Within the pseudopotential approximation, the core energy is assumed to be constant and is usually subtracted out. All interaction between the core and valence electrons is transfered to the pseudopotential. This implies a linearisation of the interaction which is only an approximation to the kinetic and exchange-correlation energies, the latter of which is explicitly non-linear. If the valence and core charge densities are well separated, as is often the case, this will introduce no serious errors. However, if there is significant overlap between the two densities, this approximation will lead to systematic errors in the total energy and reduced transferability of the pseudopotential. In particular, spin density calculations introduce additional non-linearity and this necessitates the explicit consideration of the non-linear dependence of the energy on the core charge density.

The reason for these difficulties is that the ionic pseudopotential is conventionally defined as the potential given by subtracting the Coulomb and exchange-correlation potentials due to the valence charge density, , and spin polarisation,

from the neutral atomic pseudopotential. The implicit assumptions in this are the independence of the core electrons on the surrounding environment and the decoupling of the core charge in determining the exchange-correlation potential seen by the valence electrons. However, within the pseudopotential approximation the total exchange-correlation potential is implicitly written

where

and is the core charge density. Notice that can differ significantly from the valence polarisation defined in Equation . Since is a non-linear function of the charge density, the valence charge does not cancel and the ionic pseudopotentials are dependent on the valence configuration.

The solution to this, proposed by Louie *et al.* [30],
is to redefine the ionic pseudopotential as that given by subtracting
the Coulomb potential of the valence charge density and the total
exchange-correlation potential . The core
charge is then stored and used to reconstruct the full
exchange-correlation potential in the calculation.

There are two potential problems associated with this
procedure. Firstly, there are inevitably small errors in the
calculated valence charge density which usually lead to negligible
errors in the total energy. However, when considering the core
charge, any error in the valence charge within the core region is
multiplied by the core charge and could lead to significant error in
the total energy. The second concern relates specifically to the
implementation of this correction within a plane-wave expansion
method. The high Fourier components of the full core charge density
makes it impractical to represent the core charge density in Fourier
space. Louie *et al.* propose a method to overcome these
difficulties. They point out that the core charge has a significant
effect only where the core and the valence charge densities are of
similar magnitude, ie. in the regions in which there is significant
overlap. It is therefore unimportant close to the nucleus where the
majority of the core charge lies. They suggest that the full core
charge density can be replaced by a partial core charge density which is
identical to the true charge density outside some radius and is
an arbitrary, smooth function within this radius. Their tests show that
may be chosen as the radius at which the core charge density is
from 1 to 2 times larger than the valence charge density.

Wed Sep 24 12:24:18 BST 1997