  We study a system of atoms that are laser-driven to $nD_{3/2}$ Rydberg states
and assess how accurately they can be mapped onto spin-$1/2$ particles for the
quantum simulation of anisotropic Ising magnets. Using non-perturbative
calculations of the pair interaction potentials between two atoms in the
presence of both electric and magnetic fields, we emphasize the importance of a
careful selection of the experimental parameters in order to maintain the
Rydberg blockade and avoid excitation of unwanted Rydberg states. We then
benchmark these theoretical observations against experiments using two atoms.
Finally, we show that in these conditions, the experimental dynamics observed
after a quench is in good agreement with numerical simulations of spin-1/2
Ising models in systems with up to 49 spins, for which direct numerical
simulations become intractable.
