Norm conserving pseudopotentials

A particularly efficient method for the inclusion of relativity in electronic structure calculations is the pseudopotential (PP) approach. In ihe framework of DFT usually norm-conserving PPs (Bachelet etal. 1982 Hamann etal. 1979 Troullier and Martins 1991) are applied for this purpose. The standard form of norm-conserving PPs is given by 

Among the various schemes for the construction of V] presently the Troullier-Martins (TM) form (Troullier and Martins 1991) seems to be most widely used. However, in contrast to the original approach (Bachelet et al. 1982 Hamann et al. 1979) the TM scheme has been formulated for nonrelativistic situations. A consistent relativistic extension (Engel et al. 2001b) is presented in the next section. 

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Kleinman, L. (1980) Relativistic norm-conserving pseudopotential. Physical Review B - Condensed Matter, 21, 2630-2631,... 

Bachelet, G.B. and Schliiter, M. (1982) Relativistic norm-conserving pseudopotentials. Physical Review B -Condensed Matter, 25, 2103-2108. 

A further simplication often used in density-functional calculations is the use of pseudopotentials. Most properties of molecules and solids are indeed determined by the valence electrons, i.e., those electrons in outer shells that take part in the bonding between atoms. The core electrons can be removed from the problem by representing the ionic core (i.e., nucleus plus inner shells of electrons) by a pseudopotential. State-of-the-art calculations employ nonlocal, norm-conserving pseudopotentials that are generated from atomic calculations and do not contain any fitting to experiment (Hamann et al., 1979). Such calculations can therefore be called ab initio, or first-principles. ... 

For H at T in Ge, Pickett et al. (1979) carried out empirical-pseudopotential supercell calculations. Their band structures showed a H-induced deep donor state more than 6 eV below the valence-band maximum in a non-self-consistent calculation. This binding energy was substantially reduced in a self-consistent calculation. However, lack of convergence and the use of empirical pseudopotentials cast doubt on the quantitative accuracy. More recent calculations (Denteneer et al., 1989b) using ab initio norm-conserving pseudopotentials have shown that H at T in Ge induces a level just below the valence-band maximum, very similar to the situation in Si. The arguments by Pickett et al. that a spin-polarized treatment would be essential (which would introduce a shift in the defect level of up to 0.5 Ry), have already been refuted in Section II.2.d. 

In most LDA studies reported in this article, the Ceperley-Alder exchange-correlation formula is used [10,11]. Also the norm-conserving pseudopotentials of Troullier and Martins are used [12]. Therefore, one only has to deal with the valence electrons in solving the self-consistent Kohn-Sham equations in the LDA. As for basis functions, plane waves with the cutoff energy of 50 Ryd are used. 

At this stage, the formalism can be implemented in a computer program. The applications described below [15-21] rely on the expansion of the electronic wavefunctions in terms of a large number of plane waves, as well as on the replacement of nuclear bare potentials by accurate norm-conserving pseudopotentials. The Local Density Approximation was used, with the Ceperley and Alder data for the exchange-correlation energy of the homogeneous electron gas. 

A theoretical analysis based on ab-initio molecular dynamics has been reported [10], The study employed a plane wave basis, and soft core, norm conserving pseudopotentials were used to describe the ions. The supercells consisted of 10 - 12 layers of AIN with 4-16 atoms in each layer. For most calculations, a 12 A vacuum region separated the surfaces. One side of each slab was terminated by hydrogen atoms to reduce charge transfer caused by the finite width of the slab. The electron affinities of different surface configurations of AIN are listed in TABLE 1, where prior results for the diamond (111) surface are also listed. 

In the present work, we employ the Kleinman and Bylander [29] separable form for the norm-conserving pseudopotential vps(r) in Eq. (17-2) ... 

In norm-conserving pseudopotentials a third restraint— that integrating the charge in the core region must give the same answer as for the all-electron case— is applied. This ensures that scattering properties remain correct to linear order. ... 

In early implementations of CP, norm-conserving pseudopotentials have been used. [70] In such a pseudopotential, pseudowavefunctions match the all-electron wavefunctions beyond a specified matching radius (core-radius) rc. Inside the r. ... 

We use the DFT SIESTA code [13], which implements the generalized gradient approximation (GGA), Perdew-Burke-Emzerhof (PBE) density functional [14], norm-conserving pseudopotentials and periodic boundary conditions. A localized double- polarized (DZP) basis set was used for valence electrons. 

For these norm-conserving pseudopotentials, a different potential needs to be applied on each orbital depending on its angular momentum. These pseudopotentials then have a semi-local form ... 

D. Vanderbilt (1985) Optimally smooth norm-conserving pseudopotentials. Phys. Rev. B 32, p. 8412... 

SIESTA code, the interactions of valence electrons with the atomic ionic cores are described by the norm-conserving pseudopotentials with the partial core correction of 0.6 au. on the oxygen atom. We used the optimized-zeta plus polarization (DZP) basis sets with medium localization in the SIESTA code. A mesh cutloff energy of 350 Ry, which defines the equivalent plane wave cut-off for the grid, was used. The forces on atomic ions are obtained by the Heilman IFeynman theorem and were used to relax atomic ionic positions to the minimum energy. The atomic forces within the supercell were minimized to within 0.035 eV/A and 0.05 eV/A in the SIESTA and CASTEP codes respectively. 

Early applications of pseudopotentials in cluster models [62,63], which dealt with impurities in alkali halide crystals, used Hartree-Fock (HF) based model potentials [64] and complete-cation norm-conserving pseudopotentials [65]. A similar technique was found valuable to describe bulk properties of alkaline-earth oxides [66-68]. A general procedure for calculating embedded clusters under the assumption of a frozen environment and orthogonality requirements for the wave function of the cluster and the environment was also discussed... 

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