PageRank as a collective variable to study complex chemical transformations and their energy landscapes

A reduced set of reaction coordinates is often employed in chemistry to describe the collective change between reactants and products within the context of rare event theories and the exploration of energy landscapes. Yet selecting the proper collective variable becomes increasingly challenging as the systems under study become more complex. Recent advancement of new descriptions of collective molecular coordinates has included graph-theoretical metrics, including social permutation invariant and PageRank (PR) coordinates, based upon the network of interactions about molecules and atoms within a system. Herein we continue the development of PR by (1) presenting a new formulation that is continuous along a reaction path, (2) illustrating that the fluctuations in PR are demonstrative of the fundamental motions of the atoms/molecules, and (3) providing the analytical derivatives with respect to atomic coordinates. The latter is subsequently combined with a harmonic bias to create the potential of mean force (PMF). As an example, we first consider the transformation of tetrahedral $[Al(OH)4](aq)−$ to octahedral $[Al(OH)4(H2O)2](aq)−$ using the PR PMF. Second, we explore the interchange of contact ion pair and solvent separated ion pairs of aqueous Na⋯OH, where the distance-biased PMF is projected onto PR space. In turn, this reveals where solvent rearrangement has the most impact upon the reaction pathway.