Extended Thomas-Fermi density functional for the unitary Fermi gas

We determine the energy density xi(3/5)n epsilon(F) and the gradient correction lambda h(2)(del n)(2)/(8mn) of the extended Thomas-Fermi (ETF) density functional, where n is the number density and epsilon(F) is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that xi=0.455 and lambda=0.13 give the best fit of the DMC data with an even number N of particles. We also study the odd-even splitting gamma N(1/9)h omega of the ground-state energy for the unitary gas in a harmonic trap of frequency omega determining the constant gamma. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.