Pseudopotential self-consistent solution

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. 

Here Ho is the kinetic energy operator of valence electrons Vps is the pseudopotential [40,41] which defines the atomic core. V = eUn(r) is the Hartree energy which satisfies the Poisson equation ArUn(r) = —4nep(r) with proper boundary conditions as discussed in the previous subsection. The last term is the exchange-correlation potential Vxc [p which is a functional of the density. Many forms of 14c exist and we use the simplest one which is the local density approximation [42] (LDA). One may also consider the generalized gradient approximation (GGA) [43,44] which can be implemented for transport calculations without too much difficulty [45]. Importantly a self-consistent solution of Eq. (2) is necessary because Hks is a functional of the charge density p. One constructs p from the KS states Ts, p(r) = (r p r) = ns Fs(r) 2, where p is the density matrix,... 

Given an ion core pseudopotential, the valence electrons are allowed to respond to the potentials to form a self-consistent total potential. Initially, an approximate potential is used to solve the one-electron Schrodinger equation. The wavefunetions from the solution of this approximate potential are then used to construct anew potential which in turn can generate new wave functions from which the potential can be updated again. When the input potential agrees with the output potential, a self-consistent, field has been obtained. 

A technique of the solid simulation starting from the first principles ah initio theory) is the subject of Chapter 8. We study milestones in solution of the many-body problem. We describe the density functional theory as an essence of the technique. The Kohn-Sham approach, pseudopotential method, iterative technique of calculations are described here. These methods enable one to determine and calculate the equilibrium structure of a solid quantitatively and self-consistently. 

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