Projector Augmented Wave Formalism Research Articles

Covariant Jacobi-Legendre expansion for total energy calculations within the projector augmented wave formalism

Covariant Jacobi-Legendre expansion for total energy calculations within the projector augmented wave formalism

Stochastic and mixed density functional theory within the projector augmented wave formalism for simulation of warm dense matter.

Stochastic density functional theory (DFT) and mixed stochastic-deterministic DFT are burgeoning approaches for the calculation of the equationof state and transport properties in materials under extreme conditions. In the intermediate warm dense matter regime, a state between correlated condensed matter and kinetic plasma, electrons can range from being highly localized around nuclei to delocalized over the whole simulation cell. The plane-wave basis pseudopotential approach is thus the typical tool of choice for modeling such systems at the DFT level. Unfortunately, stochastic DFT methods scale as the square of the maximum plane-wave energy in this basis. To reduce the effect of this scaling and improve the overall description of the electrons within the pseudopotential approximation, we present stochastic and mixed DFT approaches developed and implemented within the projector augmented wave formalism. We compare results between the different DFT approaches for both single-point and molecular dynamics trajectories and present calculations of self-diffusion coefficients of solid density carbon from 1 to 50eV.

Orbital magnetism and chemical shielding in the projector augmented-wave formalism

Orbital magnetism and chemical shielding in the projector augmented-wave formalism

Nonrad: Computing nonradiative capture coefficients from first principles

Point defects in semiconductor crystals provide a means for carriers to recombine nonradiatively. This recombination process impacts the performance of devices. We present the Nonrad code that implements the first-principles approach of Alkauskas et al. (2014) [8] for the evaluation of nonradiative capture coefficients based on a quantum-mechanical description of the capture process. An approach for evaluating electron-phonon coupling within the projector augmented wave formalism is presented. We also show that the common procedure of replacing Dirac delta functions with Gaussians can introduce errors into the resulting capture rate, and implement an alternative scheme to properly account for vibrational broadening. Lastly, we assess the accuracy of using an analytic approximation to the Sommerfeld parameter by comparing with direct numerical evaluation. Program summaryProgram Title: NonradCPC Library link to program files:https://doi.org/10.17632/xmfj4zxmn3.1Developer's repository link:https://doi.org/10.5281/zenodo.4274317Code Ocean capsule:https://codeocean.com/capsule/5582062Licensing provisions: MIT LicenseProgramming language: Python 3Nature of problem: Nonradiative carrier capture at point defects in semiconductors and insulators significantly impacts the performance of devices. A first-principles approach to calculating nonradiative capture rates is necessary to guide materials design and provide atomistic insight into the nonradiative capture process.Solution method: Our code, written in the Python language, implements the first-principles methodology developed by Alkauskas et al. (2014) [8]. Nonradiative capture rates are calculated using a one-dimensional approach, which maps the prohibitively large phonon problem onto a single, effective phonon mode. Harmonic phonon matrix elements may then be computed using an analytic technique or by direct numerical integration of the harmonic oscillator wavefunctions. The electron-phonon coupling is evaluated using the VASP code [Kresse and Furthmüller (1996) [27]] to linear order for the effective mode.

Calculation of optical properties with spin-orbit coupling for warm dense matter

Optical properties of warm dense matter are computed within linear response theory using the Kubo-Greenwood formula in the framework of Density Functional Theory and the Projector Augmented–Wave (PAW) method. For transition metals, the effect of spin-orbit coupling (SOC) has to be taken into account. We present the theoretical framework to introduce SOC in the PAW formalism and then to compute optical properties including SOC using the Kubo-Greenwood formula. This formulation has been implemented in the abinit software. Moreover, for high energy absorption spectra, we use Dirac relativistic core wave functions from the atompaw code to obtain the correct SOC energy splitting. This implementation enables a description of optical properties for warm dense metals including SOC for both core and valence orbitals. Example calculations are presented for warm dense gold and copper.

Smart local orbitals for efficient calculations within density functional theory and beyond.

Localized basis sets in the projector augmented wave formalism allow for computationally efficient calculations within density functional theory (DFT). However, achieving high numerical accuracy requires an extensive basis set, which also poses a fundamental problem for the interpretation of the results. We present a way to obtain a reduced basis set of atomic orbitals through the subdiagonalization of each atomic block of the Hamiltonian. The resulting local orbitals (LOs) inherit the information of the local crystal field. In the LO basis, it becomes apparent that the Hamiltonian is nearly block-diagonal, and we demonstrate that it is possible to keep only a subset of relevant LOs that provide an accurate description of the physics around the Fermi level. This reduces to some extent the redundancy of the original basis set, and at the same time, it allows one to perform post-processing of DFT calculations, ranging from the interpretation of electron transport to extracting effective tight-binding Hamiltonians, very efficiently and without sacrificing the accuracy of the results.

Accurate Computation of Nonadiabatic Coupling with Projector Augmented-Wave Pseudopotentials

Synergy of nonadiabatic molecular dynamics with real-time time-dependent density functional theory has led to significant progress in modeling excited-state dynamics in nanoscale and condensed matter systems over the past decade. Nonadiabatic coupling (NAC) is the central quantity in such simulations, and its accurate and efficient evaluation is an enduring challenge in time-dependent Kohn-Sham theory, particularly in conjunction with planewave basis sets and projector augmented-wave (PAW) pseudopotentials because of the complexity of the PAW "all-electron" wave function. We report a method for rigorous evaluation of the NAC with PAW wave functions and demonstrate an efficient approximation to the rigorous NAC that gives comparable accuracy. As a validation, we intensely examine the NAC matrix elements calculated using both pseudo- and all-electron wave functions under the PAW formalism in six representative systems. The approximate NAC obtained with pseudowave functions is close to the exact all-electron NAC, with the largest deviations observed when subshell d-electrons are involved in the transitions. The developed approach provides a rigorous and convenient methodology for the numerical computation of NAC in the Kohn-Sham theory framework.

Crystal field coefficients for yttrium analogues of rare-earth/transition-metal magnets using density-functional theory in the projector-augmented wave formalism

We present a method of calculating crystal field coefficients of rare-earth/transition-metal (RE-TM) magnets within density-functional theory (DFT). The principal idea of the method is to calculate the crystal field potential of the yttrium analogue (‘Y-analogue’) of the RE-TM magnet, i.e. the material where the lanthanide elements have been substituted with yttrium. The advantage of dealing with Y-analogues is that the methodological and conceptual difficulties associated with treating the highly-localized 4f electrons in DFT are avoided, whilst the nominal valence electronic structure principally responsible for the crystal field is preserved. In order to correctly describe the crystal field potential in the core region of the atoms we use the projector-augmented wave formalism of DFT, which allows the reconstruction of the full charge density and electrostatic potential. The Y-analogue crystal field potentials are combined with radial 4f charge densities obtained in self-interaction-corrected calculations on the lanthanides to obtain crystal field coefficients. We demonstrate our method on a test set of ten materials comprising nine RE-TM magnets and elemental Tb. We show that the calculated easy directions of magnetization agree with experimental observations, including a correct description of the anisotropy within the basal plane of Tb and NdCo5. We further show that the Y-analogue calculations generally agree quantitatively with previous calculations using the open-core approximation to treat the 4f electrons, and argue that our simple approach may be useful for large-scale computational screening of new magnetic materials.

Analytic Interatomic Forces in the Random Phase Approximation.

We discuss that in the random phase approximation (RPA) the first derivative of the energy with respect to the Green's function is the self-energy in the GW approximation. This relationship allows us to derive compact equations for the RPA interatomic forces. We also show that position dependent overlap operators are elegantly incorporated in the present framework. The RPA force equations have been implemented in the projector augmented wave formalism, and we present illustrative applications, including abinitio molecular dynamics simulations, the calculation of phonon dispersion relations for diamond and graphite, as well as structural relaxations for water on boron nitride. The present derivation establishes a concise framework for forces within perturbative approaches and is also applicable to more involved approximations for the correlation energy.

Theory and Applications of Generalized Pipek-Mezey Wannier Functions.

The theory for the generation of Wannier functions within the generalized Pipek-Mezey approach (Lehtola, S.; Jónsson, H. J. Chem. Theory Comput. 2014, 10, 642) is presented and an implementation thereof is described. Results are shown for systems with periodicity in one, two, and three dimensions as well as isolated molecules. The generalized Pipek-Mezey Wannier functions (PMWF) are highly localized orbitals consistent with chemical intuition where a distinction is maintained between σ- and π-orbitals. The PMWF method is compared with the so-called maximally localized Wannier functions (MLWFs) that are frequently used for the analysis of condensed matter calculations. Whereas PMWFs maximize the localization criterion of Pipek and Mezey, MLWFs maximize that of Foster and Boys and have the disadvantage of mixing σ- and π-orbitals in many cases. The PMWF orbitals turn out to be as localized as the MLWF orbitals as evidenced by cross-comparison of the values of the PMWF and MLWF objective functions for the two types of orbitals. Our implementation in the atomic simulation environment (ASE) is compatible with various representations of the wave function, including real-space grids, plane waves, and linear combinations of atomic orbitals. The projector-augmented wave formalism for the representation of atomic core electrons is also supported. Results of calculations with the GPAW software are described here, but our implementation can also use output from other electronic structure software such as ABINIT, NWChem, and VASP.

Finite homogeneous electric fields in the projector augmented wave formalism: Applications to linear and nonlinear response

Computation of the effect of finite homogeneous electric fields in periodic systems is described in the context of the projector augmented wave formalism. Generalizing previous norm-conserving implementations, we compute the polarization using the approach of King-Smith and Vanderbilt [Phys. Rev. B 47 (1993) 1651] fully within the projector augmented wave framework, and then minimizing the electric field enthalpy. Applications to simple systems are provided, including both linear and nonlinear field effects, and photoelastic response.

Comment on “Interplay of defect cluster and the stability of xenon in uranium dioxide from density functional calculations”

Geng et al. [H. Y. Geng, Y. Chen, Y. Kaneta, M. Kinoshita, and Q. Wu, Phys. Rev. B 82, 094106 (2010)] recently reported $\mathrm{DFT}+U$ calculations of xenon behavior in uranium dioxide UO${}_{2}$. One of the main conclusions of their work is that the quasiannealing (QA) procedure allowed them to avoid metastable states created by the $\text{DFT}+U$ approximation. However, based on a comparison of total energies, they stated that an incomplete implementation of occupation matrix control (OMC) had been done in our previous work [B. Dorado, G. Jomard, M. Freyss, and M. Bertolus, Phys. Rev. B 82, 035114 (2010)] and that we failed to reach the ground state of the perfect UO${}_{2}$ fluorite structure. In this Comment, we show that the discrepancy they observed does not stem from an incomplete implementation of OMC, but from the calculation of the compensation charge used in the projector augmented-wave formalism.

Hybrid functional for correlated electrons in the projector augmented-wave formalism: Study of multiple minima for actinide oxides

Exact (Hartree-Fock) exchange for correlated electrons is implemented to describe correlated orbitals in the projector augmented-waves (PAW) framework, as suggested recently in another context [P. Nov\'ak et al., Phys. Status Solidi B 243, 563 (2006)]. Hartree-Fock exchange energy is applied to strongly correlated electrons only inside the PAW atomic spheres. This allows the use of PBE0 hybrid exchange-correlation functional for correlated electrons. This method is tested on NiO and results agree well with already published results and generalized gradient approximation, $\text{GGA}+U$ calculations. It is then applied to plutonium oxides and ${\text{UO}}_{2}$ for which the results are comparable with the ones of $\text{GGA}+U$ calculations but without adjustable parameter. As evidenced in the uranium oxide case, the occurrence of multiple energy minima may lead to very different results depending on the initial electronic configurations and on the symmetries taken into account in the calculation.

Computation of Mössbauer isomer shifts from first principles

Computation of the observables of a Mössbauer spectrum, primarily the isomershift, from a first-principles approach is described. The framework used isdensity functional theory using the projector augmented wave formalism (DFTPAW), which enables efficient computation even of many-electron solids such asSnCl2. The proper PAW version of the isomer shift is derived and shown to be correct throughcomparison of computed shifts and experiment in a variety of compounds basedon tin, germanium and zinc. The effects of pressure are considered as well asmotional effects including the Lamb–Mössbauer factor and the second-order Dopplershift.

First-principles study of the nuclear quadrupole resonance parameters and orbital ordering inLaTiO3

The electronic structure of the distorted perovskite ${\text{LaTiO}}_{3}$ is studied using density-functional theory, implemented with a projector-augmented wave formalism together with the local-density approximation plus Hubbard $U$-parameter $(\text{LDA}+U)$ model. Various experimentally determined quantities, including the antiferromagnetic ordering and spin moments and electric field gradients at the lanthanum and titanium sites, are well reproduced. It is found that the excess spin density on the titanium sites, which accounts for the antiferromagnetism, is very well described by the conjectured ${d}_{xy}+{d}_{yz}+{d}_{zx}$ orbital configuration.

Projector augmented-wave approach to density-functional perturbation theory

The density-functional theory total energy within the projector-augmented wave formalism is expressed in a form suitable for application of the variation-perturbation formalism. We derive the corresponding expressions up to the third order. The much deeper complexity of the projector-augmented wave formalism, compared to the norm-conserving pseudopotential case, implies the introduction of several new notations. However, the structure of the resulting formalism is quite similar, and should be as useful, accurate, and widely applicable.

Geometry optimization of periodic systems using internal coordinates

An algorithm is proposed for the structural optimization of periodic systems in internal (chemical) coordinates. Internal coordinates may include in addition to the usual bond lengths, bond angles, out-of-plane and dihedral angles, various "lattice internal coordinates" such as cell edge lengths, cell angles, cell volume, etc. The coordinate transformations between Cartesian (or fractional) and internal coordinates are performed by a generalized Wilson B-matrix, which in contrast to the previous formulation by Kudin et al. [J. Chem. Phys. 114, 2919 (2001)] includes the explicit dependence of the lattice parameters on the positions of all unit cell atoms. The performance of the method, including constrained optimizations, is demonstrated on several examples, such as layered and microporous materials (gibbsite and chabazite) as well as the urea molecular crystal. The calculations used energies and forces from the ab initio density functional theory plane wave method in the projector-augmented wave formalism.

First-principles study on thermal vibration effects ofMgH2

We report the results of the first-principles calculation on the atomic displacement parameters, the crystal structure factors, and the charge density distributions of $\mathrm{Mg}{\mathrm{H}}_{2}$ at finite temperature. Our calculations have been performed with the ultrasoft pseudopotential method and the linear response approach based on the density-functional perturbation theory (DFPT). The thermal vibration effects have been taken into consideration using phonon dispersions predicted by the DFPT and the harmonic approximation. To obtain the crystal structure factors, the pseudo charge density has been converted to the all electron one by utilizing the projector augmented wave formalism. The computed atomic displacement parameters of the hydrogen atom in $\mathrm{Mg}{\mathrm{H}}_{2}$ are in good agreement with the inelastic neutron data. The crystal structure factors at $300\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ predicted from the atomic form factors and the Debye-Waller factors have been compared with those obtained from the synchrotron x-ray diffraction measurement. They are in good agreement for lower diffraction vector indices $hkl$, but there are slightly differences in higher $hkl$. We also discuss the temperature dependence of the charge density distribution. It is confirmed that thermal vibration is more effective to the H atom than the Mg atom. The ionic charge which is estimated from the number of electrons within the sphere around the H atom is well in agreement with the experimental result by the maximum entropy method.

First-principles characterization of a heteroceramic interface:ZrO2(001)deposited on anα−Al2O3(11¯02)substrate

We have studied an alumina/zirconia interface using the all-electron projector augmented wave formalism within density functional theory. We present the electronic, structural, and energetic properties of the ${\mathrm{ZrO}}_{2}(001)/\ensuremath{\alpha}\ensuremath{-}{\mathrm{Al}}_{2}{\mathrm{O}}_{3}(11\ifmmode\bar\else\textasciimacron\fi{}02)$ interface as well as of the free $\ensuremath{\alpha}\ensuremath{-}{\mathrm{Al}}_{2}{\mathrm{O}}_{3}(11\ifmmode\bar\else\textasciimacron\fi{}02)$ and ${\mathrm{ZrO}}_{2}(001)$ surfaces. We find that the generalized gradient correction significantly lowers the oxide surface energies, compared to values obtained by the local density approximation. The monoclinic-tetragonal transition in ${\mathrm{ZrO}}_{2}(001)$ thin films is discussed as well as strain effects involved in the interface formation. The stoichiometric alumina/zirconia interface is found to be weakly bonded, regardless of the film thickness, and the ${\mathrm{ZrO}}_{2}(001)/\ensuremath{\alpha}\ensuremath{-}{\mathrm{Al}}_{2}{\mathrm{O}}_{3}(11\ifmmode\bar\else\textasciimacron\fi{}02)$ interface has a rather epitaxial character, due to a low lattice mismatch of $\ensuremath{\sim}4%.$ The impact of such weak interactions on ceramic coating stability is discussed.

Comparison of the projector augmented-wave, pseudopotential, and linearized augmented-plane-wave formalisms for density-functional calculations of solids

The projector augmented-wave (PAW) method was developed by Bl\ochl as a method to accurately and efficiently calculate the electronic structure of materials within the framework of density-functional theory. It contains the numerical advantages of pseudopotential calculations while retaining the physics of all-electron calculations, including the correct nodal behavior of the valence-electron wave functions and the ability to include upper core states in addition to valence states in the self-consistent iterations. It uses many of the same ideas developed by Vanderbilt in his ``soft pseudopotential'' formalism and in earlier work by Bl\ochl in his ``generalized separable potentials,'' and has been successfully demonstrated for several interesting materials. We have developed a version of the PAW formalism for general use in structural and dynamical studies of materials. In the present paper, we investigate the accuracy of this implementation in comparison with corresponding results obtained using pseudopotential and linearized augmented-plane-wave (LAPW) codes. We present results of calculations for the cohesive energy, equilibrium lattice constant, and bulk modulus for several representative covalent, ionic, and metallic materials including diamond, silicon, SiC, ${\mathrm{CaF}}_{2}$, fcc Ca, and bcc V. With the exception of ${\mathrm{CaF}}_{2}$, for which core-electron polarization effects are important, the structural properties of these materials are represented equally well by the PAW, LAPW, and pseudopotential formalisms.