Orbital-free Density Functional Theory Research Articles

Detailed analysis of deformation potentials with application in orbital-free density functional theory

A detailed analysis of the recently published deformation potentials for application in orbital-free density functional theory is given. Since orbital-free density functional theory is a purely density-based description of quantum mechanics, it may in the future provide itself useful in quantum crystallography as it establishes a direct link between experiment and theory via a single meaningful quantity: the electron density. In order to establish this goal, sufficiently accurate approximations for the kinetic energy have to be found. The present work is a further step in this direction. The so-called deformation potentials allow the interaction between the atoms to be taken into account through the help of their electron density only. It is shown that the present ansatz provides a systematic pathway beyond the recently introduced atomic fragment approach.

Revised Huang-Carter nonlocal kinetic energy functional for semiconductors and their surfaces

Nonlocal kinetic energy functionals with a density-dependent kernel are the most accurate functionals available for carrying out orbital-free density functional theory simulations. Among them, the Huang and Carter (HC) functional [Huang and Carter, Phys. Rev. B 81, 045206 (2010)] is the most accurate for bulk semiconductors. A major hurdle in applying HC to nonbulk systems (such as clusters and surfaces which have at least one nonperiodic dimension where the density decays to zero) lies in its numerical instability for large values of the reduced density gradient, $s(\mathbf{r})\ensuremath{\propto}\frac{|\mathbf{\ensuremath{\nabla}}\mathbf{n}(\mathbf{r})|}{{\mathbf{n}}^{\mathbf{4}/\mathbf{3}}(\mathbf{r})}$, where $n$ is the electron density. We propose a revision to the HC functional, revHC, that allows it to achieve dramatically improved numerical stability, efficiency (in terms of timing to solution), and applicability. Not only does revHC reproduce all previously presented results for HC, but it extends them to a crucially important class of materials: surfaces. We show that surface energy trends of clean-cut and reconstructed surfaces of group IV and III-V semiconductors are recovered and, where available semiquantitatively, reproduce the experimental results.

Time-dependent orbital-free density functional theory: Background and Pauli kernel approximations

Time-dependent orbital-free DFT is an efficient method for calculating the dynamic properties of large scale quantum systems due to the low computational cost compared to standard time-dependent DFT. We formalize this method by mapping the real system of interacting fermions onto a fictitious system of non-interacting bosons. The dynamic Pauli potential and associated kernel emerge as key ingredients of time-tependent orbital-free DFT. Using the uniform electron gas as a model system, we derive an approximate frequency-dependent Pauli kernel. Pilot calculations suggest that space nonlocality is a key feature for this kernel. Nonlocal terms arise already in the second order expansion with respect to unitless frequency and reciprocal space variable ($\frac{\omega}{q\, k_F}$ and $\frac{q}{2\, k_F}$, respectively). Given the encouraging performance of the proposed kernel, we expect it will lead to more accurate orbital-free DFT simulations of nanoscale systems out of equilibrium. Additionally, the proposed path to formulate nonadiabatic Pauli kernels presents several avenues for further improvements which can be exploited in future works to improve the results.

Machine Learning Kinetic Energy Functional for a One-Dimensional Periodic SystemSupported by the Hong Kong Research Grants Council (Project No. GRF16300918), the National Key R&D Program of China (Grant Nos. 2016YFA0300603 and 2016YFA0302400), and the National Natural Science Foundation of China (Grant No. 11774398).

Kinetic energy (KE) functional is crucial to speed up density functional theory calculation. However, deriving it accurately through traditional physics reasoning is challenging. We develop a generally applicable KE functional estimator for a one-dimensional (1D) extended system using a machine learning method. Our end-to-end solution combines the dimensionality reduction method with the Gaussian process regression, and simple scaling method to adapt to various 1D lattices. In addition to reaching chemical accuracy in KE calculation, our estimator also performs well on KE functional derivative prediction. Integrating this machine learning KE functional into the current orbital free density functional theory scheme is able to provide us with expected ground state electron density.

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.

Efficient Embedded Cluster Density Approximation Calculations with an Orbital-Free Treatment of Environments.

The computational cost of the Kohn-Sham density functional theory (KS-DFT), employing advanced orbital-based exchange-correlation (XC) functionals, increases quickly for large systems. To tackle this problem, we recently developed a local correlation method in the framework of KS-DFT: the embedded cluster density approximation (ECDA). The aim of ECDA is to obtain accurate electronic structures in an entire system. With ECDA, for each atom in a system, we define a cluster to enclose that atom, with the rest atoms treated as the environment. The system's electron density is then partitioned among the cluster and the environment. The cluster's XC energy density is then calculated based on its electron density using an advanced orbital-based XC functional. The system's XC energy is obtained by patching all clusters' XC energy densities in an atom-by-atom manner. In our previous formulation of ECDA, environments were treated by KS-DFT, which makes the following two tasks computationally expensive for large systems. The first task is to partition the system's electron density among a cluster and its environment. The second task is to solve the environments' Sternheimer equations for calculating the system's XC potential. In this work, we remove these two computational bottlenecks by treating the environments with the orbital-free (OF) DFT. The new method is called ECDA-envOF. The performance of ECDA-envOF is examined in two systems: ester and Cl-tetracene, for which the exact exchange (EXX) is used as the advanced XC functional. We show that ECDA-envOF gives results that are very close to the previous formulation in which the environments were treated by KS-DFT. Therefore, ECDA-envOF can be used for future large-scale simulations. Another focus of this work is to examine ECDA-envOF's performance on systems having different bond types. With ECDA-envOF, we calculate the energy curves for stretching/compressing some covalent, metallic, and ionic systems. ECDA-envOF's predictions agree well with the benchmarks by using reasonably large clusters. These examples demonstrate that ECDA-envOF is nearly a black-box local correlation method for investigating heterogeneous materials in which different bond types exist.

Artificial neural networks for the kinetic energy functional of non-interacting fermions.

A novel approach to find the fermionic non-interacting kinetic energy functional with chemical accuracy using machine learning techniques is presented. To that extent, we apply machine learning to an intermediate quantity rather than targeting the kinetic energy directly. We demonstrate the performance of the method for three model systems containing three and four electrons. The resulting kinetic energy functional remarkably accurately reproduces self-consistently the ground state electron density and total energy of reference Kohn-Sham calculations with an error of less than 5 mHa. This development opens a new avenue to advance orbital-free density functional theory by means of machine learning.

Random Structure Searching with Orbital-Free Density Functional Theory.

The properties of a material depend on how its atoms are arranged, and predicting these arrangements from first principles is a longstanding challenge. Orbital-free density functional theory provides a quantum-mechanical model based solely on the electron density, not individual wave functions. The resulting speedups make it attractive for random structure searching, whereby random configurations of atoms are relaxed to local minima in the energy landscape. We use this strategy to map the low-energy crystal structures of Li, Na, Mg, and Al at zero pressure. For Li and Na, our searching finds numerous close-packed polytypes of almost-equal energy, consistent with previous efforts to understand their low-temperature forms. For Mg and Al, the searching identifies the expected ground state structures unambiguously, in addition to revealing other low-energy structures. This new role for orbital-free density functional theory—particularly as continued advances make it accurate for more of the periodic table—will expedite crystal structure prediction over wide ranges of compositions and pressures.

Learning to Approximate Density Functionals

ConspectusDensity functional theory (DFT) calculations are used in over 40,000 scientific papers each year, in chemistry, materials science, and far beyond. DFT is extremely useful because it is computationally much less expensive than ab initio electronic structure methods and allows systems of considerably larger size to be treated. However, the accuracy of any Kohn-Sham DFT calculation is limited by the approximation chosen for the exchange-correlation (XC) energy. For more than half a century, humans have developed the art of such approximations, using general principles, empirical data, or a combination of both, typically yielding useful results, but with errors well above the chemical accuracy limit (1 kcal/mol). Over the last 15 years, machine learning (ML) has made major breakthroughs in many applications and is now being applied to electronic structure calculations. This recent rise of ML begs the question: Can ML propose or improve density functional approximations? Success could greatly enhance the accuracy and usefulness of DFT calculations without increasing the cost.In this work, we detail efforts in this direction, beginning with an elementary proof of principle from 2012, namely, finding the kinetic energy of several Fermions in a box using kernel ridge regression. This is an example of orbital-free DFT, for which a successful general-purpose scheme could make even DFT calculations run much faster. We trace the development of that work to state-of-the-art molecular dynamics simulations of resorcinol with chemical accuracy. By training on ab initio examples, one bypasses the need to find the XC functional explicitly. We also discuss how the exchange-correlation energy itself can be modeled with such methods, especially for strongly correlated materials. Finally, we show how deep neural networks with differentiable programming can be used to construct accurate density functionals from very few data points by using the Kohn-Sham equations themselves as a regularizer. All these cases show that ML can create approximations of greater accuracy than humans, and is capable of finding approximations that can deal with difficult cases such as strong correlation. However, such ML-designed functionals have not been implemented in standard codes because of one last great challenge: generalization. We discuss how effortlessly human-designed functionals can be applied to a wide range of situations, and how difficult that is for ML.

Extended Lagrangian Born–Oppenheimer molecular dynamics for orbital-free density-functional theory and polarizable charge equilibration models

Extended Lagrangian Born-Oppenheimer molecular dynamics (XL-BOMD) [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008)] is formulated for orbital-free Hohenberg-Kohn density-functional theory and for charge equilibration and polarizable force-field models that can be derived from the same orbital-free framework. The purpose is to introduce the most recent features of orbital-based XL-BOMD to molecular dynamics simulations based on charge equilibration and polarizable force-field models. These features include a metric tensor generalization of the extended harmonic potential, preconditioners, and the ability to use only a single Coulomb summation to determine the fully equilibrated charges and the interatomic forces in each time step for the shadow Born-Oppenheimer potential energy surface. The orbital-free formulation has a charge-dependent, short-range energy term that is separate from long-range Coulomb interactions. This enables local parameterizations of the short-range energy term, while the long-range electrostatic interactions can be treated separately. The theory is illustrated for molecular dynamics simulations of an atomistic system described by a charge equilibration model with periodic boundary conditions. The system of linear equations that determines the equilibrated charges and the forces is diagonal, and only a single Ewald summation is needed in each time step. The simulations exhibit the same features in accuracy, convergence, and stability as are expected from orbital-based XL-BOMD.

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.

Nonparametric Local Pseudopotentials with Machine Learning: A Tin Pseudopotential Built Using Gaussian Process Regression

We present novel nonparametric representation math for local pseudopotentials (PP) based on Gaussian Process Regression (GPR). Local pseudopotentials are needed for materials simulations using Orbital-Free Density Functional Theory (OF-DFT) to reduce computational cost and to allow kinetic energy functional (KEF) application only to the valence density. Moreover, local PPs are important for the development of accurate KEFs for OF-DFT, but they are only available for a limited number of elements. We optimize local PPs of tin (Sn) represented with GPR to reproduce the experimental lattice constants of α- and β-Sn and the energy difference between these two phases as well as their electronic structure and charge density distributions which are obtained with Kohn-Sham Density Functional Theory employing semilocal PPs. The use of a nonparametric GPR-based PP representation avoids difficulties associated with the use of parametrized functions and has the potential to construct an optimal local PP independent of prior assumptions. The GPR-based Sn local PP results in well-reproduced bulk properties of α- and β-tin and electronic valence densities similar to those obtained with semilocal PP.

Prismatic Slip in Magnesium

Magnesium is the lowest-density structural metal but has low ductility that limits applications. The low ductility is related to the hexagonally close-packed crystal structure where activation of nonbasal slip is required for general plasticity. Here, our recent neural network potential (NNP) for Mg, trained using Kohn–Sham density functional theory (DFT), is used to examine slip of ⟨a⟩ dislocations on the prismatic plane. The generalized stacking fault surface energies (GSFEs) for basal and prismatic slip are computed and agree better with Kohn–Sham density functional theory (KS-DFT) than orbital-free density functional theory (OF-DFT) and modified embedded atom method (MEAM), which predict spurious minima. Consistent with the generalized stacking fault energy (GSFE), direct simulations of the prismatic ⟨a⟩ screw dislocation show it is unstable to dissociate into the ⟨a⟩ basal screw dislocation; this is mostly consistent with OF-DFT while MEAM predicts stability. Prismatic slip is thus achieved by a double-cross-slip process of the stable basal dislocations driven by a resolved shear stress on the orthogonal prismatic plane; this is consistent with the process deduced from experiments. The Nudged Elastic Band method is used with the NNP to examine the atomistic path and the stress-dependent enthalpy barrier for this mechanism; this requires many tens of thousands of atoms. The basal-prismatic cross-slip occurs in increments of c/2 via basal constriction, cross-slip on the prism plane, cross-slip back onto the basal plane, and lateral motion of the created jogs to extend the new basal dislocation. Comparisons with experimental deductions show some agreement and some notable disagreement. Resolution of the differences points toward further large-scale studies that require the accuracy and efficiency of KS-DFT-trained NNP, an approach that is also naturally extendable to the important domain of Mg alloys.

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.

Looking at local classical and quantum forces in stable crystals using multipole-model refined electron densities and orbital-free DFT approximations.

Two distinct approaches, that of energy and that of force, are adopted in quantum mechanics to get insights on chemical processes. In the second one, the net forces acting on the electrons and nuclei in a system (Ehrnefest and Hellmann-Feynman forces, respectively) are determined and a local version of the approach, in terms of force density fields rather than forces, has also been proposed for electrons. This is the path followed by Tsirelson & Stash (2020) in this issue of Acta Crystallographica Section B, to study for the first time the spatial distribution of the electronic forces of different nature acting in stable crystals. Interestingly, by relying on approximations taken from orbital-free DFT, all components of the inner-crystal force can be easily retrieved from multipole-model refined experimental electron densities and their derivatives. No less important is that these calculations are becoming easily doable for any X-ray density crystallographer thanks to a new version of the computer program WinXPRO, purposely developed in the study which is discussed in this commentary.

Orbital-free quantum crystallography: view on forces in crystals.

Quantum theory of atoms in molecules and the orbital-free density functional theory (DFT) are combined in this work to study the spatial distribution of electrostatic and quantum electronic forces acting in stable crystals. The electron distribution is determined by electrostatic electron mutual repulsion corrected for exchange and correlation, their attraction to nuclei and by electron kinetic energy. The latter defines the spread of permissible variations in the electron momentum resulting from the de Broglie relationship and uncertainty principle, as far as the limitations of Pauli principle and the presence of atomic nuclei and other electrons allow. All forces are expressed via kinetic and DFT potentials and then defined in terms of the experimental electron density and its derivatives; hence, this approach may be considered as orbital-free quantum crystallography. The net force acting on an electron in a crystal at equilibrium is zero everywhere, presenting a balance of the kinetic Fkin(r) and potential forces F(r). The critical points of both potentials are analyzed and they are recognized as the points at which forces Fkin(r) and F(r) individually are zero (the Lagrange points). The positions of these points in a crystal are described according to Wyckoff notations, while their types depend on the considered scalar field. It was found that F(r) force pushes electrons to the atomic nuclei, while the kinetic force Fkin(r) draws electrons from nuclei. This favors formation of electron concentration bridges between some of the nearest atoms. However, in a crystal at equilibrium, only kinetic potential vkin(r) and corresponding force exhibit the electronic shells and atomic-like zero-flux basins around the nuclear attractors. The force-field approach and quantum topological theory of atoms in molecules are compared and their distinctions are clarified.

Data-driven kinetic energy density fitting for orbital-free DFT: Linear vs Gaussian process regression.

We study the dependence of kinetic energy densities (KEDs) on density-dependent variables that have been suggested in previous works on kinetic energy functionals for orbital-free density functional theory. We focus on the role of data distribution and on data and regressor selection. We compare unweighted and weighted linear and Gaussian process regressions of KEDs for light metals and a semiconductor. We find that good quality linear regression resulting in good energy-volume dependence is possible over density-dependent variables suggested in previous literature studies. This is achieved with weighted fitting based on the KED histogram. With Gaussian process regressions, excellent KED fit quality well exceeding that of linear regressions is obtained as well as a good energy-volume dependence, which was somewhat better than that of best linear regressions. We find that while the use of the effective potential as a descriptor improves linear KED fitting, it does not improve the quality of the energy-volume dependence with linear regressions but substantially improves it with Gaussian process regression. Gaussian process regression is also able to perform well without data weighting.

Machine Learning Approaches toward Orbital-free Density Functional Theory: Simultaneous Training on the Kinetic Energy Density Functional and Its Functional Derivative.

Orbital-free approaches might offer a way to boost the applicability of density functional theory by orders of magnitude in system size. An important ingredient for this endeavor is the kinetic energy density functional. Snyder et al. [Phys. Rev. Lett.2012, 108, 25300223004593] presented a machine learning approximation for this functional achieving chemical accuracy on a one-dimensional model system. However, a poor performance with respect to the functional derivative, a crucial element in iterative energy minimization procedures, enforced the application of a computationally expensive projection method. In this work we circumvent this issue by including the functional derivative into the training of various machine learning models. Besides kernel ridge regression, the original method of choice, we also test the performance of convolutional neural network techniques borrowed from the field of image recognition.

DFTpy: An efficient and object‐oriented platform for orbital‐free DFT simulations

AbstractIn silico materials design is hampered by the computational complexity of Kohn–Sham DFT, which scales cubically with the system size. Owing to the development of new‐generation kinetic energy density functionals (KEDFs), orbital‐free DFT (OFDFT) can now be successfully applied to a large class of semiconductors and such finite systems as quantum dots and metal clusters. In this work, we present DFTpy, an open‐source software implementing OFDFT written entirely in Python 3 and outsourcing the computationally expensive operations to third‐party modules, such as NumPy and SciPy. When fast simulations are in order, DFTpy exploits the fast Fourier transforms from PyFFTW. New‐generation, nonlocal and density‐dependent‐kernel KEDFs are made computationally efficient by employing linear splines and other methods for fast kernel builds. We showcase DFTpy by solving for the electronic structure of a million‐atom system of aluminum metal which was computed on a single CPU. The Python 3 implementation is object‐oriented, opening the door to easy implementation of new features. As an example, we present a time‐dependent OFDFT implementation (hydrodynamic DFT) which we use to compute the spectra of small metal clusters recovering qualitatively the time‐dependent Kohn–Sham DFT result. The Python codebase allows for easy implementation of application programming interfaces. We showcase the combination of DFTpy and ASE for molecular dynamics simulations of liquid metals. DFTpy is released under the MIT license.This article is categorized under: Software > Quantum Chemistry Electronic Structure Theory > Density Functional Theory Data Science > Computer Algorithms and Programming

Single-photon photoionization of highly charged ions under warm- and hot-dense plasmas using a unified description of screening

We present a detailed investigation of the spectral properties and photoionization of highly charged ions subjected to warm- and hot-dense plasma environments. A unified framework, based on the finite-temperature orbital-free density functional theory, is adopted to describe (warm- and hot-dense) plasma screening effects on the nucleus-electron interaction. Self-consistent solving the coupled radial Diarc equations for both bound and continuum wave functions is used through the model, EPPP, we developed, to account the relativistic and plasma shielding effects, which is based on the flexible atomic code. The key features of the proposed model are the capacities to systematically include gradient corrections and exchange-correlation effects. Additionally, numerical tests are conducted to validate the accuracy and stability of the model. The transition energies, radiative transition parameters, ionization potentials, and photoionization (radiation recombination) cross sections of an exemplary ion Al12+are studied under finite temperature and density conditions. The present work offers an insight into electron screening of ions and is expected to be beneficial for plasma modeling.