A three-component decomposition of pairwise Coulomb terms in finite crystals produces a direct-summation formula for Madelung constants that converges at p=1 for cubic lattices.
Using this symmetry andx 2 +y 2 = r2 −z 2, the combinationx 2x2 1+y 2x2 2 may be replaced—upon integration—by(r 2 −z 2)x2 2 (or equivalently by(r 2 −z 2)x2 1)
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The Madelung Problem of Finite Crystals
A three-component decomposition of pairwise Coulomb terms in finite crystals produces a direct-summation formula for Madelung constants that converges at p=1 for cubic lattices.