  We study spin-1/2 fermions in spin dependent potentials under the \emph{spin
model approximation}, in which interatomic collisions that change the total
occupation of single-particle modes are ignored. The spin model approximation
maps the interacting fermion problem to an ensemble of lattice spin models in
energy space, where spin-spin interactions are long-ranged and
spin-anisotropic. We show that the spin model approximation is accurate for
weak interactions compared to the harmonic oscillator frequency, and captures
the collective spin dynamics to timescales much longer than would be expected
from perturbation theory. We explore corrections to the spin model, and the
relative importance of corrections when realistic anharmonic potential
corrections are taken into account. Additionally, we present numerical
techniques that are useful for analysis of spin models on an energy lattice,
including enacting a change of single-particle basis on a many-body state as an
effective time evolution, and fitting of spatially inhomogeneous long-range
interactions with exponentials. This latter technique is useful for
constructing matrix product operators for use in DMRG analyses, and may have
broader applicability within the tensor network community.
