  We present a real-space formulation and higher-order finite-difference
implementation of periodic Orbital-free Density Functional Theory (OF-DFT).
Specifically, utilizing a local reformulation of the electrostatic and kernel
terms, we develop a generalized framework for performing OF-DFT simulations
with different variants of the electronic kinetic energy. In particular, we
propose a self-consistent field (SCF) type fixed-point method for calculations
involving linear-response kinetic energy functionals. In this framework,
evaluation of both the electronic ground-state as well as forces on the nuclei
are amenable to computations that scale linearly with the number of atoms. We
develop a parallel implementation of this formulation using the
finite-difference discretization. We demonstrate that higher-order
finite-differences can achieve relatively large convergence rates with respect
to mesh-size in both the energies and forces. Additionally, we establish that
the fixed-point iteration converges rapidly, and that it can be further
accelerated using extrapolation techniques like Anderson's mixing. We validate
the accuracy of the results by comparing the energies and forces with
plane-wave methods for selected examples, including the vacancy formation
energy in Aluminum. Overall, the suitability of the proposed formulation for
scalable high performance computing makes it an attractive choice for
large-scale OF-DFT calculations consisting of thousands of atoms.
