Chemical reactions are often understood as molecules crossing a high-energy saddle point, known as a transition state, before following a downhill pathway toward a product. Conventional quantum chemical calculations typically trace this single pathway to determine how a reaction proceeds. In recent years, however, it has become increasingly evident that some chemical reactions can follow multiple pathways after passing through the same transition state, producing two or even three different products. This phenomenon, known as post-transition-state bifurcation (PTSB), challenges the traditional understanding that one transition state leads to one product.
Although molecular dynamics simulations can reveal such branching behavior, they often require extensive calculations involving hundreds or thousands of molecular trajectories. Moreover, conventional reaction-path calculations may overlook alternative pathways entirely. Developing a reliable and practical method to identify these hidden connections is therefore important for understanding complex chemical reactions.
A research team led by Professor Toshiyuki Takayanagi of the Graduate School of Science and Engineering at Saitama University aimed to identify alternative reaction pathways that conventional calculations fail to detect. The researchers applied the double-ended string method, a computational technique that optimizes pathways connecting two molecular structures, to 33 reaction systems, including reactions that produce two or three different products. Using machine-learning-based energy calculations and quantum chemical methods, they examined whether alternative products could be reached from the same transition state without crossing an additional energy barrier higher than that transition state. The study was published online on 28 September 2026 in the Journal of Computational Chemistry. Through this approach, the team successfully identified previously overlooked reaction pathways and distinguished downhill connections from those involving small additional energy barriers.
Key findings of the study include:
- In all 27 intramolecular cycloaddition systems examined, the method identified downhill connections between a transition state and alternative products overlooked by conventional reaction-path calculations.
- In another cycloaddition reaction, the researchers identified a small energy barrier of just 0.27 kcal/mol above the original transition state, demonstrating the method's ability to distinguish different types of reaction pathways.
- In two reactions capable of producing three distinct products, the method clarified which alternative pathways were downhill and which involved an additional energy barrier.
- The approach also proved applicable to three non-pericyclic reactions, including radical addition and nitrene insertion, demonstrating its usefulness beyond cycloaddition reactions.
The study provides a new perspective on how computational chemistry can investigate complex reaction mechanisms. "Conventional calculations generally follow only one pathway from a transition state, but our results show that important alternative connections can remain hidden, " explains Keita Yamaki, a doctoral student at the Graduate School of Science and Engineering at Saitama University and the first author of the paper. "By examining these alternative pathways directly, we can build a more complete picture of how chemical reactions proceed and better understand why a single transition state may lead to different molecular products."
Importantly, the method identifies pathways on a calculated energy landscape rather than predicting how frequently each product will form. It therefore complements, rather than replaces, molecular dynamics simulations.
Looking ahead, a more complete understanding of competing reaction pathways could contribute to the development of more selective and efficient chemical synthesis. Such advances may eventually benefit pharmaceutical manufacturing, fine chemicals, and advanced materials, where controlling reaction outcomes is particularly important.
We hope that integrating this approach with automated reaction-path searches and molecular dynamics simulations will make it easier to understand and predict complex chemical transformations," says Professor Takayanagi. "By identifying alternative reaction pathways that might otherwise be overlooked, this methodology could ultimately support the design of more selective chemical processes, contributing to the development of new medicines and functional materials. Further research will be needed to translate these computational insights into practical applications."
