Low-dimensional projection of reactivity classes in chemical reaction dynamics using supervised dimensionality reduction.

Transition state theory (TST) provides a framework to estimate the rate of chemical reactions. Despite its great success with many reaction systems, the underlying assumptions such as local equilibrium and nonrecrossing do not necessarily hold in all cases. Although dynamical systems theory can provide the mathematical foundation of reaction tubes existing in phase space that enables us to predict the fate of reactions free from the assumptions of TST, numerical demonstrations for large systems have been yet one of the challenges. Here, we propose a dimensionality reduction algorithm to demonstrate structures in phase space (called reactive islands) that predict reactivity in systems with many degrees of freedom. The core of this method is the application of supervised principal component analysis, where a coordinate transformation is performed to preserve the dynamical information on reactivity (i.e., to which potential basin the system moves from a region of interest) as much as possible. The reactive island structures are expected to be reflected in the transformed, low-dimensional phase space. As an illustrative example, the algorithm is scrutinized using a modified Hénon-Heiles Hamiltonian system extended to many degrees of freedom, which has three channels leading to three different products from one stable potential basin. It is shown that our algorithm can predict the reactivity in the transformed, low-dimensional coordinate system better than a naïve coordinate system and that the reactivity distribution in the transformed low-dimensional space is considered to reflect the underlying reactive islands.