Local pseudopotential theory

Here kp> denotes a plane wave with wave vector of magnitude kp, and and are contributions to the effective mass from energy dependence and nonlocality. Such effective masses have been calculated by Weaire " and others (Ref. 51, pp. 64-69 Ref 75). The total effective mass m is almost always close to unity, which to some extent justifies simple local pseudopotential theories—on the other hand, any property which involves and in some other combination than i V iiot be accurately treated without their inclusion in the theory (Ref 51, pp. 64-69). 

How is the molecular geometry of the liquid affected by the presence of an excess electron Two extreme cases may be immediately distinguished the free electron case and the localized electron case. The free electron may be adequately described by using a plane wave state. The physical significance of such a simple description is not trivial and will be clarified after the discussion of the pseudopotential theory. Here we... 

This matrix clement is the Fourier component of the pseudopotential with wave number equal to the difference q. In the more complete pseudopotential theory of Appendix D, the pseudopotential becomes an operator, so that IT (r) and c cannot be interchanged in the final step of Eq. (16-2), and the matrix element depends upon k also. This is not a major complication, but we shall utilize the simpler form given in the last step of Eq. (16-2), called the local approximation to the pseudopotential. 

Shape-Consistent Pseudopotentials. - While with model potentials the wavefunction is (ideally) not changed with respect to the valence part of an AE frozen-core wavefunction, such a change is desirable for computational reasons. The nodal structure of the valence orbitals in the core region requires highly localized basis functions these are not really needed for the description of bonding properties in molecules but rather for the purpose of core-valence orthogonalization. The idea to incorporate this Pauli repulsion of the core into the pseudopotential is as old as pseudopotential theory itself.62,63 Modem ab initio pseudopotentials of this type have been developed since the end of the seventies, cf. e.g. refs. 64-68. 

Chai, J. D. Weeks, J. D. Orbital-free density functional theory kinetic potentials and ab initio local pseudopotentials. Phys. Rev. B 2007, 75, 205122. 

Here we will not go into the details of the origin and the theoretical foundation of pseudopotential methods, as they can be found in [68]. Suffice it to say that, in terms of plane-wave applications, the use of pseudopotentials has a twofold aim (1) to exclude the highly localized core electrons and (2) to cancel the rapid oscillations of wave functions close to the nucleus that are present even in valence states owing to an orthogonalization to the core states. A long evolution of the pseudopotential theory and practice converged... 

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. 

We will describe the calculation of the total energy and its relation to the band structure in the framework of Density Functional Theory (DFT in the following) see chapter 2. The reason is that this formulation has proven the most successful compromise between accuracy and efficiency for total-energy calculations in a very wide range of solids. In the following we will also adopt the pseudopotential method for describing the ionic cores, which, as we discussed in chapter 2, allows for an efficient treatment of the valence electrons only. Furthermore, we will assume that the ionic pseudopotentials are the same for all electronic states in the atom, an approximation known as the local pseudopotential this will help keep the discussion simple. In realistic calculations the pseudopotential typically depends on the angular momentum of the atomic state it represents, which is known as a non-local pseudopotential . 

Some further remarks are in order on the subject of truncation of the pseudopotential. Most current semiempirical studies involve quite large secular determinants (say 50 x 50) but set f(g) equal to zero for g 2kp. However, a somewhat cruder procedure, that of truncating the basis set at g = 2kp, resulting in a smaller secular determinant, has also been widely used. This procedure may be put on a formal basis by the use of Lowdin perturbation theory, by which a larger secular determinant is reexpressed as a smaller one, with correction terms. For a local pseudopotential the correction terms are given by (Ref 51, pp. 78-83)... 

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. 

Two assumptions are made in this choice. Core orbitals are deemed to have negligible influence on bonding, and the shape of atomic orbitals is used to describe the molecular orbitals. More complete ab initio calculations often allow for orbital variation so the latter assumption is a possible source of error. The neglect of core orbitals is justified by their localized nature, which excludes significant participation in bond formation. A recent pseudopotential formulation by Cusachs (5), in which core orbitals were included, has shown that the form of the equations used in MO theory is unchanged although the input parameters may require some modification. Thus, most workers do not consider core orbital effects significant. 

The delocalized (right-hand) side of Fig. 1.1 involves some form of calculation on the full lattice such as a band-theory calculation. Again, the Hartree-Fock wave function may be employed in an ab initio method or some approximate method such as Huckel band theory, or the local-exchange approximations employed leading to augmented-plane-wave or ab initio pseudopotential (PP) methods. As an alternative to band theory, the development of the ionic approach using pair potentials or modified electron-gas (MEG) theory has proved effective for certain crystalline species. 

According to past investigations into the structure of oxyapatite, there exists a linear chain of O2- ions parallel to the c-axis, each one followed by a vacancy (Alberius Henning et al., 1999) (Figure 6.9b). Calculations by density-functional theory with local-density approximation (DFT-LDA) and first-principles pseudopotentials (Calderin, Stott and Rubio, 2003) postulated a hexagonal "c empty ... 

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