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

Kohn-Sham accuracy from orbital-free density functional theory via Δ-machine learning

We present a Δ-machine learning model for obtaining Kohn-Sham accuracy from orbital-free density functional theory (DFT) calculations. In particular, we employ a machine-learned force field (MLFF) scheme based on the kernel method to capture the difference between Kohn-Sham and orbital-free DFT energies/forces. We implement this model in the context of on-the-fly molecular dynamics simulations and study its accuracy, performance, and sensitivity to parameters for representative systems. We find that the formalism not only improves the accuracy of Thomas-Fermi-von Weizsäcker orbital-free energies and forces by more than two orders of magnitude but is also more accurate than MLFFs based solely on Kohn-Sham DFT while being more efficient and less sensitive to model parameters. We apply the framework to study the structure of molten Al0.88Si0.12, the results suggesting no aggregation of Si atoms, in agreement with a previous Kohn-Sham study performed at an order of magnitude smaller length and time scales.

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.

Adaptive Subsystem Density Functional Theory

Subsystem density functional theory (DFT) is emerging as a powerful electronic structure method for large-scale simulations of molecular condensed phases and interfaces. Key to its computational efficiency is the use of approximate nonadditive noninteracting kinetic energy functionals. Unfortunately, currently available nonadditive functionals lead to inaccurate results when the subsystems interact strongly such as when they engage in chemical reactions. This work disrupts the status quo by devising a workflow that extends subsystem DFT's applicability also to strongly interacting subsystems. This is achieved by implementing a fully automated adaptive definition of subsystems which is realized during geometry optimizations or ab initio molecular dynamics simulations. The new method prescribes subsystem merging and splitting events redistributing the resources (both for work and data) in an efficient way making use of modern parallelization strategies and object-oriented programming. We showcase the method with examples probing from moderate-to-strong inter-subsystem interactions, opening the door to using subsystem DFT for modeling chemical reactions in molecular condensed phases with a black box computational tool.

Orbital-Free Methods for Plasmonics: Linear Response

Plasmonic systems, such as metal nanoparticles, are widely used in different application areas, going from biology to photovoltaics.The modeling of the optical response of such systems is of fundamental importance to analyze their behavior and to design new systems with required properties.When the characteristic sizes/distances reach a few nanometers, non-local and spill-out effects become relevant and conventional classical electrodynamics models are no more appropriate. Methods based on the Time-Dependent Density-Functional Theory (TD-DFT) represent the current reference for the description of quantum effects. However, TD-DFT is based on knowledge of all occupied orbitals whose calculation is computationally prohibitive to model large plasmonic systems of interest for applications.On the other hand, methods based on the Orbital-Free (OF) formulation of TD-DFT, can scale linearly with the system size.In this Review, OF methods ranging from semiclassical models to the quantum hydrodynamic theory, will be derived from the linear response TD-DFT, so that the key approximations and properties of each method can be clearly highlighted. The accuracy of the various approximations will be then validated for the linear optical properties of jellium nanoparticles, the most relevant model system in plasmonics. OF methods can describe the collective excitations in plasmonic systems with great accuracy andwithout system-tuned parameters. The accuracy on these methods depends only on the accuracy on the (universal) kinetic energy functional of the ground-state electronic density. Current approximations and future development directions will be indicated.

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.

Approximate Analytical Solutions for the Euler Equation for Second-Row Homonuclear Dimers

This work presents a new method how to obtain approximate analytical solutions for the Euler equation for second-row homonuclear dimers. In contrast to the well-known Kohn-Sham method where a system of N nonlinear coupled differential equations must be solved iteratively, orbital-free density functional theory allows to access the minimizing electron density directly via the Euler equation. For simplified models, here, an atom-centered monopole expansion with one free parameter, solutions of the electron density can be obtained analytically by solving the Euler equation at the bond critical point. The procedure is exemplarily carried out for N₂, C₂, and B₂, yielding bound molecules with an internuclear distance of 2.01, 2.43, and 3.07 bohr, respectively.

Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory

This work presents a method to move beyond the recently introduced atomic fragment approximation. Like the bare atomic fragment approach, the new method is an ab initio, parameter-free, orbital-free implementation of density functional theory based on the bifunctional formalism that treats the potential and the electron density as two separate variables, and provides access to the Kohn–Sham Pauli kinetic energy for an appropriately chosen Pauli potential. In the present ansatz, the molecular Pauli potential is approximated by the sum of the bare atomic fragment approach, and a so-called deformation potential that takes the interaction between the atoms into account. It is shown that this model can reproduce the bond-length contraction due to multiple bonding within the list of second-row homonuclear dimers. The present model only relies on the electron densities of the participating atoms, which themselves are represented by a simple monopole expansion. Thus, the bond-length contraction can be rationalized without referring to the angular quantum numbers of the participating atoms.

Analysis of the kinetic energy functional in the generalized gradient approximation

A new density functional for the total kinetic energy in the generalized gradient approximation is developed through an enhancement factor that leads to the correct behavior in the limits when the reduced density gradient tends to 0 and to infinity and by making use of the conjoint conjecture for the interpolation between these two limits, through the incorporation, in the intermediate region of constraints that are associated with the exchange energy functional. The resulting functional leads to a reasonable description of the kinetic energies of atoms and molecules when it is used in combination with Hartree-Fock densities. Additionally, in order to improve the behavior of the kinetic energy density, a new enhancement factor for the Pauli kinetic energy is proposed by incorporating the correct behavior into the limits when the reduced density gradient tends to 0 and to infinity, together with the positivity condition, and imposing through the interpolation function that the sum of its integral over the whole space and the Weiszacker energy must be equal to the value obtained with the enhancement factor developed for the total kinetic energy.

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.

Equilibrium Bond Lengths from Orbital-Free Density Functional Theory

This work presents an investigation to model chemical bonding in various dimers based on the atomic fragment approach. The atomic fragment approach is an ab-initio, parameter-free implementation of orbital-free density functional theory which is based on the bifunctional formalism, i.e., it uses both the density and the Pauli potential as two separate variables. While providing the exact Kohn-Sham Pauli kinetic energy when the orbital-based Kohn-Sham data are used, the bifunctional formalism allows for approximations of the functional derivative which are orbital-free. In its first implementation, the atomic fragment approach uses atoms in their ground state to model the Pauli potential. Here, it is tested how artificial closed-shell fragments with non-integer electron occupation perform regarding the prediction of bond lengths of diatomics. Such fragments can sometimes mimic the electronic structure of a molecule better than groundstate fragments. It is found that bond lengths may indeed be considerably improved in some of the tested diatomics, in accord with predictions based on the electronic structure.

The first order atomic fragment approach-An orbital-free implementation of density functional theory

An orbital-free implementation of the original Hohenberg-Kohn theorems is presented, making use of the scaling properties from a fictitious Kohn-Sham system, but without reintroducing orbitals. The first order fragment approach does not contain data or parameters that are fitted to the final outcome of the molecular orbital-free calculation and thus represents a parameter-free implementation of orbital-free density functional theory, although it requires the precalculation of atomic data. Consequently, the proposed method is not limited to a specific type of molecule or chemical bonding. The different approximation levels arise from including (first order) or neglecting (zeroth order) the dependency between the potential and the electron density, which in the bifunctional approach are formally treated as independent variables.

A thermal orbital-free density functional approach

A generating function σ is defined for spherically symmetric systems. Compared to the density, the generating functional has two extra variables and reduces to the density if these variables are equal to zero. It is proved that σ satisfies a differential equation that contains only the derivatives of σ and the Kohn-Sham potential. A Schrödinger-like equation for the square root of σ is also derived. The effective potential of this equation is the sum of the Kohn-Sham potential and a term that is expressed with an integral containing the derivatives of σ. The noninteracting kinetic energy can be calculated in the knowledge of σ. The theory is valid in case of zero and nonzero temperatures as well. For nonspherically symmetric systems, the muffin-tin approximation can be applied.

Semilocal Pauli-Gaussian Kinetic Functionals for Orbital-Free Density Functional Theory Calculations of Solids

Kinetic energy (KE) approximations are key elements in orbital-free density functional theory. To date, the use of nonlocal functionals, possibly employing system-dependent parameters, has been considered mandatory in order to obtain satisfactory accuracy for different solid-state systems, whereas semilocal approximations are generally regarded as unfit to this aim. Here, we show that, instead, properly constructed semilocal approximations, the Pauli-Gaussian (PG) KE functionals, especially at the Laplacian level of theory, can indeed achieve similar accuracy as nonlocal functionals and can be accurate for both metals and semiconductors, without the need for system-dependent parameters.

Nonlocal kinetic energy functionals by functional integration

Since the seminal studies of Thomas and Fermi, researchers in the Density-Functional Theory (DFT) community are searching for accurate electron density functionals. Arguably, the toughest functional to approximate is the noninteracting kinetic energy, T_s[ρ], the subject of this work. The typical paradigm is to first approximate the energy functional and then take its functional derivative, δT_s[ρ]δρ(r), yielding a potential that can be used in orbital-free DFT or subsystem DFT simulations. Here, this paradigm is challenged by constructing the potential from the second-functional derivative via functional integration. A new nonlocal functional for T_s[ρ] is prescribed [which we dub Mi-Genova-Pavanello (MGP)] having a density independent kernel. MGP is constructed to satisfy three exact conditions: (1) a nonzero "Kinetic electron" arising from a nonzero exchange hole; (2) the second functional derivative must reduce to the inverse Lindhard function in the limit of homogenous densities; (3) the potential is derived from functional integration of the second functional derivative. Pilot calculations show that MGP is capable of reproducing accurate equilibrium volumes, bulk moduli, total energy, and electron densities for metallic (body-centered cubic, face-centered cubic) and semiconducting (crystal diamond) phases of silicon as well as of III-V semiconductors. The MGP functional is found to be numerically stable typically reaching self-consistency within 12 iterations of a truncated Newton minimization algorithm. MGP's computational cost and memory requirements are low and comparable to the Wang-Teter nonlocal functional or any generalized gradient approximation functional.

Kinetic Energy of Hydrocarbons as a Function of Electron Density and Convolutional Neural Networks

We demonstrate a convolutional neural network trained to reproduce the Kohn-Sham kinetic energy of hydrocarbons from an input electron density. The output of the network is used as a nonlocal correction to conventional local and semilocal kinetic functionals. We show that this approximation qualitatively reproduces Kohn-Sham potential energy surfaces when used with conventional exchange correlation functionals. The density which minimizes the total energy given by the functional is examined in detail. We identify several avenues to improve on this exploratory work, by reducing numerical noise and changing the structure of our functional. Finally we examine the features in the density learned by the neural network to anticipate the prospects of generalizing these models.

Local conditions for the Pauli potential in order to yield self-consistent electron densities exhibiting proper atomic shell structure

The local conditions for the Pauli potential that are necessary in order to yield self-consistent electron densities from orbital-free calculations are investigated for approximations that are expressed with the help of a local position variable. It is shown that those local conditions also apply when the Pauli potential is given in terms of the electron density. An explicit formula for the Ne atom is given, preserving the local conditions during the iterative procedure. The resulting orbital-free electron density exhibits proper shell structure behavior and is in close agreement with the Kohn-Sham electron density. This study demonstrates that it is possible to obtain self-consistent orbital-free electron densities with proper atomic shell structure from simple one-point approximations for the Pauli potential at local density level.

A real-space stochastic density matrix approach for density functional electronic structure

A novel stochastic approach aimed at solving for the ground-state one-particle density matrix in density functional theory is developed.