Hydrogen-bond (HB) patterns correspond to topologically distinct isomers of water clusters, and can be expressed by digraphs. The HB pattern is used to divide the configuration space of water cluster at a finite temperature. The populations of the HB patterns are transformed into the relative Helmholtz energies. The method is based on the combination of molecular simulation with graph theory. At a finite temperature it can be observed that other isomers than local minimum structures on the

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... tial energy surface are highly populated. The dipole moment of a constituent molecule in a water cluster is enhanced depending on the local HB network around the water molecule. Rooted digraph is used to represent topologically distinct isomers of protonated water (PW) clusters. O–H bonds of PW clusters are classified into 10 topological types based on the combination of the local HB types of the contributing water molecules to the O–H bond. If the topological type is the same, vibrational frequencies of those O–H bonds of PW clusters are similar even in different isomers; i.e. vibrational frequency of O–H bond is transferable, and can be used as a vibrational spectral signature of PW clusters.