Predicting Energies of Small Clusters from the Inhomogeneous Unitary Fermi Gas

We investigate the inhomogeneous unitary Fermi gas and use the long-wavelength properties to predict the energies of small clusters of unitary fermions trapped in harmonic potentials. The large pairing gap and scale invariance place severe restrictions on the form of the density functional. We determine the relevant universal constants needed to constrain the functional from calculations of the bulk in oscillating external potentials. Comparing with exact Quantum Monte Carlo calculations, we find that the same functional correctly predicts the lack of shell closures for small clusters of fermions trapped in harmonic wells as well as their absolute energies. A rapid convergence to the bulk limit in three dimensions, where the surface to volume ratio is quite large, is demonstrated. The resulting functional can be tested experimentally, and is a key ingredient in predicting possible polarized superfluid phases and the properties of the unitary Fermi gas in optical lattices.