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A simple general proof of the Krazer–Prym theorem and related famous formulae resolving convergence properties of Coulomb series in crystals

The Krazer–Prym theorem extending the Poisson summation formula to oblique lattices is proved as a direct consequence of translational symmetry. A generalized Ewald approach based on this theorem is reproduced. The two other famous approaches proposed by Nijboer and De Wette, and by Harris and Monkhorst, are derived in the same fashion. The absence of any uniform contribution to bulk electrostatic potentials in each of those treatments is emphasized as a property of locally neutral systems subject to translational symmetry. It is found that an analytic evaluation of the Ewald variable parameter naturally follows from the treatment of the Coulomb lattice characteristic in terms of the Krazer–Prym theorem. The extension of those characteristics to off-lattice points is also discussed.