Ewald summations and series convergence Does anyone have a explanation of why the direct summation of the coulomb interactions between the ions of a crystal structure which might be conditionally convergent is mathematically allowed to be converted to an absolutely convergent series by using the Ewald summation method?
How do we know that the Ewald summation is the best way to make this conversion? Could there be other methods that could convert the direct summation into a different absolutely convergent series which might give us a different answer?
 A: This is a topic with a long history. Even though the Ewald formula is just this year becoming a one-century old method to evaluate the electrostatic energy of periodic systems, it continues to motivate new studies. Interestingly, a diffuse confusion exists about the real role of the Ewald formula in connection with the conditional convergence of the direct summation of all the pairwise potential energy terms. The false idea that the Ewald formula alone cures the ill-defined convergence of the direct sum is quite widespread.
Things are different. Firstly, one has to identify the source of the conditional convergence, then finding a way to suitably isolate a physically relevant absolutely convergent series. Only at this stage is it possible to introduce Ewald's method to improve the speed of convergence of the isolated absolutely convergent series.
From the logical point of view, it is not directly Ewald's formula that transforms a conditionally convergent series into an absolutely convergent one. Instead, Ewald's formula hinges on choosing a well definite and physically motivated way of summing all the contributions.
Probably, one of the most up-to-date and terse discussions of such issues is a paper by V. Ballenegger,  J. Chem. Phys. 140, 161102 (2014) https://doi.org/10.1063/1.4872019. Even though the first clear discussion of the problem started with some older paper by A. Redlack and J. Grindlay in the early seventies and by S. W. de Leeuw, J. W. Perram, and coworkers in the late seventies or beginning of eighties. Here, I'll try to sketch the main ideas behind this issue.

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*Where the conditional convergence comes from?
It is a mathematical fact, connected with the slow decaying of the pair-wise Coulomb interaction. Still, it has a nice physical interpretation when we realize that, due to the long-range character of the Coulomb interaction, there is a surface contribution to the energy that does not become negligible compared to the bulk contribution when the size of the system grows. If we build up the total energy as the sum of the interactions between neutral cells, if one of those cells has an electric dipole moment, a surface cell interacts with a bulk cell via dipole-dipole energy, which is still borderline between diverging and converging interaction laws. The physical effect is that interaction energy does depend on the shape of the finite system. The conditional convergence is physically due to the possibility of increasing the system's size by suitably varying the shape of the system to keep the sequence of energies as close as possible to any real number arbitrarily chosen. This way, the series may converge to whatever real number we like.


*How to get meaningful results?
Once understood the origin of the problem, the recipe is quite obvious. We should work with a sequence of increasing volumes with a shape suitably chosen to isolate the bulk contribution to the energy from the shape-dependent surface term. In this way, we can get meaningful shape-independent Coulomb energies. Once we have such a contribution, we can always add back the surface dependent term. The procedure is equivalent to choosing a well-defined sequence of finite size systems to perform the thermodynamic limit. For example, a sequence of increasingly large systems keeps the same average potential over each cell's volume.


*Where Ewald algorithm appears?
As I said, only after the absolutely converging sequence has been selected one can ask the question of accelerating the slow convergence of Coulomb interactions. That is the point where Ewald's method enters this game.
It is not the only possible algorithm, and during the years, a few alternative methods have been proposed. However, they cannot provide results different from Ewald's formula if they correspond to the same method to add contributions from different cells.
