  In this work we demonstrate the robustness of a real-space approach for the
treatment of infinite systems described with periodic boundary conditions that
we have recently proposed [J. Phys. Chem. Lett. 17, 7090]. In our approach we
extract a fragment, i.e., a supercell, out of the infinite system, and then
modifying its topology into the that of a Clifford torus which is a flat,
finite and border-less manifold. We then renormalize the distance between two
points by defining it as the Euclidean distance in the embedding space of the
Clifford torus. With our method we have been able to calculate the reference
results available in the literature with a remarkable accuracy, and at a very
low computational effort. In this work we show that our approach is robust with
respect to the shape of the supercell. In particular, we show that the Madelung
constants converge to the same values but that the convergence properties are
different. Our approach scales linearly with the number of atoms. The
calculation of Madelung constants only takes a few seconds on a laptop computer
for a relative precision of about 10$^{-6}$.

Materials Science