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

Clifford Boundary Conditions for Periodic Systems: the Madelung Constant of Cubic Crystals in 1, 2 and 3 Dimensions