  The Feshbach resonance provides precise control over the scattering length
and effective range of interactions between ultracold atoms. We propose the
ultratransferable pseudopotential to model effective interaction ranges $-1.5
\leq k_\mathrm{F}^2 R_\mathrm{eff}^2 \leq 0$, here $R_\mathrm{eff}$ is the
effective range and $k_\mathrm{F}$ is the Fermi wave vector, describing narrow
to broad Feshbach resonances. We develop a mean-field treatment and exploit the
pseudopotential to perform a variational and diffusion Monte Carlo study of the
ground state of the two-dimensional Fermi gas, reporting on the ground-state
energy, contact, condensate fraction, momentum distribution, and
pair-correlation functions as a function of the effective interaction range
across the BEC-BCS crossover. The limit $k_\mathrm{F}^2 R_\mathrm{eff}^2 \to
-\infty$ is a gas of bosons with zero binding energy, whereas $\ln(k_\mathrm{F}
a) \to -\infty$ corresponds to noninteracting bosons with infinite binding
energy.
