  We present a comparison of methods for treating the electrostatic
interactions of finite, isolated systems within periodic boundary conditions
(PBCs), within Density Functional Theory (DFT), with particular emphasis on
linear-scaling (LS) DFT. Often, PBCs are not physically realistic but are an
unavoidable consequence of the choice of basis set and the efficacy of using
Fourier transforms to compute the Hartree potential. In such cases the effects
of PBCs on the calculations need to be avoided, so that the results obtained
represent the open rather than the periodic boundary. The very large systems
encountered in LS-DFT make the demands of the supercell approximation for
isolated systems more difficult to manage, and we show cases where the open
boundary (infinite cell) result cannot be obtained from extrapolation of
calculations from periodic cells of increasing size. We discuss, implement and
test three very different approaches for overcoming or circumventing the
effects of PBCs: truncation of the Coulomb interaction combined with padding of
the simulation cell, approaches based on the minimum image convention, and the
explicit use of Open Boundary Conditions (OBCs). We have implemented these
approaches in the ONETEP LS-DFT program and applied them to a range of systems,
including a polar nanorod and a protein. We compare their accuracy, complexity,
and rate of convergence with simulation cell size. We demonstrate that
corrective approaches within PBCs can achieve the OBC result more efficiently
and accurately than pure OBC approaches.
