If all of the electrons in a system were explicitly included when
performing a calculation and 
constructed from the full
Coulombic potential of the nuclei, the computational cost would still
be prohibitive using a plane wave basis set. The rapid oscillations of
the wavefunctions near to the nucleus, due to the very strong potential
in the region and the orthogonality condition between different
states, mean that a very large cut-off energy, and hence basis set,
would be necessary. Fortunately, the study of Physics and Chemistry
shows that the core electrons on different atoms are almost
independent of the environment surrounding the atom and that only the
valence electrons participate strongly in interactions between
atoms. Thus, the core electron states may be assumed to be fixed and a
pseudopotential may be constructed for each atomic species which takes
into account the effects of the nucleus and core
electrons [23, 24, 25]. The pseudowavefunctions
corresponding to this modified potential do not exhibit the rapid
oscillations of the true wavefunctions, dramatically reducing the
number of plane waves needed for their representation (See Figure
). The calculations then need only explicitly consider
the valence electrons, offering a further saving in effort.
Figure: A
schematic illustration of all-electron (solid lines) and
pseudo- (dashed lines) potentials and their corresponding
wavefunctions. The radius at which all-electron and pseudopotential
values match is 
. Source [5].
A pseudopotential is constructed such that it matches the true potential outside a given radius, designated the core radius. Similarly, each pseudowavefunction must match the corresponding true wavefunction beyond this distance. In addition, the charge densities obtained outside the core region must be identical to the true charge density. Thus, the integral of the squared amplitudes of the real and pseudowavefunctions over the core region must be identical. This condition is known as norm-conservation [26].
The atomic properties of the element must be preserved, including phase shifts on scattering across the core. These phase shifts will be different for different angular momentum states and so, in general, a pseudopotential must be non-local, with projectors for different angular momentum components. The pseudopotential is often represented using the form [27],
where 
are the projectors which
project the electronic wavefunctions onto the eigenfunctions of
different angular momentum states. The choice of 
is
arbitrary and if it is made equal to one of the 
this avoids the
need for the corresponding set of angular momentum projectors. Work
has also been done by Lee et al. [28] on further
reducing the number of projectors needed. The evaluation of the
non-local potential in reciprocal space requires a computational time
which is proportional to the cube of the system size. The projections
may instead by carried out in real-space, using the method of
King-Smith et al. [29], which reduces the
computational cost to the order of the system size squared.
Pseudopotentials are constructed using an ab initio procedure. The `true' wavefunctions are calculated for an isolated atom using an all-electron DFT approach. The resulting valence wavefunctions are then modified in the core region to remove the oscillations while obeying the norm-conservation constraint. The Schrödinger equation is then inverted to find the pseudopotential which will reproduce the pseudowavefunctions. This procedure produces a pseudopotential which may be transfered between widely varying systems. This contrasts with semi-empirical potentials which are constructed to describe a particular atomic environment and may not be simply transferred to different environments.