refers to the object of which we want to obtain the atomic charges, and can thus refer to bulk materials, wires, molecules or even single ions. The free atoms/ions on the other hand refer to the single atoms which are used for the generation of the reference radial CDD $\rho_{X}^{i}$ of eqn.. For both types there are two kinds of 3D grids involved:

1.  Linear grids: Instead of using the analytical expression for the underlying wave function, we use the CDDs stored by <span style="font-variant:small-caps;">VASP</span> on a finite numerical grid. These grids span a single unit cell and use uniformly spaced grid points in direct coordinates.[@fn:CHGCAR] (*cf.* Sec.\[ssc:chargedensitygrid\]) In the remainder, the following notation is used:

    -   V$_{atom}$ : The linear grid for the reference atom density distributions as obtained from the atomic calculations.

    -   V$_{sys}$ : The linear grid for the (poly)atomic system under study

2.  Spherical grids: These are atom centered grids which are not limited to a single unit cell. The spherical grids decompose into a radial and a shell grid. In our current setup, a logarithmic grid is used as radial grid, such that closer to the core the grid is sufficiently dense to describe this 