/Biox)-catalyzed C–H alkoxylation, with DFT calculations also corroborating the ion pair-type bond formation mechanism20. Fu and Peters conducted a series of thorough mechanistic studies on Cu-catalyzed asymmetrical kylation of oxygen and nitrogen nucleophiles21–23, which led to collective evidence from EPR, stoichiometric experiments on organometallic intermediates, and DFT calculations supporting a radical-substitution-type C–N bond formation mode23. Our collaborations with Liu indicated that anionic copper catalyst with tertiary radicals tend to favor an outer-sphere radical substitution mechanism24. These studies have established the mechanistic foundation for the Cu-catalyzed radical transformation, affording chemists a detailed understanding of how copper catalysts facilitate radical bond formation processes.

Although the previous investigations have provided a solid basis for understanding the operative mechanisms, it is worth noting that there remains a lack of comprehensive insights into the controlling factors that determine the preference between the three bond formation modes. Consequently, systematic rules that clarify how variations of bond formation mechanisms manifest in this catalysis are absent. To the best of our knowledge, among the myriad combinations of ligands, nucleophiles, and radicals within Cu-catalyzed radical transformations, there are no established protocols to predict the mechanistic pathway for each combination. Currently, knowledge about the operative bond formation mode under a given radical transformation predominantly stems from time-consuming post factum DFT calculations13, which cannot support the proactive design of new radical transformation reactions, posing challenges in further development and exploration within this catalytic system.

The elucidation of mechanistic principles in Cu-catalyzed radical transformations requires a precise and systematic understanding of the structure-activity relationships (SARs) for each radical bond formation pathway, thereby enabling the determination of mechanistic trends. In this context, linear free energy relationships (LFERs) emerge as a powerful strategy. As a milestone discovery in physical organic chemistry, LFER has unveiled fundamental correlations between reaction thermodynamics and kinetics, providing an effective means for quantifying substituent effects. Classical LFER models like the Hammett equation have played a pivotal role in mechanistic interpretation and rational reaction design, and contemporary LFER research has further expanded its scope and impact. A prominent example of modern LFER discoveries is the Mayr equation25. It establishes quantitative scales for nucleophilicity and electrophilicity, offering a robust and practical framework for predicting rate constants across millions of polar reactions. Furthermore, Sigman’s pioneering work has revolutionized the application of multivariate linear regression in organic synthesis, underscoring the importance and practicality of a multidimensional approach in SAR analysis26–28. These advancements have collectively contributed to the development of predictive linear models for reactivity and selectivity, profoundly shaping the rationalization and design of synthetic transformations in contemporary chemical research.

Inspired by the Mayr equation, we envisioned employing LFERs to build SARs for each bond formation mode in Cu-catalyzed radical transformations, aiming to systematically unravel the mechanistic trends for these transformations(Fig.1c). Herein we report DFT calculations on hundreds of diverse catalytic combinations of ligands, radicals, and nucleophiles, which revealed that the influence of each component on the barriers of Cu(II)-mediated radical C–C bond formation is linearly additive. This enabled the development of quantitative scale for radicals, nucleophiles, and especially ligands, allowing the precise evaluation of all three bond formation modes with a high level of accuracy (R² about 0.99, MAE within 1.8kcal/mol). This methodology has rigorously quantified shifts in mechanistic pathways of Cu(II)-mediated radical C–C bond formation, predicting bond formation modes across tens of thousands of Cu-catalyzed radical transformation reactions. Selected predictions were further corroborated with transition state calculations, yielding compelling mechanistic revelations. This study not only offers a comprehensive and quantitative perspective on the radical bond formation mechanisms in Cu-catalyzed radical transformations but also broadens the scope of LFER application, providing valuable insights for developing quantitative SARs in transition metal catalysis.

Results
Workflow of mechanism space mapping
Figure 2 illustrates the workflow of mapping the mechanism space of radical bond formation in Cu-catalyzed radical transformations through predictive LFER modelling. Following the methodological principles of the Mayr equation and other classical LFER studies, our approach integrated three key steps: (1) DFT Calculation and LFER Evaluation: Starting with a statistically significant and chemically diverse array of catalytic combinations, transition states, and reaction barriers for all three bond formation modes were computed. Ternary linear equations were constructed to evaluate whether the LFER approach can quantitatively describe the reaction barriers of radical bond formation process and to assess the combined equations’ accuracy in determining mechanistic preferences; (2) Building Quantitative Scale: Based on the ternary LFER equations, the most suitable reference reactions were identified. From these ternary LFER equations, the quantitative scales for commonly encountered radicals, nucleophiles, and ligands in Cu-catalyzed radical transformations were established by comparing the barrier differences with the selected references; (3) Mapping the Mechanism Space: Utilizing these scales, the reaction mechanisms of conceivable Cu-catalyzed radical transformations are predicted. This not only provides a systematic guide map for an extensive chemical space of the titled reactions but also offers a quantitative analysis of the influence of each catalytic component on bond formation mechanism, especially the ligand effect.

Definition of the target chemical space
Driven by the objectives of this study, we first delineated a broad target chemical space with the aim of establishing a generalizable and predictive LFER framework to understand and characterize Cu-catalyzed radical bond formation. To this end, we systematically selected representative molecular sets from the three reaction components—radicals, nucleophiles, and ligands—based on their structural features and physicochemical properties. Specifically, the radical set includes 126 carbon radicals, spanning primary, secondary, and tertiary substitution patterns with variations in electronic effects and steric environments; the nucleophile set comprises 75 representative molecules, covering different hybridization states and diverse electronic and steric characteristics; and the ligand set contains 14 distinct cases, accounting for variations in denticity, charge state, and donor atoms. Together, these selections constitute approximately 132,300 possible radical bond formation reactions, reflecting the complexity and breadth of the system while providing sufficient theoretical scope and applicability for constructing the LFER model. Detailed component lists are provided in Supplementary Tables 4, 6 and 8.

In addition, assessing the reliability of the LFER also requires evaluating all linear relationships within a complete set, which necessitates focusing on a manageable subset. Therefore, we further selected a subset that is both representative and computationally feasible within the defined broader space. As shown in Fig. 3a, we ultimately chose 250 reactions comprising 5 ligands, 10 radicals, and 5 nucleophiles as the data set to systematically evaluate whether LFER applies. This carefully curated subset captures the diversity of reaction components relevant to Cu-catalyzed radical transformations while ensuring statistical robustness and applicability of the model, thereby laying a solid foundation for the subsequent construction of the LFER.

Evaluation of the LFERs in the radical bond formation process
For the selected 250 reaction combinations, DFT calculations were performed on a total of 750 bond formation transition states. Following previous mechanistic studies by Bao29, Lin30, Liu18 et al., our mechanistic model focuses on the radical bond formation process through a neutral transition state. Based on this, an additional chloride anion is introduced to coordinate with the LCu(II) complex when L is an electrically neutral ligand(Fig. 3b).

Due to inherent mechanistic preferences, optimizations intended for one bond formation mode may converge to another in certain cases, particularly with RS and IP transitioning to RE. Nevertheless, we were eventually able to locate 199 RE, 62 RS, 52 IP transition states (see Supplementary Tables 14 to 16 for corresponding statistics). Quantitative scales of ligands, radicals, and nucleophiles were constructed by subjecting these computed free energy changes to a least-square minimization based on the equation of ΔG‡ = Σ(L+R+N). To avoid the complexity associated with ternary combinations during the scale establishment, we did not choose the actual pre-intermediate copper complexes of the bond formation process as the energy reference for the transition state barriers. Instead, independent radicals, nucleophiles, and the ligand-coordinated Cu(II) catalysts served as references (Fig. 3b). Although this choice of reference did not allow for assessing the intrinsic bond formation efficiency of each pathway (due to the general instability of these reference species compared to the pre-intermediate copper complexes), the relative free energy differences between the isomeric transition states can still be accurately described, thereby reliably predicting mechanistic preferences and facilitating efficient mapping of mechanism space.

Subsequently, we formulated ternary linear equations under the LFER assumption to describe the energies of the bond formation transition states. To our delight, the LFER equations for the RE, RS, and IP bond formation modes demonstrated high precision on the subset used for constructing the LFER. (Fig. 3c–e). The R2 values for these equations were 0.992, 0.983, and 0.988, respectively, with mean absolute errors of less than 1.8kcal/mol across an energetic range of up to over one hundred kcal/mol.

Building upon the results obtained for the selected subset, we further assessed the reliability of the LFER constructed from the DFT-computed free energy barriers from multiple perspectives to ensure that the derived statistical relationships are not only numerically accurate but also chemically meaningful. To this end, we first examined the 250 reaction combinations using additional density functionals. In addition to the original calculations with B3LYP-D3(BJ), single-point energy evaluations were carried out with three widely used functionals, M06, PBE0-D3(zero), and ωB97X-D. The resulting LFER equations retained a high degree of robustness, with R² values ranging from 0.972 to 0.994 and mean absolute errors of 1.3–1.7kcal/mol (Supplementary Table 9). More importantly, when the LFER equations parameterized with different functionals were used to predict the operative mechanism of the same reaction, the predictions were largely consistent, with only 9 reactions showing discrepancies in the assigned mechanism.

In addition to this validation across different theoretical levels, we also evaluated the reliability of the LFER through dataset partitioning. Using a modified 5-fold cross-validation protocol, the mechanistic data were divided into five subsets, with each subset in turn serving as the set for constructing LFER while the remaining reactions were used for testing. Under these data splittings, the accuracy of mechanistic predictions remained excellent, with R² values consistently above 0.95 and MAEs within 2.0kcal/mol(Supplementary Table 13). To further rigorously probe the extrapolative capacity and reliability of the model, we designed an additional test set consisting of 67 completely new reactions, incorporating 9 ligands, 51 radicals, and 32 nucleophiles not included in the reactions for constructing LFER. The parameters for these new components were derived from their barrier differences relative to predefined reference compounds, as detailed in Supplementary Fig. 14. Encouragingly, the LFER established on the 250 reactions also performed excellently on this extrapolative test set, with MAEs of 1.8, 1.6, and 1.7kcal/mol for the RE, RS, and IP pathways, respectively (red crosses, Fig. 3c–e), thereby demonstrating the robustness and reliability of the proposed LFER model of Cu-catalyzed radical bond formation.

We want to emphasize that these linear equations should not be seen merely as interpolation within a limited dataset; rather, they elucidated the underlying LFER principles governing the bond formation SARs in this catalysis, indicating that the influences of radicals, nucleophiles, and catalysts on the stability of bond formation transition states are linearly additive. Due to the high accuracy of these equations, when used collectively, they can precisely describe the energy differences between bond formation transition states, enabling the accurate prediction of the favored reaction mechanisms. In Fig. 4, the ΔΔG‡ LFER equation showed an RMSE of 2.3kcal/mol and an MAE of 1.8kcal/mol, with a 93.9% accuracy in predicting the correct mechanism of the reactions for constructing LFERs, and an RMSE of 2.9kcal/mol and an MAE of 2.3kcal/mol, with a 92.3% accuracy in predicting the correct mechanism of additional reactions for testing LFERs, thus providing a robust framework for determining the favored bond formation mechanism in Cu-catalyzed radical transformations.

To assess the applicability of the model to real-world synthetic systems, solvation effects were also examined. Transition states were computed in solvents with varying polarities, and corresponding LFER models were constructed. Consistent LFER models were obtained for each solvent, all exhibiting R² values above 0.98 (Supplementary Tables 10–12 of the supplementary information). However, variations in LFER parameters were observed as solvent polarity changed, reflecting differential stabilization effects on radical bond formation transition states. These results suggest that the LFER model established in this study are directly applicable to nonpolar and low-polar solvents, while additional solvation corrections may be required for highly polar solvents to account for their specific stabilization effects, which is the subject of ongoing research in our laboratory.

For the chemical interpretability of our LFER parameters, our strategy essentially reflects the quantitative contribution of each reaction component in the three-component LFER equations describing Cu(II)-catalyzed radical bond formation, which is analogous to Mayr’s LFER approach for defining nucleophilicity and electrophilicity. In this framework, the parameters L, R, and N each carry explicit chemical meaning: they represent the contributions of the ligand, radical, and nucleophile, respectively, to the stabilization of the transition state. In other words, these parameters provide a quantitative measure of how each component modulates reactivity and influences the mechanistic preference within this catalytic system.

Building quantitative LFER scales for the reaction components
Encouraged by the success of the LFERs established on the above 250-reaction subset, we further extended this framework to the entire target chemical space and constructed LFER parameters for all ligands, radicals, and nucleophiles (details provided in Supplementary Fig. 14). This extension enables us to quantitatively discuss how ligands, radicals, and nucleophiles influence mechanistic preferences in Cu(II)-mediated radical bond formation.

Figure 5 illustrates the LFER scales (LRE, LRS, and LIP) for 14 widely applied ligand scaffolds in Cu-catalyzed radical transformations. The scales derived from these ligands are plotted on a mechanistic distribution diagram to represent their preferences, revealing intriguing ligand effects. Bidentate neutral nitrogen ligands (L2 to L6) are located near the anchor point of the distribution diagram, exhibiting unremarkable mechanistic preferences. The bidentate phosphorus ligand L1, due to its significant σ-donor capability, is positioned far from the zone of bidentate neutral nitrogen ligands, prominently favoring the IP mechanism. Compared to bidentate neutral ligands, bidentate anionic ligands significantly shift towards the RE direction, particularly acetylacetonate L9, which show a propensity for the RE mechanism. This is because the copper coordinated by anionic ligands does not have additional chloride coordination, freeing up coordination sites to support the RE pathway. As the ligand denticity increases, the tridentate neutral ligand PyBox L10 aligns with the region of bidentate neutral nitrogen ligands, suggesting they possess similar mechanistic tendencies. However, when it comes to tridentate Pincer-type anionic ligands L11 to L14, they consistently show a preference for the RS mechanism, likely because the increased coordination on Cu is unfavorable for RE processes and the oxidative capacity of anionic copper is insufficient to oxidize radicals for the IP mechanism. The pinpoint positioning of these ligands on the mechanism distribution diagram allowed a quantitative comparison on mechanistic trends, providing a clear understanding of their role in directing reaction pathways.

We further investigated the influence of radicals on mechanism determination. Scales for 126 common carbon radicals were established, with 13 representative examples showcased in Fig. 6. The secondary benzyl radical R1, positioned near the anchor point, demonstrates negligible tendency towards the RE mechanism. Similarly, other conjugatively stabilized secondary radicals (R2 to R4) are located in the zone where RE and IP mechanisms compete. The degree of substitution at the radical center markedly impacts the inclination towards the RE mechanism; compared to secondary radical R1, the primary benzyl radical R5 distinctly favors RE, whereas the corresponding tertiary radical R6 tends towards the opposite direction. Electronic effects also play a key role, mainly influencing the inclination towards IP, with notable shifts along the vertical axis of the mechanistic distribution diagram. Comparing to benzyl radical R1, electron-deficient α-(hetero)aryl radicals (R7 and R8), as well as the α-acyl radical R11, are positioned away from IP, while electron-rich radicals (R9 and R10) lean towards IP. Furthermore, these substituent effects are cumulative. The tertiary, conjugatively electron-withdrawing radicals R12 and R13, distanced from both RE and IP mechanisms for the above-discussed reasons, thereby favor the RS mechanism. While these substituent effects are understandable through classical structural analogies in organic chemistry, it is important to note that these LFER scales do not align with traditional physical organic parameters of radicals. Efforts to correlate RRE and RRS with common radical parameters such as globalelectrophilicity ω31,32 and SOMO energy revealed in apparent correlations, while RIP can be approximately obtained by multivariate linear regression (MLR), as shown in Supplementary Table 22 and Supplementary Fig. 16. This observation indicated that known physical organic understandings on radical behavior may not fully apply to Cu(II)-mediated radical bond formation and highlights the new mechanistic insights embedded in these LFER scales.

In addition, the LFER scales of nucleophiles also unveiled new mechanistic insights. We examined 75 common carbon nucleophiles relevant to the target catalysis, with 12 representative examples displayed in Fig. 7. As the LFER scale was constructed utilizing three different reaction components, the corresponding anionic species are adequate to quantitatively evaluate the impact of nucleophiles on the bond formation transition states. Aryl and vinyl anions N1 and N2 are closely positioned on the mechanistic distribution diagram, both showing a preference for the RE mechanism. The electronic effects on C(sp2) nucleophiles manifest in their propensity towards the RE mechanism, with the electron-deficient aryl N3 and vinyl N4 more distinctly disfavoring RE, whereas the electron-rich aryl anion N5 and vinyl anion N6 demonstrate opposite trends. Intriguingly, for sp-hybridized nucleophiles, the electronic effects primarily affect the trend towards the IP mechanism. Compared to phenylethynyl N7, the electron-deficient pyridyl N8 and the electron-rich thiophenyl N9 shifts the corresponding nucleophiles along the IP axis on the trend diagram. Introducing a bulky p-tBu group on the benzene ring results in N12, markedly disfavoring IP and positioned in the RS-favored region. Similar to the LFER scales of radical, the LFER scales of nucleophiles do not correlate with known nucleophilicity parameters. When compared with typical nucleophilicity indices such as electro-donating power33 and HOMO energy, these scales show limited correlations (results are shown in Supplementary Tables 23 and 24), further emphasizing the distinctive structure-selectivity relationships of nucleophiles in Cu-catalyzed radical transformations as revealed by our LFER scales.

Mapping of the bond formation mechanism space
By applying the LFER scales and equations in combination, we can predict the bond formation mechanisms for all conceivable combinations of the studied ligands, radicals, and nucleophiles. This predictive scope encompassed a vast space involving 132,300 potential Cu-catalyzed radical transformations (Fig. 8a), with key statistical results presented in Fig. 8b. Among all the reaction combinations evaluated, 82.9% exhibited a single dominant mechanism, indicating that the favored pathway was energetically more favorable by at least 3kcal/mol, while the remaining 17.1% showed competition between two or even three bond formation modes. Among the reactions with a single dominant mechanism, 68.2% were predominantly governed by the RE mechanism, affirming its role as the principal operative C–C bond formation mechanism in Cu(II)-catalyzed radical transformations. Additionally, 9.7% and 5.1% of reactions demonstrated a propensity toward the RS and IP mechanisms, respectively, with certain guiding principles detailed below. In the 17.1% where mechanism competition occurred, competitions involving the RE mechanism predominated, consistent with the overall tendency for RE across the chemical space; only 0.9% of reactions exhibited competition between RS and IP mechanisms. Furthermore, based on the parameter ranges for ligands, radicals, and nucleophiles (Supplementary Fig. 18), the influence of ligands on the bond formation mechanism is the most substantial, followed by nucleophiles, while the effect of radicals is markedly smaller than the other two components. All prediction results are available in a SMILES-queryable spreadsheet on GitHub (https://github.com/XuGuoXiong/LFER-on-Mechanism-Prediction.git).

The multitude of mechanism predictions reveals insightful statistical patterns. Perhaps the most practically valuable of these insights is the identification of characteristic structural features that, studied C–C bond formations, we indeed identified a series of mechanistic principles, with key findings displayed in Fig. 8c. Among all ligands, the presence of bidentate anionic ligands consistently leads to nearly 90% of mechanisms being RE-dominant. The presence of primary carbon radicals or C(sp3) nucleophiles also implies that the RE mechanism is operative. However, for tertiary α-amino radicals, nearly 80% of C–C bond formations proceed through the IP mechanism. When structural features promoting RE or IP are both present (i.e., tertiary α-amino radicals with C(sp3) nucleophiles), conflicting rules lead to competitive mechanisms, displaying a competition between IP and RE without clear dominance. However, binary combinations still demonstrate clear mechanistic preferences, as detailed in Fig. 8d. When tridentate anionic ligands L13 and L14 are paired with C(sp) nucleophiles, the majority of reactions undergo the RS mechanism, with a minor portion of RE and IP mechanisms arising from the radical effects. Moreover, when the strong σ-donating bidentate phosphorus ligand L1 is combined with tertiary electron-rich radicals, 66.9% of the corresponding transformations favor the IP pathway.

It should be emphasized that our LFER model is not intended to replace DFT transition state calculations for individual reactions. For the mechanistic elucidation of a specific transformation, direct DFT calculations are often more accurate and straightforward. The value of the LFER approach lies in its ability to efficiently uncover overarching patterns and trends across large chemical spaces. Compared with the O(n³) complexity of calculating every transition state one by one, the LFER framework requires only limited DFT data and reduces the problem to O(n), thereby enabling systematic prediction and comparison over a vast number of reaction combinations. This approach not only captures mechanistic changes induced by substituent effects but also provides a more comprehensive and reliable assessment of mechanistic trends than traditional strategies that rely on a few model systems, thus serving as a powerful complement to individual DFT calculations aimed at clarifying specific mechanisms.

After completing the prediction and distribution analysis of 132,300 reaction combinations in the chemical space, we further selected a series of catalytic systems that are either closely related to the frontiers of synthetic chemistry or mechanistically representative, in order to validate and challenge the established LFER model. A total of 20 representative cases were examined, including 10 reactions reported in the literature whose competing radical bond formation mechanisms had been thoroughly investigated by other groups using DFT calculations, as well as an additional 10 reactions predicted by our model to favor RS or IP pathways, thereby ensuring a balanced representation of all three mechanistic classes in the evaluation. Among these 20 cases, the LFER model correctly identified the operative bond formation mechanism in 17 reactions, with only 3 cases showing discrepancies compared to the reference DFT results. These findings further demonstrate that the LFER model not only uncovers general trends across large chemical spaces but also provides reliable and applicable mechanistic predictions for specific catalytic reactions. Detailed results and the full list of reactions are provided in Supplementary Figs. 19 and 20.

One representative example for each bond formation mechanism is presented in Fig. 9, with more detailed results available in Supplementary Fig. 19. Liu et al. previously reported a copper-catalyzed radical-mediated cyanotrifluoromethylation of alkenes (Fig. 9a), where the LFER model predicted a dominant RE mechanism34; our DFT calculations indeed confirmed that the corresponding RE transition state was more favorable by 7.8kcal/mol. In the Sonogashira coupling of cyclic gem-dihaloketones developed by the Zhang group35 (Fig. 9b), characterized by a tridentate anionic ligand and an alkynyl nucleophile, our model anticipated a dominant RS mechanism, the prediction was supported by a significantly favorable RS transition state. Interestingly, in the hydrocyanation of olefins developed by Xiao, Chen, and colleagues, although the previous DFT mechanistic study on model reaction suggested a preference for RE mechanism36, our analysis indicated potential reversals in the mechanism for certain substrates within the reported substrate scope. Following this prediction, we specifically studied the bond formation process between the electron-rich para-methoxybenzyl radical and the cyanide nucleophile, confirming that this combination indeed proceeds through an IP mechanism, as suggested by our LFER model (Fig. 9c).

Moreover, our review of previous mechanism studies on Cu-catalyzed radical transformations revealed that the RS mechanism is often overlooked in transition state calculations, yet our LFER model and DFT calculations have identified that many cases favor the RS mechanism as the predominant bond formation pathway (see also Supplementary Fig. 19 and Supplementary Tables 25 and 26). Therefore, evaluating the RS mechanism is a noteworthy issue for Cu-catalyzed radical transformations, and we recommend referencing our LFER scales and predictions in future mechanism studies to provide a more comprehensive understanding.

Methods
Computational Protocol
All DFT calculations were performed using Gaussian 16 program at B3LYP-D3(BJ)/6-311+G(d, p)-SDD// B3LYP-D3(BJ)/6-31G(d)-LANL2DZ level of theory.

A computational workflow was developed to reliably identify the transition states of each bond formation process (see Supplementary Fig. 1 for more details).

Conformers of each bond formation transition state were thoroughly considered, and the lowest-energy conformer was selected as the optimal result for DFT calculations (see Supplementary Fig. 2 for more details).

Generation of LFER parameters
For components of the reactions for constructing LFERs, quantitative scales were constructed by subjecting these computed free energy changes to a least-square minimization based on the equation of ΔG‡ = Σ(L+R+N).

For components of the reactions for testing LFERs, quantitative scales were derived from their barrier differences relative to predefined reference compounds. Detailed scaling method is shown in Supplementary Fig. 14.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper describes a computational study using transition state energy (TSE) calculations to analyze Cu-catalyzed coupling reactions between carbon-based radical species and nucleophiles. The authors systematically vary the nucleophile, radical partner, and ligand, and compute TSEs for three prevalent mechanistic pathways. These computed energies are then used to develop linear free energy relationship (LFER) models, in which the TSE is approximated as a linear combination of three parameters—one for each reaction component. However, these parameters are not physically interpretable, and the models are not validated against experimental data. Instead, the predicted TSEs are only benchmarked against the DFT-calculated values, which is a notable weakness of the study.\nDespite this, the paper offers value in using the descriptor-based LFER models to rationalize mechanistic preferences across different substrate combinations. This insight is important and potentially impactful enough to merit publication. However, instead of relying on non-interpretable descriptors, the authors could have more directly compared the DFT-computed transition state energies for each mechanistic pathway across the substrate combinations. This would provide a more straight forward analysis without the added abstraction of building and interpreting an LFER model based on opaque parameters.\nAs such, while the study is useful, its novelty and depth may not justify publication in a high-impact venue and could be better suited to a more specialized journal.\nMinor Comments:\n1. In the introduction, contributions from the Sigman lab are acknowledged, but no references are provided—please add appropriate citations.\n2. In Figure 3b, the nucleophile shown in the reaction scheme is an alkyne, but the data actually corresponds to CN⁻. This should be corrected.\n3. Consider adding a brief footnote or caption to figure 3 explaining the meaning of the parameters used in the LFER model. As it stands, this section is difficult to follow without additional context.", "reviewer_2": "In this study, Zhang, Hong and co-workers present a statistical model to elucidate mechanistic patterns in copper-catalyzed radical bond formation by decomposing the activation free energy barrier (ΔG‡) into three components L, R, and N using a linear free energy relationship (LFER) approach. While the study introduces an innovative and promising framework, several concerns arise that suggest the work would benefit from further refinement in order to strengthen its conclusions. Although the methodology is thorough and leverages a wide array of computational and statistical tools, I recommend publication only after the following major concerns are addressed:\n1. Chemical Space Representation and Data Selection\nLack of Chemical Space Visualization:\nThe paper does not provide a clear visualization of the chemical space covered by the chosen substrates and ligands. It would be extremely valuable to see how the chemical space is mapped and whether the combinations have been selected based on a defined, systematic criterion.\nUnclear Data Splitting Criteria:\nThe paper lacks a detailed explanation of how the dataset was split into training and test sets. Ensuring the test set is representative of the entire data distribution is critical for validating model generalizability. Figures 3C and 3D show a notable absence of test points in the lower ΔG‡ range. A more thorough and transparent description of the splitting criteria, including how the test set captures the full data variation, is necessary to evaluate the robustness of the statistical model.\n2. MLR as an Alternative for Deriving RIP Parameters\nSince the MLR approach is used to derive RIP parameters (supplementary figure 6) for a majority of the radicals (74 out of 126), a more robust assessment of this model's predictive capability is essential. I strongly suggest applying 5-fold cross-validation or similar approaches to rigorously evaluate its performance. Given the importance of these parameters in shaping the LFER-based mechanistic insights, a deeper validation is needed to support the reliability of the MLR approach.\n3. Inconsistencies Between Figures/Tables and Manuscript\nThere are several inconsistencies in ligand labeling between the manuscript and the figures. For example, on page 5, the acetylacetone ligand is incorrectly labeled as L10, whereas it should be L9. Similarly, the Pybox ligand is listed as L9 but should be L10. Moreover, confusion appears to persist with the Pincer-type anionic ligands (L11–L14). A comprehensive review and correction of ligand labels across the manuscript is warranted for clarity and reproducibility.\n4. Model Accuracy May Be Inflated by Reaction Imbalance\nAccording to the reaction space described, ~70% of the reactions are dominated by the reductive elimination (RE) pathway. Therefore, a naive model that predicts RE as the operative mechanism for all cases would already achieve ~70% accuracy. The current model predicts the correct mechanism in 8 out of 10 test cases (~80%), but notably, the two misclassified examples correspond to non-RE pathways. To more rigorously test the model’s predictive utility, I recommend including a new validation set containing at least 10 transformations where the operative mechanism is exclusively not RE (i.e., RS or IP), in order to assess the model’s performance across the full mechanistic landscape.\n5. LFER Predictive Robustness and Extrapolation Ability\nThe LFER models presented in Figure 3C should be subjected to 5-fold cross-validation or similar resampling methods to assess their extrapolative power. This would help determine whether the models are truly generalizable or simply overfit to the current dataset.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "I find this study very interesting for both computational and experimental colleagues. I believe it can be published in Nature Communications, even though the results are not extremely groundbreaking and the applied methods are fairly standard. Nevertheless, I appreciate the paper for the way it bridges theory and experiment. The manuscript is also well-written.\nHowever, I do have one question regarding the computational approach. The systems studied involve flexible organic molecules, and it is crucial to perform a thorough exploration of the conformational space for both transition states (TSs) and minima. How was this aspect addressed? Including a comment on this in the computational details of the Supporting Information (SI), and perhaps even in the main text, would be desirable.\nI recommend the publication with minor revisions."}, "author_response": "Response to Referee 1\nComment: This paper describes a computational study using transition state energy\n(TSE) calculations to analyze Cu-catalyzed coupling reactions between carbon-based\nradical species and nucleophiles. The authors systematically vary the nucleophile,\nradical partner, and ligand, and compute TSEs for three prevalent mechanistic pathways.\nThese computed energies are then used to develop linear free energy relationship\n(LFER) models, in which the TSE is approximated as a linear combination of three\nparameters—one for each reaction component. However, these parameters are not\nphysically interpretable, and the models are not validated against experimental data.\nResponse: We sincerely thank the reviewer for the thoughtful evaluation and valuable\ncomments on our work. The reviewer pointed out that the LFER parameters in our study\nmay lack physical interpretability and that further validation is missing. We fully\nrecognize that both aspects are critical for highlighting the scientific significance and\npotential applicability of our model. During the revision, we not only clarified these\npoints but also performed additional, systematic computational analyses and\ncomparative benchmarks to substantiate our conclusions. The reviewer’s suggestions\nhave greatly improved the completeness and clarity of our manuscript.\nFigure R1. Conceptual comparison of LFER models: (a) Hammett equation for\nsubstituent effects on ionization thermodynamics; (b) Mayr equation for\nnucleophilicity/electrophilicity in polar reactions; (c) the present LFER framework\nquantifying ligand, radical, and nucleophile contributions to transition state\nstabilization in Cu-mediated radical bond formation.\nFirst, regarding the interpretability of the LFER parameters, we fully agree with the\nreviewer. The parameters we introduced for ligands (L), radicals (R), and nucleophiles\n(N) are indeed not as straightforward as traditional physical organic descriptors such as\nelectronic charges, steric volumes, or geometric indices. The reason lies in the very\nnature of LFER parameters: they are reaction-specific constructs. For example, the\nclassical Hammett equation quantifies the influence of substituents on the\nthermodynamics of benzoic acid ionization (Figure R1a), and the Mayr equation\ndescribes variation of electrophile/nucleophile on the kinetics of electrophile-\nnucleophile combination reactions (Figure R1b). These parameters are not direct\n“physical properties” either, but rather quantitative measures of the overall substituent\neffect in a given reaction context. Over time, chemists have developed intuition and\ninterpretative traditions for these descriptors through repeated application. By analogy,\nthe LFER parameters we developed are also reaction-specific: they quantify the\ncumulative effects of substituents on the stabilization of the transition state—and thus\nthe relative reactivity—in copper-mediated radical–nucleophile bond formation\nreactions (Figure R1c). While they may appear less familiar than widely used electronic\nor steric constants, their scientific meaning is entirely consistent with established LFER\nframeworks.\nIn addition, we would like to clarify that defining LFER parameters as reaction-specific\ndoes not mean they are “not physically interpretable.” The parameters L, R, and N each\ncarry explicit chemical meaning: they represent the contribution of the ligand, radical,\nand nucleophile, respectively, to the stabilization of the transition state in copper-\nmediated radical bond formation. In other words, they provide a quantitative description\nof the relative reactivity contributions of the three reaction components. This\nformulation is physically interpretable and inherently verifiable. The main challenge is\nexperimental: the active copper intermediates are difficult to isolate, and measuring the\nelementary rate constants directly is extremely challenging. Therefore, we were only\nable to rely on high-level quantum chemical calculations to generate the LFER\nparameters, which is different to the Mayr equation constructed from experimentally\nmeasured rate constants. However, this methodological choice does not diminish their\nphysical interpretability, but rather enables systematic and high-throughput analysis\nthat is currently inaccessible experimentally.\nIn summary, we are truly grateful for the reviewer’s comments and suggestions. They\nprompted us to expand the discussion in the revised manuscript and to emphasize the\nanalogy between our parameters and classical physical organic models. We believe\nthese revisions will help readers better appreciate the meaning of the parameters and\ntheir value in probing mechanistic patterns in radical bond formation.\nComment: Instead, the predicted TSEs are only benchmarked against the DFT-\ncalculated values, which is a notable weakness of the study.\nResponse: We sincerely thank the reviewer for this valuable comment. We fully\nunderstand the reviewer’s concern and, prompted by this suggestion, have recognized\nthe importance of this issue. In the revised manuscript, we have provided a systematic\nand comprehensive set of additional analyses to address this critical issue. As discussed\nin our above response, it is experimentally very challenging to synthesize the reactive\ncopper intermediates and to directly characterize the rate constants of their elementary\nbond formation steps with different radicals and nucleophiles. For this reason, our\nLFER parameters cannot be constructed in the same way as the Mayr or Hammett\nequations, which are based directly on experimental kinetics or thermodynamics.\nNevertheless, in the revision we have undertaken an extensive set of theoretical\ncalculations to rigorously evaluate the reliability of our LFER models.\nTo assess the accuracy of our computational protocol, we first benchmarked the DFT\nmethod employed in this study against the high-level DLPNO-CCSD(T) approach,\nwhich is widely regarded as the gold standard for reaction energetics. For two\nrepresentative radical bond formation reactions involving distinctive ligands, radicals\nand nucleophiles, we calculated the free energy differences between the reductive\nelimination (RE) and ion pair-type (IP) pathways and directly compared the DFT results\nwith those from DLPNO-CCSD(T) (Figure R2). The two levels of theory show\nexcellent agreement, with deviations of only 0.8 kcal/mol and 1.6 kcal/mol, respectively.\nThis close correspondence strongly supports the reliability of our chosen DFT protocol\nfor capturing the mechanistic selectivity in these copper-catalyzed radical bond\nformation reactions.\nFigure R2. Benchmarking of DFT against DLPNO-CCSD(T) for representative\ncopper-catalyzed radical bond formation reactions, transition states optimized at the\nB3LYP-D3(BJ)/6-31G(d)-LANL2DZ level of theory.\nSubsequently, we carried out a large-scale functional assessment across the entire scope\nof 250 reactions to confirm that the observed LFER relationships are robust and not\ndependent on a single functional choice. In addition to the original B3LYP-D3(BJ)\nresults, we recalculated the transition states for all three bond formation pathways (RE,\nRS, IP) using three other common functionals: M06, PBE0-D3(zero), and ωB97X-D.\nBased on the activation free energies obtained from each functional, we reconstructed\nthe LFER models and performed mechanism predictions separately for each case.\nAs summarized in Table R1, the LFER models derived from all four functionals show\nconsistently high accuracy: for RE, RS, and IP steps, the correlation coefficients (R²)\nfall in the range of 0.972–0.994, with MAEs between 1.3 and 1.7 kcal/mol. Furthermore,\nwhen applying the LFER parameters derived from each functional to predict the\noperative mechanism for the same reaction, models parameterized with different\nfunctionals typically returned the same operative pathway. Among the 250 reactions\nexamined, only 9 exhibited discrepancies due to highly competing pathways (see\nSupplementary Table 8 in Page S23 of the revised Supplementary Information for\ndetails)\nTable R1. Comparison of LFER model performance across four different DFT\nfunctionals for copper-catalyzed radical bond formation reactions, transition states\noptimized at the B3LYP-D3(BJ)/6-31G(d)-LANL2DZ level of theory.\nTherefore, these high-level and multi-functional computations not only confirm the\nreliability of our chosen DFT method but also provide strong validation of the\nrobustness and broad applicability of the proposed LFER model.\nComment: Despite this, the paper offers value in using the descriptor-based LFER\nmodels to rationalize mechanistic preferences across different substrate combinations.\nThis insight is important and potentially impactful enough to merit publication.\nHowever, instead of relying on non-interpretable descriptors, the authors could have\nmore directly compared the DFT-computed transition state energies for each\nmechanistic pathway across the substrate combinations. This would provide a more\nstraightforward analysis without the added abstraction of building and interpreting an\nLFER model based on opaque parameters.\nResponse: We sincerely thank the reviewer for the positive evaluation of our work and\ngreatly appreciate the recognition of the insights provided by our LFER model. On one\nhand, this model provides a systematic understanding of mechanistic trends in copper-\nmediated radical bond formation, a class of elementary steps that are actively studied\nat present. These insights offer a solid foundation for the rational design of future\ncatalytic systems based on bond forming mechanisms. On the other hand, the work\ndemonstrates the potential of LFER, a classical and powerful tool in physical organic\nchemistry, to address complex transition metal catalysis. We believe that this\nperspective is equally important, as it suggests a way to disentangle substituent effects\nin such systems and to build predictive models that can manage the combinatorial\ngrowth of substituent space.\nWe also fully understand the reviewer’s concern that, in practical catalytic studies,\ndirectly computing the transition states of competing pathways by DFT is often more\nstraightforward and more accurate than relying on an LFER model. As the reviewer\nnoted, the parameters in our LFER framework are derived from DFT energetics, so the\nupper bound of their accuracy is inherently limited by the chosen level of theory. In\naddition, if the target reaction is not among the 132,300 combinations already included\nin our predictions, it would still require dedicated DFT calculations to obtain the\ncorresponding parameters.\nHowever, we would like to emphasize that the central purpose of the LFER model\npresented here is not to replace DFT in resolving the mechanism of a single, specific\nreaction, but rather to uncover general trends and governing principles of mechanistic\nchanges when all components of the reaction are subject to substituent effects. This\nLFER framework provides a new, efficient, and reliable way to evaluate mechanisms\nacross a very large number of possible combinations. For example, the current\napplication dataset used for demonstration in this study already encompasses 132,300\ndistinct reactions, involving 126 different radicals, 75 nucleophiles, and 14 ligands.\nTogether, these combinations span a significant portion of the chemical space relevant\nto contemporary copper-catalyzed radical coupling reactions. Performing one-by-one\nDFT transition-state calculations for a dataset of this magnitude would require decades,\nwhich is clearly infeasible. Traditional strategies that rely on a handful of model\nsystems implicitly assume that substituent variation does not alter the mechanism, an\nassumption that is often risky and prone to error when extrapolated.\nBy contrast, our LFER model reduces the computational complexity from O(n³) to O(n),\ndramatically enhancing the tractability of exploring broad chemical space. Within the\n132,300 combinations studied, only limited DFT calculations for each dimension\n(radical, nucleophile, and ligand) are needed to derive the parameters, which can then\nbe applied to predict the operative mechanism across all combinations. This approach\nhas enabled us not only to uncover several previously unrecognized mechanistic trends\n(as discussed in Figure 8 of the main text), but also to correct potential pitfalls\nassociated with extrapolations from traditional model-system-based transition state\ncalculations.\nFigure R3. Complementarity of LFER model and DFT transition state calculation in\ncopper-catalyzed radical cyanation: (a) reproduction of the reported preference for\nreductive elimination; (b) prediction and validation of a substituent-driven mechanistic\nswitch, where a methoxy-substituted benzyl radical shifts the favored pathway from\nreductive elimination to ion pair-type bond formation. Distances are given in Å.\nTo more concretely demonstrate the complementarity between our LFER framework\nand conventional DFT studies, we added in the revision an application to the copper-\ncatalyzed radical coupling previously reported by Chen et al. (J. Am. Chem. Soc. 2024,\n146, 1410). In their study, DFT calculations showed that cyanation of a biphenyl benzyl\nradical favors the reductive elimination pathway by 2.8 kcal/mol, and our LFER model\nreproduces this preference with a value of 1.0 kcal/mol (Figure R3a). While this\nconclusion applies to benzyl radicals with similar electronic character to the model\nsubstrate, the reaction scope described by Chen et al. includes many substituents with\npronounced electronic effects. Applying our LFER model to these broader substrates\nrevealed that even a modest substitution, such as a methoxy substitution on the benzyl\nradical, switches the favored pathway from reductive elimination to ion pair-type bond\nformation by 1.1 kcal/mol. Our subsequent DFT transition state calculations indeed\nconfirmed this prediction, showing that the ion pair pathway is more favorable by 1.3\nkcal/mol (Figure R3b). This example highlights that the LFER model is not only\ncapable of capturing the outcome of individual model reactions but, more importantly,\ncan also expose substituent-driven mechanistic switches that are extremely difficult to\nsystematically identify by sequential DFT calculations alone.\nTherefore, our LFER model and traditional DFT studies serve complementary roles:\nthe latter excels in pinpointing the mechanism of a specific reaction, whereas the former\nenables scalable, predictive analysis of mechanistic landscapes across vast chemical\nspaces. Inspired by the reviewer’s constructive suggestion, we have further underscored\nthis complementary relationship in the revised manuscript and illustrated it with the\nabove case study.\nComment: Minor Comments. 1. In the introduction, contributions from the Sigman\nlab are acknowledged, but no references are provided—please add appropriate citations.\nResponse: We sincerely thank the reviewer for pointing out this important issue. In the\noriginal submission, we did not cite Sigman's work in the main text but included it as\nref. 16 in the Supplementary Information. As the reviewer correctly noted, our\nreferencing of the Sigman lab’s contributions was not comprehensive. Following the\nreviewer’s suggestion, we carefully reviewed the relevant literature and have now\nincluded the key citations of multivariate linear regression in the revised manuscript\n(refs. 26 to 28). We believe these additions provide appropriate acknowledgment of\nprior work and improve the completeness of the introduction.\nComment: 2. In Figure 3b, the nucleophile shown in the reaction scheme is an alkyne,\nbut the data actually corresponds to CN⁻. This should be corrected.\nResponse: We thank the reviewer for noting this error. In the revised manuscript, Figure\n3b has been corrected so that the nucleophile shown in the reaction scheme is consistent\nwith the corresponding data (CN⁻).\nComment: 3. Consider adding a brief footnote or caption to figure 3 explaining the\nmeaning of the parameters used in the LFER model. As it stands, this section is difficult\nto follow without additional context.\nResponse: We thank the reviewer for this helpful suggestion. In the revised manuscript,\nwe have added explanatory notes to Figure 3 to clarify the meaning of the parameters\nused in the LFER model, which indeed improves readability and context for the reader.\nResponse to Referee 2\nComment: In this study, Zhang, Hong and co-workers present a statistical model to\nelucidate mechanistic patterns in copper-catalyzed radical bond formation by\ndecomposing the activation free energy barrier (ΔG‡) into three components L, R, and\nN using a linear free energy relationship (LFER) approach. While the study introduces\nan innovative and promising framework, several concerns arise that suggest the work\nwould benefit from further refinement in order to strengthen its conclusions. Although\nthe methodology is thorough and leverages a wide array of computational and statistical\ntools, I recommend publication only after the following major concerns are addressed.\nResponse: We sincerely thank the reviewer for the positive evaluation of our work and\nfor the insightful suggestions and comments. We especially appreciate the reviewer’s\nperspective on the statistical modeling aspects, which provided us with valuable\nguidance for strengthening our study. In the revision, we carefully addressed these\npoints and performed comprehensive additional analyses. These efforts not only\nreinforce the reliability of our LFER framework but also substantially improve the\ncompleteness and clarity of the manuscript.\nComment: 1. Chemical Space Representation and Data Selection. Lack of Chemical\nSpace Visualization: The paper does not provide a clear visualization of the chemical\nspace covered by the chosen substrates and ligands. It would be extremely valuable to\nsee how the chemical space is mapped and whether the combinations have been selected\nbased on a defined, systematic criterion.\nResponse: We greatly appreciate the reviewer’s insightful comment. As rightly pointed\nout, the systematic and rational construction of the chemical space is critical for the\nreliability and applicability of the LFER model. In the revised manuscript, we have\nprovided a detailed explanation of the principles guiding the construction of the\nchemical space and the specific ligands, radicals, and nucleophiles selected. In addition,\nfollowing the reviewer’s suggestion, we have visualized the chemical set from multiple\nphysical organic perspectives to validate its representativeness and scientific soundness.\nIn this study, our goal was to include radicals, nucleophiles, and ligands broadly\nrepresentative of Cu-catalyzed radical coupling reactions. Accordingly, we constructed\nthe molecular sets systematically from structural and physicochemical perspectives.\nFor the ligands, we considered denticity (bidentate, tridentate), charge state (neutral,\nanionic), and coordination atoms (N, P, O), selecting 14 ligands to ensure broad\ncoverage (Figure R4a). In light of the reviewer’s guidance, we further evaluated the\ndistribution of the ligand set using two quantitative descriptors: the SOMO energy level\nof the LCu(II) active catalyst (electronic effect) and its buried volume (steric effect).\nThe resulting distributions (Figure R4b and R4c) are highly uniform, underscoring the\nrepresentativeness of the chosen ligand set.\nFigure R4. Details and distributions of the ligand (a–c), radical (d–f), and nucleophile\n(g–i) sets constructed for LFER modeling.\nFor the radicals, we considered radical class (primary, secondary, tertiary), the\nelectronic character, and steric environment of the reactive site, selecting 126 radicals\nrepresentative of those encountered in Cu-catalyzed radical couplings (Figure R4d).\nThis collection comprehensively covers the substrate variations relevant to the reaction\nscope. To further map the radical set, we analyzed the distributions of their SOMO\nenergy levels (electronic effect) and the buried volume at the radical site (steric effect).\nAs shown in Figure R4e and R4f, the distributions follow typical Gaussian profiles,\ndemonstrating the soundness of our sampling strategy.\nFor the nucleophiles, we accounted for differences in hybridization, electronic effects,\nand steric features, ultimately selecting 75 representative nucleophiles (Figure R4g).\nThese nucleophiles cover the types typically involved in Cu-catalyzed radical couplings.\nIn the revised manuscript, we also evaluated their distributions using the HOMO energy\nlevel (electronic effect) and the buried volume at the bond formation site (steric effect).\nThe results (Figure R4h and R4i) again show smooth and symmetric distributions,\nfurther confirming the representativeness of the sampling.\nIn summary, the reviewer’s constructive suggestion prompted us to systematically\nsupplement both the logic and visualization of our chemical space construction. These\nadditions significantly enhance the clarity and persuasiveness of the chemical space\nrepresentation in our study. The corresponding discussions have been incorporated into\nthe revised manuscript and supplementary information.\nComment: Unclear Data Splitting Criteria. The paper lacks a detailed explanation of\nhow the dataset was split into training and test sets. Ensuring the test set is\nrepresentative of the entire data distribution is critical for validating model\ngeneralizability. Figures 3C and 3D show a notable absence of test points in the lower\nΔG‡ range. A more thorough and transparent description of the splitting criteria,\nincluding how the test set captures the full data variation, is necessary to evaluate the\nrobustness of the statistical model.\nResponse: We sincerely thank the reviewer for this important reminder. We fully agree\nthat the criteria for splitting the dataset into training and test sets are critical to ensuring\nfairness and scientific rigor in model evaluation, and should be clearly articulated in the\nmanuscript. In the revised version, we have substantially expanded the discussion and\nprovided a detailed description of the splitting logic together with visual validation\nanalyses. We believe these revisions significantly enhance the robustness and\ncredibility of the LFER model.\nFor the training set, since the construction of the LFER model requires fitting over a\ncomplete set of three-component combinations of ligands, radicals, and nucleophiles to\nsystematically evaluate the feasibility of linear correlations, it was necessary to restrict\nthe number of ligands, radicals, and nucleophiles included. Specifically, we selected 5\nligands, 10 radicals, and 5 nucleophiles (Figure R5a). Following the same rationale\nused in constructing the broader chemical space, we ensured that the training set\ncaptured key variations along relevant chemical dimensions: the ligands covered\ndifferences in denticity and charge state; the radicals included primary, secondary, and\ntertiary classes as well as diverse electronic and steric substituent effects; and the\nnucleophiles represented sp, sp², and sp³ hybridization types. These species combined\nto form 250 reactions, corresponding to 750 possible bond formation transition states.\nWe computed the associated activation barriers and used them to construct the training\nset for the LFER model.\nFor the test set, our goal was to rigorously challenge the extrapolative power and\nreliability of the model. To this end, we intentionally avoided reusing molecular\nstructures already present in the training set. Considering the combinatorial explosion\nof possible species, we did not attempt exhaustive enumeration; instead, we designed\nthe test set to maximize chemical diversity by varying ligands, radicals, and\nnucleophiles simultaneously within each reaction. In the original version, this test set\ncomprised 59 entirely new reactions. In the revision, following the reviewer’s\nsuggestion, we further supplemented the test set with 8 additional reactions to improve\nthe coverage of the barrier distribution (vide infra), resulting in a total of 67 reactions\nas the final test set, incorporating 9 new ligands, 51 new radicals, and 32 new\nnucleophiles not included in the reactions for constructing LFER (Supplementary Table\n18 to 20). The structures of the selected ligands, radicals and nucleophiles in the revised\ntest set are elaborated in Figure R5b to R5d. This design ensures that the test set is\nstructurally and physicochemically distinct from the training set, while also improving\nits coverage across the full energetic range, thereby providing a stringent and balanced\nevaluation of the model’s predictive performance.\nTo further illustrate the representativeness of the training and test sets, we employed\nUMAP dimensionality reduction to visualize and compare their distributions with those\nof the complete set of 132,300 predicted reactions. Based on six key physical organic\ndescriptors (including the SOMO energy of radicals, the buried volume of radicals, the\nHOMO energy of nucleophiles, the buried volume of nucleophiles, the SOMO energy\nof the LCu(II) active catalyst, and the buried volume of the LCu(II) active catalyst),\nFigure R5e shows that the training (blue) and test (red) sets are evenly distributed across\nthe overall chemical space and together span the major regions of interest. In addition,\nwe performed the same two-dimensional projection using Morgan fingerprints (Figure\nR5f), which again confirmed that the training and test sets are well balanced in\nstructural diversity.\nFigure R5. Structural details of the training set (a) and the test set (b–d), along with\ntheir visualization within the overall chemical space (e–f).\nRegarding the reviewer’s observation that Figures 3c and 3d showed a lack of test points\nin the lower ΔG‡ range, we are very grateful for this important suggestion. It highlighted\nto us that structural diversity does not necessarily guarantee diversity across the\nenergetic domain. To address this gap, in the revision we rationally selected 8 additional\nreactions based on their structural diversity and, importantly, on LFER predictions\nindicating low barriers. The structures of these reactions are shown in Figure R6a.\nAs expected, these added reactions effectively populate the lower ΔG‡ region, and the\nLFER model’s predictions remain in excellent agreement with the corresponding DFT\nresults (Figure R6b). With the revised and more comprehensive test set, the LFER\nmodel maintains excellent agreement with the corresponding DFT results, with mean\nabsolute errors of only 1.8, 1.6, and 1.7 kcal/mol for the reductive elimination, radical\nsubstitution, and ion pair-type pathways, respectively. These results confirm that the\nmodel remains robust and reliable over the full energetic range.\nThanks to the reviewer’s suggestion, these additions have substantially improved the\ntransparency and rationale of our dataset construction. The corresponding analyses and\nfigures have been incorporated into the revised manuscript.\nFigure R6. Additional reactions supplementing the test set: (a) structures of ligands,\nradicals and nucleophiles in the eight newly included reactions; (b) their placement in\nthe lower ΔG‡ region (highlighted in red) and corresponding LFER versus DFT results,\nshowing maintained accuracy across pathways.\nComment: MLR as an Alternative for Deriving RIP Parameters. Since the MLR\napproach is used to derive RIP parameters (supplementary figure 6) for a majority of\nthe radicals (74 out of 126), a more robust assessment of this model's predictive\ncapability is essential. I strongly suggest applying 5-fold cross-validation or similar\napproaches to rigorously evaluate its performance. Given the importance of these\nparameters in shaping the LFER-based mechanistic insights, a deeper validation is\nneeded to support the reliability of the MLR approach.\nResponse: We sincerely thank the reviewer for this insightful comment. As the\nreviewer noted, for the ion pair-type bond formation pathway, the corresponding\ntransition states often collapse during geometry optimization into either reductive\nelimination or radical substitution when the carbocation intermediates are unstable.\nThis makes it difficult to directly obtain R parameters for a substantial subset of\nIP\nradicals. Inspired by the pioneering work from the Sigman group on the application of\nmultivariate linear regression (MLR) in physical organic chemistry, we adopted this\nstrategy in our study to generate R parameters for cases where transition state\nIP\noptimization was not feasible. We fully agree with the reviewer that a more systematic\nand rigorous validation of the MLR procedure is essential.\nFigure R7. Multivariate linear regression analysis for R parameter prediction.\nIP\nDuring the revision, we performed 5-fold cross validation on the dataset of 48 radicals\nused for building the MLR model. The results show that our original MLR equation (eq\n1) achieves good statistical performance in predicting R parameters, with R² = 0.879\nIP\nand MAE = 2.03 kcal/mol (Figure R7b). Given the limited dataset size, however,\napplying 5-fold cross validation may raise concerns of potential underfitting. To further\naddress this, we also carried out leave-one-out (LOO) cross validation, which yielded\nvery similar results (R² = 0.927, MAE = 1.98 kcal/mol, Figure R7c), demonstrating that\nthe model remains robust under different cross-validation schemes.\nIn addition, in the revision we expanded the set of input descriptors for the MLR,\nleading to the development of an improved R predictive equation (eq 2). This refined\nIP\nmodel exhibited enhanced statistical performance under both validation approaches.\nThe model achieved an R2 of 0.882 and an MAE of 2.00 kcal/mol under 5-fold cross\nvalidation (Figure R7e), and an R2 of 0.933 with an MAE of 1.91 kcal/mol under leave-\none-out validation (Figure R7f). Together, these results support the reliability of the\nMLR approach in predicting R parameters and provide a more solid foundation for\nIP\nthe LFER model. All corresponding analyses and figures have been incorporated into\nthe revised Supplementary Information.\nComment: 3. Inconsistencies Between Figures/Tables and Manuscript. There are\nseveral inconsistencies in ligand labeling between the manuscript and the figures. For\nexample, on page 5, the acetylacetone ligand is incorrectly labeled as L10, whereas it\nshould be L9. Similarly, the Pybox ligand is listed as L9 but should be L10. Moreover,\nconfusion appears to persist with the Pincer-type anionic ligands (L11-L14). A\ncomprehensive review and correction of ligand labels across the manuscript is\nwarranted for clarity and reproducibility.\nResponse: We thank the reviewer for pointing out this important issue. We have\nthoroughly reviewed all figures, tables, and references in the text, and have corrected\nthe labeling errors. The revised manuscript and Supplementary Information now uses\nconsistent ligand labels throughout.\nComment: Model Accuracy May Be Inflated by Reaction Imbalance. According to the\nreaction space described, ~70% of the reactions are dominated by the reductive\nelimination (RE) pathway. Therefore, a naive model that predicts RE as the operative\nmechanism for all cases would already achieve ~70% accuracy. The current model\npredicts the correct mechanism in 8 out of 10 test cases (~80%), but notably, the two\nmisclassified examples correspond to non-RE pathways. To more rigorously test the\nmodel’s predictive utility, I recommend including a new validation set containing at\nleast 10 transformations where the operative mechanism is exclusively not RE (i.e., RS\nor IP), in order to assess the model’s performance across the full mechanistic landscape.\nResponse: We sincerely thank the reviewer for this valuable comment. In the original\nversion of the manuscript, the reactions we selected for evaluation were primarily cases\nwhere the mechanism had already been studied in detail by other groups using DFT.\nThis was intended to provide an independent and fair assessment of our model’s\npredictive reliability. However, as the reviewer pointed out, this selection criterion\ninadvertently led to a dataset dominated by RE pathways, which in turn inflated the\nbaseline accuracy and limited the extent to which the predictive advantage of our model\ncould be clearly demonstrated. Thanks to the reviewer’s suggestion, in the revision we\nhave explicitly expanded the distribution of RS and IP pathways in the test set, thereby\nproviding a more balanced and rigorous evaluation of the LFER model.\nSpecifically, we added 10 new reactions that our LFER model predicts to proceed via\neither the RS or IP pathway (Figure R8). These additional cases involve 7 distinct\nligands, 9 radicals, and 9 nucleophiles, thus posing a substantial challenge in terms of\nstructural diversity. Comparison of the model predictions with DFT validation shows\nthat the model correctly identified the operative pathway in 9 out of 10 cases. The only\nmisclassified case was the reaction of R110 with N32, where the model predicted RS\nas the favored pathway, while DFT calculations indicated that RE is more favorable.\nFigure R8. Additional validations with RS- and IP-dominated reactions.\nThis additional validation demonstrates that the LFER model retains strong predictive\npower not only in datasets dominated by RE but also in scenarios where RS and IP\npathways are operative. It thus confirms the robustness and broad applicability of the\nmodel across the full mechanistic landscape. Based on these results, we have\nconsolidated the two sets of test cases and updated the corresponding discussions.\nComment: 5. LFER Predictive Robustness and Extrapolation Ability. The LFER\nmodels presented in Figure 3C should be subjected to 5-fold cross-validation or similar\nresampling methods to assess their extrapolative power. This would help determine\nwhether the models are truly generalizable or simply overfit to the current dataset.\nResponse: We thank the reviewer for this valuable suggestion. Indeed, cross-validation\nprovides a more rigorous way to evaluate the robustness of predictive models. However,\nbecause construction of the LFER equation requires a fixed set of reactions as the\nreference, the validation cannot be carried out through a conventional random split of\nthe full dataset. If certain reference reactions were excluded from the training set, it\nwould not be possible to construct the corresponding LFER parameters, and as a result,\npredictions for some test cases could not be performed.\nTherefore, in the revised manuscript we implemented a validation scheme inspired by\nk-fold cross-validation, while keeping the reference reactions fixed (Figure R9a).\nSpecifically, the full dataset was randomly divided into five folds. In each iteration, four\nfolds were combined with the reference reactions to construct the LFER model, and the\nremaining fold was used for validation. This strategy preserves the reference\nrequirement intrinsic to the LFER framework, while maintaining the breadth and\nindependence emphasized in k-fold approaches.\nAs shown in Figure R9b, the LFER models trained under this protocol exhibit\nconsistently strong predictive performance across all three pathways: for RE, the model\nachieved an average R² of 0.991 with an average MAE of 1.74 kcal/mol; for RS, an\naverage R² of 0.957 with an average MAE of 1.89 kcal/mol; and for IP, an average R²\nof 0.959 with an average MAE of 2.12 kcal/mol. These results clearly demonstrate that\nthe LFER model maintains strong predictive power under extrapolative validation.\nFigure R9. Validation of the LFER model using a modified 5-fold cross-validation\nscheme, showing (a) the validation protocol and (b) average predicted performances of\nten trials.\nResponse to Referee 3\nComment: I co-reviewed this manuscript with one of the reviewers who provided the\nlisted reports. This is part of the Nature Communications initiative to facilitate training\nin peer review and to provide appropriate recognition for Early Career Researchers who\nco-review manuscripts.\nResponse: We sincerely thank Reviewer 3 for co-reviewing this manuscript together\nwith Reviewer 4, and we greatly appreciate the valuable feedback provided. In\nparticular, the comment regarding the importance of thoroughly exploring\nconformational space for flexible molecules offered us crucial guidance. In the revised\nmanuscript, we have provided a detailed description of our conformational search\nprotocol, including a representative case study to illustrate the workflow. These\nadditions substantially improve the transparency and reliability of our computational\napproach.\nResponse to Referee 4\nComment: I find this study very interesting for both computational and experimental\ncolleagues. I believe it can be published in Nature Communications, even though the\nresults are not extremely groundbreaking and the applied methods are fairly standard.\nNevertheless, I appreciate the paper for the way it bridges theory and experiment. The\nmanuscript is also well-written.\nResponse: We really appreciate the reviewer’s recognition of our efforts to bridge\ntheory and experiment, which was one of our central motivations in this study. Our goal\nin developing the LFER framework is to provide a systematic way to connect\nmechanistic insights from high-level computations with concepts familiar to\nexperimental chemists, thereby enabling a more predictive understanding of substituent\neffects in copper-mediated radical bond formation. We hope that this approach will not\nonly aid in interpreting complex mechanistic trends but also serve as a practical tool to\nguide the rational design of new catalytic systems.\nComment: However, I do have one question regarding the computational approach.\nThe systems studied involve flexible organic molecules, and it is crucial to perform a\nthorough exploration of the conformational space for both transition states (TSs) and\nminima. How was this aspect addressed? Including a comment on this in the\ncomputational details of the Supporting Information (SI), and perhaps even in the main\ntext, would be desirable. I recommend the publication with minor revisions.\nResponse: We sincerely thank the reviewer for this valuable comment. As the reviewer\npointed out, the chemical space studied in this work indeed involves organic molecules\nwith flexible substituents. A thorough exploration of conformational space for both\ntransition states and intermediates is therefore essential. This step is not only critical for\nensuring the statistical reliability of the LFER models but also directly impacts the\napplicability and accuracy of the LFER parameters when predicting new reactions.\nIn practice, we systematically examined the conformational factors associated with the\ncatalytic system. For each reaction, we constructed a comprehensive set of initial guess\nstructures and performed geometry optimizations at the selected DFT level of B3LYP-\nD3(BJ)/6-31G(d)-LANL2DZ. Among the optimized conformers, the lowest-energy\nstructure was consistently chosen as the representative for subsequent analysis.\nTo make our conformational search workflow more transparent, we included in the\nrevision a representative case study involving a flexible substrate combination (Figure\nR10). In this example, we first considered conformational factors from the nucleophile\nand radical individually: the vinyl fragment of the nucleophile has two key orientations,\nwhile the α-naphthyl radical features two distinct conformers due to ring orientation.\nThe ligand is a rigid tridentate system and thus does not introduce additional\nconformational complexity at the transition state.\nFigure R10. Representative conformational search for a flexible substrate combination,\nshowing the selected system (a), key conformational factors considered (b), and the\nresulting ensemble of 24 conformers with the DFT-optimized lowest-energy transition\nstate structure (c). Distances are given in Å.\nWhen combining the radical and nucleophile, the bond formation region relative to the\nLCu(II) catalyst can adopt two possible orientations along the z-axis. In addition, the\nradical can freely rotate around the Cu–C bond, leading to three further relative\norientations. Together, this introduced six additional conformational possibilities. When\ncombined with the intrinsic conformers of the radical and nucleophile, a total of 6 × 2\n× 2 = 24 initial transition state guesses were generated. Optimizations of these structures\nyielded 24 unique transition states (Table R2), from which the lowest-energy TS-17\nwas selected for inclusion in the LFER analysis.\nThrough this procedure, we ensured a systematic treatment of conformational space.\nDetails have been provided in the revised Supplementary Information, and a concise\ndescription has also been added to the main text to address the reviewer’s concern.\nTable R2. Summary of the 24 initial transition state conformers generated for the\nrepresentative case and corresponding optimization results.\nConformations Conformations Bond Formation Radical ΔΔG‡\nNo\nof Nucleophile of Radical Orientations Rotations (kcal/mol)\nTS-1 A A Down A 5.2\nTS-2 A A Down B 4.4\nTS-3 A A Down C 1.4\nTS-4 A A Up A 3.8\nTS-5 A A Up B 4.9\nTS-6 A A Up C 1.9\nTS-7 A B Down A 1.7\nTS-8 A B Down B 4.7\nTS-9 A B Down C 3.1\nTS-10 A B Up A 5.2\nTS-11 A B Up B 1.9\nTS-12 A B Up C 4.1\nTS-13 B A Down A 6.2\nTS-14 B A Down B 3.2\nTS-15 B A Down C 3.2\nTS-16 B A Up A 6.4\nTS-17 (Optimal) B A Up B 0.0\nTS-18 B A Up C 2.2\nTS-19 B B Down A 4.5\nTS-20 B B Down B 1.6\nTS-21 B B Down C 5.9\nTS-22 B B Up A 8.9\nTS-23 B B Up B 4.3\nTS-24 B B Up C 6.1"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am happy with the changes and support the publication of this manuscript in its current form", "reviewer_2": "In their revision, I believe that the authors have successfully addressed my comments.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "I find the revised version of the paper much stronger than the initial one. I congratulate the authors on the substantial additional work they have done. My criticisms were adequately addressed. Overall, I recommend the paper for publication as is."}, "author_response": ""}]

Decision letter