Density dependent atomic pseudopotentials

An ultrasoft type of pseudopotential was introduced by Vanderbilt (1990) and Laasonen et al. [1993] to deal with nodeless valence states which are strongly localized in the core region. In this scheme the normconserving condition is lifted and only a small portion of the electron density inside the cutoff radius is recovered by the pseudo-wavefunction, the remainder is added in the form of so-called augmentation charges. Complications arising from this scheme are the nonorthogonality of Kohn-Sham orbitals, the density dependence of the nonlocal pseudopotential, and need to evaluate additional terms in atomic force calculations. 

Since and depend only on die valence charge densities, they can be detennined once the valence pseudo- wavefiinctions are known. Because the pseudo-wavefiinctions are nodeless, the resulting pseudopotential is well defined despite the last temi in equation Al.3.78. Once the pseudopotential has been constructed from the atom, it can be transferred to the condensed matter system of interest. For example, the ionic pseudopotential defined by equation Al.3.78 from an atomistic calculation can be transferred to condensed matter phases without any significant loss of accuracy. 

The energy-wave-number characteristic, ( )> depends only on the density of the free-electron gas and the nature of the pseudopotential core but not on the structural arrangement of the atoms. Its behaviour as a function of the wave vector, q, is illustrated in Fig. 6.6, where we see that it vanishes at q0 as expected. It also has a weak logarithmic singularity in its slope at q = 2kF. 

Pseudopotentials are usually derived from all-electron atomic calculations. The valence electron pseudopotential is then required to reproduce the behaviour and properties of the valence electrons in the full calculation. For example, the energy levels with the pseudopotential should be the same as for the all-electron calculation. In addition, the pseudopotential will often depend upon the orbital angular momentum of the wavefimction (i.e. for s, p, d, etc. orbitals) and it will be desired that the total valence electron density within the core radius equals that in the all-electron situation. Such pseudopotentials are... 

Inserting the relation between the induced density and the electronic part of the pseudopotential, one again realizes that the structure factor can be factored out, and the structure independent energy-wavenumber characteristic F(q) only depends on the atomic properties ... 

The NFE theory describes a simple metal as a collection of ions that are weakly coupled through the electron gas. The potential energy is volume-dependent but is independent of the position of the electrons. This is valid for both solids and dense liquids. At densities well above that of the MNM transition, we can use effective pair potentials and find the thermophysical properties of metallic liquids with the thermodynamic variational methods usually employed in theoretical treatments of normal insulating liquids. One approach is a variational method based on hard sphere reference systems (Shimoji, 1977 Ashcroft and Stroud, 1978). The electron system is assumed to be a nearly-free-electron gas in which electrons interact weakly with the ions via a suitable pseudopotential. It is also assumed that the Helmholtz free energy per atom can be expressed in terms of the following contributions ... 

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