Electronic Collective Variables for Chemical Reactions

Abstract: Chemical reaction sampling critically depends on collective variables (CVs) that capture the slow degrees of freedom governing reactive transformations. However, existing reaction CVs are often defined in geometric space or learned in a system-specific manner, which limits their transferability and leaves open the more fundamental question of how reaction progress should be represented. From a physical perspective, chemical reactions are defined by electron redistribution. Here, we introduce a charge-space electronic collective variable that describes the electronic component of reaction progress in a common linear form based on atomic charges. To enable its use in enhanced sampling, atomic charges and the corresponding CV gradients are provided by a neural-network model trained on QM/MM data within an iterative sampling-training workflow. Across multiple reactions in aqueous and enzymatic environments, we show that this electronic CV can be constructed in a common charge-space form, with the corresponding coefficients assigned in a simple manner from charge differences between relevant states. Our simulations further show that reaction progress generally involves coupled electronic and conformational components, and that the same framework can also be extended to restrain side reactions. These findings support charge-based electronic CVs as a physically motivated framework for describing the electronic component of chemical reaction progress with reduced reliance on handcrafted geometric descriptors.

• The paper introduces a charge-based electronic CV that leverages quantum atomic charges to capture electron redistribution as a robust marker of reaction progress.
• It employs a neural network surrogate model with iterative enhanced sampling to accurately compute CV gradients and ensure transferability across different environments.
• The study demonstrates that integrating electronic and conformational CVs enhances sampling efficiency, improves reaction selectivity, and clarifies mechanistic insights.

Electronic Collective Variables in Chemical Reaction Sampling

Motivation and Conceptual Framework

The problem of defining robust and transferable collective variables (CVs) for sampling chemical reactions in molecular simulations remains open, primarily due to the strong dependence of prevailing CVs on geometric descriptors and system-specific construction. Geometric CVs, typically formulated as bond distances, angles, and dihedrals, demand extensive manual curation, constraining their transferability and limiting their capacity to encapsulate the fundamental physics of reactive events. This work addresses the representation issue at a more intrinsic level by positing that electronic redistribution, rather than mere geometric rearrangement, is the essential feature of chemical reactions.

To operationalize this perspective, the authors propose a charge-based electronic CV, constructed as a linear combination of quantum atomic charges. By defining the CV in "charge space," this approach aims to capture the dominant component of reaction progress in a physically motivated and system-agnostic manner. Importantly, the coefficients of the linear combination are determined in a straightforward fashion—either by charge differences between reactant and product states or using linear discriminant analysis (LDA), circumventing the need for intricate geometric parameterization.

Methodological Implementation

Electronic CV Construction

The CV is expressed as a linear combination:

$CV_{\text{elec}} = \sum_i C_i q_i$

where $q_i$ is the quantum atomic charge of atom $i$, and $C_i$ is the assigned coefficient. For most systems, $C_i$ is proportional to the signed difference between product and reactant charges, with a threshold to ignore negligible contributions. This concise formulation ensures the reaction CV predominantly follows the main direction of electron flow, while suppressing noise from minor, unreactive atoms.

Neural Network Model for Charges and Gradients

To provide $q_i$ and their analytic gradients for enhanced sampling algorithms (e.g., metadynamics), a dedicated neural surrogate model is trained on QM/MM-generated datasets. The architecture builds on an equivariant message-passing framework, efficiently encoding the influence of the MM electrostatic environment as external-potential descriptors acting on the QM region. Feature updates and the external field are coupled to ensure the model's predictions properly reflect both intra-region and environmental effects.

Training follows an iterative workflow. An initial data set is harvested from unbiased sampling, the neural model is trained, and then enhanced sampling is performed with the new CV. New configurations sampled outside the original set trigger retraining. The charge model is regularized using energies and forces from QM/MM calculations, further improving the reliability of CV gradients needed for efficient biasing.

Integration with Enhanced Sampling Protocols

The approach distinguishes between electronic and conformational reactors. While the charge-based electronic CV robustly captures electron redistribution, conformational changes necessary to access reactive configurations are either accelerated via methods such as integrated tempering sampling (ITS) or included as an explicit complementary CV. The total biasing variable used in simulations thus commonly takes the form:

$$CV = C_{\text{conf}} \times CV_{\text{conf}} + C_{\text{elec}} \times CV_{\text{elec}}$$

where $CV_{\text{conf}}$ typically involves linear combinations of forming/breaking bond distances.

Results Across Reaction Classes

Michael Addition: Aqueous and Enzymatic

The electronic CV, constructed from quantum charges, was assessed in the Michael addition of 6'-deoxychalcone both in solution and within chalcone isomerase (CHI). A purely geometry-based CV showed inefficient sampling, requiring extensive chemical intuition and still failing to capture the full reaction coordinate. By contrast, the electronic CV alone correctly resolved electron redistribution but insufficiently accelerated sampling due to the orthogonality of structural adjustment. Efficient sampling was only achieved by combining the electronic CV with ITS (for solution) or by explicitly adding the forming bond as a conformational CV (for CHI’s active site).

Notably, the electronic CV and its coefficients, calibrated in aqueous solution, transferred without modification to the enzyme environment, indicating strong robustness and transferability within a reaction family. The free-energy barrier and reaction profile were distinctly better resolved along the electronic CV than any geometric coordinate.

Claisen Rearrangements: Distinct Reaction Mechanisms

Assessment on the Claisen rearrangement of cis-2-vinylcyclopropanecarboxaldehyde established that the framework generalizes across reaction classes. In this system, geometric and electronic components are strongly coupled, and even a simple conformational CV (difference of breaking and forming bond distances) sufficed when combined with the electronic CV to sample the pathway efficiently. Free-energy surfaces projected on the electronic and conformational CVs were highly congruent.

Suppression of Competing Pathways: Chorismate Rearrangement

The extension to the chorismate-to-prephenate rearrangement demonstrates the utility of charge-based CVs in controlling reaction selectivity. By constructing an additional restraining electronic CV targeting the side reaction, the simulation protocol suppressed the undesired channel without ad hoc geometric restraints. This selective biasing exploits the orthogonality of electronic features distinguishing the competing mechanisms, emphasizing a principal advantage over classical geometric approaches.

Implications and Future Perspectives

The overarching contribution is the formalization and practical demonstration of charge-space electronic CVs as a generalizable, physically motivated descriptor of reaction progress. This framework systematically reduces reliance on handcrafted, system-specific geometric variables, shifting the focus toward data-driven, transferable electronic representations. The robust transfer of CV construction across environments (solution, enzyme) and reaction classes establishes its utility for high-throughput, automated studies in catalysis, enzymology, and machine-learned potentials.

Practically, the integration with enhanced sampling modules is seamless, as the neural model delivers both charges and analytic gradients. Theoretical implications extend to disentangling electronic and conformational contributions along the reaction coordinate, which has relevance for mechanistic studies and for design of improved biasing strategies in rare event sampling.

Looking forward, the framework opens several avenues:

• Extension to higher-order electronic descriptors (beyond atomic charges) could further refine CV resolution.
• Embedding the protocol within workflows for automated reaction discovery and dataset generation for ML potentials is natural.
• Systematic exploration of the coupling and orthogonality between conformational and electronic coordinates may yield new strategies for controlling or predicting chemoselectivity and regioselectivity.

Conclusion

Defining collective variables in charge space by exploiting electronic structure information offers a transferable framework to resolve chemical reaction progress, sharply reducing dependence on geometric intuition and manual engineering. The demonstrated accuracy and versatility of the approach across diverse reaction environments and mechanisms positions charge-based electronic CVs as a foundational tool for enhanced sampling, reaction mechanism elucidation, and next-generation data-driven simulation in chemistry and biophysics.