The predictions of kinetics and mechanisms of complex chemical reactions are severely limited by the high dimensionality of potential energy surfaces (PES). While most reactive processes are rationalized in terms of low-dimensional structures, such as internal reaction coordinates, these structures are obtained by means of energy minimizations and transition state (TS) searches on the full, high-dimensional PES. It is the strong and nonlinear coupling between reactive and nonreactive degrees of freedom (DOFs) that prevents dimensionality reduction by linear methods, for example, principal component analysis (PCA).

Here, we propose a method for constructing a low-dimensional geometric structure within the PES, which locally approximates the reactive DOFs, specifically, bond breaks and bond formations. Due to the nonlinear coupling between reactive and nonreactive DOFs, this structure cannot be simply described as a subspace of the internal coordinate space R^3n-6 of an n-atomic system but as its submanifold. Generally, a manifold as a subset of points in the Euclidean space R^D that has a local one-to-one mapping to a lower-dimensional Euclidean space R^d. The manifold of distance displacements (DD) is defined algorithmically using a random walk in R^3n-6 projected locally onto the set of interatomic distances. Since this construction does not rely on energy evaluations or TS searches, it can be carried out at minimal computational cost.

We show that DD manifolds have low local dimension d^loc, about an order of magnitude smaller than the PES dimension d^PES or the number of interatomic distances d^int. Moreover, their dimension increases only slowly with the number of degrees of freedom. DD manifolds are thus convenient low-dimensional approximations to study chemical reaction dynamics using the methods of manifold learning.

Local dimension d<sup>loc</sup> of distace displacement (DD) manifolds, number of interatomic distances d<sup>dist</sup>, and PES dimension d<sup>PES</sup> in C<i><sub>x</sub></i>H<i><sub>y</sub></i>O<i><sub>z</sub></i> with <i>n</i> = <i>x</i> + <i>y</i> + <i>z</i> = 3–6.

Local dimension d^loc of distace displacement (DD) manifolds, number of interatomic distances d^dist, and PES dimension d^PES in C_xH_yO_z with n = x + y + z = 3–6.

Dmitrij Rappoport

Assistant Researcher, University of California, Irvine

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