Non-local Core Corrections

It is tempting to assume that the Kohn-Sham potential depends linearly on the density, so that the unscreening of the pseudo-potential can be performed as in (6.60). Unfortunately, even though the Hartree contribution is indeed linearly dependent on the density, the xc term is not 

In some cases, like the alkali metals, the use of a nonlinear core-valence xc scheme may be necessary to obtain a transferable pseudo-potential. In these 

The polynomial P r) decays monotonically and has vanishing first and second derivatives at the origin. At r ic it joins smoothly the true core density (it is continuous up to the third derivative). The core cutoff radius, rnic, is typically chosen to be the point where the true atomic core density becomes smaller that the atomic valence density. It can be chosen to be larger than this value but if it is too large the description of the non-linearities may suffer. Note that, as the word partial suggests, 

These corrections are more important for the alkali metals and other elements with few valence electrons and core orbitals extending into the tail of the valence density (e.g., Zn and Cd). 

In some cases, the use of the generalized gradient approximation (GGA) for exchange and correlation leads to the appearance of very short-ranged oscillations in the pseudo-potentials (see Fig. 6.4). These oscillations are artifacts of the GGA that usually disappear when non-local core corrections are considered. Nevertheless, they do not pose a real threat for plane-wave calculations, since they are mostly filtered out by the energy cutoff. 

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