  We determine the energy density $\xi (3/5) n \epsilon_F$ and the gradient
correction $\lambda \hbar^2(\nabla n)^2/(8m n)$ of the extended Thomas-Fermi
(ETF) density functional, where $n$ is number density and $\epsilon_F$ is Fermi
energy, for a trapped two-components Fermi gas with infinite scattering length
(unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC)
calculations [Phys. Rev. Lett. {\bf 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 $\gamma
N^{1/9} \hbar \omega$ 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.
