Ewald summation versus direct summation of shifted-force potentials for the calculation of electrostatic interactions in solids: A quantitative study

The calculated Madelung energies and Madelung forces of the electrostatic interaction for nine crystal structures are reported. The method of direct summation with two different shifted-force potentials is compared to the Ewald summation. There is a considerable difference in the convergence of the energy and the force for the two shifted-force potentials regarding the cutoff radius. The convergence depends not only on the potential itself, but also on the crystal structure. One of the shifted-force potentials used is implemented in the CHARMM force field. The energy calculated with this potential shows a good convergence for small cutoff radii. With the other shifted-force potential, the force shows a better convergence for small cutoff radii. The number of pair interactions for obtaining the Madelung limit using the Ewald summation and the direct summation of a shifted-force potential is also reported. For complex structures like zeolites, the number of relevant pair interactions is smaller using the direct summation of a shifted-force potential. For simple structures such as cesium chloride, the number of significant pair interactions is smaller using the Ewald summation.