Optimum Atomic Shape for Bertaut Series

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Abstract
Calculations are made of the error in the Madelung constant when computed by Bertaut's method with a finite number of terms and with various shapes for the charge distribution within the atoms. A charge density which is proportional to the distance from the edge of the spherical atom gives most rapid convergence for the range of accuracy one percent to 0.003 percent. For higher accuracy a better shape is charge density proportional to the square of the distance from the edge. If Gaussian atoms are used and overlap neglected, no choice of parameter allows an accuracy for a given number of terms as great as for the above functions. If the overlap effect is included by a second series (double series of Ewald), the convergence can be made more rapid than for any of the single series methods, but at the price of a more complicated program.

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