- 9.1. The basic idea behind the pseudopotential method.
- 9.1.1. Key terms and points to keep in mind in implementing the pseudopotential method.
- 9.1.2. Schrödinger-like approximation to the Dirac equation for a central field .
- 9.1.3. The nonrelativistic approximation.
- 9.1.4. A guide to some relevant articles in the development of modern pseudopotential theory.

- 9.2. A summary of the Troullier-Martins pseudopotential (TM) .
- 9.3. A summary of the Hartwigsen-Goedecker-Hutter pseudopotential (HGH) .

Development note | |
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Development of this chapter has been postponed. |

Norm-conserving [132]: 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 [131] 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 [134].

In most formulations both the pseudopotential and the pseudo wavefunctions
are equal to their all-electron counterparts beyond some cut-off radius

For hydrogenlike atoms, the root-mean-square velocity is given by [Gasiorowicz,p.271;Levine,p.525]

both
*4×4* dimensional matrices, and
*2×2* dimensional Pauli spin matrices [Eq. (2.21)].
A solution to the Dirac equation can be written as

where

where, following Bachelet, Hamann, and Schüter (BHS) [137], the
rest mass m is not included in the energy

In the case of hydrogen-like atoms, an exact solution to the Dirac equations exist [Itzykson,p.79]. Here however, we will consider the nonrelativistic limit in which case we can write Eq. (9.8) as

valid for valence electrons (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 [135] [136]
[137]

which is valid up to, but not including, terms of order

where we have used the identities

then we can use Eq. (9.13) to find the following relationship between the eigenvalues of

Although relativistically *l* is no longer a good quantum
number it can be shown that in the nonrelativistic limit the quantum number,
*l*. The two components of

**(9.20)**

Next, we will need to find solutions to Eq. (9.10) with Eq. (9.22) using pseudopotential methods.

There are several nice review articles of which Fuchs and Scheffler [128] is one of the most recent and up-to-date. In that review, Fuchs and Scheffler [128] also introduce their ab initio pseudopotential package fhi98PP, which can be used to produce pseudopotentials. Other nice review articles include ones by Pickett [127], Heine [125], Cohen [126], 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 [147] and Frenking et al. [149].

Articles specific to this implementation:

In 1959 Phillips and Kleinman (PK) [130] presented a pseudopotential of the form

Norm-conserving pseudopotentials are introduced by Topp and Hopfield in 1973 [131] and further developed by Hamann, Schülter, and Chiang [132] in 1979.

The notion of fully separable pseudopotentials are introduced by Kleinman and Bylander in 1982 [134] and generalized by Blöchl [139] and Vanderbilt [140] and others. The Kleinman-Bylander pseudopotential is of the form

Separable dual-space Gaussian pseudopotentials [Goedecker,1996;Hartwigsen,1998]:

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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 [133] they wrote the radial pseudo wavefunction as

where the polynomial *p(r)* is given by

The order *n* of the *p(r)* was taken to be
*n = 10* by Troullier and Martins [141] compared
to *n = 4* in Kerker's work [133].

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The total HGH pseudopotential is given by

where the local part of the pseudopotential is

The contribution from the spin-orbit coupling is

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

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