The sampled structure ( X), reactant ( R), and product ( P ) in Fig. can be represented using either Cartesian, internal, or distance matrix coordinates. By utilizing these geometric vectors, one can define the tangent vector T , perpendicular vector X ⊥ , and the advancement of reaction (ζ). The tangent vector T represents the one-dimensional direction of the reaction while the remaining vibrational degrees of freedom are contained in X ⊥ . The advancement of the reaction (ζ) represents the magnitude of X projected onto the direction of the reaction. For convenience, ζ is redefined such that its values lie between 0 and 1, where ζ = 0 corresponds to the reactant and ζ = 1 corresponds to the product.