Further orbital-free kinetic-energy functionals for ab initio molecular dynamics

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.

Automatic differentiation for orbital-free density functional theory

Differentiable programming has facilitated numerous methodological advances in scientific computing. Physics engines supporting automatic differentiation have simpler code, accelerating the development process and reducing the maintenance burden. Furthermore, fully differentiable simulation tools enable direct evaluation of challenging derivatives-including those directly related to properties measurable by experiment-that are conventionally computed with finite difference methods. Here, we investigate automatic differentiation in the context of orbital-free density functional theory (OFDFT) simulations of materials, introducing PROFESS-AD. Its automatic evaluation of properties derived from first derivatives, including functional potentials, forces, and stresses, facilitates the development and testing of new density functionals, while its direct evaluation of properties requiring higher-order derivatives, such as bulk moduli, elastic constants, and force constants, offers more concise implementations than conventional finite difference methods. For these reasons, PROFESS-AD serves as an excellent prototyping tool and provides new opportunities for OFDFT.

Incremental solver for orbital-free density functional theory

First-principle calculations are still a challenge since they require a great amount of computational time. In this article, we introduce a new algorithm to perform orbital-free density functional theory (OF-DFT) calculations. Our new algorithm focuses computational efforts on important parts of the particle system, which, in the context of adaptively restrained particle simulations (ARPS) allows us to accelerate particle simulations. © 2019 Wiley Periodicals, Inc.

Enhanced von Weizsäcker Wang-Govind-Carter kinetic energy density functional for semiconductors

We propose a new form of orbital-free (OF) kinetic energy density functional (KEDF) for semiconductors that is based on the Wang-Govind-Carter (WGC99) nonlocal KEDF. We enhance within the latter the semi-local von Weizsäcker KEDF term, which is exact for a single orbital. The enhancement factor we introduce is related to the extent to which the electron density is localized. The accuracy of the new KEDF is benchmarked against Kohn-Sham density functional theory (KSDFT) by comparing predicted energy differences between phases, equilibrium volumes, and bulk moduli for various semiconductors, along with metal-insulator phase transition pressures. We also compare point defect and (100) surface energies in silicon for a broad test of its applicability. This new KEDF accurately reproduces the exact non-interacting kinetic energy of KSDFT with only one additional adjustable parameter beyond the three parameters in the WGC99 KEDF; it exhibits good transferability between semiconducting to metallic silicon phases and between various III-V semiconductors without parameter adjustment. Overall, this KEDF is more accurate than previously proposed OF KEDFs (e.g., the Huang-Carter (HC) KEDF) for semiconductors, while the computational efficiency remains at the level of the WGC99 KEDF (several hundred times faster than the HC KEDF). This accurate, fast, and transferable new KEDF holds considerable promise for large-scale OFDFT simulations of metallic through semiconducting materials.

Angular-momentum-dependent orbital-free density functional theory

Orbital-free (OF) density functional theory (DFT) directly solves for the electron density rather than the wave function of many electron systems, greatly simplifying and enabling large scale first principles simulations. However, the required approximate noninteracting kinetic energy density functionals and local electron-ion pseudopotentials severely restrict the general applicability of conventional OFDFT. Here, we present a new generation of OFDFT called angular-momentum-dependent (AMD)-OFDFT to harness the accuracy of Kohn-Sham DFT and the simplicity of OFDFT. The angular momenta of electrons are explicitly introduced within atom-centered spheres so that the important ionic core region can be accurately described. In addition to conventional OF total energy functionals, we introduce a crucial nonlocal energy term with a set of AMD energies to correct errors due to the kinetic energy density functional and the local pseudopotential. We find that our AMD-OFDFT formalism offers substantial improvements over conventional OFDFT, as we show for various properties of the transition metal titanium.

Partial ionization in dense plasmas: comparisons among average-atom density functional models

Nuclei interacting with electrons in dense plasmas acquire electronic bound states, modify continuum states, generate resonances and hopping electron states, and generate short-range ionic order. The mean ionization state (MIS), i.e, the mean charge Z of an average ion in such plasmas, is a valuable concept: Pseudopotentials, pair-distribution functions, equations of state, transport properties, energy-relaxation rates, opacity, radiative processes, etc., can all be formulated using the MIS of the plasma more concisely than with an all-electron description. However, the MIS does not have a unique definition and is used and defined differently in different statistical models of plasmas. Here, using the MIS formulations of several average-atom models based on density functional theory, we compare numerical results for Be, Al, and Cu plasmas for conditions inclusive of incomplete atomic ionization and partial electron degeneracy. By contrasting modern orbital-based models with orbital-free Thomas-Fermi models, we quantify the effects of shell structure, continuum resonances, the role of exchange and correlation, and the effects of different choices of the fundamental cell and boundary conditions. Finally, the role of the MIS in plasma applications is illustrated in the context of x-ray Thomson scattering in warm dense matter.

O(N) methods in electronic structure calculations

Linear-scaling methods, or O(N) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N, in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high-performance computers. The linear-scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas are then discussed. The applications of linear-scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear-scaling methods are discussed.

Structural and dynamical properties of liquid Mg. An orbital-free molecular dynamics study

Several static and dynamic properties of liquid magnesium near melting have been evaluated by the orbital-free ab initio molecular dynamics method. The calculated static structure shows good agreement with recent experimental data, including an asymmetric second peak in the structure factor which has been linked to the existence of an important icosahedral short-range order in the liquid. As for the dynamic structure, we obtain collective density excitations with an associated dispersion relation which closely follows recent experimental results. Accurate estimates have also been obtained for several transport coefficients.

Total energy evaluation in the Strutinsky shell correction method

We analyze the total energy evaluation in the Strutinsky shell correction method (SCM) of Ullmo et al. [Phys. Rev. B 63, 125339 (2001)], where a series expansion of the total energy is developed based on perturbation theory. In agreement with Yannouleas and Landman [Phys. Rev. B 48, 8376 (1993)], we also identify the first-order SCM result to be the Harris functional [Phys. Rev. B 31, 1770 (1985)]. Further, we find that the second-order correction of the SCM turns out to be the second-order error of the Harris functional, which involves the a priori unknown exact Kohn-Sham (KS) density, rho(KS)(r). Interestingly, the approximation of rho(KS)(r) by rho(out)(r), the output density of the SCM calculation, in the evaluation of the second-order correction leads to the Hohenberg-Kohn-Sham functional. By invoking an auxiliary system in the framework of orbital-free density functional theory, Ullmo et al. designed a scheme to approximate rho(KS)(r), but with several drawbacks. An alternative is designed to utilize the optimal density from a high-quality density mixing method to approximate rho(KS)(r). Our new scheme allows more accurate and complex kinetic energy density functionals and nonlocal pseudopotentials to be employed in the SCM. The efficiency of our new scheme is demonstrated in atomistic calculations on the cubic diamond Si and face-centered-cubic Ag systems.

Energetics and kinetics of vacancy diffusion and aggregation in shocked aluminium via orbital-free density functional theory

A possible mechanism for shock-induced failure in aluminium involves atomic vacancies diffusing through the crystal lattice and agglomerating to form voids, which continue to grow, ultimately resulting in ductile fracture. We employ orbital-free density functional theory, a linear-scaling first-principles quantum mechanics method, to study vacancy formation, diffusion, and aggregation in aluminium under shock loading conditions of compression and tension. We calculate vacancy formation and migration energies, and find that while nearest-neighbor vacancy pairs are unstable, next-nearest-neighbor vacancy pairs are stable. As the number of nearby vacancies increases, we predict that vacancy clusters preferentially grow through next-nearest-neighbor vacancies. The energetics are found to be greatly affected by expansion and compression, leading to insight as to how vacancies behave under shock conditions.

Orbital-corrected orbital-free density functional theory

A new implementation of density functional theory (DFT), namely orbital-corrected orbital-free (OO) DFT, has been developed. With at most two non-self-consistent iterations, OO-DFT accomplishes the accuracy comparable to fully self-consistent Kohn-Sham DFT as demonstrated by its application on the cubic-diamond Si and the face-centered-cubic Ag systems. Our work provides a new impetus to further improve orbital-free DFT method and presents a robust means to significantly lower the cost associated with general applications of linear-scaling Kohn-Sham DFT methods on large systems of thousands of atoms within different chemical bonding environment.

Orbital-Free Density Functional Theory Applied to NaAlH4

We present the application of orbital-free density functional theory (OF-DFT) to NaAlH(4), a potential hydrogen storage material, and related systems. Although the simple Al and NaH structures are reproduced reasonably well by OF-DFT, the approach fails for the more complex NaAlH(4) structure. Calculations on AlH(3) show that the failure to describe the Al-H interaction is related to the kinetic energy functionals used rather than the local pseudopotentials which are required within the OF-DFT approach. Thus, systems such as NaAlH(4) present a challenge which awaits the development of more reliable orbital-free kinetic energy functionals.

Nonlinear instability of density-independent orbital-free kinetic-energy functionals

We study in this article the mathematical properties of a class of orbital-free kinetic-energy functionals. We prove that these models are linearly stable but nonlinearly unstable, in the sense that the corresponding kinetic-energy functionals are not bounded from below. As a matter of illustration, we provide an example of an electronic density of simple shape, the kinetic energy of which is negative.

First principles local pseudopotential for silver: Towards orbital-free density-functional theory for transition metals

Orbital-free density-functional theory (OF-DFT) with modern kinetic-energy density functionals (KEDFs) is a linear scaling technique that accurately describes nearly-free-electron-like (main group) metals. In an attempt towards extending OF-DFT to transition metals, here we consider whether OF-DFT can be used effectively to study Ag, a metal with a localized d shell. OF-DFT has two approximations: use of a KEDF and local pseudopotentials (LPSs). This paper reports construction of a reasonably accurate LPS for Ag by means of inversion of the Kohn-Sham (KS) DFT equations in a bulk crystal environment. The accuracy of this LPS is determined within KS-DFT (where the exact noninteracting kinetic energy is employed) by comparing its predictions of bulk properties to those obtained from a conventional (orbital-based) nonlocal pseudopotential (NLPS). We find that the static bulk properties of fcc and hcp Ag predicted within KS-DFT using this LPS compare fairly well to those predicted by an NLPS. With the transferability of the LPS established, we then use this LPS in OF-DFT, where several approximate KEDFs were tested. We find that a combination of the Thomas-Fermi (T(TF)) and von Weizsacker (T(vW)) functionals (T(vW)+0.4T(TF)) produces better densities than those from the linear-response-based Wang-Teter KEDF. However, the equations of state obtained from both KEDFs in OF-DFT contain unacceptably large errors. The lack of accurate KEDFs remains the final barrier to extending OF-DFT to treat transition metals.

Conjugate-gradient optimization method for orbital-free density functional calculations

Orbital-free density functional theory as an extension of traditional Thomas-Fermi theory has attracted a lot of interest in the past decade because of developments in both more accurate kinetic energy functionals and highly efficient numerical methodology. In this paper, we developed a conjugate-gradient method for the numerical solution of spin-dependent extended Thomas-Fermi equation by incorporating techniques previously used in Kohn-Sham calculations. The key ingredient of the method is an approximate line-search scheme and a collective treatment of two spin densities in the case of spin-dependent extended Thomas-Fermi problem. Test calculations for a quartic two-dimensional quantum dot system and a three-dimensional sodium cluster Na216 with a local pseudopotential demonstrate that the method is accurate and efficient.

Surface structure of liquid Li and Na: an ab initio molecular dynamics study

Molecular dynamics simulations of the liquid-vapor interfaces of liquid metals have been performed using first principles methods. Results are presented for liquid lithium and sodium near their respective triple points, for samples of 2000 particles in a slab geometry. The atomic density profiles show a pronounced stratification extending several atomic diameters into the bulk, which is similar to that already experimentally observed in liquid K, Ga, In, and Hg.

Collective ionic dynamics in the liquid Na-Cs alloy: an ab initio molecular dynamics study

We present results for several structural and dynamical properties of the liquid Na-Cs alloy. The study has been carried out by means of the orbital-free ab initio molecular dynamics method, combined with local ionic pseudopotentials constructed within the same framework. The results show good agreement with the available experimental data, reproducing the homocoordinating tendency exhibited by this alloy.

AB INITIO DYNAMICS OF SURFACE CHEMISTRY

▪ Abstract  We review the young field of ab initio molecular dynamics applied to molecule-surface reactions. The techniques of ab initio molecular dynamics include methods that use an analytic potential energy function fit to ab initio data and those that are fully ab initio. In this review, we focus on the insights provided by ab initio–based molecular dynamics that are currently unavailable from experimental studies and discuss current techniques and limitations. As an example of how different aspects of a problem can be tackled with state-of-the-art theoretical tools, we consider the well-studied case of H2 desorption and adsorption from the Si(100)-2 × 1 surface.