Pseudopotential normconserving

Ultrasoft pseudopotentials have now been constructed for all elements for all elements from H to Bi [10]. It has been shown that without any loss of accuracy even for first-row and transition-elements small PW basis-sets comparable in size to those necessai y for soft normconserving pseudopotentials for A1 or Si can be used. For any further details, see [9,. 37]. 

A crucial development in pseudopotential theory is the formulation of normconserving pseudopotentials (Hamann et al., 1979 Kerker, 1980). From a local-density calculation of the allelectron free atom, the relatively weak pseudopotentials which bind only the valence electrons are constructed. The valence pseudo-wavefunctions do not contain the oscillations necessary to orthogonalize to the core, but are instead smooth functions which are much easier to handle in calculations on real solids. The features of such potentials are discussed in detail by, e.g., Bachelet et al. (1982), who also present pseudopotentials for all atoms from H to Pu. 

The normconserving pseudopotentials have proven to work well for many systems, notably semiconductors (see, e.g., Kune, this volume) and their surfaces (see, e.g., Morthrup and Cohen, 1982), ionic compounds (Froyen and Cohen, 1984), and simple metals (see, e.g., Lam and Cohen, 1981). Applications to transition metals also exist (see, e.g., Greenside and Schluter, 1983). The pseudopotential approximation becomes less satisfactory when valence and core electrons begin to have large overlap, both because of the pseudo-wavefunctions lacking nodes, and because the x-c potential ih the core region should also account for the presence of the core electrons. The latter problem can in many cases be treated well by "nonlinear" pseudopotentials (Louie et al., 1982). 

The electronic structure for the MgO crystal was calculated in [608] both in the LCAO approximation and in the PW basis. In both cases the calculations were done by the density-functional theory (DFT) method in the local density approximation (LD A). The Monkhorst Pack set of special points of BZ, which allows a convergence to be obtained (relating to extended special-points sets) in the calculations of electronic structure, was used in both cases. For the LCAO calculations the Durand Barthelat pseudopotential [484] was used. In the case of the PW calculations the normconserving pseudopotential and a PW kinetic energy cutoff of 300 eV were used. 

The crystal orbitals were calculated by the DFT method in the plane-wave basis set with the CASTEP code [377] in the GGA density functional. A set of special points k in the Brillouin zone for all the crystals was generated by the snperceU method (see Chap. 3) with a 5 x 5 x 5 diagonal symmetric extension, which corresponds to 125 points. In all cases, the pseudopotentials were represented by the normconserving optimized atomic pseudopotentials [621], which were also used to calculate the atomic potentials of free atoms. In the framework of both techniques (A, B), the population analysis was performed in the minimal atomic basis set i.e. the basis set involved only occupied or partially occupied atomic orbitals of free atoms. It is well known that the inclusion of diffuse vacant atomic orbitals in the basis set can substantially change the results of the population analysis. For example, if the Mg 2p vacant atomic orbitals are included in the basis set, the charge calculated by technique A for the Mg... 

Normconserving pseudopotentials are generated subject to the condition that the pseudo-wavefunction has the same norm as the all-electron wavefunction and thus gives rise to the same electron density. Although normconserving pseudopotentials have to fulfill a (small) number of mathematical conditions, there remains considerable freedom in how to create them. Hence several different recipes exist (Bachelet et al. 1982 Goedecker et al. 1996 Hamann et al. 1979 Hartwigsen et al. 1998 Kerker 1980 TrouUier and Martins 1990,1991 Vanderbilt 1985). 

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. 

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