  We show an $\textit{ab initio}$ construction of the energy density functional
(EDF) for electron systems using the functional renormalization group. The
correlation energies of the homogeneous electron gas given in our framework
reproduce the exact behavior at high density and agree with the Monte-Carlo
data in a wide range of densities. Our analytic technique enables us to get the
correlation energies efficiently for various densities, which realizes the
determination of EDF in the local density approximation (LDA) without any
fitting for physically relevant densities. Applied to the Kohn-Sham calculation
for the noble gas atoms, our EDF shows comparable results to those of other
conventional ones in LDA.
