Fourth-order Gradient Expansion Research Articles

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.

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.

Assessing the performance of the Tao-Mo semilocal density functional in the projector-augmented-wave method.

We assess the performance of the recently proposed Tao-Mo (TM) semilocal exchange-correlation functional [J. Tao and Y. Mo, Phys. Rev. Lett. 117, 073001 (2016)] using the projector-augmented-wave method with the plane wave basis set. The meta-generalized gradient approximation level semilocal functional constructed by Tao-Mo is an all-purpose exchange-correlation functional for the quantum chemistry and solid-state physics. The exchange of the TM functional is based on the density matrix expansion technique together with the slowly varying fourth order gradient expansion. The correlation functional corresponding to the exchange is based on the one-electron self-interaction-free Tao-Perdew-Staroverov-Scuseria functional. Our test includes solid-state lattice constants, bulk moduli, bandgaps, cohesive energies, magnetic moments and vacancy-formation energies of transition metals. It is observed that in the plane wave basis, the TM functional performs accurately in predicting all the solid state properties at the semilocal level.

Assessment of the Tao-Mo nonempirical semilocal density functional in applications to solids and surfaces

Recently, Tao and Mo developed a new nonempirical semilocal exchange-correlation density functional. The exchange part of this functional is derived from a density matrix expansion corrected to reproduce the fourth-order gradient expansion in the slowly varying limit, while the correlation part is based on the TPSS correlation model with a modification for the low-density limit. In the present work, the Tao-Mo functional is assessed by calculations on a variety of solids and jellium surfaces. This includes 22 lattice constants and bulk moduli, 7 cohesive energies, and jellium surface exchange and correlation energies for the density parameter rs in the range from 2 to 3 bohrs. Our calculations show that this meta-generalized gradient approximation can yield consistently remarkable accuracy for the properties considered here, with mean absolute errors of 0.017 {\AA} for lattice constants, 7.0 GPa for bulk moduli, 0.08 eV for cohesive energies, and 35 erg/cm2 for surface exchange-correlation energies, substantially improving upon existing nonempirical semilocal density functionals.

Improved analytical representation of combinations of Fermi–Dirac integrals for finite-temperature density functional calculations

Smooth, highly accurate analytical representations of Fermi–Dirac (FD) integral combinations important in free-energy density functional calculations are presented. Specific forms include those that occur in the local density approximation (LDA), generalized gradient approximation (GGA), and fourth-order gradient expansion of the non-interacting free energy as well as in the LDA and second-order gradient expansion for exchange. By construction, all the representations and their derivatives of any order are continuous on the full domains of their independent variables. The same type of technique provides an analytical representation of the function inverse to the FD integral of order 1/2. It plays an important role in physical problems related to the electron gas at finite temperature. From direct evaluation, the quality of these improved representations is shown to be substantially superior to existing ones, many of which were developed before the era of large-scale computation or early in the era.

Global hybrids from the semiclassical atom theory satisfying the local density linear response.

We propose global hybrid approximations of the exchange-correlation (XC) energy functional which reproduce well the modified fourth-order gradient expansion of the exchange energy in the semiclassical limit of many-electron neutral atoms and recover the full local density approximation (LDA) linear response. These XC functionals represent the hybrid versions of the APBE functional [Phys. Rev. Lett. 2011, 106, 186406] yet employing an additional correlation functional which uses the localization concept of the correlation energy density to improve the compatibility with the Hartree-Fock exchange as well as the coupling-constant-resolved XC potential energy. Broad energetic and structural testing, including thermochemistry and geometry, transition metal complexes, noncovalent interactions, gold clusters and small gold-molecule interfaces, as well as an analysis of the hybrid parameters, show that our construction is quite robust. In particular, our testing shows that the resulting hybrid, including 20% of Hartree-Fock exchange and named hAPBE, performs remarkably well for a broad palette of systems and properties, being generally better than popular hybrids (PBE0 and B3LYP). Semiempirical dispersion corrections are also provided.

Laplacian-Level Kinetic Energy Approximations Based on the Fourth-Order Gradient Expansion: Global Assessment and Application to the Subsystem Formulation of Density Functional Theory.

We tested Laplacian-level meta-generalized gradient approximation (meta-GGA) noninteracting kinetic energy functionals based on the fourth-order gradient expansion (GE4). We considered several well-known Laplacian-level meta-GGAs from the literature (bare GE4, modified GE4, and the MGGA functional of Perdew and Constantin (Phys. Rev. B 2007,75, 155109)), as well as two newly designed Laplacian-level kinetic energy functionals (L0.4 and L0.6). First, a general assessment of the different functionals is performed to test them for model systems (one-electron densities, Hooke's atom, and different jellium systems) and atomic and molecular kinetic energies as well as for their behavior with respect to density-scaling transformations. Finally, we assessed, for the first time, the performance of the different functionals for subsystem density functional theory (DFT) calculations on noncovalently interacting systems. We found that the different Laplacian-level meta-GGA kinetic functionals may improve the description of different properties of electronic systems, but no clear overall advantage is found over the best GGA functionals. Concerning the subsystem DFT calculations, the here-proposed L0.4 and L0.6 kinetic energy functionals are competitive with state-of-the-art GGAs, whereas all other Laplacian-level functionals fail badly. The performance of the Laplacian-level functionals is rationalized thanks to a two-dimensional reduced-gradient and reduced-Laplacian decomposition of the nonadditive kinetic energy density.

Comment on “More accurate generalized gradient approximation for solids”

We point out an issue in the derivation of a nonempirical parameter in the Wu-Cohen exchange functional. With the value of the parameter that yields the correct fourth-order gradient expansion of the functional, the performance of the Wu-Cohen exchange functional is worse than the Perdew-Burke-Ernzerhof functional.

Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy

We construct a Laplacian-level meta-generalized gradient approximation (meta-GGA) for the non-interacting (Kohn-Sham orbital) positive kinetic energy density $\tau$ of an electronic ground state of density $n$. This meta-GGA is designed to recover the fourth-order gradient expansion $\tau^{GE4}$ in the appropiate slowly-varying limit and the von Weizs\"{a}cker expression $\tau^{W}=|\nabla n|^2/(8n)$ in the rapidly-varying limit. It is constrained to satisfy the rigorous lower bound $\tau^{W}(\mathbf{r})\leq\tau(\mathbf{r})$. Our meta-GGA is typically a strong improvement over the gradient expansion of $\tau$ for atoms, spherical jellium clusters, jellium surfaces, the Airy gas, Hooke's atom, one-electron Gaussian density, quasi-two dimensional electron gas, and nonuniformly-scaled hydrogen atom. We also construct a Laplacian-level meta-GGA for exchange and correlation by employing our approximate $\tau$ in the Tao, Perdew, Staroverov and Scuseria (TPSS) meta-GGA density functional. The Laplacian-level TPSS gives almost the same exchange-correlation enhancement factors and energies as the full TPSS, suggesting that $\tau$ and $\nabla^2 n$ carry about the same information beyond that carried by $n$ and $\nabla n$. Our kinetic energy density integrates to an orbital-free kinetic energy functional that is about as accurate as the fourth-order gradient expansion for many real densities (with noticeable improvement in molecular atomization energies), but considerably more accurate for rapidly-varying ones.

Energy of step formation on metal surfaces from the stabilized-jellium model

We describe stepped metal surfaces by the stabilized-jellium model, and calculate the step formation energy within the local density approximation of the density-functional theory. Using fourth-order gradient expansion for the kinetic energy, we determine the electron density variationally from a parameterized trial density. We compare our numerical results for a monatomic step on the Al(1 1 1) surface with other theoretical calculations and estimates.

Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation

We approximate the exchange-correlation energy of density functional theory as a controlled extrapolation from the slowly varying limit. While generalized gradient approximations (GGA's) require only the local density and its first gradient as input, our meta-GGA also requires the orbital kinetic energy density. Its exchange energy component recovers the fourth-order gradient expansion, while its correlation energy is free of self-interaction error. Molecular atomization energies and metal surface energies are significantly improved over GGA, while lattice constants are little changed.

Fourth-order gradient corrections to the exchange-only energy functional: Importance of

The pertinence of four-order gradient corrections to the exchange-only energy functional ${\mathit{E}}_{\mathit{x}}$[n] for producing accurate exchange potentials ${\mathit{v}}_{\mathit{x}}$(r) is examined utilizing the exact ${\mathit{v}}_{\mathit{x}}$(r)'s obtained from the optimized-potential model (OPM) for spherical atoms and jellium spheres (whose inner regions have slowly varying densities for which the gradient expansion should be valid). It is found that the fourth-order contributions containing ${\mathrm{\ensuremath{\nabla}}}^{2}$n represent important ingredients of ${\mathit{E}}_{\mathit{x}}$[n] which should be included in gradient-based nonlocal extensions of the local-density approximation. In accordance with this observation the ${\mathit{v}}_{\mathit{x}}$(r)'s resulting from the fourth-order gradient expansion are closer to the exact ${\mathit{v}}_{\mathit{x}}^{\mathrm{OPM}}$(r)'s than those from generalized gradient approximations which do not contain ${\mathrm{\ensuremath{\nabla}}}^{2}$n contributions.

Fourth-order gradient expansion of the fermion kinetic energy: Extra terms for nonanalytic densities

The fourth-order density-gradient expansion of the fermion kinetic energy contains terms which integrate over space to zero foranalytic densities, butnot for the densities of atoms or of those metal-surface models for which the expansion has been tested. Evaluation of these terms for spin-unpolarized atoms and in spin-density functional calculations for atoms, as well as for metal surfaces, show that the fourth-order expansion is not quite as accurate as it has been believed to be.

Energies of curved metallic surfaces from the stabilized-jellium model.

In the liquid-drop model, the total energy of a system is expanded as a sum of volume, surface, and curvature terms. We derive an expression for the curvature energy of a metal in terms of the electron-density profile for a planar surface, and show that the resulting values agree with the fits of calculated or measured total energies to the liquid-drop expansion. In particular, this expansion accurately describes the formation energies of microscopic voids (including monovacancies) in metals. In our calculations, the curvature energy is determined by the bulk density. It is nearly the same for restricted trial density profiles as for self-consistent Kohn-Sham profiles, for the fourth-order gradient expansion as for the exact kinetic energy, and for jellium as for stabilized jellium. We also report Kohn-Sham results for the surface energy and work function. The stabilized-jellium model, while retaining the simplicity and nonempirical character of jellium, gives a significantly more realistic description of the simple metals, especially those with high bulk densities.

Fourth-order gradient expansion of the fermion kinetic energy: Extra terms for nonanalytic densities.

The fourth-order density-gradient expansion of the fermion kinetic energy contains terms which integrate over space to zero for analytic densities, but not for the densities of atoms or of those metal-surface models for which the expansion has been tested. Evaluation of these terms shows that the fourth-order expansion is not quite as accurate as it has been believed to be.

Fourth order gradient corrections to the free energy of noninteracting many-electron systems at finite temperature

The fourth order gradient expansion corrections to the free energy of a noninteracting many-electron system at finite temperature in an external potential are obtained. The required coefficients are conveniently expressed in terms of Fermi-Dirac integrals.

Long-range part of the interaction between two hydrogen impurities in jellium

The interaction of two hydrogen impurities in jellium is calculated in the "overlapping-densities" approximation where the total displaced charge is the sum of two frozen single-ion screening densities. Using the fourth-order gradient expansion of the kinetic-energy functional, we obtain the results at various densities, corresponding to Al, Mg, and Na. The results provide an extension for the full treatment of the short-range proton-proton interaction to the asymptotic region where such exact calculation are as yet not available.