Treatment of Coulomb interactions in Hartree-Fock calculations of periodic systems

A computational scheme for the treatment of Coulomb sums in the Hartree-Fock approach to periodic systems in one, two, and three dimensions is presented. The philosophy is as follows: (a) The interaction of two charge distributions contributing to the total charge in cells $\overrightarrow{0}$ and $\overrightarrow{\mathrm{m}}$ is treated exactly at short range; (b) when the reciprocal penetration of the distributions is sufficiently small, the charge distribution at $\overrightarrow{\mathrm{m}}$ is partitioned into "shell-charge distributions" which are then expanded in a multipole series; (c) for $\left|\overrightarrow{\mathrm{m}}\right|$ larger than a given threshold, a Madelung sum of atomic charges is performed. Results are reported for the $\mathrm{S}{\mathrm{N}}_x$ polymer, the graphite, boron nitride, and beryllium monolayers, for the beryllium monolayer with hydrogen chemisorbed thereon, and for three-dimensional silicon, with a view to compare two charge-partitioning schemes, and the convergence of the results with respect to the order of the multipole expansions. It is shown that the inclusion of all terms to hexadecapole confines the error in the Coulomb contribution to the total energy for all the systems considered to within 0.001 a.u./atom, with the exactly treated zone reduced to a few neighbors, and that at this level the results are essentially independent of the particular charge-partitioning scheme adopted.