Electrostatic energy of ions in a uniform background

The electrostatic energy per ion of point ions with valence 2 in a uniform compensating background of electrons is given by  

For the background interaction, the same decomposition can be used, which gives  

Because / d re — CtS, o, the last term exactly cancels the divergent term at 9 = 0 in the sum over the ion positions. The remaining integral can 

Differentiation with respect to tj confirms that this expression is indeed independent of rf as it should be. 

The choice of ri determines the convergence of the summations. If one requires that the maximum contribution is given by the structure independent terms, one finds  

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