6.7 The pseudopotential method 145
Figure 6.11which shows two different potentials and their solutions (for the same
energy). These solutions differ strongly within the core region but they coincide in
the valence region.
What the pseudopotential does is to remove nodes from the core region of the
valence wave function while leaving it unchanged in the valence region. The nodes
in the core region are necessary in order to make the valence wave functions ortho-
gonal to the core states. If there are no core states for a givenl, the valence wave
function is nodeless and the pseudopotential method is less effective. Such is the
case in 3d transition metals, such as copper.
The phase shift depends on the angular momentumland on the energy. A pseudo-
potential that gives the correct phase shift will therefore also depend on these
quantities. The energy dependence is particularly inconvenient, as we have seen in
the discussion of the APW method. InSection 6.7.2we shall see that this depend-
ence disappears automatically when solving another problem associated with the
pseudopotential: that of the distribution of the charge inside and outside the core
region. More details will be given in that section, and we restrict ourselves here to
energy-independent pseudopotentials.
There is a considerable freedom in choosing the pseudopotential as it only has
to yield the correct phase shift outside the core region, and several simple para-
metrised forms of pseudopotentials have been proposed. These are fitted either to
experimental data for the material in question (thesemi-empiricalapproach), or
to data obtained usingab initiomethods for ions and atoms of the same material,
obtained using full-potential calculations.
We give two examples of pseudopotentials.
- The Ashcroft empty-core pseudopotential[19]:
V(r)=
−
Ze
r
, r>rc
0 r<rc
. (6.51)
Zis the valence of the ion and there is only one parameter to be adjusted: the
cut-off lengthrc. Although its simplictity is very attractive, this potential does
not perform very well for wide energy ranges, although it reproduces some
material properties reasonably well.
- The Fourier-component parametrisation
V(r)=
∑′
KVKe
iK·R. (6.52)
where the sum
∑′
is over a limited set ofK-vectors. This parametrisation is
convenient for the plane wave basis set which is (nearly) always used in
pseudopotential calculations. In the next subsection we shall use this form of
the pseudopotential in a band structure program for silicon.
