  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.
