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Liquid-liquid phase transitions have been found experimentally or proposed toexist based on computer simulations in many compounds such as water, hydrogen,sulfur, phosphorus, carbon, silica, and silicon. Recently introduced maximalvalence model implemented via discrete molecular dynamics (DMD) provides simplegeneric mechanism of the liquid-liquid phase transitions for all these diversecases. Here we will show that certain variants of the maximal valence modelhave a well defined Hamiltonian i.e a unique algorithm by which the potentialenergy of the system of particles can be computed solely from the coordinatesof the particles and is thus equivalent to a complex multi-body potential. Wewill present several examples of the model which can be used to reproduceliquid-liquid phase transition in systems with maximal valence $z=1$(hydrogen), $z=2$ (sulfur) and $z=4$ (water), where $z$ is the maximal valence,i.e. the maximal number of bonds an atom is allowed to have. The maximalvalence model can be modified to forbid not only too large valences but alsotoo low valences. In the case of sulfur, we forbid also the formation ofmonomers, thus restricting the valence $v$ of an atom to be within an interval$1=v_{min}\leq v\leq v_{max}\equiv z=2$.

Hamiltonian formulation of the maximal valence model for liquid polyamorphism