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

We present a comprehensive study of single-point kinetic energy density functionals (KEDFs) to be used in orbital-free density functional theory (DFT) calculations. We first propose a form of KEDFs based on a pointwise Kohn-Sham (KS) kinetic energy density (KED) and electron localization function (ELF) analysis. We find that the ELF and modified enhancement factor have a very strong and transferable correlation with the reduced density in various bulk metals. The non-self-consistent kinetic energy errors predicted by our KEDF models are decreased greatly compared to previously reported generalized gradient approximation (GGA) KEDFs. Second, we perform self-consistent calculations with various single-point KEDFs and investigate their numerical convergence behavior. We find striking numerical instabilities for previous GGA KEDFs; most of the GGA KEDFs fail to converge and show unphysical densities during the optimization. In contrast, our KEDFs demonstrate stable convergence, and their self-consistent results of various bulk properties agree reasonably well with KSDFT. A further detailed KED analysis reveals an interesting bifurcation phenomenon in defective metals and alloys, which may shed light on directions for future KEDF development.

Both model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single-particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one-shell harmonic confinement is generalized to include Coulomb repulsion in the two-electron Hookean atom and some results on the Pauli potentials for many-Fermion systems are obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

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.

We propose a kinetic energy density functional scheme with nonlocal terms based on the von Weizs\acker functional, instead of the more traditional approach where the nonlocal terms have the structure of the Thomas-Fermi functional. The proposed functionals recover the exact kinetic energy and reproduce the linear response function of homogeneous electron systems. In order to assess their quality, we have tested the total kinetic energies as well as the kinetic energy density for atoms. The results show that these nonlocal functionals give as good results as the most sophisticated functionals in the literature. The proposed scheme for constructing the functionals means a step ahead in the field of fully nonlocal kinetic energy functionals, because they are capable of giving better local behavior than the semilocal functionals, yielding at the same time accurate results for total kinetic energies. Moreover, the functionals enjoy the possibility of being evaluated as a single integral in momentum space if an adequate reference density is defined, and then quasilinear scaling for the computational cost can be achieved.

Kinetic energy density functionals (KEDFs) approximate the kinetic energy of a system of electrons directly from its electron density. They are used in electronic structure methods that lack direct access to orbitals, for example, orbital-free density functional theory (OFDFT) and certain embedding schemes. In this contribution, we introduce libKEDF, an accelerated library of modern KEDF implementations that emphasizes nonlocal KEDFs. We discuss implementation details and assess the performance of the KEDF implementations for large numbers of atoms. We show that using libKEDF, a single computing node or (GPU) accelerator can provide easy computational access to mesoscale chemical and materials science phenomena using OFDFT algorithms. © 2017 Wiley Periodicals, Inc.

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.

The formation and migration energies for various point defects, including vacancies and self-interstitials, in aluminium are systematically reinvestigated using the supercell approximation in the framework of orbital-free density functional theory. In particular, the finite-size effects and the accuracy of various kinetic energy density functionals are examined. It is demonstrated that as the supercell size increases, the finite-size errors asymptotically decrease as . Notably, the formation energies of self-interstitials converge much more slowly than that of vacancies. With carefully chosen kinetic energy density functionals, the calculated results agree quite well with the available experimental data and those obtained through Kohn–Sham density functional theory, which has an exact kinetic term.

Orbital-free density functional theory (OFDFT) is a quantum-mechanics-based method that utilizes electron density as its sole variable. The main computational cost in OFDFT is the ubiquitous use of the fast Fourier transform (FFT), which is mainly adopted to evaluate the kinetic energy density functional (KEDF) and electron-electron Coulomb interaction terms. We design and implement a small-box FFT (SBFFT) algorithm to overcome the parallelization limitations of conventional FFT algorithms. We also propose real-space truncation of the nonlocal Wang-Teter KEDF kernel. The scalability of the SBFFT is demonstrated by efficiently simulating one full optimization step (electron density, energies, forces, and stresses) of 1,024,000 lithium (Li) atoms on up to 65,536 cores. We perform other tests using Li as a test material, including calculations of physical properties of different phases of bulk Li, geometry optimizations of nanocrystalline Li, and molecular dynamics simulations of liquid Li. All of the tests yield excellent agreement with the original OFDFT results, suggesting that the OFDFT-SBFFT algorithm opens the door to efficient first-principles simulations of materials containing millions of atoms.

Frozen density embedding (FDE) has become a popular subsystem density functional theory (DFT) method for systems with weakly overlapping charge densities. The failure of this method for strongly interacting and covalent systems is due to the approximate kinetic energy density functional (KEDF), although the need for approximate KEDFs may be eliminated if each subsystem's Kohn-Sham (KS) orbitals are orthogonal to the other, termed external orthogonality (EO). We present an implementation of EO into the FDE framework within the Amsterdam density functional program package, using the level-shift projection operator method. We generalize this method to remove the need for orbital localization schemes and to include multiple subsystems, and we show that the exact KS-DFT energies and densities may be reproduced through iterative freeze-and-thaw cycles for a number of systems, including a charge delocalized benzene molecule starting from atomic subsystems. Finally, we examine the possibility of a truncated basis for systems with and without charge delocalization, and found that subsystems require a basis that allows them to correctly describe the supermolecular delocalized orbitals.

In the beginning of quantum mechanical Density-Functional Theory (DFT), there was the Thomas-Fermi (TF) model, which uses the electron density p(r) (a function of only 3 coordinates) as the only physical variable. Calculations with this model were inexpensive but yielded poor numerical results due to a lack of understanding of exchange-correlation effects and the kinetic-energy density functional. Many years later, Hohenberg and Kohn (HK) established the formal foundation for DFT; Kohn and Sham (KS) devised a practical implementation and brought DFT into mainstream calculations of electronic structure. Although the KS formulation allows exact evaluation of the KS kinetic energy (Ts [p]), the one-electron orbitals introduced by the KS scheme inevitably encumber the formulation in three ways: (i) 3N (vs. 3) degrees of freedom, (ii) orbital orthonormalization, and (iii) Brillouin-zone (k-point) sampling in condensed phases. Given the accuracy of DFT with present exchange-correlation density functionals, it is logical to conclude that the last frontier in DFT is a better representation of the kinetic energy solely in terms of the density. If this is true, KS orbitals will be completely eliminated from DFT formulation, and the density can be solved directly from the TF-HK equation. This is certainly superior to the KS scheme because all energy terms can be computed in momentum space with an effectively linear scaling, O(MlnM), where M is the integration grid size. This work reviews major ideas in the design of such optimal orbital-free kinetic-energy density functionals and their applications.

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.

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

A gradient expansion for the total kinetic energy was derived. We imposed the condition that the densities should be a slowly varying function. It is applicable up to the fourth order in ▿n. The leading terms are the same as those of Thomas-Fermi and Weizsacker. The fourth-order term is in good agreement with the nonlinear response part of Hodges correction. A small term,Y[n], has been added to the derived gradient expansion. A new formula for the total kinetic energy with full nonlinear response is achieved as a result. The same method is applied to the exchange energy functional, where the well-known gradient expansion has been found but it differs only in the numerical coefficients. Divergent terms have been found in both expansions.

To begin the topic of DFT, this chapter overviews how the Kohn–Sham method was developed historically and then introduces various extensions of this method. The Thomas–Fermi method, which is the first form of DFT, is first explained, focusing on the local density approximation of kinetic and exchange energy density functionals, in Sect.4.1. Then, the Hohenberg–Kohn theorem, which is the basic theory of DFT, is reviewed, with a mention of the constrained search formulation used to solve the V -representability problem, in Sect. 4.2. The Kohn–Sham method based on this theorem is introduced, along with the corresponding computational methods, in Sect. 4.3. As the extension of the Kohn–Sham method to include general functionals, the generalized Kohn–Sham method is surveyed in Sect. 4.4. The constrained search method, which directly constructs a Kohn–Sham potential from the electron density, is explained, and as a consequence of this method, it is clarified why the Kohn–Sham method can accurately reproduce chemical behavior in Sect. 4.5. Finally, the time-dependent and coupled-perturbed Kohn–Sham methods are reviewed as methods with which to apply the Kohn–Sham method to calculations of photoexcitation spectra and response properties, respectively, in Sects. 4.6 and 4.7.

Identities and approximations are proposed for the current-density-functional theory. Based on the virial theorem and the scaling properties of the kinetic and exchange energy density functionals, when local and variable-separation assumptions are made, local formulas for these functionals are obtained. One also obtains hierarchies of equations, which combine to give series expansions of the functionals in terms of their functional derivatives with respect to both the density and the paramagnetic current density. In addition, a general formulation is investigated in the form of the constrained search of a coupled Hamiltonian. Subject to the validity of a Taylor series, both the correlation energy density functional and its kinetic component can be expanded into series, each of which, if locality density approximations are assumed, consists of homogeneous functionals of degree (4{minus}{ital n})/3 in density scaling, (4{minus}{ital n})/2 with respect to {bold v}({bold r}), a gauge-invariant variable. {copyright} {ital 1996 The American Physical Society.}