Table of Contents
Development of this chapter has been postponed.
The inner core of electrons is tightly bound to the nucleus and have a relatively high kinetic energy. Indeed, for atoms with high atomic number relativistic effects can be substantial. Tightly bound electrons with large velocities translates into wavefunctions with a high degree of curvature and a broad spectrum in momentum space. To represent these wavefunctions would require a large number of basis functions (especially in the case of plane-waves), but at the same time these states have only an indirect role in chemical processes and are insensitive to the chemical environment. The valence state wavefunctions also have a high degree of curvature in the core region, which manifests itself as oscillatory behavior. Unlike the core states, however, the amplitude of the valence wavefunctions decrease rapidly near the core. A key assumption of the pseudopotential method is that the core states in a chemical environment are the same as in a free atom. The main objective is to replace the need for calculating the core electron states with a pseudopotential that reproduces the effects of the core electrons.
Before we start a general discussion we would like to mention a few key terms and points to keep in mind, which we hope will become clearer as one progresses. These are:
Transferability: the idea that a pseudopotential should reproduce all-electron results in a variety of chemical environments.
Norm-conserving : the notion that outside the core region the pseudo valence wavefunctions and the all-electron wavefunctions should be identical; and the integrated charge within the "core" region calculated with the pseudo wavefunctions should equal the real charge within that region. An idea introduced by Topp and Hopfield  to improve transferability.
Consistency: pseudopotentials constructed in the framework of the local-density approximation (LDA) should not be used in calculations using the generalized gradient approximation (GGA) and vice versa.
Local, semilocal and nonlocal potentials: A local potential is only a function of the distance from the nucleus, r. Both the nonlocal and the semilocal pseudopotentials are angular momentum dependent, but a semilocal pseudopotential is only nonlocal in angular coordinates, not in the radial coordinate .
In most formulations both the pseudopotential and the pseudo wavefunctions are equal to their all-electron counterparts beyond some cut-off radius . At radial distances less the cut-off radius , the oscillatory behavior of the all-electron (AE) valence wavefunction is replaced by a smoothly varying function. Thus, must be large enough that all the nodes found in the AE valence wavefunction lie within this radius. At the same time, however, with a smaller it is easier to replicate the behavior of the AE valence wavefunction and therefore improve transferability.
where is related to the Lorentz factor , a measure of the importance of relativistic effects, as follows
Thus, for atoms with a large nuclear charge Z relativistic effects cannot be ignored if accurate results are required. In 1926, Schrödinger presented a simple relativistic extension to his wave equation valid for spin zero particles. In 1928, Dirac presented a new relativistic formulation of quantum mechanics that incorporated four component wavefunctions (spinors), which accurately predicted the existence of positrons and provided a stepping stone to the development of quantum field theory. The Dirac equation with a central field can be written
both (no relation to the used previously) and are 4×4 dimensional matrices, and are the 2×2 dimensional Pauli spin matrices [Eq. (2.21)]. A solution to the Dirac equation can be written as
where is a four-component spinor and are r-independent two-component spinors. By writing in terms of spherical harmonics [Sakurai,p.124] and applying the Dirac equation to one can obtain the following radial wave equations [Schiff,p.484; Sakurai,p.125; Itzykson,p.78], 
where, following Bachelet, Hamann, and Schüter (BHS) , the rest mass m is not included in the energy . We have also used atomic units with and .
valid for valence electrons (small ) outside the core region [small V(r)]. By substituting Eq. (9.8) into Eq. (9.7) we obtain the Schrödinger-like approximation to the Dirac equation for a central field   
The operator commutes with the relativistic Hamiltonian and the total momentum operator and is a good quantum number. describes the extent that the spin and orbital momentum are aligned. The values of can be related to the total angular momentum quantum number j by writing
where we have used the identities
If we recall that
then we can use Eq. (9.13) to find the following relationship between the eigenvalues of and ,
which has the two solutions
Although relativistically l is no longer a good quantum number it can be shown that in the nonrelativistic limit the quantum number, , is related to the orbital angular momentum quantum number, l. The two components of in Eq. (9.6), and , are separately eigenfunctions of the operators . Applying the two block diagonal elements of Eq. (9.11) to the two components of ,
The minus sign in the last equation corresponds to the upper component of the wave function (i.e. ) and the plus sign corresponds to the lower part (i.e. ). For G(r) we are interested in the solution with the minus sign and thus
There are several nice review articles of which Fuchs and Scheffler  is one of the most recent and up-to-date. In that review, Fuchs and Scheffler  also introduce their ab initio pseudopotential package fhi98PP, which can be used to produce pseudopotentials. Other nice review articles include ones by Pickett , Heine , Cohen , which are not as up-to-date but offer a good background discussion. There are also several good reviews that are directed a bit more toward those with a chemistry background. Among these are ones by Krauss and Stevens  and Frenking et al. .
Articles specific to this implementation:
In 1959 Phillips and Kleinman (PK)  presented a pseudopotential of the form
where V is the true potential and and are the exact core eigenstates and eigen-energies, respectively. Many modern pseudopotentials are descendants of this work.
The notion of fully separable pseudopotentials are introduced by Kleinman and Bylander in 1982  and generalized by Blöchl  and Vanderbilt  and others. The Kleinman-Bylander pseudopotential is of the form
where is the semilocal potential (i.e., dependent on angular momentum quantum number l, but not on both r and r') and is the atomic reference pseudo-wave-function. The motivation for this form of pseudopotential was to dramatically improve the computational speed in reciprocal space.
Troullier and Martins introduced additional constraints to produce smoother pseudopotentials, which require a smaller set of basis functions resulting in greatly reduced computation times. Following Kerker  they wrote the radial pseudo wavefunction as
where the polynomial p(r) is given by
where the local part of the pseudopotential is
and the nonlocal part is
The contribution from the spin-orbit coupling is
where are the spherical harmonics, l is the angular momentum quantum number, gives the range of the Gaussian ionic charge distribution, is the ionic charge (the charge of the nucleus minus the charge of the valence electrons), and the projectors are given by
The parameters for the various chemical elements are given in tables with the following format
For example, the LDA pseudopotential parameters for hydrogen in atomic units are
and the LDA pseudopotential parameters for carbon are
In the octopus program these tables are stored in files labeled element.hgh, or element_sc.hgh for semicore pseudopotentials . ABINIT has six options available for pseudopotentials including Hartwigsen-Goedecker-Hutter LDA pseudopotentials, and Hartwigsen-Goedecker-Hutter GGA (PBE) pseudopotentials for a limited number of chemical elements. The parameters are stored in files labeled element.psphgh and element.pbe_hgh, respectively, and a couple examples are included with the installation and can be found in the ~ABINIT/Psps_for_tests directory.