Norm conserving

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

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. 

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

The method employed was similar to that of Ref. 35, but with several improvements. ab initio, norm-conserving, nonlocal pseudopotential were used to represent the metal ions. This capability enables reliably realistic representation of the metal s electronic structure. Thus the cadmium pseudopotential was able, for example, to reproduce the experimental cadmium-vacuum work function using no adjustable parameters (unlike the procedure followed in Ref. 35). Pseudopotentials of the Troullier and Martins form [53] were used with the Kleinman-Bylander [54] separable form, and a real space... 

The relaxation of the structure in the KMC-DR method was done using an approach based on the density functional theory and linear combination of atomic orbitals implemented in the Siesta code [97]. The minimum basis set of localized numerical orbitals of Sankey type [98] was used for all atoms except silicon atoms near the interface, for which polarization functions were added to improve the description of the SiOx layer. The core electrons were replaced with norm-conserving Troullier-Martins pseudopotentials [99] (Zr atoms also include 4p electrons in the valence shell). Calculations were done in the local density approximation (LDA) of DFT. The grid in the real space for the calculation of matrix elements has an equivalent cutoff energy of 60 Ry. The standard diagonalization scheme with Pulay mixing was used to get a self-consistent solution. In the framework of the KMC-DR method, it is not necessary to perform an accurate optimization of the structure, since structure relaxation is performed many times. 

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. 

The calculated and measured electron effective mass m c and its k-dependency for WZ and ZB GaN and AIN are summarised in TABLES 1 and 2, respectively. Suzuki et al derived them with a full-potential linearised augmented plane wave (FLAPW) band calculation [4,5], Miwa et al used a pseudopotential mixed basis approach to calculate them [6]. Kim et al [7] determined values for WZ nitrides by the full-potential linear muffin-tin orbital (FP-LMTO) method. Majewski et al [8] and Chow et al [9,10] used the norm-conserving pseudo-potential plane-wave (PPPW) method. Chen et al [11] also used the FLAPW method to determine values for WZ GaN, and Fan et al obtained values for ZB nitrides by their empirical pseudo-potential (EPP) calculation [12],... 

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

Each pseudopotential is defined within a cut-off radius from the atom center. At the cut-off, the potential and wavefunctions of the core region must join smoothly to the all-electron-like valence states. Early functional forms for pseudopotentials also enforced the norm-conserving condition so that the integral of the charge density below the cut-off equals that of the aU-electron calculation [42, 43]. However, smoother, and so computationally cheaper, functions can be defined if this condition is relaxed. This idea leads to the so called soft and ultra-soft pseudopotentials defined by Vanderbilt [44] and others. The Unk between the pseudo and real potentials was formaUzed more clearly by Blochl [45] and the resulting... 

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