# Analyse:Ym/Fragment 022 02

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3.1.1 The APW Method

Within the APW approach, space is divided into spheres centered at each atom site, the so-called muffin-tins, and the remaining interstitial region (cf. Fig. 3.1). Inside the muffin-tins the potential is approximated by a spherically symmetric shape, and in many implementations the interstitial potential is set to a constant. The restrictions to the potential are commonly called shape-approximations. Noting that plane-waves solve the Schro ̈dinger equation in a constant potential, while spherical harmonics times radial functions are the solutions in a spherical potential, suggests to expand the single particle wavefunctions in terms of the following basis functions: [...] Where k is the Bloch vector, Ω is the cell volume, G is a reciprocal lattice vector, L abbreviates the quantum numbers l and m and ul is the regular solution of the radial Schro ̈dinger equation [...] V (r) is the spherical component of the potential. The coefficients AμG(k) are determined from the requirement, that the wavefunctions have to be continuous at the boundary of the muffin-tin spheres. Hence, the APW’s form a set of continuous basis functions that cover all space. Each basis function consists of a plane-wave in the interstitial region plus a sum of functions, which are solutions of the Schro ̈dinger equation for a given set of angular momentum quantum numbers (lm) and a given parameter El inside the muffin-tin spheres. If the El’s were fixed, used only as a parameter during the construction of the basis, the Hamiltonian could be set up in terms of this basis. This would lead to a standard secular equation for the band energies. Unfortunately, it turns out, that the APW basis does not offer enough variational freedom if the El’s are kept fixed. An accurate description can only be achieved if they are set to the corresponding band energies. However, requiring the El’s to be equal to the band energies, the latter can no longer be determined by a simple diagonalization of the Hamiltonian matrix. Since the ul’s depend on the band energies, the solution of the secular equation becomes a nonlinear problem, which is computationally much more demanding than a secular problem. Another disadvantage of the APW method is, that it is difficult to extend beyond the spherically averaged muffin-tin potential approximation, because in the case of a general potential the optimal choice of El is no longer the band energy. And finally, but less serious, if, for a given choice of El’s, the radial functions ul vanish at the muffin tin radius, the boundary conditions on the spheres cannot be satisfied, i.e. the plane-waves and the radial functions become decoupled. This is called the asymptote problem. It can already cause numerical difficulties if ul becomes very small at the sphere boundary. Further information about the APW method can be found in the book by Loucks [60], which also reprints several early papers including Slater’s original publication [100].

3.1.1 The APW Method

Within the APW approach, space is divided into spheres centered at each atom site, the so-called muffin-tins, and the remaining interstitial region (cf. Fig. 3.1). Inside the muffin-tins the potential is approximated to be spherically symmetric, and in many implementations the interstitial potential is set constant. The restrictions to the potential are commonly called shape-approximations. Noting that plane-waves solve the Schro ̈dinger equation in a constant potential, while spherical harmonics times a radial function are the solution in a spherical potential, suggests to expand the single particle wavefunctions in terms of the following basis functions: [...] Where k is the Bloch vector, Ω is the cell volume, G is a reciprocal lattice vector, L abbreviates the quantum numbers l and m and ul is the regular solution of the radial Schro ̈dinger equation [...] V (r) is the spherical component of the potential. The coefficients AμG(k) are determined from the requirement, that the wavefunctions have to be continuous at the boundary of the muffin-tin spheres. Hence, the APW’s form a set of continuous basis functions that cover all space. Where each function consists of a planewave in the interstitial region plus a sum of functions, which are solutions of the Schro ̈dinger equation to a given set of angular momentum quantum numbers lm and a given parameter El inside the muffin-tin spheres. If the El were kept fixed, used only as a parameter during the construction of the basis, the hamiltonian could be set up in terms of this basis. This would lead to a standard secular equation for the band energies. Unfortunately, it turns out, that the APW basis does not offer enough variational freedom if the El are kept fixed. An accurate description can only be achieved if they are set to the corresponding band energies. However, requiring the El’s to be equal to the band energies, the latter can no longer be determined by a simple diagonalization of the Hamiltonian matrix. Since the ul’s depend on the band energies, the solution of the secular equation becomes a nonlinear problem, which is computationally much more demanding than a secular problem. Another disadvantage of the APW method is, that it is difficult to extend beyond the spherically averaged muffin-tin potential approximation, because in the case of a general potential the optimal choice of El is no longer the band energy. And finally, but less serious, if, for a given choice of El, the radial functions ul vanish at the muffin tin radius, the boundary conditions on the spheres cannot be satisfied, i.e. the planewaves and the radial functions become decoupled. This is called the asymptote problem. It can already cause numerical difficulties if ul becomes very small at the sphere boundary. Further information about the APW method can be found in the book by Loucks [Lou67], which also reprints several early papers including Slater’s original publication [Sla37].

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