Bond-counting potentials - A classical many-body model of covalent bonding with exact solutions in one dimension

We introduce "bond-counting" potentials, which provide an elementary description of covalent bonding. These simplistic potentials are intended for studies of the mechanisms behind a variety of phase transitions in elemental melts, including the liquid-liquid phase transitions (LLPT) in phosphorus and bismuth. As a first study employing such potentials, an analytic solution of a one-dimensional model system is presented, including its thermodynamic properties and its structure factor. In the simplest case, the chemical valency of each atom is 1, and either single atoms or diatomic molecules are present. At low temperatures and moderate pressures, the system consists almost exclusively of molecules, and single atoms act as topological defects. A slightly more complicated case involves a valency of 2, with either single or double bonding. This system exhibits a first-order LLPT from a molecular to a polymeric phase, as in phosphorus. In this case, the one-dimensional model system exhibits phase separation for finite-sized systems at low temperatures. A variant of this system also exhibits a non-equilibrium phase transformation upon heating the molecular condensed phase, qualitatively similar to boiling in white phosphorus.