Orbital-free Density Functional Theory Calculations Research Articles

Semilocal Kinetic Energy Density Functionals on Atoms and Diatoms.

An accurate semilocal kinetic energy density functional (KEDF) is crucial for reliable orbital-free density functional theory calculations. In our study, we assessed the performance of representative semilocal KEDFs using a more stringent indicator. Our findings highlight the superiority of the Perdew-Constantin (PC) functional in delivering energies close to the reference values. Upon analysis of the PC functional, we identified that enhancing its performance can be achieved through a more effective region selection regime. Experimenting with various region selection indicators, we discovered that the Laplacian-dependent reduced density gradient proves to be helpful. Subsequently, we empirically constructed an augmented variant of the PC functional, which not only yields energies close to the references but also, more importantly, demonstrates qualitative predictions for stable molecules and provides reasonable quantitative estimates for bond lengths in diatomic systems.

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.

Accurate parameterization of the kinetic energy functional for calculations using exact-exchange.

Electronic structure calculations based on Kohn-Sham density functional theory (KSDFT) that incorporate exact-exchange or hybrid functionals are associated with a large computational expense, a consequence of the inherent cubic scaling bottleneck and large associated prefactor, which limits the length and time scales that can be accessed. Although orbital-free density functional theory (OFDFT) calculations scale linearly with system size and are associated with a significantly smaller prefactor, they are limited by the absence of accurate density-dependent kinetic energy functionals. Therefore, the development of accurate density-dependent kinetic energy functionals is important for OFDFT calculations of large realistic systems. To this end, we propose a method to train kinetic energy functional models at the exact-exchange level of theory by using a dictionary of physically relevant terms that have been proposed in the literature in conjunction with linear or nonlinear regression methods to obtain the fitting coefficients. For our dictionary, we use a gradient expansion of the kinetic energy nonlocal models proposed in the literature and their nonlinear combinations, such as a model that incorporates spatial correlations between higher order derivatives of electron density at two points. The predictive capabilities of these models are assessed by using a variety of model one-dimensional (1D) systems that exhibit diverse bonding characteristics, such as a chain of eight hydrogens, LiF, LiH, C4H2, C4N2, and C3O2. We show that by using the data from model 1D KSDFT calculations performed using the exact-exchange functional for only a few neutral structures, it is possible to generate models with high accuracy for charged systems and electron and kinetic energy densities during self-consistent field iterations. In addition, we show that it is possible to learn both the orbital dependent terms, i.e., the kinetic energy and the exact-exchange energy, and models that incorporate additional nonlinearities in spatial correlations, such as a quadratic model, are needed to capture subtle features of the kinetic energy density that are present in exact-exchange-based KSDFT calculations.

Order- N orbital-free density-functional calculations with machine learning of functional derivatives for semiconductors and metals

Orbital-free density functional theory (OFDFT) offers a challenging way of electronic-structure calculations scaled as $\mathcal{O}(N)$ computation for system size $N$. We here develop a scheme of the OFDFT calculations based on the accurate and transferrable kinetic-energy density functional (KEDF) which is created in an unprecedented way using appropriately constructed neural network (NN). We show that our OFDFT scheme reproduces the electron density obtained in the state-of-the-art DFT calculations and then provides accurate structural properties of 24 different systems, ranging from atoms, molecules, metals, semiconductors and an ionic material. The accuracy and the transferability of our KEDF is achieved by our NN training system in which the kinetic-energy functional derivative (KEFD) at each real-space grid point is used. The choice of the KEFD as a set of training data is essentially important, because first it appears directly in the Euler equation which one should solve and second, its learning assists in reproducing the physical quantity expressed as the first derivative of the total energy. More generally, the present development of KEDF $T[\rho]$ is in the line of systematic expansion in terms of the functional derivatives $\delta^{\ell_1} T/\delta \rho^{\ell_1}$ through progressive increase of $\ell_1$. The present numerical success demonstrates the validity of this approach. The computational cost of the present OFDFT scheme indeed shows the $\mathcal{O}(N)$ scaling, as is evidenced by the computations of the semiconductor SiC used in power electronics.

Two-temperature warm dense hydrogen as a test of quantum protons driven by orbital-free density functional theory electronic forces

We consider a steady-state (but transient) situation in which a warm dense aggregate is a two-temperature system with equilibrium electrons at temperature Te, ions at Ti, and Te ≠ Ti. Such states are achievable by pump–probe experiments. For warm dense hydrogen in such a two-temperature situation, we investigate nuclear quantum effects (NQEs) on structure and thermodynamic properties, thereby delineating the limitations of ordinary ab initio molecular dynamics. We use path integral molecular dynamics (PIMD) simulations driven by orbital-free density functional theory (OFDFT) calculations with state-of-the-art noninteracting free-energy and exchange-correlation functionals for the explicit temperature dependence. We calibrate the OFDFT calculations against conventional (explicit orbitals) Kohn–Sham DFT. We find that when the ratio of the ionic thermal de Broglie wavelength to the mean interionic distance is larger than about 0.30, the ionic radial distribution function is meaningfully affected by the inclusion of NQEs. Moreover, NQEs induce a substantial increase in both the ionic and electronic pressures. This confirms the importance of NQEs for highly accurate equation-of-state data on highly driven hydrogen. For Te > 20 kK, increasing Te in the warm dense hydrogen has slight effects on the ionic radial distribution function and equation of state in the range of densities considered. In addition, we confirm that compared with thermostatted ring-polymer molecular dynamics, the primitive PIMD algorithm overestimates electronic pressures, a consequence of the overly localized ionic description from the primitive scheme.

Performance of Semilocal Kinetic Energy Functionals for Orbital-Free Density Functional Theory

We assess several generalized gradient approximations (GGAs) and Laplacian-level meta-GGAs (LL-MGGA) kinetic energy (KE) functionals for orbital-free density functional theory calculations of bulk metals and semiconductors, considering equilibrium distances, bulk moduli, total and kinetic energies, and the electron densities. We also considered the effects of the pseudopotentials, the vacancy formation energies, and the bond lengths of molecular dimers. We found that LL-MGGA KE functionals are distinctively superior to GGA functionals, showing the importance of the Laplacian of the density in the functional construction. We extended the recently developed Pauli-Gaussian second-order and Laplacian (PGSL) functional ( J. Phys. Chem. Lett. 2018 , 9 , 4385 , DOI: 10.1021/acs.jpclett.8b01926 ) including high-order corrections, achieving higher transferability and accuracy than conventional nonlocal functionals based on the Lindhard response function.

Orbital-free density functional theory study of the energetics of vacancy clustering and prismatic dislocation loop nucleation in aluminium

In the present work, we conduct large-scale orbital-free density functional theory calculations to study the energetics of vacancy clustering in aluminium from electronic structure calculations by accurately accounting for both the electronic structure and long-ranged elastic fields. Our results show positive binding energies for a range of vacancy clusters considered. However, the binding energies for the various quad-vacancy clusters considered in this study are vastly different, and only some of these clusters are found to be stable with respect to dissociation into divacancies. This suggests that, while vacancy clustering is an energetically feasible mechanism, this happens preferentially through the formation of certain vacancy clusters. Among the vacancy clusters considered in this study, the 19 vacancy hexagonal cluster lying in plane has a very large binding energy with the relaxed atomic structure representative of a prismatic dislocation loop. This suggests that vacancy prismatic loops as small as those formed from 19 vacancies are stable, thus providing insights into the nucleation sizes of these defects in aluminium.

Higher-order finite-difference formulation of periodic Orbital-free Density Functional Theory

We present a real-space formulation and higher-order finite-difference implementation of periodic Orbital-free Density Functional Theory (OF-DFT). Specifically, utilizing a local reformulation of the electrostatic and kernel terms, we develop a generalized framework for performing OF-DFT simulations with different variants of the electronic kinetic energy. In particular, we propose a self-consistent field (SCF) type fixed-point method for calculations involving linear-response kinetic energy functionals. In this framework, evaluation of both the electronic ground-state and forces on the nuclei are amenable to computations that scale linearly with the number of atoms. We develop a parallel implementation of this formulation using the finite-difference discretization. We demonstrate that higher-order finite-differences can achieve relatively large convergence rates with respect to mesh-size in both the energies and forces. Additionally, we establish that the fixed-point iteration converges rapidly, and that it can be further accelerated using extrapolation techniques like Anderson's mixing. We validate the accuracy of the results by comparing the energies and forces with plane-wave methods for selected examples, including the vacancy formation energy in Aluminum. Overall, the suitability of the proposed formulation for scalable high performance computing makes it an attractive choice for large-scale OF-DFT calculations consisting of thousands of atoms.

ATLAS: A real-space finite-difference implementation of orbital-free density functional theory

Orbital-free density functional theory (OF-DFT) is a promising method for large-scale quantum mechanics simulation as it provides a good balance of accuracy and computational cost. Its applicability to large-scale simulations has been aided by progress in constructing kinetic energy functionals and local pseudopotentials. However, the widespread adoption of OF-DFT requires further improvement in its efficiency and robustly implemented software. Here we develop a real-space finite-difference (FD) method for the numerical solution of OF-DFT in periodic systems. Instead of the traditional self-consistent method, a powerful scheme for energy minimization is introduced to solve the Euler–Lagrange equation. Our approach engages both the real-space finite-difference method and a direct energy-minimization scheme for the OF-DFT calculations. The method is coded into the ATLAS software package and benchmarked using periodic systems of solid Mg, Al, and Al3Mg. The test results show that our implementation can achieve high accuracy, efficiency, and numerical stability for large-scale simulations.

Generalized-gradient-approximation noninteracting free-energy functionals for orbital-free density functional calculations

We develop a framework for orbital-free generalized gradient approximations (GGAs) for the noninteracting free energy density and its components (kinetic energy, entropy) based upon analysis of the corresponding gradient expansion. From that we obtain a new finite-temperature GGA (ftGGA) pair. We discuss implementation of the finite-temperature Thomas-Fermi, second-order gradient expansion, and our new ftGGA free energy functionals in an orbital-free density functional theory (OF-DFT) code, including the construction and validation of required local pseudopotentials. Then we compare results of self-consistent OF-DFT calculations on hydrogen using those noninteracting free energy functionals (in combination with the zero-temperature local density approximation (LDA) for exchange-correlation) with results from conventional finite-temperature Kohn-Sham calculations and the same LDA. As an aid to implementation, we provide analytical expressions for the temperature-dependent scaling factors involved.

Quantum simulation of materials at micron scales and beyond

We present a multiscale modeling approach that can simulate multimillion atoms effectively via density-functional theory. The method is based on the framework of the quasicontinuum (QC) approach with orbital-free density-functional theory (OFDFT) as its sole energetics formulation. The local QC part is formulated by the Cauchy-Born hypothesis with OFDFT calculations for strain energy and stress. The nonlocal QC part is treated by an OFDFT-based embedding approach, which couples OFDFT nonlocal atoms to local region atoms. The method---QCDFT---is applied to a nanoindentation study of an Al thin film, and the results are compared to a conventional QC approach. The results suggest that QCDFT represents a new direction for the quantum simulation of materials at length scales that are relevant to experiments.

Transferable local pseudopotentials for magnesium, aluminum and silicon

One obstacle in orbital-free density functional theory (OF-DFT) is the lack of accurate and transferable local pseudopotentials (LPSs). In this work, we build high quality LPSs by inverting Kohn-Sham (KS) equations on bulk valence electron densities to obtain an atom-centered local pseudopotential. With this approach, we build LPSs for Mg, Al, and Si, and then test them in KS DFT calculations of static bulk properties for several Mg, Al, and Si bulk structures as well as beta''-Al3Mg. Our Mg, Al, and Si LPSs produce correct ground state properties and phase orderings. These LPSs are then tested in KS-DFT calculations of surface energies for several low-index Mg and Al surfaces, point defect properties in hexagonal-close-packed (hcp) Mg, face-centered cubic (fcc) Al, and diamond Si, and stacking fault energies in fcc Al. All of these LPS results agree quantitatively with the results from nonlocal pseudopotentials with errors less than or equal to 40 meV per atom. Finally, we perform OF-DFT calculations for various Mg and Al structures, employing the Wang-Govind-Carter (WGC) nonlocal kinetic energy density functional (KEDF). The OF-DFT results generally agree well with the corresponding KS-DFT results. With our new Mg and Al LPSs and the WGC KEDF, OF-DFT now provides a practical method for accurate, large-scale first principles simulations of main group metals and their alloys.

Non-periodic finite-element formulation of orbital-free density functional theory

We propose an approach to perform orbital-free density functional theory calculations in a non-periodic setting using the finite-element method. We consider this a step towards constructing a seamless multi-scale approach for studying defects like vacancies, dislocations and cracks that require quantum mechanical resolution at the core and are sensitive to long range continuum stresses. In this paper, we describe a local real-space variational formulation for orbital-free density functional theory, including the electrostatic terms and prove existence results. We prove the convergence of the finite-element approximation including numerical quadratures for our variational formulation. Finally, we demonstrate our method using examples.

Improving the orbital-free density functional theory description of covalent materials

The essential challenge in orbital-free density functional theory (OF-DFT) is to construct accurate kinetic energy density functionals (KEDFs) with general applicability (i.e., transferability). During the last decade, several linear-response (LR)-based KEDFs have been proposed. Among them, the Wang-Govind-Carter (WGC) KEDF, containing a density-dependent response kernel, is one of the most accurate that still affords a linear scaling algorithm. For nearly-free-electron-like metals such as Al and its alloys, OF-DFT employing the WGC KEDF produces bulk properties in good agreement with orbital-based Kohn-Sham (KS) DFT predictions. However, when OF-DFT, using the WGC KEDF combined with a recently proposed bulk-derived local pseudopotential (BLPS), was applied to semiconducting and metallic phases of Si, problems arose with convergence of the self-consistent density and energy, leading to poor results. Here we provide evidence that the convergence problem is very likely caused by the use of a truncated Taylor series expansion of the WGC response kernel. Moreover, we show that a defect in the ansatz for the first-order reduced density matrix underlying the LR KEDFs limits the accuracy of these KEDFs. By optimizing the two free parameters involved in the WGC KEDF, the two-body Fermi wave vector mixing parameter gamma and the reference density rho* used in the Taylor expansion, OF-DFT calculations with the BLPS can achieve semiquantitative results for nine phases of bulk silicon. These new parameters are recommended whenever the WGC KEDF is used to study nonmetallic systems.

Orbital-free density functional theory calculations of the properties of Al, Mg and Al–Mg crystalline phases

We have used density functional theory in an orbital-free, implementation to calculate the properties of pure fcc Al, hcp and bcc Mg, and the meta-stable alloy phase β′′ (Al3Mg). Five linear-response-based kinetic energy density functionals have been used, one of which has a density-dependent (DD) response kernel. We demonstrate that orbital-free density functional theory (OF-DFT) can produce physically accurate properties for Al–Mg alloys, if the kinetic energy density functional employed has a DD-kernel.