The Atomic Simulation Environment (ASE) is a software package written in the Python programming language with the aim of setting up, steering, and analyzing atomistic simula- tions. In ASE, tasks are fully scripted in Python. The powerful syntax of Python combined with the NumPy array library make it possible to perform very complex simulation tasks. For example, a sequence of calculations may be performed with the use of a simple "for-loop" construction. Calculations of energy, forces, stresses and other quantities are performed through interfaces to many external electronic structure codes or force fields using a uniform interface. On top of this calculator interface, ASE provides modules for performing many standard simulation tasks such as structure optimization, molecular dynamics, handling of constraints and performing nudged elastic band calculations.

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The Atomic Simulation Environment - A Python library for working with atoms

The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or "Boys orbitals" as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.

Maximally-localized Wannier Functions: Theory and Applications

Molybdenum disulphide is a promising non-precious catalyst for hydrogen evolution because it contains active edge sites and an inert basal plane. Introducing sulphur vacancies and strain now leads to activation and optimization of the basal plane. As a promising non-precious catalyst for the hydrogen evolution reaction (HER; refs 1,2,3,4,5), molybdenum disulphide (MoS2) is known to contain active edge sites and an inert basal plane1,6,7,8. Activating the MoS2 basal plane could further enhance its HER activity but is not often a strategy for doing so. Herein, we report the first activation and optimization of the basal plane of monolayer 2H-MoS2 for HER by introducing sulphur (S) vacancies and strain. Our theoretical and experimental results show that the S-vacancies are new catalytic sites in the basal plane, where gap states around the Fermi level allow hydrogen to bind directly to exposed Mo atoms. The hydrogen adsorption free energy (ΔGH) can be further manipulated by straining the surface with S-vacancies, which fine-tunes the catalytic activity. Proper combinations of S-vacancy and strain yield the optimal ΔGH = 0 eV, which allows us to achieve the highest intrinsic HER activity among molybdenum-sulphide-based catalysts.

Activating and optimizing MoS2 basal planes for hydrogen evolution through the formation of strained sulphur vacancies

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.

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Reproducibility in density functional theory calculations of solids

The discovery of graphene and other two-dimensional (2D) materials together with recent advances in exfoliation techniques have set the foundations for the manufacturing of single layered sheets from any layered 3D material. The family of 2D materials encompasses a wide selection of compositions including almost all the elements of the periodic table. This derives into a rich variety of electronic properties including metals, semimetals, insulators and semiconductors with direct and indirect band gaps ranging from ultraviolet to infrared throughout the visible range. Thus, they have the potential to play a fundamental role in the future of nanoelectronics, optoelectronics and the assembly of novel ultrathin and flexible devices. We categorize the 2D materials according to their structure, composition and electronic properties. In this review we distinguish atomically thin materials (graphene, silicene, germanene, and their saturated forms; hexagonal boron nitride; silicon carbide), rare earth, semimetals, transition metal chalcogenides and halides, and finally synthetic organic 2D materials, exemplified by 2D covalent organic frameworks. Our exhaustive data collection presented in this Atlas demonstrates the large diversity of electronic properties, including band gaps and electron mobilities. The key points of modern computational approaches applied to 2D materials are presented with special emphasis to cover their range of application, peculiarities and pitfalls.

https://www.rsc.org/suppdata/cs/c4/c4cs00102h/c4cs00102h1.pdf

An atlas of two-dimensional materials

Electronic structure calculations have become an indispensable tool in many areas of materials science and quantum chemistry. Even though the Kohn-Sham formulation of the density-functional theory (DFT) simplifies the many-body problem significantly, one is still confronted with several numerical challenges. In this article we present the projector augmented-wave (PAW) method as implemented in the GPAW program package (https://wiki.fysik.dtu.dk/gpaw) using a uniform real-space grid representation of the electronic wavefunctions. Compared to more traditional plane wave or localized basis set approaches, real-space grids offer several advantages, most notably good computational scalability and systematic convergence properties. However, as a unique feature GPAW also facilitates a localized atomic-orbital basis set in addition to the grid. The efficient atomic basis set is complementary to the more accurate grid, and the possibility to seamlessly switch between the two representations provides great flexibility. While DFT allows one to study ground state properties, time-dependent density-functional theory (TDDFT) provides access to the excited states. We have implemented the two common formulations of TDDFT, namely the linear-response and the time propagation schemes. Electron transport calculations under finite-bias conditions can be performed with GPAW using non-equilibrium Green functions and the localized basis set. In addition to the basic features of the real-space PAW method, we also describe the implementation of selected exchange-correlation functionals, parallelization schemes, Delta SCF-method, x-ray absorption spectra, and maximally localized Wannier orbitals.

https://iopscience.iop.org/article/10.1088/0953-8984/22/25/253202/pdf

Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method

We model a Kohn-Sham potential with the discontinuity at integer particle numbers starting from the approximation by (GLLB) Gritsenko et al. [Phys. Rev. A 51, 1944 (1995)]. We evaluate the Kohn-Sham gap and the discontinuity to obtain the quasiparticle gap. This allows us to compare the Kohn-Sham gaps to those obtained by accurate many-body perturbation-theory-based optimized potential methods. In addition, the resulting quasiparticle band gap is compared to experimental gaps. In the GLLB model potential, the exchange-correlation hole is modeled using a generalized gradient approximation (GGA) energy density and the response of the hole-to-density variations is evaluated by using the common-denominator approximation and homogeneous electron-gas-based assumptions. In our modification, we have chosen the PBEsol potential as the GGA to model the exchange hole and add a consistent correlation potential. The method is implemented in the GPAW code, which allows efficient parallelization to study large systems. A fair agreement for Kohn-Sham and the quasiparticle band gaps with semiconductors and other band gap materials is obtained with a potential which is as fast as GGA to calculate.

Kohn-Sham potential with discontinuity for band gap materials

We have carried out electronic structure calculations on N interstitials in GaAs alloys using the first-principles plane-wave pseudopotential (PWPP) and projector augmented-wave (PAW) electronic structure methods in the framework of the density functional theory (DFT). Both the ultrasoft pseudopotential (USPP) method in connection with the generalized gradient approximation (GGA) and the PAW method in connection with the local density approximation (LDA) have been employed. Effects of the single nitrogen atom and nitrogen dimer related interstitial defects on the atomic and electronic structures of GaAs have been studied. Total energies, electronic band structures, and local densities of states have been evaluated. In general, energies of the defects with the NN dimer at the center of the Ga or As tetrahedron are more than 2 eV lower per nitrogen atom than those with a single N impurity at the same sites. We have also found that there are metastable defect candidates with a single N impurity in the middle of a particular edge of the Ga or As tetrahedra. Considering the modifications of the atomic structure of GaAs, our calculations show that the relaxations of the nearest neighbor atoms around the NN dimer at the center of the Ga or As tetrahedron, are essentially smaller than those around the single N impurity at the center of an edge of the Ga tetrahedron. Finally, these defect states induce drastic modifications into the electronic structure of GaAs. Interestingly, the NN dimer related defects cause noticeable changes only to the conduction bands near the conduction band edge, while the single N impurity related defects mainly modify the valence band edge and also induce localized and delocalized states into the band gap. Notably, the NN dimer and N impurity related defects lead to redshift and blueshift behavior of the band gap, respectively. The blueshift behavior is tentatively supported by the recent photoluminescence experiments.

Atomic and electronic structures of N interstitials in GaAs

An azobenzene molecule Disperse Red 1 with strongly delocalised frontier orbitals has been studied with a number of density-functional theory (DFT) related approaches. The purpose is two-fold: to interpret observed photoabsorption and to compare the performance of various DFT-based approximations. The planarity of the vacuum conformation of the lower energy trans conformation is found to be significantly distorted when transformed to the higher energy cis conformation, and both isomers are found to absorb in the experimentally observed wavelengths in solutions. A common feature in both is essentially forbidden lowest energy absorption because of a negligible overlap of HOMO and LUMO due to symmetry unmatch. We find the time-dependent DFT to be the best approach for quantitative evaluation of photoabsorption energies and intensities, whereas Kohn-Sham eigenenergies and orbitals give good description only qualitatively. Isomerisation and dimerisation energetics have also been evaluated, and trans dimers have found to be up to 1 eV per molecule more stable than single molecules.

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Jussi Ojanen

Papers

Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method

Kohn-Sham potential with discontinuity for band gap materials

Atomic and electronic structures of N interstitials in GaAs

Electronic Structure and Absorption Spectrum of Disperse Red 1: Comparison of Computational Approaches