Elimination of the linearization error and improved basis-set convergence within the FLAPW method

We analyze in detail the error that arises from the linearization in linearized augmented-plane-wave (LAPW) basis functions around predetermined energies $E_l$ and show that it can lead to undesirable dependences of the calculated results on method-inherent parameters such as energy parameters $E_l$ and muffin-tin sphere radii. To overcome these dependences, we evaluate approaches that eliminate the linearization error systematically by adding local orbitals (LOs) to the basis set. We consider two kinds of LOs: (i) constructed from solutions $u_l(r,E)$ to the scalar-relativistic approximation of the radial Dirac equation with $E>E_l$ and (ii) constructed from second energy derivatives $\partial^2 u_l(r,E) / \partial E^2$ at $E=E_l$. We find that the latter eliminates the error most efficiently and yields the density functional answer to many electronic and materials properties with very high precision. Finally, we demonstrate that the so constructed LAPW+LO basis shows a more favorable convergence behavior than the conventional LAPW basis due to a better decoupling of muffin-tin and interstitial regions, similarly to the related APW+lo approach, which requires an extra set of LOs to reach the same total energy, though.Comment: 12 pages, 15 figure

Comparison between exact and semilocal exchange potentials: An all-electron study for solids

The exact-exchange (EXX) potential, which is obtained by solving the optimized-effective potential (OEP) equation, is compared to various approximate semilocal exchange potentials for a set of selected solids (C, Si, BN, MgO, Cu$_{2}$O, and NiO). This is done in the framework of the linearized augmented plane-wave method, which allows for a very accurate all-electron solution of electronic structure problems in solids. In order to assess the ability of the semilocal potentials to approximate the EXX-OEP, we considered the EXX total energy, electronic structure, electric-field gradient, and magnetic moment. An attempt to parameterize a semilocal exchange potential is also reported

Precise response functions in all-electron methods: Application to the optimized-effective-potential approach

The optimized-effective-potential (OEP) method is a special technique to construct local Kohn-Sham potentials from general orbital-dependent energy functionals. In a recent publication [M. Betzinger, C. Friedrich, S. Bl\"ugel, A. G\"orling, Phys. Rev. B 83, 045105 (2011)] we showed that uneconomically large basis sets were required to obtain a smooth local potential without spurious oscillations within the full-potential linearized augmented-plane-wave method (FLAPW). This could be attributed to the slow convergence behavior of the density response function. In this paper, we derive an incomplete-basis-set correction for the response, which consists of two terms: (1) a correction that is formally similar to the Pulay correction in atomic-force calculations and (2) a numerically more important basis response term originating from the potential dependence of the basis functions. The basis response term is constructed from the solutions of radial Sternheimer equations in the muffin-tin spheres. With these corrections the local potential converges at much smaller basis sets, at much fewer states, and its construction becomes numerically very stable. We analyze the improvements for rock-salt ScN and report results for BN, AlN, and GaN, as well as the perovskites CaTiO3, SrTiO3, and BaTiO3. The incomplete-basis-set correction can be applied to other electronic-structure methods with potential-dependent basis sets and opens the perspective to investigate a broad spectrum of problems in theoretical solid-state physics that involve response functions.Comment: 17 pages, 7 figures, 3 table

Local exact exchange potentials within the all-electron FLAPW method and a comparison with pseudopotential results

We present a general numerical approach to construct local Kohn-Sham potentials from orbital-dependent functionals within the all-electron full-potential linearized augmented-plane-wave (FLAPW) method, in which core and valence electrons are treated on an equal footing. As a practical example, we present a treatment of the orbital-dependent exact-exchange (EXX) energy and potential. A formulation in terms of a mixed product basis, which is constructed from products of LAPW basis functions, enables a solution of the optimized-effective-potential (OEP) equation with standard numerical algebraic tools and without shape approximations for the resulting potential. We find that the mixed product and LAPW basis sets must be properly balanced to obtain smooth and converged EXX potentials without spurious oscillations. The construction and convergence of the exchange potential is analyzed in detail for diamond. Our all-electron results for C, Si, SiC, Ge, GaAs semiconductors as well as Ne and Ar noble-gas solids are in very favorable agreement with plane-wave pseudopotential calculations. This confirms the adequacy of the pseudopotential approximation in the context of the EXX-OEP formalism and clarifies a previous contradiction between FLAPW and pseudopotential results.Comment: 12 pages, 7 figures, 5 table

Hybrid functionals within the all-electron FLAPW method: implementation and applications of PBE0

We present an efficient implementation of the PBE0 hybrid functional within the full-potential linearized augmented-plane-wave (FLAPW) method. The Hartree-Fock exchange term, which is a central ingredient of hybrid functionals, gives rise to a computationally expensive nonlocal potential in the one-particle Schroedinger equation. The matrix elements of this exchange potential are calculated with the help of an auxiliary basis that is constructed from products of FLAPW basis functions. By representing the Coulomb interaction in this basis the nonlocal exchange term becomes a Brillouin-zone (BZ) sum over vector-matrix-vector products. We show that the Coulomb matrix can be made sparse by a suitable unitary transformation of the auxiliary basis, which accelerates the computation of the vector-matrix-vector products considerably. Additionally, we exploit spatial and time-reversal symmetry to identify the nonvanishing exchange matrix elements in advance and to restrict the k summations for the nonlocal potential to an irreducible set of k points. Favorable convergence of the self-consistent-field cycle is achieved by a nested density-only and density-matrix iteration scheme. We discuss the convergence with respect to the parameters of our numerical scheme and show results for a variety of semiconductors and insulators, including oxide materials, where the PBE0 hybrid functional improves the band gaps and the description of localized states in comparison with the PBE functional. Furthermore, we find that in contrast to conventional local exchange-correlation functionals ferromagnetic EuO is correctly predicted to be a semiconductor.Comment: 15 pages, 6 figures, 2 table

Renormalization of effective interactions in a negative charge-transfer insulator

We compute from first principles the effective interaction parameters appropriate for a low-energy description of the rare-earth nickelate LuNiO$_{3}$ involving the partially occupied $e_g$ states only. The calculation uses the constrained random-phase approximation and reveals that the effective on-site Coulomb repulsion is strongly reduced by screening effects involving the oxygen-$p$ and nickel-$t_{2g}$ states. The long-range component of the effective low-energy interaction is also found to be sizeable. As a result, the effective on-site interaction between parallel-spin electrons is reduced down to a small negative value. This validates effective low-energy theories of these materials proposed earlier. Electronic structure methods combined with dynamical mean-field theory are used to construct and solve an appropriate low-energy model and explore its phase diagram as a function of the on-site repulsion and Hund's coupling. For the calculated values of these effective interactions we find, in agreement with experiments, that LuNiO$_{3}$ is a metal without disproportionation of the $e_g$ occupancy when considered in its orthorhombic structure, while the monoclinic phase is a disproportionated insulator.Comment: 10 pages, 4 figure

Phonons from Density-Functional Perturbation Theory using the All-Electron Full-Potential Linearized Augmented Plane-Wave Method FLEUR

Phonons are quantized vibrations of a crystal lattice that play a crucial role in understanding many properties of solids. Density functional theory (DFT) provides a state-of-the-art computational approach to lattice vibrations from first-principles. We present a successful software implementation for calculating phonons in the harmonic approximation, employing density-functional perturbation theory (DFPT) within the framework of the full-potential linearized augmented plane-wave (FLAPW) method as implemented in the electronic structure package FLEUR. The implementation, which involves the Sternheimer equation for the linear response of the wave function, charge density, and potential with respect to infinitesimal atomic displacements, as well as the setup of the dynamical matrix, is presented and the specifics due to the muffin-tin sphere centered LAPW basis-set and the all-electron nature are discussed. As a test, we calculate the phonon dispersion of several solids including an insulator, a semiconductor as well as several metals. The latter are comprised of magnetic, simple, and transition metals. The results are validated on the basis of phonon dispersions calculated using the finite displacement approach in conjunction with the FLEUR code and the phonopy package, as well as by some experimental results. An excellent agreement is obtained.Comment: 44 pages, 6 figure

The HSE hybrid functional within the FLAPW method and its application to GdN

We present an implementation of the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional within the full-potential linearized augmented-plane-wave (FLAPW) method. Pivotal to the HSE functional is the screened electron-electron interaction, which we separate into the bare Coulomb interaction and the remainder, a slowly varying function in real space. Both terms give rise to exchange potentials, which sum up to the screened nonlocal exchange potential of HSE. We evaluate the former with the help of an auxiliary basis, defined in such a way that the bare Coulomb matrix becomes sparse. The latter is computed in reciprocal space, exploiting its fast convergence behavior in reciprocal space. This approach is general and can be applied to a whole class of screened hybrid functionals. We obtain excellent agreement of band gaps and lattice constants for prototypical semiconductors and insulators with electronic-structure calculations using plane-wave or Gaussian basis sets. We apply the HSE hybrid functional to examine the ground-state properties of rocksalt GdN, which have been controversially discussed in literature. Our results indicate that there is a half-metal to insulator transition occurring between the theoretically optimized lattice constant at 0 K and the experimental lattice constant at room temperature. Overall, we attain good agreement with experimental data for band transitions, magnetic moments, and the Curie temperature.Comment: 13 pages, 4 figures, 6 table

Orbital-dependent exchange-correlation functionals in density-functional theory realized by the FLAPW method

In this thesis, we extended the applicability of the full-potential linearized augmented-planewave (FLAPW) method, one of the most precise, versatile and generally applicable electronic structuremethods for solids working within the framework of density-functional theory (DFT), to orbital-dependent functionals for the exchange-correlation (xc) energy. In contrast to the commonly applied local-density approximation (LDA) and generalized gradient approximation (GGA) for the xc energy, orbital-dependent functionals depend directly on the Kohn-Sham (KS) orbitals and only indirectly on the density. Two different schemes that deal with orbital-dependent functionals, the KS and the generalized Kohn-Sham (gKS) formalism, have been realized. While the KS scheme requires a local multiplicative xc potential, the gKS scheme allows for a non-local potential in the oneparticle Schrödinger equations. Hybrid functionals, combining some amount of the orbital-dependent exact exchange energy with local or semi-local functionals of the density, are implemented within the gKS scheme. We work in particular with the PBE0 hybrid of Perdew, Burke, and Ernzerhof. Our implementation relies on a representation of the non-local exact exchange potential – its calculation constitutes the most time consuming step in a practical calculation – by an auxiliary mixed product basis (MPB). In this way, thematrix elements of theHamiltonian corresponding to the non-local potential become a Brillouin-zone (BZ) sum over vector-matrix-vector products. Several techniques are developed and explored to further accelerate our numerical scheme. We show PBE0 results for a variety of semiconductors and insulators. In comparison with experiment, the PBE0 functional leads to improved band gaps and an improved description of localized states. Even for the ferromagnetic semiconductor EuO with localized 4 f electrons, the electronic andmagnetic properties are correctly described by the PBE0 functional. Subsequently, we discuss the construction of the local,multiplicative exact exchange (EXX) potential from the non-local, orbital-dependent exact exchange energy. For this purpose we employ the optimized effective potential (OEP) method. Central ingredients of the OEP equation are the KS wave-function response and the single-particle density response function. A formulation in terms of a slightly modified MPB enables to solve the OEP integral [...