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.