Determining point charge arrays that produce accurate ionic crystal fields for atomic cluster calculations

In performing atomic cluster calculations of local electronic structure defects in ionic crystals, the crystal is often modeled as a central cluster of 5–50 ions embedded in an array of point charges. For most crystals, however, a finite three-dimensional repeated array of unit cells generates electrostatic potentials that are in significant disagreement with the Madelung (infinite crystal) potentials computed by the Ewald method. This is illustrated for the cubic crystal ${\mathrm{CaF}}_{\mathrm{2}}.$ We present a novel algorithm for solving this problem for any crystal whose unit cell information is known: (1) the unit cell is used to generate a neutral array containing typically 10 000 point charges at their normal crystallographic positions; (2) the array is divided into zone 1 (a volume defined by the atomic cluster of interest), zone 2 (several hundred additional point charges that together with zone 1 fill a spherical volume), and zone 3 (all other point charges); (3) the Ewald formula is used to compute the site potentials at all point charges in zones 1 and 2; (4) a system of simultaneous linear equations is solved to find the zone 3 charge values that make the zone 1 and zone 2 site potentials exactly equal to their Ewald values and the total charge and dipole moments equal to zero, and (5) the solution is checked at 1000 additional points randomly chosen in zone 1. The method is applied to 33 different crystal types with 50–71 ions in zone 1. In all cases the accuracy determined in step 5 steadily improves as the sizes of zones 2 and 3 are increased, reaching a typical rms error of 1 μV in zone 1 for 500 point charges in zone 2 and 10 000 in zone 3.