Here are the notations used in this paper. We consider a system of n atoms atoms, with coordinates represented by a 3n atoms -dimensional vector x, which can also be written as n atoms 3-dimensional vectors r 1 , r 2 , • • • , r natoms . The reactant and product structures are denoted by x r and x p . The interatomic distances between atoms i and j in these structures are d r,ij and d p,ij . The path with fixed endpoints x r and x p is x(t), where t ∈ [0, 1], x(0) = x r , and x(1) = x p .