Density functional theory beyond the Born–Oppenheimer approximation: exact mapping onto an electronically non-interacting Kohn–Sham molecule

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.

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.

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

Kohn-Sham (KS) density functional theory (DFT) is a very efficient method for calculating various properties of solids as, for instance, the total energy, the electron density, or the electronic band structure. The KS-DFT method leads to rather fast calculations; however, the accuracy depends crucially on the chosen approximation for the exchange and correlation (xc) functional Exc and/or potential vxc. Here, an overview of xc methods to calculate the electronic band structure is given, with the focus on the so-called semilocal methods that are the fastest in KS-DFT and allow to treat systems containing up to thousands of atoms. Among them, there is the modified Becke-Johnson potential that is widely used to calculate the fundamental bandgap of semiconductors and insulators. The accuracy for other properties like magnetic moment or electron density, that are also determined directly by vxc, is also discussed.

We employed an explicit expression for the dispersion (D) energy in conjunction with Kohn-Sham (KS) density functional theory and frozen-density embedding (FDE) to calculate interaction energies between DNA base pairs and a selected set of amino acid pairs in the hydrophobic core of a small protein Rubredoxin. We use this data to assess the accuracy of an FDE-D approach for the calculation of intermolecular interactions. To better analyze the calculated interaction energies we furthermore propose a new energy decomposition scheme that is similar to the well-known KS bond formation analysis [F. M. Bickelhaupt and E. J. Baerends, Rev. Comput. Chem. 15, 1 (2000)], but differs in the electron densities used to define the bond energy. The individual subsystem electron densities of the FDE approach sum to the total electron density which makes it possible to define bond energies in terms of promotion energies and an explicit interaction energy. We show that for the systems considered only a few freeze-and-thaw cycles suffice to reach convergence in these individual bond energy components, illustrating the potential of FDE-D as an efficient method to calculate intermolecular interactions.

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

Probing the Interplay between Electronic and Geometric Degrees-of-Freedom in Molecules and Reactive Systems

Degenerate perturbation theory from quantum mechanics is inadequate in density functional theory (DFT) because of nonlinearity in the Kohn-Sham potential. Herein, we develop the fully general perturbation theory for open-shell, degenerate systems in Kohn-Sham DFT, without assuming the presence of symmetry or equal occupation of degenerate orbitals. To demonstrate the resulting methodology, we apply it to the iron atom in the central field approximation, perturbed by an electric quadrupole. This system was chosen because it displays both symmetry required degeneracy, between the five 3\textit{d} orbitals, as well as accidental degeneracy, between the 3\textit{d} and 4\textit{s} orbitals. The quadrupole potential couples the degenerate 3\textit{d} and 4\textit{s} states, serving as an example of the most general perturbation.

Simulating electron transfer at reactive solid-liquid interfaces under constant electrochemical potentials of the constituents (electrons, ions, solvent etc.) is crucial to understanding the formation, functional and failure of electrochemical devices. Albeit being sufficiently accurate in decreasing breaking and formation of chemical bond at solid surfaces, existing methods based on Kohn-Sham density functional theory (DFT) are unsatisfactory in system consistency, namely, simulating the open solid-liquid interface under grand-canonical conditions, as well as in scaling up the simulation due to its high computational cost. Herein, to improve the system consistency and computational efficiency, we develop density-potential functional theoretic (DPFT) schemes out of Kohn-Sham DFT, drawing upon ideas of orbital-free DFT, frozen density embedding theory, and tight-binding DFT. The DPFT transforms an all-atom, Kohn-Sham DFT description of the nonreactive electrolyte solution into a coarse-grained, field-based description, while retaining a Kohn-Sham DFT description for the reactive subsystem. In the absence of surface reactions, the solid electrode can be described using orbital-free DFT, reducing the computational cost further. On the conceptual level, the physical meaning of potential in DPFT is examined. Recent applications of DPFT to simulate EDLs with mesoscopic roughness and EDLs at supported nanoparticles will be introduced. Related papers Zhang, Y., Binninger, T., Huang, J., & Eikerling, M. H. (2025). Theory of Electro-Ionic Perturbations at Supported Electrocatalyst Nanoparticles. Physical Review Letters, 134(6), 066201.Zhang, M., Chen, Y., Eikerling, M., & Huang, J. (2025). Structured solvent on a split electron tail: A semiclassical theory of electrified metal-solution interfaces. Physical Review Applied, 23(2), 024009.Huang, J., Domínguez-Flores, F., & Melander, M. (2024). Variants of surface charges and capacitances in electrocatalysis: Insights from density-potential functional theory embedded with an implicit chemisorption model. PRX Energy, 3(4), 043008. Acknowledgements Our research is supported financially by the Initiative and Networking Fund of the Helmholtz Association (no. VH-NG-1709), and European Research Council (ERC) Starting Grant (MESO-CAT, Grant agreement No. 101163405).

ABACUS (Atomic-orbital Based Ab initio Computation at USTC) is an open-source software for first-principles electronic structure calculations and molecular dynamics simulations. It mainly features density functional theory (DFT) and molecular dynamics functions and is compatible with both plane wave basis sets and numerical atomic orbital basis sets. ABACUS serves as a platform that facilitates the integration of various electronic structure methods, such as Kohn-Sham DFT, stochastic DFT, orbital-free DFT, real-time time-dependent DFT, etc. In addition, with the aid of high-performance computing, ABACUS is designed to perform efficiently and provide massive amounts of first-principles data for generating general-purpose machine learning potentials, such as deep potential with attention models. Furthermore, ABACUS serves as an electronic structure platform that interfaces with several artificial intelligence-assisted algorithms and packages, such as DeePKS-kit, DeePMD, DP-GEN, DeepH, DeePTB, HamGNN, etc.

A quantitative description of the excited electronic states of point defects and impurities is crucial for understanding materials properties, and possible applications of defects in quantum technologies. This is a considerable challenge for computational methods, since Kohn-Sham density functional theory (DFT) is inherently a ground-state theory, while higher-level methods are often too computationally expensive for defect systems. Recently, embedding approaches have been applied that treat defect states with many-body methods, while using DFT to describe the bulk host material. We implement such an embedding method, based on Wannierization of defect orbitals and the constrained random-phase approximation approach, and perform systematic characterization of the method for three distinct systems with current technological relevance: a carbon dimer replacing a B and N pair in bulk hexagonal BN (${\mathrm{C}}_{\text{B}}{\mathrm{C}}_{\text{N}}$), the negatively charged nitrogen-vacancy center in diamond (${\mathrm{NV}}^{\ensuremath{-}}$), and an Fe impurity on the Al site in wurtzite AlN (${\text{Fe}}_{\text{Al}}$). In the context of these test-case defects, we demonstrate that crucial considerations of the methodology include convergence of the bulk screening of the active-space Coulomb interaction, the choice of exchange-correlation functional for the initial DFT calculation, and the treatment of the ``double-counting'' correction. For ${\mathrm{C}}_{\text{B}}{\mathrm{C}}_{\text{N}}$ we show that the embedding approach gives many-body states in agreement with analytical results on the Hubbard dimer model, which allows us to elucidate the effects of the DFT functional and double-counting correction. For the ${\mathrm{NV}}^{\ensuremath{-}}$ center, our method demonstrates good quantitative agreement with experiments for the zero-phonon line of the triplet-triplet transition. Finally, we illustrate challenges associated with this method for determining the energies and orderings of the complex spin multiplets in ${\text{Fe}}_{\text{Al}}$.

Adsorbed transition metal atoms can have partially filled $d$- or $f$-shells due to strong on-site Coulomb interaction. Capturing all effects originating from electron correlation in such strongly correlated systems is a challenge for electronic structure methods. It requires a sufficiently accurate description of the atomistic structure (in particular bond distances and angles), which is usually obtained from Kohn-Sham density functional theory (DFT), which due to the approximate nature of the exchange-correlation functional may provide an unreliable description of strongly correlated systems. To elucidate the consequences of this popular procedure, we apply a combination of DFT with the Anderson impurity model (AIM), as well as DFT+U for a calculation of the potential energy surface along the Co/Cu(001) adsorption coordinate, and compare the results with those obtained from DFT. The adsorption minimum is shifted towards larger distances by applying DFT+AIM, or the much cheaper DFT+U method, compared to the corresponding spin-polarized DFT results, by a magnitude comparable to variations between different approximate exchange-correlation functionals (0.08 to 0.12 Angstrom). This shift originates from an increasing correlation energy at larger adsorption distances, which can be traced back to the Co 3$d_{xy}$ and 3$d_{z2}$ orbitals being more correlated as the adsorption distance is increased.

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.

In traditional finite-temperature Kohn–Sham density functional theory (KSDFT), the partial occupation of a large number of high-energy KS eigenstates restricts the use of first-principles molecular dynamics methods at extremely high temperatures. However, stochastic density functional theory (SDFT) can overcome this limitation. Recently, SDFT and the related mixed stochastic–deterministic density functional theory, based on a plane-wave basis set, have been implemented in the first-principles electronic structure software ABACUS [Q. Liu and M. Chen, Phys. Rev. B 106, 125132 (2022)]. In this study, we combine SDFT with the Born–Oppenheimer molecular dynamics method to investigate systems with temperatures ranging from a few tens of eV to 1000 eV. Importantly, we train machine-learning-based interatomic models using the SDFT data and employ these deep potential models to simulate large-scale systems with long trajectories. Subsequently, we compute and analyze the structural properties, dynamic properties, and transport coefficients of warm dense matter.

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.