Orbital-free Density Functional Theory Research Articles

High-Quality Local Pseudopotentials for Metals.

A major obstacle hindering the application of orbital-free density functional theory (OF-DFT) to all metals is the lack of accurate local pseudopotentials (LPSs), especially for transition metals. In this work, we developed high-quality LPSs for all simple and transition metals by fitting the atomic eigenvalues and orbital norms beyond the cutoff radii. Due to the lack of nonlocality in LPSs, it is very challenging to simultaneously fit the semicore and outermost valence orbitals of transition metals. We overcame this issue by excluding the semicore orbitals from the LPS optimizations. This allows us to achieve excellent fittings of the outermost valence orbitals, which are responsible for chemical bonding. The norm-conserving condition is then satisfied, leading to high-quality LPSs. To construct LPSs for magnetic systems, we introduce an additional metric: the atomic spin-polarization energy. By including this metric in the fitting, the LPSs reasonably reproduced many properties of magnetic metals and alloys. The high-quality LPSs developed in this work bring us one step closer to large-scale, reliable OF-DFT simulations of all metals and their alloys.

Machine learning nuclear orbital-free density functional based on Thomas–Fermi approach

Orbital-free density functional theory (DFT) is much more efficient than the orbital-dependent Kohn–Sham DFT due to the avoidance of the auxiliary one-body orbitals. The machine learning approach has been applied to build nuclear orbital-free DFT recently [Wu et al., Phys. Rev. C 105 (2022) L031303] and achieved more precise descriptions for nuclei than existing orbital-free DFTs. Here, improved machine learning nuclear orbital-free density functional is built by including the Thomas–Fermi approach as a basement. Performances of the functional are compared in detail with the ones based on the pure machine learning approach. It is found that with the Thomas–Fermi functional included, the machine-learning-based functional can achieve better performance in directly predicting the kinetic energies and in providing the ground-state properties by the self-consistent procedures.

Overcoming the barrier of orbital-free density functional theory for molecular systems using deep learning.

Orbital-free density functional theory (OFDFT) is a quantum chemistry formulation that has a lower cost scaling than the prevailing Kohn-Sham DFT, which is increasingly desired for contemporary molecular research. However, its accuracy is limited by the kinetic energy density functional, which is notoriously hard to approximate for non-periodic molecular systems. Here we propose M-OFDFT, an OFDFT approach capable of solving molecular systems using a deep learning functional model. We build the essential non-locality into the model, which is made affordable by the concise density representation as expansion coefficients under an atomic basis. With techniques to address unconventional learning challenges therein, M-OFDFT achieves a comparable accuracy to Kohn-Sham DFT on a wide range of molecules untouched by OFDFT before. More attractively, M-OFDFT extrapolates well to molecules much larger than those seen in training, which unleashes the appealing scaling of OFDFT for studying large molecules including proteins, representing an advancement of the accuracy-efficiency trade-off frontier in quantum chemistry.

Density functional theory of material design: fundamentals and applications—II

Abstract This is the second and the final part of the review on density functional theory (DFT), referred to as DFT-II. In the first review, DFT-I, we have discussed wavefunction-based methods, their complexity, and basics of density functional theory. In DFT-II, we focus on fundamentals of DFT and their implications for the betterment of the theory. We start our presentation with the exact DFT results followed by the concept of exchange-correlation (xc) or Fermi-Coulomb hole and its relationship with xc energy functional. We also provide the exact conditions for the xc-hole, xc-energy and xc-potential along with their physical interpretation. Next, we describe the extension of DFT for non-integer number of electrons, the piecewise linearity of total energy and discontinuity of chemical potential at integer particle numbers, and derivative discontinuity of the xc potential, which has consequences on fundamental gap of solids. After that, we present how one obtains more accurate xc energy functionals by going beyond the LDA. We discuss the gradient expansion approximation (GEA), generalized gradient approximation (GGA), and hybrid functional approaches to designing better xc energy functionals that give accurate total energies. However, these functionals fail to predict properties like the ionization potential and the band gap. Thus, we next describe different methods of modelling these potentials and results of their application for calculation of the band gaps of different solids to highlight accuracy of different xc potentials. Finally, we conclude with a glimpse on orbital-free density functional theory and the machine learning approach.

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.

Imposing correct jellium response is key to predict the density response by orbital-free DFT

Imposing correct jellium response is key to predict the density response by orbital-free DFT

A Density Potential Functional Approach for Mesoscopic Metal-Solution Interfaces

Mesoscopic metal-solution interfaces (MSI) are found at nanostructured electrodes in myriad electrochemical devices. Resolving the local reaction environment for electrochemical reactions occurring in mesoscopic MSIs requires a constant-potential, computationally economic method that properly treat both quantum mechanical metal electrons and almost classical electrolyte components. We have been developing a density potential functional theory (DPFT) to meet these harsh requirements [1-4]. Our DPFT method combines an orbital-free density functional theory of quantum mechanical metal electrons, a statistical field theoretic description of classical electrolyte ions and solvent, and Morse potentials describing short-range interactions between the two parts that are parameterized using first-principles based computations. In this talk, I will introduce the theoretical framework, its relationship to other similar methods including the jellium model and implicit solvation model, its parameterization and benchmark with experimental and ab initio simulation data, and recent applications to understand solvent effects on potential of zero charge, mechanoelectrochemistry of MSIs, and supported nanoparticle catalysts. The emerging opportunities, future development needs, and important caveats of the DPFT method will be discussed. Reference Huang J, Zhang Y, Li M, Groß A, Sakong S. Comparing Ab Initio Molecular Dynamics and a Semiclassical Grand Canonical Scheme for the Electric Double Layer of the Pt (111)/Water Interface. The Journal of Physical Chemistry Letters. 2023 Feb 27;14:2354-63.Huang J. Density-Potential Functional Theory of Electrochemical Double Layers: Calibration on the Ag (111)-KPF6 System and Parametric Analysis. Journal of Chemical Theory and Computation. 2023 Jan 18.Huang J, Chen S, Eikerling M. Grand-canonical model of electrochemical double layers from a hybrid density–potential functional. Journal of chemical theory and computation. 2021 Mar 31;17(4):2417-30.Huang J. Hybrid density-potential functional theory of electric double layers. Electrochimica Acta. 2021 Sep 1;389:138720. J.H. is supported by the Helmholtz Young Investigators Group Leader Program.

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 for metal slabs.

Orbital-Free Density-Functional Theory (OF-DFT) is known to represent a promising alternative to the standard Kohn-Sham (KS) DFT, as it relies on the electron density alone, without the need to calculate all KS single-particle orbitals and energies. Here, we investigate the behavior of the main ingredients of this theory, which are the non-interacting kinetic-energy density (KED) and the Pauli potential, for metal slabs. We derive explicit density functionals for these quantities in the quantum limit where all electrons are in the same slab discrete level of energy, and we present numerical calculations beyond this quantum limit for slabs of various widths. We have found the first explicit KED functional for a realistic many-particle fermionic system, which we prove to be generally valid with no assumption about the KS potential. We also discuss the total non-interacting kinetic energy and the corresponding enhancement factor, which represent basic quantities for the practical implementation of OF-DFT.

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.

Single-particle-exact density functional theory

We introduce ‘single-particle-exact density functional theory’ (1pEx-DFT), a novel density functional approach that represents all single-particle contributions to the energy with exact functionals. Here, we parameterize interaction energy functionals by utilizing two new schemes for constructing density matrices from ‘participation numbers’ of the single-particle states of quantum many-body systems. These participation numbers play the role of the variational variables akin to the particle densities in standard orbital-free density functional theory. We minimize the total energies with the help of evolutionary algorithms and obtain ground-state energies that are typically accurate at the one-percent level for our proof-of-principle simulations that comprise interacting Fermi gases as well as the electronic structure of atoms and ions, with and without relativistic corrections. We thereby illustrate the ingredients and practical features of 1pEx-DFT and reveal its potential of becoming an accurate, scalable, and transferable technology for simulating mesoscopic quantum many-body systems.

KineticNet: Deep learning a transferable kinetic energy functional for orbital-free density functional theory.

Orbital-free density functional theory (OF-DFT) holds promise to compute ground state molecular properties at minimal cost. However, it has been held back by our inability to compute the kinetic energy as a functional of electron density alone. Here, we set out to learn the kinetic energy functional from ground truth provided by the more expensive Kohn-Sham density functional theory. Such learning is confronted with two key challenges: Giving the model sufficient expressivity and spatial context while limiting the memory footprint to afford computations on a GPU and creating a sufficiently broad distribution of training data to enable iterative density optimization even when starting from a poor initial guess. In response, we introduce KineticNet, an equivariant deep neural network architecture based on point convolutions adapted to the prediction of quantities on molecular quadrature grids. Important contributions include convolution filters with sufficient spatial resolution in the vicinity of nuclear cusp, an atom-centric sparse but expressive architecture that relays information across multiple bond lengths, and a new strategy to generate varied training data by finding ground state densities in the face of perturbations by a random external potential. KineticNet achieves, for the first time, chemical accuracy of the learned functionals across input densities and geometries of tiny molecules. For two-electron systems, we additionally demonstrate OF-DFT density optimization with chemical accuracy.

Neural network learned Pauli potential for the advancement of orbital-free density functional theory.

The Pauli kinetic energy functional and its functional derivative, termed Pauli potential, play a crucial role in the successful implementation of orbital-free density functional theory for electronic structure calculations. However, the exact forms of these two quantities are not known. Therefore, perforce, one employs the approximate forms for the Pauli functional or Pauli potential for performing orbital-free density functional calculations. In the present study, we developed a feed-forward neural network-based representation for the Pauli potential using a 1-dimensional (1-D) model system. We expanded density in terms of basis functions, and the coefficients of the expansion were used as input to a feed-forward neural network. Using the neural network-based representation of the Pauli potential, we calculated the ground-state densities of the 1-D model system by solving the Euler equation. We calculated the Pauli kinetic energy using the neural network-based Pauli potential employing the exact relation between the Pauli kinetic energy functional and the potential. The sum of the neural network-based Pauli kinetic energy and the von Weizsäcker kinetic energy resulted in an accurate estimation of the total kinetic energy. The approach presented in this paper can be employed for the calculation of Pauli potential and Pauli kinetic energy, obviating the need for a functional derivative. The present study is an important step in the advancement of application of machine learning-based techniques toward the orbital-free density functional theory-based methods.

An efficient zero-order evolutionary method for solving the orbital-free density functional theory problem by direct minimization.

A differential evolution (DE) global optimization method for all-electron orbital-free density functional theory (OF-DFT) is presented. This optimization method does not need information about function derivatives to find extreme solutions. Results for a series of known orbital-free energy functionals are presented. Ground state energies of atoms (H to Ar) are obtained by direct minimization of the energy functional without using either Lagrange multipliers or damping procedures for reaching convergence. Our results are in agreement with previous OF-DFT calculations obtained using the standard Newton-Raphson and trust region methods. Being a zero-order method, the DE method can be applied to optimization problems dealing with non-differentiable functionals or functionals with non-closed forms.

Orbital-free density functional theory: Differences and similarities between electronic and nuclear systems

Abstract Orbital-free density functional theory (OF-DFT) has been used when studying atoms, molecules, and solids. In nuclear physics, applications of OF-DFT have been quite scarce so far, as DFT has been widely applied to the study of many nuclear properties mostly within the Kohn–Sham (KS) scheme. There are many realizations of nuclear KS-DFT, but computations become very demanding for heavy systems, such as superheavy nuclei and the inner crust of neutron stars, and it is hard to describe exotic nuclear shapes using a finite basis made with a limited number of orbitals. These bottlenecks could, in principle, be overcome by an orbital-free formulation of DFT. This work is a first step towards the realistic application of OF-DFT to nuclei. In particular, we have implemented possible choices for an orbital-free kinetic energy and solved the associated Schrödinger equation either with simple potentials or with simplified nuclear density functionals. While the former choice sheds light on the differences between electronic and nuclear systems, the latter choice allows us to discuss the practical applications to nuclei as well as open questions.

Dragon: A multi-GPU orbital-free density functional theory molecular dynamics simulation package for modeling of warm dense matter

As progress in electronic structure theoretical methods is made, ab initio molecular dynamics (MD) based on orbital-free density functional theory (OF-DFT) is becoming increasingly more successful at substituting the traditional, very accurate but computationally costly Kohn-Sham (KS) approach for simulations of matter at the challenging warm dense matter (WDM) regime. However, despite the significant cost alleviation of eliminating the dependence on the KS orbitals, OF-DFT MD runs require ∼102 to 103 CPU cores running for days, or even weeks, for simulations of systems comprised of 102 to 103 atoms, depending on thermodynamic conditions. We present Dragon, a multi-GPU OF-DFT MD code for fast and efficient simulations of WDM. With a relatively small allocation of resources (4 to 8 GPU devices) it can provide an order of magnitude speedup for simulations containing O(104) atoms and target systems composed of O(105) atoms at conditions within the WDM regime, which is currently outside the capabilities of CPU codes.

Vortex-lattice structures in rotating Bose-Fermi superfluid mixtures

The system of Bose-Fermi superfluid mixture offers a playground to explore rich macroscopic quantum phenomena. In a recent experiment of Yao et al. (2016) [35], 41K-6Li superfluid mixture is implemented. Coupled quantized vortices are generated via rotating the superfluid mixture, and a few unconventional behaviors on the formations of vortex numbers are observed, which can be traced to boson-fermion interactions. Here we provide a theoretical insight into the unconventional behaviors observed in the experiment. To this end, the orbital-free density functional theory is hired, and its utility is validated by making comparison of the numerical results and a microscopic theory for vortex lattices in strongly interacting Fermi superfluids alone. We also predict interesting phenomena which can be readily explored experimentally, including the novel structures of bosonic-fermionic vortices, and attractive interactions between the bosonic and fermionic vortices.

A modified cusp condition for the single density equations of DFT and orbital-free DFT for atoms

A modified cusp condition for the single density equations of DFT and orbital-free DFT for atoms

Assessing the source of error in the Thomas-Fermi-von Weizsäcker density functional.

We investigate the source of error in the Thomas-Fermi-von Weizsäcker (TFW) density functional relative to Kohn-Sham density functional theory (DFT). In particular, through numerical studies on a range of materials, for a variety of crystal structures subject to strain and atomic displacements, we find that while the ground state electron density in TFW orbital-free DFT is close to the Kohn-Sham density, the corresponding energy deviates significantly from the Kohn-Sham value. We show that these differences are a consequence of the poor representation of the linear response within the TFW approximation for the electronic kinetic energy, confirming conjectures in the literature. In so doing, we find that the energy computed from a non-self-consistent Kohn-Sham calculation using the TFW electronic ground state density is in very good agreement with that obtained from the fully self-consistent Kohn-Sham solution.

Efficient single-grid and multi-grid solvers for real-space orbital-free density functional theory

To improve the computational efficiencies of the real-space orbital-free density functional theory, this work develops a new single-grid solver by directly providing the closed-form solution to the inner iteration and using an improved bisection method to accelerate the line search process in the outer iteration, and extended the single-grid solver to a multi-grid solver. Numerical examples show that while maintaining high level of accuracy, the proposed single-grid solver can improve the computational efficiencies by two orders of magnitude comparing with the methods in the literature (For example, the ground state energy of a 45×45×45 FCC Aluminum cluster (376786 atoms) can be computed within 4h27min on a desktop computer.) and the multi-grid solver can improve the computational efficiencies even once for the cases where high-resolution electron densities are needed.