This application paper investigates the stability of hydrogen bonds (H-bonds), as characterized by the Quantum Theory of Atoms in Molecules (QTAIM). First, we contribute a database of 4544 electron densities associated to four isomers of water hexamers (the so-called Ring, Book, Cage and Prism), generated by distorting their equilibrium geometry under various structural perturbations, modeling the natural dynamic behavior of molecular systems. Second, we present a new stability measure, called bond occurrence rate, associating each bond path present at equilibrium with its rate of occurrence within the input ensemble. We also provide an algorithm, called BondMatcher, for its automatic computation, based on a tailored, geometry-aware partial isomorphism estimation between the extremum graphs of the considered electron densities. Our new stability measure allows for the automatic identification of densities lacking H-bond paths, enabling further visual inspections. Specifically, the topological analysis enabled by our framework corroborates experimental observations and provides refined geometrical criteria for characterizing the disappearance of H-bond paths. Our electron density database and our C++ implementation are available at this address: https://github.com/thom-dani/BondMatcher.