Chemical reaction pathways estimated by a transformation between manifolds

Chemical reaction pathways are often estimated by linearly interpolating a set of internal coordinates, such as bond lengths, between reactant and product structures. However, these internal coordinates are not generally isomorphic to atomic positions. For example, the number of bond lengths between N atoms is N(N−1)/2, which is usually greater than the 3N -6 coordinates that specify unique molecular geometries. If arbitrarily chosen, the bond lengths may not be physically realizable because they violate the triangle inequality. The solution used in the method known as linear synchronous transit is to employ a least squares minimization to approximate the desired set of generalized coordinates. However, this method is prone to generating discontinuous reaction pathways and outright failure at generating transition states. As an alternative, we treat the spaces of generalized coordinates and Cartesian coordinates as separate manifolds and locally transform a velocity vector from the former into the latter. This results in a first-order differential equation for the reaction path which is numerically integrable even when linear synchronous transit fails.