Coulomb Potential in Ionic Crystals by Direct Summation

A method is proposed to determine the shape and size of the finite crystal piece to be used in the calculations of the Coulomb potential at points in the unit cell of ionic crystals. The Coulomb potential at a point is calculated by direct summation of the potentials of all the ions in this finite crystal piece. A theorem was established which helps to make the resulting series in the calculations converge much faster, thus cutting the computing time by a factor of millions or more, depending on the accuracy desired. With the help of this theorem, it was possible to calculate the Coulomb potential at a grid of points in the unit cells of NaCl and CsCl structures to 10 decimal places. The Madelung constant for NaCl structure was calculated to 25 decimal places as an illustration of the power of this theorem.