Artificial neural networks for the kinetic energy functional of non-interacting fermions

Variational principle to regularize machine-learned density functionals: The non-interacting kinetic-energy functional

Practical density functional theory (DFT) owes its success to the groundbreaking work of Kohn and Sham that introduced the exact calculation of the non-interacting kinetic energy of the electrons using an auxiliary mean-field system. However, the full power of DFT will not be unleashed until the exact relationship between the electron density and the non-interacting kinetic energy is found. Various attempts have been made to approximate this functional, similar to the exchange-correlation functional, with much less success due to the larger contribution of kinetic energy and its more non-local nature. In this work, we propose a new and efficient regularization method to train density functionals based on deep neural networks, with particular interest in the kinetic-energy functional. The method is tested on (effectively) one-dimensional systems, including the hydrogen chain, non-interacting electrons, and atoms of the first two periods, with excellent results. For atomic systems, the generalizability of the regularization method is demonstrated by training also an exchange-correlation functional, and the contrasting nature of the two functionals is discussed from a machine-learning perspective.

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.

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.

Accurate parameterization of the kinetic energy functional for calculations using exact-exchange

Electronic structure calculations based on Kohn-Sham density functional theory (KSDFT) that incorporate exact-exchange or hybrid functionals are associated with a large computational expense, a consequence of the inherent cubic scaling bottleneck and large associated prefactor, which limits the length and time scales that can be accessed. Although orbital-free density functional theory (OFDFT) calculations scale linearly with system size and are associated with a significantly smaller prefactor, they are limited by the absence of accurate density-dependent kinetic energy functionals. Therefore, the development of accurate density-dependent kinetic energy functionals is important for OFDFT calculations of large realistic systems. To this end, we propose a method to train kinetic energy functional models at the exact-exchange level of theory by using a dictionary of physically relevant terms that have been proposed in the literature in conjunction with linear or nonlinear regression methods to obtain the fitting coefficients. For our dictionary, we use a gradient expansion of the kinetic energy nonlocal models proposed in the literature and their nonlinear combinations, such as a model that incorporates spatial correlations between higher order derivatives of electron density at two points. The predictive capabilities of these models are assessed by using a variety of model one-dimensional (1D) systems that exhibit diverse bonding characteristics, such as a chain of eight hydrogens, LiF, LiH, C₄H₂, C₄N₂, and C₃O₂. We show that by using the data from model 1D KSDFT calculations performed using the exact-exchange functional for only a few neutral structures, it is possible to generate models with high accuracy for charged systems and electron and kinetic energy densities during self-consistent field iterations. In addition, we show that it is possible to learn both the orbital dependent terms, i.e., the kinetic energy and the exact-exchange energy, and models that incorporate additional nonlinearities in spatial correlations, such as a quadratic model, are needed to capture subtle features of the kinetic energy density that are present in exact-exchange-based KSDFT calculations.

Accurate parameterization of the kinetic energy functional

The absence of a reliable formulation of the kinetic energy density functional has hindered the development of orbital free density functional theory. Using the data-aided learning paradigm, we propose a simple prescription to accurately model the kinetic energy density of any system. Our method relies on a dictionary of functional forms for local and nonlocal contributions, which have been proposed in the literature, and the appropriate coefficients are calculated via a linear regression framework. To model the nonlocal contributions, we explore two new nonlocal functionals-a functional that captures fluctuations in electronic density and a functional that incorporates gradient information. Since the analytical functional forms of the kernels present in these nonlocal terms are not known from theory, we propose a basis function expansion to model these seemingly difficult nonlocal quantities. This allows us to easily reconstruct kernels for any system using only a few structures. The proposed method is able to learn kinetic energy densities and total kinetic energies of molecular and periodic systems, such as H₂, LiH, LiF, and a one-dimensional chain of eight hydrogens using data from Kohn-Sham density functional theory calculations for only a few structures.

Toward Orbital-Free Density Functional Theory with Small Data Sets and Deep Learning

We use voxel deep neural networks to predict energy densities and functional derivatives of electron kinetic energies for the Thomas-Fermi model and Kohn-Sham density functional theory calculations. We show that the ground-state electron density can be found via direct minimization for a graphene lattice without any projection scheme using a voxel deep neural network trained with the Thomas-Fermi model. Additionally, we predict the kinetic energy of a graphene lattice within chemical accuracy after training from only two Kohn-Sham density functional theory (DFT) calculations. We identify an important sampling issue inherent in Kohn-Sham DFT calculations and propose future work to rectify this problem. Furthermore, we demonstrate an alternative, functional derivative-free, Monte Carlo based orbital-free density functional theory algorithm to calculate an accurate two-electron density in a double inverted Gaussian potential with a machine-learned kinetic energy functional.