A note on calculating volume integrals

In many cases (e.g. when calculating a projected DOS (LDOS), atomic charges or when analysing the density as to be described in later sections) real space integrals are calculated numerically. Two methods are implemented for those which are referred to as ``conserving'' and ``non-conserving'' algorithms:

• the conserving method: provides the correct charge inside a sphere or a layer, each grid point in the cell is scanned once and only those grid points contribute which are positioned inside the volume; when checking whether a grid point is inside the volume, lattice translations are also applied. As the region of integration increases (e.g. the radius of a sphere), the integration volume approaches that of the cell. Still, the integration volume will never exceed that of the cell. Thus, this method ``conserves'' the volume of the cell.
• the non-conserving method: a finer grid is constructed in the region of interest (a sphere or a layer) and the integrals are calculated by summing up contributions on this finer grid (an interpolation is used). This method does not give the volume conservation as the volume increases, howewer, it is more efficient provided the integration volumes are sufficiently small.

If the size of the cell is large enough with respect to the region of integration, than both methods should give close results.

The conserving method is extremely demanding and scales linearly with the size of the system. The non-conserving method scales linearly with the size of its own grid. However, since this grid is limited to the size of the integration region, the time of the calculation does not depend on the size of the system at all, so that this method is extremely efficient.

In the conserving method the original grid is used with the directions along the cell basic vectors $\mathbf{a}_{i}$. That is why this method is inappropriate for calculations of the angular momenta projected DOS for non-simple-cubic cells or for the calculation of the dipole/quadrupole momenta of a molecule. On the contrary, the non-conserving method has its own grid which is always cubic (along the Cartesian axes) so that it has the atomic symmetry and is ideal for these calculations.