We determine the energy density $\xi (3∕5)n{\epsilon }_F$ and the gradient correction $\lambda {\hslash }^2{(\mathbf{\nabla }n)}^2∕\left(8mn\right)$ 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 $\gamma N^{1∕9}\hslash \omega$ 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.

• Received 9 September 2008

DOI:https://doi.org/10.1103/PhysRevA.78.053626