  We perform a large suite of $N=4$ numerical scattering experiments between
two identical binaries consisting of identical point particles in a
(continuous) background potential. For investigative purposes, albeit without
loss of generality, we assume that the potential corresponds to a uniform
(natal or star-forming) gas medium. We explore a range of constant gas
densities, from $n=10~ {\rm cm}^{-3}$ to $10^{5}~ {\rm cm}^{-3}$. These
densities are relevant for various astrophysical environments, including
molecular clouds (i.e., star-forming regions) and denser, fragmented cores
within these clouds. Our primary goal is to characterize the effects of the
background potential on the subsequent stellar dynamics. We consider the
outcome probabilities as well as the properties of any binaries formed during
the binary-binary encounters, such as the distributions of binary binding
energies and eccentricities. We also present the final velocity distributions
of the ejected single stars. The background potential has two important effects
on the stellar dynamics: 1) The potential acts to reset the zero-point of the
total system energy, which in turn affects the types and properties of the
products of the encounter; 2) For higher $n$ and weakly bound systems (i.e.,
large semimajor axes), the stellar dynamics are significantly affected when
stars become trapped in the potential, oscillating around the system centre of
mass (CM). This, in turn, increases the number of scattering events between
stars (single, binary or triple) near the CM and makes it harder for single
stars to escape to infinity. This ultimately leads to the preferential
ionization of triples and wide binaries and the survival of compact binaries,
with the single stars escaping at very high ejection velocities.
