Orbital-free density functional theory implementation with the projector augmented-wave method

GPAW: An open Python package for electronic structure calculations

We review the GPAW open-source Python package for electronic structure calculations. GPAW is based on the projector-augmented wave method and can solve the self-consistent density functional theory (DFT) equations using three different wave-function representations, namely real-space grids, plane waves, and numerical atomic orbitals. The three representations are complementary and mutually independent and can be connected by transformations via the real-space grid. This multi-basis feature renders GPAW highly versatile and unique among similar codes. By virtue of its modular structure, the GPAW code constitutes an ideal platform for the implementation of new features and methodologies. Moreover, it is well integrated with the Atomic Simulation Environment (ASE), providing a flexible and dynamic user interface. In addition to ground-state DFT calculations, GPAW supports many-body GW band structures, optical excitations from the Bethe-Salpeter Equation, variational calculations of excited states in molecules and solids via direct optimization, and real-time propagation of the Kohn-Sham equations within time-dependent DFT. A range of more advanced methods to describe magnetic excitations and non-collinear magnetism in solids are also now available. In addition, GPAW can calculate non-linear optical tensors of solids, charged crystal point defects, and much more. Recently, support for graphics processing unit (GPU) acceleration has been achieved with minor modifications to the GPAW code thanks to the CuPy library. We end the review with an outlook, describing some future plans for GPAW.

Construction of 2D/1D Cu7S4 nanosheets/Mn0.3Cd0.7S nanorods heterojunction for highly efficient photocatalytic hydrogen evolution

Developing cost-effective cocatalyst-modified photocatalytic systems with boosted carrier separation and rapid surface catalytic reaction is an ideal strategy for effectively converting solar energy into desired fuels. Herein, a series of Cu7S4/Mn0.3Cd0.7S hierarchical heterostructures are designed and fabricated to achieve efficient and robust photocatalytic H2 evolution by coupling one-dimensional (1D) Mn0.3Cd0.7S nanorods with two-dimensional (2D) Cu7S4 nanosheets through a facile sonochemical strategy. Benefiting from dimensionality and cocatalyst effects, the constructed 2D/1D Cu7S4/Mn0.3Cd0.7S heterojunction photocatalyst containing 1.5 wt% Cu7S4 displays excellent photostability and superior photocatalytic H2 evolution rate up to 914.3 μmol h-1, which is 4.43 and 2.22-folds increment relative to bare Mn0.3Cd0.7S and the 3 wt% Pt/Mn0.3Cd0.7S, respectively. The various characterization results reveal that the utilization of semimetallic Cu7S4 nanosheets as the cocatalyst to form a Schottky heterojunction can promote the light-harvesting capability, suppress charge carrier recombination, and provide sufficient reaction sites for hydrogen generation, thereby resulting in the dramatically improved photocatalytic performance. This work clarifies the role of Cu7S4 nanosheets as the robust and cost-effective cocatalyst in the photocatalytic reaction and opens a new horizon for designing other Cu7S4-based cocatalyst/semiconductor Schottky heterostructures for efficient solar-to-fuel conversion.

Orbital-Free Density Functional Theory: An Attractive Electronic Structure Method for Large-Scale First-Principles Simulations

Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.

An efficient zero-order evolutionary method for solving the orbital-free density functional theory problem by direct minimization

A differential evolution (DE) global optimization method for all-electron orbital-free density functional theory (OF-DFT) is presented. This optimization method does not need information about function derivatives to find extreme solutions. Results for a series of known orbital-free energy functionals are presented. Ground state energies of atoms (H to Ar) are obtained by direct minimization of the energy functional without using either Lagrange multipliers or damping procedures for reaching convergence. Our results are in agreement with previous OF-DFT calculations obtained using the standard Newton-Raphson and trust region methods. Being a zero-order method, the DE method can be applied to optimization problems dealing with non-differentiable functionals or functionals with non-closed forms.

Nonlocal pseudopotential energy density functional for orbital-free density functional theory

Orbital-free density functional theory (OF-DFT) is an electronic structure method with a low computational cost that scales linearly with the number of simulated atoms, making it suitable for large-scale material simulations. It is generally considered that OF-DFT strictly requires the use of local pseudopotentials, rather than orbital-dependent nonlocal pseudopotentials, for the calculation of electron-ion interaction energies, as no orbitals are available. This is unfortunate situation since the nonlocal pseudopotentials are known to give much better transferability and calculation accuracy than local ones. We report here the development of a theoretical scheme that allows the direct use of nonlocal pseudopotentials in OF-DFT. In this scheme, a nonlocal pseudopotential energy density functional is derived by the projection of nonlocal pseudopotential onto the non-interacting density matrix (instead of "orbitals") that can be approximated explicitly as a functional of electron density. Our development defies the belief that nonlocal pseudopotentials are not applicable to OF-DFT, leading to the creation for an alternate theoretical framework of OF-DFT that works superior to the traditional approach.

The Impact of the Surface Modification on Tin-Doped Indium Oxide Nanocomposite Properties

This review provides an analysis of the theoretical methods to study the effects of surface modification on structural properties of nanostructured indium tin oxide (ITO), mainly by organic compounds. The computational data are compared with experimental data such as X-ray diffraction (XRD), atomic force microscopy (AFM) and energy-dispersive X-ray spectroscopy (EDS) data with the focus on optoelectronic and electrocatalytic properties of the surface to investigate potential relations of these properties and applications of ITO in fields such as biosensing and electronic device fabrication. Our analysis shows that the change in optoelectronic properties of the surface is mainly due to functionalizing the surface with organic molecules and that the electrocatalytic properties vary as a function of size.

Magnetism-induced topological transition in EuAs3

The nature of the interaction between magnetism and topology in magnetic topological semimetals remains mysterious, but may be expected to lead to a variety of novel physics. We systematically studied the magnetic semimetal EuAs3, demonstrating a magnetism-induced topological transition from a topological nodal-line semimetal in the paramagnetic or the spin-polarized state to a topological massive Dirac metal in the antiferromagnetic ground state at low temperature. The topological nature in the antiferromagnetic state and the spin-polarized state has been verified by electrical transport measurements. An unsaturated and extremely large magnetoresistance of ~2 × 105% at 1.8 K and 28.3 T is observed. In the paramagnetic states, the topological nodal-line structure at the Y point is proven by angle-resolved photoemission spectroscopy. Moreover, a temperature-induced Lifshitz transition accompanied by the emergence of a new band below 3 K is revealed. These results indicate that magnetic EuAs3 provides a rich platform to explore exotic physics arising from the interaction of magnetism with topology.

Approximate Analytical Solutions for the Euler Equation for Second-Row Homonuclear Dimers

This work presents a new method how to obtain approximate analytical solutions for the Euler equation for second-row homonuclear dimers. In contrast to the well-known Kohn-Sham method where a system of N nonlinear coupled differential equations must be solved iteratively, orbital-free density functional theory allows to access the minimizing electron density directly via the Euler equation. For simplified models, here, an atom-centered monopole expansion with one free parameter, solutions of the electron density can be obtained analytically by solving the Euler equation at the bond critical point. The procedure is exemplarily carried out for N₂, C₂, and B₂, yielding bound molecules with an internuclear distance of 2.01, 2.43, and 3.07 bohr, respectively.

Efficient DFT Solver for Nanoscale Simulations and Beyond

We present the one-orbital ensemble self-consistent field (OE-SCF), an alternative orbital-free DFT solver that extends the applicability of DFT to beyond nanoscale system sizes, retaining the accuracy required to be predictive. OE-SCF treats the Pauli potential as an external potential updating it iteratively, dramatically outperforming current solvers because only few iterations are needed to reach convergence. OE-SCF enabled us to carry out the largest ab initio simulations for silicon-based materials to date by employing only 1 CPU. We computed the energy of bulk-cut Si nanoparticles as a function of their diameter up to 16 nm, and the polarization and interface charge transfer when a Si slab is sandwiched between two metal slabs where lattice matching mandated a large contact area. Additionally, OE-SCF opens the door to adopting even more accurate functionals in orbital-free DFT simulations while still tackling large system sizes.

Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory

This work presents a method to move beyond the recently introduced atomic fragment approximation. Like the bare atomic fragment approach, the new method is an ab initio, parameter-free, orbital-free implementation of density functional theory based on the bifunctional formalism that treats the potential and the electron density as two separate variables, and provides access to the Kohn–Sham Pauli kinetic energy for an appropriately chosen Pauli potential. In the present ansatz, the molecular Pauli potential is approximated by the sum of the bare atomic fragment approach, and a so-called deformation potential that takes the interaction between the atoms into account. It is shown that this model can reproduce the bond-length contraction due to multiple bonding within the list of second-row homonuclear dimers. The present model only relies on the electron densities of the participating atoms, which themselves are represented by a simple monopole expansion. Thus, the bond-length contraction can be rationalized without referring to the angular quantum numbers of the participating atoms.

Robust All-Electron Optimization in Orbital-Free Density-Functional Theory Using the Trust-Region Image Method

We present a Gaussian-basis implementation of orbital-free density-functional theory (OF-DFT) in which the trust-region image method (TRIM) is used for optimization. This second-order optimization scheme has been constructed to provide benchmark all-electron results with very tight convergence of the particle-number constraint, associated chemical potential, and electron density. It is demonstrated that, by preserving the saddle-point nature of the optimization and simultaneously optimizing the density and chemical potential, an order of magnitude reduction in the number of iterations required for convergence is obtained. The approach is compared and contrasted with a new implementation of the nested optimization scheme put forward by Chan, Cohen, and Handy. Our implementation allows for semilocal kinetic-energy (and exchange-correlation) functionals to be handled self-consistently in all-electron calculations. The all-electron Gaussian-basis setting for these calculations will enable direct comparison with a wide range of standard high-accuracy quantum-chemical methods as well as with Kohn-Sham density-functional theory. We expect that the present implementation will provide a useful tool for analyzing the performance of approximate kinetic-energy functionals in finite systems.

Inorganic photovoltaic cells based on BiFeO3: spontaneous polarization, lattice matching, light polarization and their relationship with photovoltaic performance

Inorganic ferroelectric perovskite oxides are more stable than hybrid perovskites. However, their solar energy harvesting efficiency is not so good. Here, by constructing a series of BiFeO3-based devices (solar cells), we investigated three factors that influence the photovoltaic performance, namely, spontaneous polarization, terminated ion species in the interface between BiFeO3 and the electrode, and polarized light irradiation. This work was carried out under the framework of the density functional theory combined with the non-equilibrium Green's function theory under a built-in electric field or finite bias. The results showed that (1) the photocurrent is larger only under a suitable electronic band gap rather than larger spontaneous polarization; (2) the photocurrent reaches the largest value in the Bi3+ ion-terminated interface than in the case of Fe3+ or O2- with the SrTiO3 electrode; (3) the photocurrent can be largely enhanced if the polarized direction of the monochromatic light is perpendicular to the spontaneous polarization direction. These results would deepen the understanding of some experimental results of BiFeO3-based solar cells.

Equilibrium Bond Lengths from Orbital-Free Density Functional Theory

This work presents an investigation to model chemical bonding in various dimers based on the atomic fragment approach. The atomic fragment approach is an ab-initio, parameter-free implementation of orbital-free density functional theory which is based on the bifunctional formalism, i.e., it uses both the density and the Pauli potential as two separate variables. While providing the exact Kohn-Sham Pauli kinetic energy when the orbital-based Kohn-Sham data are used, the bifunctional formalism allows for approximations of the functional derivative which are orbital-free. In its first implementation, the atomic fragment approach uses atoms in their ground state to model the Pauli potential. Here, it is tested how artificial closed-shell fragments with non-integer electron occupation perform regarding the prediction of bond lengths of diatomics. Such fragments can sometimes mimic the electronic structure of a molecule better than groundstate fragments. It is found that bond lengths may indeed be considerably improved in some of the tested diatomics, in accord with predictions based on the electronic structure.

Large-Z limit in atoms and solids from first principles

We present density functional theory (DFT) calculations of atomic ionization potentials and lattice constants of simple solids from low atomic numbers Z to the large-Z limit. We compare different kinetic energy functional approximations [Kohn-Sham (KS) vs simple orbital-free functionals] and, in the case of orbital free, also different methods for including the nuclear potential (all-electron with the projector augmented wave method vs local pseudopotentials). For both ionization potentials and lattice constants, all-electron orbital-free DFT does yield the general trend of KS DFT for moderate values of the atomic number Z. For large values of Z, all-electron orbital-free DFT deviates from the KS DFT results. Local pseudopotentials give a better qualitative description by adding shell oscillations to the orbital-free DFT model. We show that both all-electron orbital-free DFT and KS DFT have a finite value for nonrelativistic lattice constants in the large-Z limit.

Stochastic embedding DFT: Theory and application to p-nitroaniline in water

Over this past decade, we combined the idea of stochastic resolution of identity with a variety of electronic structure methods. In our stochastic Kohn-Sham density functional theory (DFT) method, the density is an average over multiple stochastic samples, with stochastic errors that decrease as the inverse square root of the number of sampling orbitals. Here, we develop a stochastic embedding density functional theory method (se-DFT) that selectively reduces the stochastic error (specifically on the forces) for a selected subsystem(s). The motivation, similar to that of other quantum embedding methods, is that for many systems of practical interest, the properties are often determined by only a small subsystem. In stochastic embedding DFT, two sets of orbitals are used: a deterministic one associated with the embedded subspace and the rest, which is described by a stochastic set. The method agrees exactly with deterministic calculations in the limit of a large number of stochastic samples. We apply se-DFT to study a p-nitroaniline molecule in water, where the statistical errors in the forces on the system (the p-nitroaniline molecule) are reduced by an order of magnitude compared with nonembedding stochastic DFT.

The first order atomic fragment approach-An orbital-free implementation of density functional theory

An orbital-free implementation of the original Hohenberg-Kohn theorems is presented, making use of the scaling properties from a fictitious Kohn-Sham system, but without reintroducing orbitals. The first order fragment approach does not contain data or parameters that are fitted to the final outcome of the molecular orbital-free calculation and thus represents a parameter-free implementation of orbital-free density functional theory, although it requires the precalculation of atomic data. Consequently, the proposed method is not limited to a specific type of molecule or chemical bonding. The different approximation levels arise from including (first order) or neglecting (zeroth order) the dependency between the potential and the electron density, which in the bifunctional approach are formally treated as independent variables.

Kinetic energy densities based on the fourth order gradient expansion: performance in different classes of materials and improvement via machine learning

We study the performance of fourth-order gradient expansions of the kinetic energy density (KED) in semi-local kinetic energy functionals depending on the density-dependent variables. The formal fourth-order expansion is convergent for periodic systems and small molecules but does not improve over the second-order expansion (the Thomas-Fermi term plus one-ninth of the von Weizsäcker term). Linear fitting of the expansion coefficients somewhat improves on the formal expansion. The tuning of the fourth order expansion coefficients allows for better reproducibility of the Kohn-Sham kinetic energy density than the tuning of the second-order expansion coefficients alone. The possibility of a much more accurate match with the Kohn-Sham kinetic energy density by using neural networks (NN) trained using the terms of the 4th order expansion as density-dependent variables is demonstrated. We obtain ultra-low fitting errors without overfitting of NN parameters. Small single hidden layer neural networks can provide good accuracy in separate KED fits of each compound, while for joint fitting of KEDs of multiple compounds multiple hidden layers were required to achieve good fit quality. The critical issue of data distribution is highlighted. We also show the critical role of pseudopotentials in the performance of the expansion, where in the case of a too rapid decay of the valence density at the nucleus with some pseudopotentials, numeric instabilities can arise.

Semilocal Pauli-Gaussian Kinetic Functionals for Orbital-Free Density Functional Theory Calculations of Solids

Kinetic energy (KE) approximations are key elements in orbital-free density functional theory. To date, the use of nonlocal functionals, possibly employing system-dependent parameters, has been considered mandatory in order to obtain satisfactory accuracy for different solid-state systems, whereas semilocal approximations are generally regarded as unfit to this aim. Here, we show that, instead, properly constructed semilocal approximations, the Pauli-Gaussian (PG) KE functionals, especially at the Laplacian level of theory, can indeed achieve similar accuracy as nonlocal functionals and can be accurate for both metals and semiconductors, without the need for system-dependent parameters.

Visualizing the molecular wave function in σ-coordinated complexes

The π-complex theory developed by Michael J. S. Dewar in 1949 has had its most profound impact as part of the Dewar-Chatt-Duncanson model, a seminal and foundational contribution to the field of organometallic chemistry. Over time it has demonstrated its utility in systems far from those originally envisaged, including σ-coordinated metal-complexes. This latter application is notable due to Dewar's original skepticism that his π-complex theory could be extended to σ-bonds. Separately it has previously been demonstrated that a one-electron wave function. can be shown to satisfy an exact one-electron Schrödinger equation describing the motion of the single electron in the average field of the remaining electrons. To celebrate the centenary of his birth this paper seeks to demonstrate that σ-coordinated metal-complexes present a perfect system to exemplify both the utility of the one-electron wave function and the power of the π-complex theory.

Kinetic Energy of Hydrocarbons as a Function of Electron Density and Convolutional Neural Networks

We demonstrate a convolutional neural network trained to reproduce the Kohn-Sham kinetic energy of hydrocarbons from an input electron density. The output of the network is used as a nonlocal correction to conventional local and semilocal kinetic functionals. We show that this approximation qualitatively reproduces Kohn-Sham potential energy surfaces when used with conventional exchange correlation functionals. The density which minimizes the total energy given by the functional is examined in detail. We identify several avenues to improve on this exploratory work, by reducing numerical noise and changing the structure of our functional. Finally we examine the features in the density learned by the neural network to anticipate the prospects of generalizing these models.

Optimizing a parametrized Thomas–Fermi–Dirac–Weizsäcker density functional for atoms

A systematic study of the parameter space of a kinetic functional is used as a route to understand the transferability problems and improve the kinetic density functionals.

Orbital free DFT versus single density equation: a perspective through quantum domain behavior of a classically chaotic system

Regular to chaotic transition takes place in a driven van der Pol oscillator in both classical and quantum domains.