Kinetic Energy Density Functionals Research Articles

Multi-channel machine learning based nonlocal kinetic energy density functional for semiconductors

Abstract The recently proposed machine learning-based physically-constrained nonlocal (MPN) kinetic energy density functional (KEDF) can be used for simple metals and their alloys [Phys. Rev. B 109, 115135 (2024)]. However, the MPN KEDF does not perform well for semiconductors. Here we propose a multi-channel MPN (CPN) KEDF, which extends the MPN KEDF to semiconductors by integrating information collected from multiple channels, with each channel featuring a specific length scale in real space. The CPN KEDF is systematically tested on silicon and binary semiconductors. We find that the multi-channel design for KEDF is beneficial for machine-learning-based models in capturing the characteristics of semiconductors, particularly in handling covalent bonds. In particular, the CPN5 KEDF, which utilizes five channels, demonstrates excellent accuracy across all tested systems. These results offer a new path for generating KEDFs for semiconductors.

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.

Triplet Excitation-Energy Transfer Couplings from Subsystem Time-Dependent Density-Functional Theory.

We present an implementation of triplet excitation-energy transfer (TEET) couplings based on subsystem-based time-dependent density-functional theory (sTDDFT). TEET couplings are systematically investigated by comparing "exact" and approximate variants of sTDDFT. We demonstrate that, while sTDDFT utilizing explicit approximate non-additive kinetic energy (NAKE) density functionals is well-suited for describing singlet EET processes, it is inadequate for characterizing TEET. However, we show that projection-based embedding (PbE)-based sTDDFT addresses the challenges faced by NAKE-sTDDFT and emerges as a promising method for accurately describing electronic couplings in TEET processes. We also introduce the mixed PbE-/NAKE-embedding procedure to investigate the TEET effects in solvated pairs of chromophores. This approach offers a good balance between accuracy and efficiency, enabling comprehensive studies of TEET processes in complex environments.

Machine learning of kinetic energy densities with target and feature smoothing: Better results with fewer training data.

Machine learning (ML) of kinetic energy functionals (KEFs), in particular kinetic energy density (KED) functionals, is a promising way to construct KEFs for orbital-free density functional theory (DFT). Neural networks and kernel methods including Gaussian process regression (GPR) have been used to learn Kohn-Sham (KS) KED from density-based descriptors derived from KS DFT calculations. The descriptors are typically expressed as functions of different powers and derivatives of the electron density. This can generate large and extremely unevenly distributed datasets, which complicates effective application of ML techniques. Very uneven data distributions require many training datapoints, can cause overfitting, and can ultimately lower the quality of an ML KED model. We show that one can produce more accurate ML models from fewer data by working with smoothed density-dependent variables and KED. Smoothing palliates the issue of very uneven data distributions and associated difficulties of sampling while retaining enough spatial structure necessary for working within the paradigm of KEDF. We use GPR as a function of smoothed terms of the fourth order gradient expansion and KS effective potential and obtain accurate and stable (with respect to different random choices of training points) kinetic energy models for Al, Mg, and Si simultaneously from as few as 2000 samples (about 0.3% of the total KS DFT data). In particular, accuracies on the order of 1% in a measure of the quality of energy-volume dependence B'=EV0-ΔV-2EV0+E(V0+ΔV)ΔV/V02 (where V0 is the equilibrium volume and ΔV is a deviation from it) are obtained simultaneously for all three materials.

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.

Truncated nonlocal kinetic energy density functionals for simple metals and silicon

Truncated nonlocal kinetic energy density functionals for simple metals and silicon

Efficient Implementation of Density Functional Theory Based Embedding for Molecular and Periodic Systems Using Gaussian Basis Functions.

A practical and effective implementation of density functional theory based embedding is reported, which allows us to treat both periodic and aperiodic systems on an equal footing. Its essence is the expansion of orbitals and electron density of the periodic system using Gaussian basis functions, rather than plane-waves, which provides a unique all-electron direct-space representation, thus avoiding the need for pseudopotentials. This makes the construction of embedding potential for a molecular active subsystem due to a periodic environment quite convenient, as transformation between representations is far from trivial. The three flavors of embedding, molecule-in-molecule, molecule-in-periodic, and periodic-in-periodic embedding, are implemented using embedding potentials based on non-additive kinetic energy density functionals (approximate) and level-shift projection operator (exact). The embedding scheme is coupled with a variety of correlated wave function theory (WFT) methods, thereby providing an efficient way to study the ground and excited state properties of low-dimensional systems using high-level methods for the region of interest. Finally, an implementation of real time-time-dependent density functional embedding theory (RT-TDDFET) is presented that uses a projection operator-based embedding potential and provides accurate results compared to full RT-TDDFT for systems with uncoupled excitations. The embedding potential is calculated efficiently using a combination of density fitting and continuous fast multipole method for the Coulomb term. The applicability of (i) WFT-in-DFT embedding, in predicting the adsorption and excitation energies, and (ii) RT-TDDFET, in predicting the absorption spectra, is explored for various test systems.

Constraint‐based analysis of a physics‐guided kinetic energy density expansion

AbstractAn approach guided by physical consistency in determining the general forms of D‐dimensional kinetic energy density functionals (KEDF) has been demonstrated previously, producing an expansion which contains the majority of the known one‐point KEDF forms. It has also been shown that any noninteracting KEDF must necessarily have a homogeneity degree of 2 in coordinate scaling, and that the ratio of the collective KED to electron density must approach the ionization energy as . This article demonstrates that the scaling condition is already satisfied in the general expansion despite not being conceived with the scaling as a constraint, and that the second condition places a restriction on the expansion terms of the KED. The discussion is extended as well for some known KEDs for comparison.

Retraction Note: Quasi-dimensional models applied to kinetic and exchange energy density functionals

Retraction Note: Quasi-dimensional models applied to kinetic and exchange energy density functionals

Kinetic Energy Density Functionals Based on a Generalized Screened Coulomb Potential: Linear Response and Future Perspectives

We consider kinetic energy functionals that depend, beside the usual semilocal quantities (density, gradient, Laplacian of the density), on a generalized Yukawa potential, that is the screened Coulomb potential of the density raised to some power. These functionals, named Yukawa generalized gradient approximations (yGGA), are potentially efficient real-space semilocal methods that include significant non-local effects and can describe different important exact properties of the kinetic energy. In this work, we focus in particular on the linear response behavior for the homogeneous electron gas (HEG). We show that such functionals are able to reproduce the exact Lindhard function behavior with a very good accuracy, outperforming all other semilocal kinetic functionals. These theoretical advances allow us to perform a detailed analysis of a special class of yGGAs, namely the linear yGGA functionals. Thus, we show how the present approach can generalize the yGGA functionals improving the HEG linear behavior and leading to an extended formula for the kinetic functional. Moreover, testing on several jellium cluster model systems allows highlighting advantages and limitations of the linear yGGA functionals and future perspectives for the development of yGGA kinetic functionals.

Scaling analysis of a physics-guided kinetic energy density expansion

An approach guided by physical consistency in determining the general forms of D-dimensional kinetic energy density functionals (KEDF) has been demonstrated previously, producing an expansion which contains the majority of the known one-point KEDF forms. It is known that any noninteracting KEDF shall necessarily have a homogeneity degree of 2 in coordinate scaling. This paper demonstrates that this condition is already satisfied in the general expansion despite not being conceived with the scaling as a constraint.

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.

Highly accurate machine learning model for kinetic energy density functional

Inspired by the remarkable ongoing progress of data-driven approaches, a very accurate predictive model is developed to estimate one-dimensional kinetic energy density functionals (KEDF) using Machine Learning (ML). Starting from possible analytical forms of kinetic energy density and by utilizing a variety of solvable models, a simple – yet highly – accurate linear regression model is statistically trained to estimate the kinetic energy as functionals of the density. The mean relative accuracy for even a small number of randomly generated potentials is found to be better than the standard KEDF (Thomas-Fermi (TF) and von Weizsäcker (vW)) by several orders of magnitudes. As more different potentials of model problems are mixed, the coefficients of the linear model significantly approach the known values of Thomas-Fermi and von Weizsäcker, suggesting the reliability of the statistical training approach. This work can provide an important step toward more accurate large-scale orbital free density functional theory (OF-DFT) calculations.

RETRACTED ARTICLE: Quasi-dimensional models applied to kinetic and exchange energy density functionals

We investigate the behavior of three-dimensional 3D exchange energy functional of density-functional theory in anisotropic systems with two-dimensional 2D character and 1D character. The local density approximation (LDA), the generalized gradient approximation (GGA), and the meta-GGA behave as functions of quantum well width. We use the infinite-barrier model (IBM) for the quantum well. In the first section, we describe the problem of three-dimensional exchange functional, in the second section we introduce the quasi-2D IBM system, in the third section we introduce the quasi-1D IBM system. Using that an exact-exchange functional provides the correct approach to the true two-dimensional limit, we want to show that the 2D limit can be considered as a constraint on approximate functionals. For the 1D limit case we also propose a new functional obtained with methods completely similar to those of 2D limit.

Pauli energy and information-theoretic approach for evaluating dynamic and nondynamic electron correlation

Even though the quantification of different correlation regimes among electrons is controversial and a robust and generally applicable approach in this context is still lacking, accurately accounting for the electron correlation is a must. As the prominent and most well-known terms used to classify correlation effects, dynamic and nondynamic electron correlation is of concern herein. In this work, we consider a different tack to qualitative and quantitative description of the dynamic and nondynamic correlations, where Pauli energy and information-theoretic approach are considered for the purpose. To do so, several versions of the Pauli energy based on approximate kinetic energy density (KED) functionals and information-theoretic (IT) quantities like Fisher information, Shannon entropy, Onicescu information energy, and Ghosh–Berkowitz–Parr entropy with the two representations of electron density and shape function have been utilized. It is shown that although some KEDs may not reproduce the accurate Pauli energy results, their prediction for dynamic and nondynamic electron correlation can be desirable. There are also KEDs providing much better Pauli energy results than others, while their shortcomings can still be unveiled when they are used to describe electron correlation regimes. On the other hand, to more balanced treatment of electron correlation effects we propose to use the IT quantities with significantly different views from the perspectives of electron density distribution, scaling properties, and physiochemical meanings in the framework of Pauli energy. The efficiency and usefulness of the developed models have been demonstrated for both the dynamic and nondynamic electron correlation in molecular systems.

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

Orbital-free density functional theory calculation applying semi-local machine-learned kinetic energy density functional and kinetic potential

This letter proposes a scheme of orbital-free density functional theory (OF-DFT) calculation for optimizing electron density based on a semi-local machine-learned (ML) kinetic energy density functional (KEDF). The electron density, which is represented by the square of the linear combination of Gaussian functions, is optimized using derivatives of electronic energy including ML kinetic potential (KP). The numerical assessments confirmed the accuracy of optimized density and total energy for atoms and small molecules obtained by the present scheme based on ML-KEDF and ML-KP.

Nonlocal kinetic energy density functionals for isolated systems obtained via local density approximation kernels

Despite a large number of nonlocal kinetic energy density functionals (KEDFs) available for large-scale calculations, most of those nonlocal KEDFs designed for the extended systems cannot be directly applied to isolated systems. In this manuscript, we proposed a generalized scheme to construct nonlocal KEDFs via the local density approximation kernels and construct a family of KEDFs for simulations of isolated systems within orbital-free density functional theory. The performance of KEDFs has been demonstrated by several clusters encompassing Mg, Si and GaAs. The results show that our constructed KEDFs can achieve high numerical accuracy and stability for random clusters, therefore, making orbital-free density functional theory accessible for practical simulations of isolated systems.

Structure and dynamics of warm dense aluminum: a molecular dynamics study with density functional theory and deep potential

We perform a systematic study on the structure and dynamics of warm dense aluminum (Al) at temperatures ranging from 0.5 to 5.0 eV with molecular dynamics utilizing both density functional theory (DFT) and the deep potential (DP) method. On one hand, unlike the Thomas–Fermi kinetic energy density functional (KEDF), we find that the orbital-free DFT method with the Wang–Teter non-local KEDF yields properties of warm dense Al that agree well with the Kohn–Sham DFT method, enabling accurate orbital-free DFT simulations of warm dense Al at relatively low temperatures. On the other hand, the DP method constructs a deep neural network that has a high accuracy in reproducing short- and long-ranged properties of warm dense Al when compared to the DFT methods. The DP method is orders of magnitudes faster than DFT and is well-suited for simulating large systems and long trajectories to yield accurate properties of warm dense Al. Our results suggest that the combination of DFT methods and the DP model is a powerful tool for accurately and efficiently simulating warm dense matter.

Nonlocal kinetic energy density functional via line integrals and its application to orbital-free density functional theory

Orbital-free density functional theory (OF-DFT) holds great promise for large-scale simulations since there is a linear scaling for the computational cost. However, OF-DFT faces an essential challenge on construction of accurate kinetic energy density functional (KEDF) for evaluating the noninteracting kinetic energy. In contrast to most of previous KEDFs constructed by satisfaction of known constraint conditions with pregiven forms, a nonlocal KEDF named Xu-Wang-Ma (XWM) is constructed from ``scratch'' by the line integrals, allowing us to ensure the inclusion of the corrected response behavior deviation from the uniform electron gas. XWM has been benchmarked on a range of model systems with different chemical environments. Numerical tests show that, in general, XWM can quantitatively reproduce the Kohn-Sham predictions of the basic bulk properties, electron density and vacancy formation energies. Particularly, the XWM functional is found to be numerically stable for random structures of both simple metals and several phases of silicon. The high accuracy and numerical stability of XWM functional yield improvements over most of KEDFs currently in use for applications, providing more insight into the development of new KEDFs.