2 DFT and the Full-Potential Local-Orbital Approach 15 The potential is decomposed according to v(r)=>v,(r-R-s),v(r-R-s)=v(r)fR(r) (2.32) Rs with use of the functions f of the previous subsection. Now,in the local items,radial dependencies are obtained numerically on an inhomogeneous grid (logarithmic equidistant),and angular dependencies are expanded into spherical harmonics (typically up to l =12).To compute the overlap and Hamiltonian matrices,one has one-center terms:ID numerical integrals, -two-center terms:2D numerical integrals, three-center terms:3D numerical integrals. 2.2.4 Basis Optimization The essential feature which allows for the use of a minimum basis is that the basis is not fixed in the course of iterations,instead it is adjusted to the actual effective crystal potential in each iteration step and it is even optimized in the course of iterations. Take s to be the total crystal potential,spherically averaged around the site center s.Core orbitals are obtained from (任+is)psLe=PaLeesLe· (2.33) Valence basis orbitals,however,are obtained from a modified equation +(2) =PsLyEsL· (2.34) The parameters rL are determined by minimizing the total energy. There are two main effects of the rsL,-potential: The counterproductive long tails of basis orbitals are suppressed. The orbital resonance energies esL,are pushed up to close to the centers of gravity of the orbital projected density of states of the Kohn-Sham band structure,providing the optimal curvature of the orbitals and avoiding insufficient completeness of the local basis. In the package FPLO the optimization is done automatically by applying a kind of force theorem during the iterations for self-consistency. 2.2.5 Examples In order to illustrate the accuracy of the approach,the simple case of fcc Al is considered.Figure 2.4 shows the dependence of the calculated total energy2 DFT and the Full-Potential Local-Orbital Approach 15 The potential is decomposed according to v(r) =  Rs vs(r − R − s), vs(r − R − s) = v(r)fRs(r) (2.32) with use of the functions f of the previous subsection. Now, in the local items, radial dependencies are obtained numerically on an inhomogeneous grid (logarithmic equidistant), and angular dependencies are expanded into spherical harmonics (typically up to l = 12). To compute the overlap and Hamiltonian matrices, one has – one-center terms: 1D numerical integrals, – two-center terms: 2D numerical integrals, – three-center terms: 3D numerical integrals. 2.2.4 Basis Optimization The essential feature which allows for the use of a minimum basis is that the basis is not fixed in the course of iterations, instead it is adjusted to the actual effective crystal potential in each iteration step and it is even optimized in the course of iterations. Take ¯vs to be the total crystal potential, spherically averaged around the site center s. Core orbitals are obtained from (t ˆ+ ¯vs)ϕsLc = ϕsLc sLc . (2.33) Valence basis orbitals, however, are obtained from a modified equation  t ˆ+ ¯vs +  r rsLv 4  ϕsLv = ϕsLv sLv . (2.34) The parameters rsLv are determined by minimizing the total energy. There are two main effects of the rsLv -potential: – The counterproductive long tails of basis orbitals are suppressed. – The orbital resonance energies sLv are pushed up to close to the centers of gravity of the orbital projected density of states of the Kohn-Sham band structure, providing the optimal curvature of the orbitals and avoiding insufficient completeness of the local basis. In the package FPLO the optimization is done automatically by applying a kind of force theorem during the iterations for self-consistency. 2.2.5 Examples In order to illustrate the accuracy of the approach, the simple case of fcc Al is considered. Figure 2.4 shows the dependence of the calculated total energy