= \int{w^{H,i}_A(\mathbf{r})\rho_{system}(\mathbf{r})\mathrm{d}\mathbf{r}},$$ where $\rho_{system}(\mathbf{r})$ is the CDD of the periodic system.

Spatial integration of the population {#ssc:spatInteg}
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In chemistry, due to the exponential decay of the charge density of atoms and molecules, and due to the sharp cusps present in CDDs at the atomic nuclei, atom centered grids are widely and successfully used. This makes them ideally suited for integrations such as given in Eq. . The multicenter integration scheme proposed by Becke splits up the full space integration into a set of overlapping atom centered spherical integrations.[@BeckeAD:1987JCP] To solve the problem of double counting in the overlapping regions a weight $h_{A}(\mathbf{r})$ is given to each point in space for every atom A in the system, such that $$\sum_{A}^{AIS}{h_{A}(\mathbf{r})}\equiv 1\qquad \forall\mathbf{r}.$$ This weight function indicates how much a point ‘belongs’ to a certain atom $A$. The weight function can be binary, when the space is split up in Voronoi or Wigner Seitz cells,[@VoronoiGeorges:JRAM1907; @WignerSeitzCell:PhysRev1933] or smoothly varying, as is the case in the Becke scheme.[@BeckeAD:1987JCP] As a result, an integrand $F(\mathbf{r})$ can be decomposed as $F(\mathbf{r})=\sum_{A}^{AIS}{h_{A}(\mathbf{r})F_{A}(\mathbf{r})}$ and the full integration becomes $$\label{eq:Becke_IntSplit}
I=\int{F(\mathbf{r})\mathrm{d}\mathbf{r}}=\sum_{A}^{AIS}{\int{h_{A}(\mathbf{r})F_{A}(\mathbf{r})\mathrm{d}\mathbf{r}}},$$ where the sum over all AIS is again an infinite sum. However, in numerical implementations for 