periodic systems, the exponential decay of the atomic CDD allows us to truncate both the infinite sum and integration region of Eq., without significant loss of numerical accuracy. The sum can be reduced to contain only the atoms included in the SoI of atom $A$ (orange circles in Fig. \[fig:Hir\_SphereOfInfluence\]), because only these atoms contribute significantly to the density in the integration region around atom $A$. In addition, the integration region for all atoms in the SoI can be reduced even further, without loss of accuracy, to only the region that overlaps with the spherical integration region of atom $A$ (blue shaded disc in Fig. \[fig:Hir\_SphereOfInfluence\]).

![(color online) Schematic representation of the integration scheme used for a linear periodic system of atoms (green discs). The black rectangle indicates the unit cell, and the circles indicate the boundaries of the spherical integration regions. The spherical integration regions for the atoms in the sphere of influence (SoI) of atom A are shown in orange, those outside are shown in grey. All integrations can be limited to the blue region (or sections thereof) around atom A (see text). []{data-label="fig:Hir_SphereOfInfluence"}](Hir_SphereOfInfluence.png "fig:"){width="8.0cm"}\

Grid stored charge densities and frozen core pseudo-potentials.
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In periodic systems, the 