Graph invariants for periodic systems: towards predicting physical properties from the hydrogen bond topology of ice

Ice-Ih consists of a disordered hydrogen-bonded network. The degree of disorder in ice-Ih, and possible phase transitions to an ordered phase have been debated in recent years. The dependence of energy, free energy, and other scalar physical properties on H-bond topology is needed to understand these phenomena. Graph invariants provide a means of linking physical properties to the topology of the H-bond network. We have previously shown the effectiveness of graph invariants for finite water clusters [J.-L. Kuo, J. V. Coe, S. J. Singer, Y. B. Band, and L. Ojamäe, J. Chem. Phys., 114, 2527 (2001)]. In this work, we develop a formalism for the graph invariants of periodic systems. We demonstrate that graph invariants for small unit cells are a subset of the graph invariants of larger unit cells, providing a hierarchy of approximations by which detailed calculations for small unit cells, such as periodic ab initio calculations as they become available, can be used to parametrize the energy of the astronomical number of H-bond arrangements present in large unit cells. We also present graph enumeration results for ice-Ih, analyzing conflicting results that have appeared previously in the literature and furnishing information on the statistical properties of the H-bond network of ice-Ih in the large system limit.