Coulomb Interactions in periodic systems

In the evaluation of the potential energy in a periodic system the interaction with all possible images has to be considered. This fact could make very inefficient the simulation of periodic systems. The Coulomb interaction ion-ion, ion-electron and electron-electron can be generally written as:

$$U = \frac{1}{2} \sum_{\vec \xi_i \neq \vec \xi_j + \vec R_s,\vec R_s} \frac{q_i q_j}{\left\vert \vec \xi_i -\vec \xi_j + \vec R_s \right\vert},$$ (4.9)

where $\xi_i$ indicates electron coordinates $\vec r$ corresponding to $q_i=-e$ and proton coordinates $R_i$ corresponding to $q_i=+e$ and$\vec{R}_s$ are the vectors of the periodic lattice associated with the simulation box. Notice that this summation converges only for neutral systems $\sum q_i = 0$ . For short range interaction it is possible to consider only the closest images, that represents an efficient and accurate way to calculate the potential energy. For long range interaction the equation eq. 4.9 cannot be used in a numerical simulation because the sum is very slowly convergent, so other approaches are necessary.
It is not possible to use a truncated Coulomb potential. In fact, large inaccuracies are introduced by neglecting the long-range part (see Ref. (78)).
In the following we present the well known Ewald method that allows to evaluate in an efficient way the potential energy in periodic systems.