Self-Consistent Convolutional Density Functional Approximations: Application to Adsorption at Metal Surfaces.

The exchange-correlation (XC) functional in density functional theory is used to approximate multi-electron interactions. A plethora of different functionals are available, but nearly all are based on the hierarchy of inputs commonly referred to as "Jacob's ladder." This paper introduces an approach to construct XC functionals with inputs from convolutions of arbitrary kernels with the electron density, providing a route to move beyond Jacob's ladder. We derive the variational derivative of these functionals, showing consistency with the generalized gradient approximation (GGA), and provide equations for variational derivatives based on multipole features from convolutional kernels. A proof-of-concept functional, PBEq, which generalizes the PBE α ${\alpha }$ framework with α ${\alpha }$ being a spatially-resolved function of the monopole of the electron density, is presented and implemented. It allows a single functional to use different GGAs at different spatial points in a system, while obeying PBE constraints. Analysis of the results underlines the importance of error cancellation and the XC potential in data-driven functional design. After testing on small molecules, bulk metals, and surface catalysts, the results indicate that this approach is a promising route to simultaneously optimize multiple properties of interest.

Strain engineering of Zeeman and Rashba effects in transition metal dichalcogenide nanotubes and their Janus variants: anab initiostudy.

We study the influence of mechanical deformations on the Zeeman and Rashba effects in transition metal dichalcogenide nanotubes and their Janus variants from first principles. In particular, we perform symmetry-adapted density functional theory simulations with spin-orbit coupling to determine the variation in the electronic band structure splittings with axial and torsional deformations. We find significant effects in molybdenum and tungsten nanotubes, for which the Zeeman splitting decreases with increase in strain, going to zero for large enough tensile/shear strains, while the Rashba splitting coefficient increases linearly with shear strain, while being zero for all tensile strains, a consequence of the inversion symmetry remaining unbroken. In addition, the Zeeman splitting is relatively unaffected by nanotube diameter, whereas the Rashba coefficient decreases with increase in diameter. Overall, mechanical deformations represent a powerful tool for spintronics in nanotubes.

Efficient real space formalism for hybrid density functionals.

We present an efficient real space formalism for hybrid exchange-correlation functionals in generalized Kohn-Sham density functional theory (DFT). In particular, we develop an efficient representation for any function of the real space finite-difference Laplacian matrix by leveraging its Kronecker product structure, thereby enabling the time to solution of associated linear systems to be highly competitive with the fast Fourier transform scheme while not imposing any restrictions on the boundary conditions. We implement this formalism for both the unscreened and range-separated variants of hybrid functionals. We verify its accuracy and efficiency through comparisons with established planewave codes for isolated as well as bulk systems. In particular, we demonstrate up to an order-of-magnitude speedup in time to solution for the real space method. We also apply the framework to study the structure of liquid water using ab initio molecular dynamics, where we find good agreement with the literature. Overall, the current formalism provides an avenue for efficient real-space DFT calculations with hybrid density functionals.

GPU acceleration of local and semilocal density functional calculations in the SPARC electronic structure code.

We present a Graphics Processing Unit (GPU)-accelerated version of the real-space SPARC electronic structure code for performing Kohn-Sham density functional theory calculations within the local density and generalized gradient approximations. In particular, we develop a modular math-kernel based implementation for NVIDIA architectures wherein the computationally expensive operations are carried out on the GPUs, with the remainder of the workload retained on the central processing units (CPUs). Using representative bulk and slab examples, we show that relative to CPU-only execution, GPUs enable speedups of up to 6× and 60× in node and core hours, respectively, bringing time to solution down to less than 30 s for a metallic system with over 14 000 electrons and enabling significant reductions in computational resources required for a given wall time.

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.

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.

On the bending of rectangular atomic monolayers along different directions: anab initiostudy.

We study the bending of rectangular atomic monolayers along different directions from first principles. Specifically, choosing the phosphorene, GeS, TiS3, and As2S3monolayers as representative examples, we perform Kohn-Sham density functional theory calculations to determine the variation in transverse flexoelectric coefficient and bending modulus with the direction of bending. We find that while the flexoelectric coefficient is nearly isotropic, there is significant and complex anisotropy in bending modulus that also differs between the monolayers, with extremal values not necessarily occurring along the principal directions. In particular, the commonly adopted orthotropic continuum plate model with uniform thickness fails to describe the observed variations in bending modulus for GeS, TiS3, and As2S3. We determine the direction-dependent effective thickness for use in such continuum models. We also show that the anisotropy in bending modulus is not associated with the rehybridization of atomic orbitals.

Real-space density kernel method for Kohn-Sham density functional theory calculations at high temperature.

Kohn-Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. We present a density matrix based method for Kohn-Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann-Feynman forces, and Hellmann-Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw-Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045 K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.

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.

Torsional moduli of transition metal dichalcogenide nanotubes from first principles.

We calculate the torsional moduli of single-walled transition metal dichalcogenide (TMD) nanotubes usingab initiodensity functional theory (DFT). Specifically, considering forty-five select TMD nanotubes, we perform symmetry-adapted DFT calculations to calculate the torsional moduli for the armchair and zigzag variants of these materials in the low-twist regime and at practically relevant diameters. We find that the torsional moduli follow the trend: MS2> MSe2> MTe2. In addition, the moduli display a power law dependence on diameter, with the scaling generally close to cubic, as predicted by the isotropic elastic continuum model. In particular, the shear moduli so computed are in good agreement with those predicted by the isotropic relation in terms of the Young's modulus and Poisson's ratio, both of which are also calculated using symmetry-adapted DFT. Finally, we develop a linear regression model for the torsional moduli of TMD nanotubes based on the nature/characteristics of the metal-chalcogen bond, and show that it is capable of making reasonably accurate predictions.

Torsional strain engineering of transition metal dichalcogenide nanotubes: anab initiostudy.

We study the effect of torsional deformations on the electronic properties of single-walled transition metal dichalcogenide (TMD) nanotubes. In particular, considering forty-five select armchair and zigzag TMD nanotubes, we perform symmetry-adapted Kohn-Sham density functional theory calculations to determine the variation in bandgap and effective mass of charge carriers with twist. We find that metallic nanotubes remain so even after deformation, whereas semiconducting nanotubes experience a decrease in bandgap with twist-originally direct bandgaps become indirect-resulting in semiconductor to metal transitions. In addition, the effective mass of holes and electrons continuously decrease and increase with twist, respectively, resulting in n-type to p-type semiconductor transitions. We find that this behavior is likely due to rehybridization of orbitals in the metal and chalcogen atoms, rather than charge transfer between them. Overall, torsional deformations represent a powerful avenue to engineer the electronic properties of semiconducting TMD nanotubes, with applications to devices like sensors and semiconductor switches.

Development of a Multiphase Beryllium Equation of State and Physics-based Variations.

We construct a family of beryllium (Be) multiphase equation of state (EOS) models that consists of a baseline ("optimal") EOS and variations on the baseline to account for physics-based uncertainties. The Be baseline EOS is constructed to reproduce a set of self-consistent data and theory including known phase boundaries, the principal Hugoniot, isobars, and isotherms from diamond-anvil cell experiments. Three phases are considered, including the known hexagonal closed-packed (hcp) phase, the liquid, and the theoretically predicted high-pressure body-centered cubic (bcc) phase. Since both the high-temperature liquid and high-pressure bcc phases lack any experimental data, we carry out ab initio density functional theory (DFT) calculations to obtain new information about the EOS properties for these two regions. At extremely high temperature conditions (>87 eV), DFT-based quantum molecular dynamics simulations are performed for multiple liquid densities using the state-of-the-art Spectral Quadrature methodology in order to validate our selected models for the ion- and electron-thermal free energies of the liquid. We have also performed DFT simulations of hcp and bcc with different exchange-correlation functionals to examine their impact on bcc compressibility, which bound the hcp-bcc transition pressure to within 4 ± 0.5 Mbar. Our baseline EOS predicts the first density maximum along the Hugoniot to be 4.4-fold in compression, while the hcp-bcc-liquid triple-point pressure is predicted to be at 2.25 Mbar. In addition to the baseline EOS, we have generated eight variations to accommodate multiple sources of potential uncertainties such as (1) the choice of free-energy models, (2) differences in theoretical treatments, (3) experimental uncertainties, and (4) lack of information. These variations are designed to provide a reasonable representation of nonstatistical uncertainties for the Be EOS and may be used to assess its sensitivity to different inertial-confinement fusion capsule designs.

Implementation of Perdew-Zunger self-interaction correction in real space using Fermi-Löwdin orbitals.

Most widely used density functional approximations suffer from self-interaction error, which can be corrected using the Perdew-Zunger (PZ) self-interaction correction (SIC). We implement the recently proposed size-extensive formulation of PZ-SIC using Fermi-Löwdin Orbitals (FLOs) in real space, which is amenable to systematic convergence and large-scale parallelization. We verify the new formulation within the generalized Slater scheme by computing atomization energies and ionization potentials of selected molecules and comparing to those obtained by existing FLOSIC implementations in Gaussian based codes. The results show good agreement between the two formulations, with new real-space results somewhat closer to experiment on average for the systems considered. We also obtain the ionization potentials and atomization energies by scaling down the Slater statistical average of SIC potentials. The results show that scaling down the average SIC potential improves both atomization energies and ionization potentials, bringing them closer to experiment. Finally, we verify the present formulation by calculating the barrier heights of chemical reactions in the BH6 dataset, where significant improvements are obtained relative to Gaussian based FLOSIC results.

Bending moduli for forty-four select atomic monolayers from first principles.

We calculate bending moduli along the principal directions for forty-four select atomic monolayers using ab initio density functional theory (DFT). Specifically, considering representative materials from each of Groups IV, III-V, V monolayers, Group IV monochalcogenides, transition metal trichalcogenides, transition metal dichalcogenides and Group III monochalcogenides, we utilize the recently developed Cyclic DFT method to calculate the bending moduli in the practically relevant but previously intractable low-curvature limit. We find that the moduli generally increase with thickness of the monolayer, while spanning three orders of magnitude between the different materials. In addition, structures with a rectangular lattice are prone to a higher degree of anisotropy relative to those with a honeycomb lattice. Exceptions to these trends are generally a consequence of unusually strong/weak bonding and/or significant structural relxation related effects.

Real-space formulation of the stress tensor for O(N) density functional theory: Application to high temperature calculations.

We present an accurate and efficient real-space formulation of the Hellmann-Feynman stress tensor for O(N) Kohn-Sham density functional theory (DFT). While applicable at any temperature, the formulation is most efficient at high temperature where the Fermi-Dirac distribution becomes smoother and the density matrix becomes correspondingly more localized. We first rewrite the orbital-dependent stress tensor for real-space DFT in terms of the density matrix, thereby making it amenable to O(N) methods. We then describe its evaluation within the O(N) infinite-cell Clenshaw-Curtis Spectral Quadrature (SQ) method, a technique that is applicable to metallic and insulating systems, is highly parallelizable, becomes increasingly efficient with increasing temperature, and provides results corresponding to the infinite crystal without the need of Brillouin zone integration. We demonstrate systematic convergence of the resulting formulation with respect to SQ parameters to exact diagonalization results and show convergence with respect to mesh size to the established plane wave results. We employ the new formulation to compute the viscosity of hydrogen at 106 K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.

On the calculation of the stress tensor in real-space Kohn-Sham density functional theory.

We present an accurate and efficient formulation of the stress tensor for real-space Kohn-Sham density functional theory calculations. Specifically, while employing a local formulation of the electrostatics, we derive a linear-scaling expression for the stress tensor that is applicable to simulations with unit cells of arbitrary symmetry, semilocal exchange-correlation functionals, and Brillouin zone integration. In particular, we rewrite the contributions arising from the self-energy and the nonlocal pseudopotential energy to make them amenable to the real-space finite-difference discretization, achieving up to three orders of magnitude improvement in the accuracy of the computed stresses. Using examples representative of static and dynamic calculations, we verify the accuracy and efficiency of the proposed formulation. In particular, we demonstrate high rates of convergence with spatial discretization, consistency between the computed energy and the stress tensor, and very good agreement with reference planewave results.

Discrete discontinuous basis projection method for large-scale electronic structure calculations.

We present an approach to accelerate real-space electronic structure methods several fold, without loss of accuracy, by reducing the dimension of the discrete eigenproblem that must be solved. To accomplish this, we construct an efficient, systematically improvable, discontinuous basis spanning the occupied subspace and project the real-space Hamiltonian onto the span. In calculations on a range of systems, we find that accurate energies and forces are obtained with 8-25 basis functions per atom, reducing the dimension of the associated real-space eigenproblems by 1-3 orders of magnitude.

Overcoming the Chemical Complexity Bottleneck in on-the-Fly Machine Learned Molecular Dynamics Simulations.

We develop a framework for on-the-fly machine learned force field molecular dynamics simulations based on the multipole featurization scheme that overcomes the bottleneck with the number of chemical elements. Considering bulk systems with up to 6 elements, we demonstrate that the number of density functional theory calls remains approximately independent of the number of chemical elements, in contrast to the increase in the smooth overlap of the atomic positions scheme.

Calculation of phonons in real-space density functional theory.

We present an accurate and efficient formulation for the calculation of phonons in real-space Kohn-Sham density functional theory. Specifically, employing a local exchange-correlation functional, norm-conserving pseudopotential in the Kleinman-Bylander representation, and local form for the electrostatics, we derive expressions for the dynamical matrix and associated Sternheimer equation that are particularly amenable to the real-space finite-difference method, within the framework of density functional perturbation theory. In particular, the formulation is applicable to insulating as well as metallic systems of any dimensionality, enabling the efficient and accurate treatment of semi-infinite and bulk systems alike, for both orthogonal and nonorthogonal cells. We also develop an implementation of the proposed formulation within the high-order finite-difference method. Through representative examples, we verify the accuracy of the computed phonon dispersion curves and density of states, demonstrating excellent agreement with established plane-wave results.

Properties of carbon up to 10 million kelvin from Kohn-Sham density functional theory molecular dynamics.

Accurately modeling dense plasmas over wide-ranging conditions of pressure and temperature is a grand challenge critically important to our understanding of stellar and planetary physics as well as inertial confinement fusion. In this work, we employ Kohn-Sham density functional theory (DFT) molecular dynamics (MD) to compute the properties of carbon at warm and hot dense matter conditions in the vicinity of the principal Hugoniot. In particular, we calculate the equation of state (EOS), Hugoniot, pair distribution functions, and diffusion coefficients for carbon at densities spanning 8 g/cm^{3} to 16 g/cm^{3} and temperatures ranging from 100 kK to 10 MK using the Spectral Quadrature method. We find that the computed EOS and Hugoniot are in good agreement with path integral Monte Carlo results and the sesame database. Additionally, we calculate the ion-ion structure factor and viscosity for selected points. All results presented are at the level of full Kohn-Sham DFT-MD, free of empirical parameters, average-atom, and orbital-free approximations employed previously at such conditions.