Karlheinz Schwarz

Full-potential, linearized augmented plane wave programs for crystalline systems

In solids, linearized augmented plane waves (LAPWs) have proven to be an effective basis for the solution of the Kohn-Sham equations, the main calculational task in the local spin density approximation (LSDA) to density functional theory. The WIEN package uses LAPWs to calculate the LSDA total energy, spin densities, Kohn-Sham eigenvalues, and the electric field gradients at nuclear sites for a broad variety of space groups. Options include retention or omission of non-muffin-tin contributions (hence WIEN is a full-potential or F-LAPW code) and relativistic corrections (full treatment for core states, Scalar-relativistic for valence states).

Electronic structure calculations of solids using the WIEN2k package for material sciences

Abstract In studies of the electronic structure of solids, the augmented plane wave (APW) method is the basis for the solution of the Kohn–Sham equations of density functional theory (DFT). The different versions and developing steps are discussed in terms of linearization, full potential, local orbitals, mixed basis sets, relativistic effects and computational aspects, as employed in the WIEN2k code.

Solid state calculations using WIEN2k

To study solid materials on the atomic scale one often starts with an ideal crystal at zero temperature and calculates its electronic structure by means of density functional theory (DFT). This allows a quantum mechanical treatment of the physics that underlines properties such as relative stability, chemical bonding, relaxation of the atoms, phase transitions, electrical, mechanical, optical or magnetic behavior, etc. For the solution of the DFT equations several methods have been developed. The linearized-augmented-plane-wave method is one of the most accurate methods. It is embodied in the computer code––WIEN2k––which is now used worldwide by more than 500 groups to solve crystal properties on the atomic scale (see www.wien2k.at). Nowadays calculations of this type can be done––on sufficiently powerful computers––for systems containing about 100 atoms per unit cell. Chromium dioxide CrO2 is selected as a representative example using both, bulk and surface structures. References to other applications are given.

Reproducibility in density functional theory calculations of solids

A comparison of DFT methods Density functional theory (DFT) is now routinely used for simulating material properties. Many software packages are available, which makes it challenging to know which are the best to use for a specific calculation. Lejaeghere et al. compared the calculated values for the equation of states for 71 elemental crystals from 15 different widely used DFT codes employing 40 different potentials (see the Perspective by Skylaris). Although there were variations in the calculated values, most recent codes and methods converged toward a single value, with errors comparable to those of experiment. Science, this issue p. 10.1126/science.aad3000; see also p. 1394 A survey of recent density functional theory methods shows a convergence to more accurate property calculations. [Also see Perspective by Skylaris] INTRODUCTION The reproducibility of results is one of the underlying principles of science. An observation can only be accepted by the scientific community when it can be confirmed by independent studies. However, reproducibility does not come easily. Recent works have painfully exposed cases where previous conclusions were not upheld. The scrutiny of the scientific community has also turned to research involving computer programs, finding that reproducibility depends more strongly on implementation than commonly thought. These problems are especially relevant for property predictions of crystals and molecules, which hinge on precise computer implementations of the governing equation of quantum physics. RATIONALE This work focuses on density functional theory (DFT), a particularly popular quantum method for both academic and industrial applications. More than 15,000 DFT papers are published each year, and DFT is now increasingly used in an automated fashion to build large databases or apply multiscale techniques with limited human supervision. Therefore, the reproducibility of DFT results underlies the scientific credibility of a substantial fraction of current work in the natural and engineering sciences. A plethora of DFT computer codes are available, many of them differing considerably in their details of implementation, and each yielding a certain “precision” relative to other codes. How is one to decide for more than a few simple cases which code predicts the correct result, and which does not? We devised a procedure to assess the precision of DFT methods and used this to demonstrate reproducibility among many of the most widely used DFT codes. The essential part of this assessment is a pairwise comparison of a wide range of methods with respect to their predictions of the equations of state of the elemental crystals. This effort required the combined expertise of a large group of code developers and expert users. RESULTS We calculated equation-of-state data for four classes of DFT implementations, totaling 40 methods. Most codes agree very well, with pairwise differences that are comparable to those between different high-precision experiments. Even in the case of pseudization approaches, which largely depend on the atomic potentials used, a similar precision can be obtained as when using the full potential. The remaining deviations are due to subtle effects, such as specific numerical implementations or the treatment of relativistic terms. CONCLUSION Our work demonstrates that the precision of DFT implementations can be determined, even in the absence of one absolute reference code. Although this was not the case 5 to 10 years ago, most of the commonly used codes and methods are now found to predict essentially identical results. The established precision of DFT codes not only ensures the reproducibility of DFT predictions but also puts several past and future developments on a firmer footing. Any newly developed methodology can now be tested against the benchmark to verify whether it reaches the same level of precision. New DFT applications can be shown to have used a sufficiently precise method. Moreover, high-precision DFT calculations are essential for developing improvements to DFT methodology, such as new density functionals, which may further increase the predictive power of the simulations. Recent DFT methods yield reproducible results. Whereas older DFT implementations predict different values (red darts), codes have now evolved to mutual agreement (green darts). The scoreboard illustrates the good pairwise agreement of four classes of DFT implementations (horizontal direction) with all-electron results (vertical direction). Each number reflects the average difference between the equations of state for a given pair of methods, with the green-to-red color scheme showing the range from the best to the poorest agreement. The widespread popularity of density functional theory has given rise to an extensive range of dedicated codes for predicting molecular and crystalline properties. However, each code implements the formalism in a different way, raising questions about the reproducibility of such predictions. We report the results of a community-wide effort that compared 15 solid-state codes, using 40 different potentials or basis set types, to assess the quality of the Perdew-Burke-Ernzerhof equations of state for 71 elemental crystals. We conclude that predictions from recent codes and pseudopotentials agree very well, with pairwise differences that are comparable to those between different high-precision experiments. Older methods, however, have less precise agreement. Our benchmark provides a framework for users and developers to document the precision of new applications and methodological improvements.

The interface between silicon and a high- k oxide

The ability of the semiconductor industry to continue scaling microelectronic devices to ever smaller dimensions (a trend known as Moores Law) is limited by quantum mechanical effects: as the thickness of conventional silicon dioxide (SiO2) gate insulators is reduced to just a few atomic layers, electrons can tunnel directly through the films. Continued device scaling will therefore probably require the replacement of the insulator with high-dielectric-constant (high-k) oxides, to increase its thickness, thus preventing tunnelling currents while retaining the electronic properties of an ultrathin SiO2 film. Ultimately, such insulators will require an atomically defined interface with silicon without an interfacial SiO2 layer for optimal performance. Following the first reports of epitaxial growth of AO and ABO3 compounds on silicon, the formation of an atomically abrupt crystalline interface between strontium titanate and silicon was demonstrated. However, the atomic structure proposed for this interface is questionable because it requires silicon atoms that have coordinations rarely found elsewhere in nature. Here we describe first-principles calculations of the formation of the interface between silicon and strontium titanate and its atomic structure. Our study shows that atomic control of the interfacial structure by altering the chemical environment can dramatically improve the electronic properties of the interface to meet technological requirements. The interface structure and its chemistry may provide guidance for the selection process of other high-k gate oxides and for controlling their growth.

DFT calculations of solids with LAPW and WIEN2k

Abstract In solids one often starts with an ideal crystal that is studied on the atomic scale at zero temperature. The unit cell may contain several atoms (at certain positions) and is repeated with periodic boundary conditions. Quantum mechanics governs the electronic structure that is responsible for properties such as relative stability, chemical bonding, relaxation of the atoms, phase transitions, electrical, mechanical, optical or magnetic behavior, etc. Corresponding first principles calculations are mainly done within density functional theory (DFT), according to which the many-body problem of interacting electrons and nuclei is mapped to a series of one-electron equations, the so-called Kohn–Sham (KS) equations. One among the most precise schemes to solve the KS equations is the linearized-augmented-plane-wave (LAPW) method that is employed for example in the computer code WIEN2k to study crystal properties on the atomic scale (see www.wien2k.at ). Nowadays such calculations can be done—on sufficiently powerful computers—for systems containing about 100 atoms per unit cell. A selection of representative examples and the references to the original literature is given.

Improving the efficiency of FP-LAPW calculations

Abstract The full-potential linearized augmented-plane wave (FP-LAPW) method is well known to enable most accurate calculations of the electronic structure and magnetic properties of crystals and surfaces. The implementation of atomic forces has greatly increased its applicability, but it is still generally believed that FP-LAPW calculations require substantial higher computational effort compared to the pseudopotential plane wave (PPW) based methods. In the present paper we analyze the FP-LAPW method from a computational point of view. Starting from an existing implementation (WIEN95 code), we identified the time consuming parts and show how some of them can be formulated more efficiently. In this context also the hardware architecture plays a crucial role. The remaining computational effort is mainly determined by the setup and diagonalization of the Hamiltonian matrix. For the latter, two different iterative schemes are compared. The speed-up gained by these optimizations is compared to the runtime of the “original” version of the code, and the PPW approach. We expect that the strategies described here, can also be used to speed up other computer codes, where similar tasks must be performed.

Band gap calculations with Becke–Johnson exchange potential

Recently, a simple analytical form for the exchange potential was proposed by Becke and Johnson. This potential, which depends on the kinetic-energy density, was shown to reproduce very well the shape of the exact exchange potential (obtained with the optimized effective potential method) for atoms. Calculations on solids show that the Becke–Johnson potential leads to a better description of band gaps of semiconductors and insulators with respect to the standard local density and Perdew–Burke–Ernzerhof approximations for the exchange–correlation potential. Comparison is also made with the values obtained with the Engel–Vosko exchange potential which was also developed using the exact exchange potential.

Interaction of water and methanol with a zeolite at high coverages

Abstract We investigate the interaction of small polar molecules such as water and methanol with a low-aluminum sodalite using first principles density functional molecular dynamics calculations and the projector augmented wave method. Whereas in the low coverage limit methanol is unprotonated, we find that at higher coverage the proton is transferred from the acid site to the adsorbed molecules. The derived vibrational spectra are consistent with infrared data. Our findings indicate that acid-catalyzed reactions in zeolites should parallel those in highly acidic solutions, while zeolites offer the additional advantages of higher acidity, temperature and selectivity due to steric effects.

Electron-density distribution in stishovite, SiO2: a new high-energy synchrotron-radiation study

The electron-density distribution of the high-pressure polymorph of SiO2, stishovite [a = 4.177 (1), c = 2.6655 (5) A, space group P4(2)/mnm, Z = 2], has been redetermined by single-crystal diffractometry using synchrotron radiation of 100.42 and 30.99 keV, respectively, in order to obtain essentially absorption- and extinction-free data. Room-temperature diffraction experiments on two samples of irregular shape were carried out on two different diffractometers installed at HASYLAB/DESY, Hamburg, Germany. The structure refinement on the high-energy data converged at R(F) = 0.0047, wR(F) = 0.0038, GoF = 0.78, for a multipole model with neutral atoms and multipole expansions up to seventh order. For each atom, the radial expansion coefficients of the multipole orders (l > 0) were constrained to a common value. The absence of extinction was indicated by a refined correction parameter equalling zero within error limit. The excellent quality of the data is also illustrated by a high-order (HO) refinement (s > 0.7 A(-1)) yielding R(F) = 0.0060, wR(F) = 0.0048, GoF = 0.85. Both static deformation electron-density distribution and structure amplitudes compare well with corresponding results obtained from band-structure calculations using the linearized-augmented-plane-wave (LAPW) method. Ensuing topological analysis of the total model electron density distribution revealed bond critical point properties for the two unique Si--O bonds, indicating a predominantly closed-shell interaction mixed with a significant shared interaction contribution that decreases with increasing interatomic distance. Calculation of atomic basins yielded charges of +3.39 e and -1.69 e for Si and O, respectively, in good agreement with the theoretically calculated values of +3.30 e and -1.65 e. The volumina of the Si and O basins are 2.32 and 10.48 A3, corresponding to spheres with radii of 0.82 and 1.36 A, respectively. The results also conform well with correlations between bond length and bond critical point properties reported in the literature for geometry-optimized hydroxyacid molecules. Estimates of the Si cation electronegativity indicate that the change of Si coordination by oxygen from 4 to 6 is accompanied by an increase of the ionicity of the Si--O bond of about 7%.