| Chapter 9. From Atoms to Solids: The Pseudopotential Method. | ||
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| Chapter 9. From Atoms to Solids: The Pseudopotential Method. | ||
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Table of Contents
![[Note]](images/note.png) | Development note |
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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 [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]
where
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
where
and
both
(9.6)
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
which is valid up to, but not including, terms of order
where
The operator
(9.13)
where we have used the identities
and
If we recall that
then we can use Eq. (9.13) to find the following relationship between the eigenvalues of
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,
together with
(9.20)
leads to
The minus sign in the last equation corresponds to the upper component of the wave
function (i.e.
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
where V is the true potential and
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
(9.24)
where
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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(9.27)
where the local part of the pseudopotential is
(9.28)
and the nonlocal part is
(9.29)
The contribution from the spin-orbit coupling is
(9.30)
where
(9.31)
The parameters for the various chemical elements are given in tables with the following format
(9.32)
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 [145][146]. 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.
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| Chapter 8. Time-Dependent Density-Functional Theory. |  | Chapter 10. Ab initio quantum chemistry packages. |