A novel splitting strategy to accelerate solving generalized eigenvalue problem from Kohn–Sham density functional theory

We explore a new formalism to study the nonlinear electronic density response based on Kohn–Sham density functional theory (KS-DFT) at partially and strongly quantum degenerate regimes. It is demonstrated that the KS-DFT calculations are able to accurately reproduce the available path integral Monte Carlo simulation results at temperatures relevant for warm dense matter research. The existing analytical results for the quadratic and cubic response functions are rigorously tested. It is demonstrated that the analytical results for the quadratic response function closely agree with the KS-DFT data. Furthermore, the performed analysis reveals that currently available analytical formulas for the cubic response function are not able to describe simulation results, neither qualitatively nor quantitatively, at small wavenumbers q < 2qF, with qF being the Fermi wavenumber. The results show that KS-DFT can be used to describe warm dense matter that is strongly perturbed by an external field with remarkable accuracy. Furthermore, it is demonstrated that KS-DFT constitutes a valuable tool to guide the development of the nonlinear response theory of correlated quantum electrons from ambient to extreme conditions. This opens up new avenues to study nonlinear effects in a gamut of different contexts at conditions that cannot be accessed with previously used path integral Monte Carlo methods.

Approximate functionals in Kohn-Sham density functional theory (KS-DFT) and reduced density matrix functional theory (RDMFT) have advantages in dealing with dynamic correlation and strong correlation, respectively; their combination can benefit from complementarity while suffering from the problem of correlation double-counting. Herein, a short-range corrected reduced density matrix (1-RDM) functional is developed to take advantage of the functionals in KS-DFT and RDMFT without double-counting. The resulting functional, denoted as ωP22, outperforms other 1-RDM functionals for the tests of thermochemistry, nonbonded interactions, and bond dissociation energy. In particular, ωP22 shows much less systematic error for systems involving fractional spins, and it can properly predict the energies at both equilibrium and dissociated distances for different single and multiple bonds, which cannot be achieved by commonly used KS-DFT and RDMFT functionals. Therefore, ωP22 is demonstrated effective in balance handling dynamic and strong correlation, and the advances in this work would create new possibilities for the development and application of approximate functionals.

Since its formal inception in 1964-1965, Kohn-Sham density-functional theory (KS-DFT) has become the most popular electronic structure method in computational physics and chemistry. Its popularity stems from its beautifully simple conceptual framework and computational elegance. The rise of KS-DFT in chemical physics began in earnest in the mid 1980s, when crucial developments in its exchange-correlation term gave the theory predictive power competitive with well-developed wave-function methods. Today KS-DFT finds itself under increasing pressure to deliver higher and higher accuracy and to adapt to ever more challenging problems. If we are not mindful, however, these pressures may submerge the theory in the wave-function sea. KS-DFT might be lost. I am hopeful the Kohn-Sham philosophical, theoretical, and computational framework can be preserved. This Perspective outlines the history, basic concepts, and present status of KS-DFT in chemical physics, and offers suggestions for its future development.

Long‐range correction for exchange functionals in Kohn–Sham density functional theory and its applicability are reviewed. Long‐range correction simply supplements the long‐range exchange effect in exchange functionals by replacing the Hartree–Fock exchange integral with the long‐range part of exchange functionals. Nevertheless, this correction has solved many problems in Kohn–Sham calculations. Using this correction, valence occupied and unoccupied orbital energies are quantitatively reproduced in a comprehensive manner for the first time. Long‐range correction has also solved the underestimations of charge transfer excitation energies and oscillator strengths in time‐dependent Kohn–Sham calculations and has clearly improved poor optical response properties such as hyperpolarizability in coupled‐perturbed Kohn–Sham and finite‐field calculations. Moreover, this correction has drastically improved the reproducibility of van der Waals bonds by simply combining with conventional van der Waals calculation methods. We, therefore, believe that the long‐range correction clearly extends the applicability of the Kohn–Sham method in future quantum chemistry calculations.This article is categorized under: Electronic Structure Theory &gt; Density Functional Theory

SQDFT: Spectral Quadrature method for large-scale parallel [formula omitted] Kohn–Sham calculations at high temperature

Deep neural networks (DNNs) have been used to successfully predict molecular properties calculated based on the Kohn-Sham density functional theory (KS-DFT). Although this prediction is fast and accurate, we believe that a DNN model for KS-DFT must not only predict the properties but also provide the electron density of a molecule. This Letter presents the quantum deep field (QDF), which provides the electron density with an unsupervised but end-to-end physics-informed modeling by learning the atomization energy on a large-scale dataset. QDF performed well at atomization energy prediction, generated valid electron density, and demonstrated extrapolation.

The accuracy of non-resonant and resonant (resonant inelastic X-ray scattering) X-ray emission spectra simulated based upon Kohn-Sham density functional theory is assessed. Accurate non-resonant X-ray emission spectra with the correct energy scale are obtained when short-range corrected exchange-correlation functionals designed for the calculation of X-ray absorption spectroscopy are used. It is shown that this approach can be extended to simulate resonant inelastic X-ray scattering by using a reference determinant that describes a core-excited state. For this spectroscopy, it is found that a standard hybrid functional, B3LYP, gives accurate spectra that reproduce the features observed in experiment. However, the ability to correctly describe subtle changes in the spectra arising from different intermediate states is more challenging and requires averaging over conformations from a molecular dynamics simulation. Overall, it is demonstrated that accurate non-resonant and resonant X-ray emission spectra can be simulated directly from Kohn-Sham density functional theory.

In this paper, we propose a new numerical method to find the ground state energy of a given physical system within the Kohn–Sham density functional theory. The h‐adaptive finite element method is adopted for spatial discretization and implemented with matrix‐free operator evaluation. The ground state energy is found by performing unconstrained minimization with non‐orthogonal orbitals using the limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. A geometric multigrid preconditioner is applied to improve the convergence. The clear advantage of the proposed approach is demonstrated on selected examples by comparing the performance to other methods such as preconditioned steepest descent minimization. The proposed method provides a solid framework toward (N) complexity for the locally adaptive real‐space solution of density functional theory with finite elements.

Strong electron correlation plays an important role in transition-metal and heavy-metal chemistry, magnetic molecules, bond breaking, biradicals, excited states, and many functional materials, but it provides a significant challenge for modern electronic structure theory. The treatment of strongly correlated systems usually requires a multireference method to adequately describe spin densities and near-degeneracy correlation. However, quantitative computation of dynamic correlation with multireference wave functions is often difficult or impractical. Multiconfiguration pair-density functional theory (MC-PDFT) provides a way to blend multiconfiguration wave function theory and density functional theory to quantitatively treat both near-degeneracy correlation and dynamic correlation in strongly correlated systems; it is more affordable than multireference perturbation theory, multireference configuration interaction, or multireference coupled cluster theory and more accurate for many properties than Kohn–Sham density functional theory. This perspective article provides a brief introduction to strongly correlated systems and previously reviewed progress on MC-PDFT followed by a discussion of several recent developments and applications of MC-PDFT and related methods, including localized-active-space MC-PDFT, generalized active-space MC-PDFT, density-matrix-renormalization-group MC-PDFT, hybrid MC-PDFT, multistate MC-PDFT, spin–orbit coupling, analytic gradients, and dipole moments. We also review the more recently introduced multiconfiguration nonclassical-energy functional theory (MC-NEFT), which is like MC-PDFT but allows for other ingredients in the nonclassical-energy functional. We discuss two new kinds of MC-NEFT methods, namely multiconfiguration density coherence functional theory and machine-learned functionals.

Kohn–Sham density functional theory (DFT) is the basis of modern computational approaches to electronic structures. Their accuracy heavily relies on the exchange-correlation energy functional, which encapsulates electron–electron interaction beyond the classical model. As its universal form remains undiscovered, approximated functionals constructed with heuristic approaches are used for practical studies. However, there are problems in their accuracy and transferability, while any systematic approach to improve them is yet obscure. In this study, we demonstrate that the functional can be systematically constructed using accurate density distributions and energies in reference molecules via machine learning. Surprisingly, a trial functional machine learned from only a few molecules is already applicable to hundreds of molecules comprising various first- and second-row elements with the same accuracy as the standard functionals. This is achieved by relating density and energy using a flexible feed-forward neural network, which allows us to take a functional derivative via the back-propagation algorithm. In addition, simply by introducing a nonlocal density descriptor, the nonlocal effect is included to improve accuracy, which has hitherto been impractical. Our approach thus will help enrich the DFT framework by utilizing the rapidly advancing machine-learning technique.

The formation and migration energies for various point defects, including vacancies and self-interstitials, in aluminium are systematically reinvestigated using the supercell approximation in the framework of orbital-free density functional theory. In particular, the finite-size effects and the accuracy of various kinetic energy density functionals are examined. It is demonstrated that as the supercell size increases, the finite-size errors asymptotically decrease as . Notably, the formation energies of self-interstitials converge much more slowly than that of vacancies. With carefully chosen kinetic energy density functionals, the calculated results agree quite well with the available experimental data and those obtained through Kohn–Sham density functional theory, which has an exact kinetic term.

The variational energies of broken-symmetry single determinants are frequently used (especially in the Kohn-Sham density functional theory) to determine the magnetic coupling between open-shell metal ions in molecular complexes or periodic lattices. Most applications extract the information from the solutions of m(s)(max) and m(s)(min) eigenvalues of S(z) magnetic spin momentum, assuming that a mapping of these energies on the energies of an Ising Hamiltonian is grounded. This approach is unable to predict the possible importance of deviations from the simplest form of the Heisenberg Hamiltonians. For systems involving s=1 magnetic centers, it cannot provide an estimate of neither the biquadratic exchange integral nor the three-body operator interaction that has recently been proven to be of the same order of magnitude [Phys. Rev. B 70, 132412 (2007)]. The present work shows that one may use other broken-symmetry solutions of intermediate values of m(s) to evaluate the amplitude of these additional terms. The here-derived equations rely on the assumption that an extended Hubbard-type Hamiltonian rules the interactions between the magnetic electrons. Numerical illustrations on a model problem of two O(2) molecules and a fragment of the La(2)NiO(4) lattice are reported. The results obtained using a variable percentage of Fock exchange in the BLYP functional are compared to those provided by elaborate wave function calculations. The relevant percentage of Fock exchange is system dependent but a mean value of 30% leads to acceptable amplitudes of the effective exchange interaction.

By examining the natural orbital occupation numbers, it was discovered that unrestricted Møller–Plesset perturbation theory (MP2) can violate N-representability by having natural occupation numbers greater than 2 or less than 0, even though the energy appears well behaved. Analytically, this problem stems from the fact that the MP2 effective one-particle density matrix is dependent on the inverse of the orbital hessian matrix (A′). When a molecular system goes through the point where spin-restricted orbitals become unstable to unrestriction, one of the eigenvalues of the A′ matrix will become 0, making it a singular matrix, thereby causing problems in the relaxed one-particle density matrix and discontinuities in the first order properties such as nuclear forces and dipole moments. Since the new ‘double-hybrid’ density functionals also involve a second-order perturbation expression for correlation using Kohn–Sham orbitals, these functionals also exhibit similar issues near the unrestriction point. The unphysical orbital occupations can be eliminated and the first derivative discontinuities removed by optimizing orbitals with inclusion of the second order correlation energy. Connections to the second derivative discontinuities in Hartree–Fock and Kohn–Sham density functional theory are also discussed.

Kohn–Sham density functional theory (DFT) is routinely used for the fast electronic structure computation of large systems and will most likely continue to be the method of choice for the generation of reliable geometries in the foreseeable future. Here, we present a hierarchy of simplified DFT methods designed for consistent structures and non-covalent interactions of large systems with particular focus on molecular crystals. The covered methods are a minimal basis set Hartree–Fock (HF-3c), a small basis set screened exchange hybrid functional (HSE-3c), and a generalized gradient approximated functional evaluated in a medium-sized basis set (B97-3c), all augmented with semi-classical correction potentials. We give an overview on the methods design, a comprehensive evaluation on established benchmark sets for geometries and lattice energies of molecular crystals, and highlight some realistic applications on large organic crystals with several hundreds of atoms in the primitive unit cell.

We present a framework for computing the shock Hugoniot using on-the-fly machine learned force field (MLFF) molecular dynamics simulations. In particular, we employ an MLFF model based on the kernel method and Bayesian linear regression to compute the free energy, atomic forces, and pressure, in conjunction with a linear regression model between the internal and free energies to compute the internal energy, with all training data generated from Kohn–Sham density functional theory (DFT). We verify the accuracy of the formalism by comparing the Hugoniot for carbon with recent Kohn–Sham DFT results in the literature. In so doing, we demonstrate that Kohn–Sham calculations for the Hugoniot can be accelerated by up to two orders of magnitude, while retaining ab initio accuracy. We apply this framework to calculate the Hugoniots of 14 materials in the FPEOS database, comprising 9 single elements and 5 compounds, between temperatures of 10 kK and 2 MK. We find good agreement with first principles results in the literature while providing tighter error bars. In addition, we confirm that the inter-element interaction in compounds decreases with temperature.